Integrand size = 30, antiderivative size = 322 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^3}{\left (e+f x^2\right )^3} \, dx=\frac {d^3 x \sqrt {a+b x^2}}{2 f^3}-\frac {(d e-c f)^3 x \sqrt {a+b x^2}}{4 e f^3 \left (e+f x^2\right )^2}-\frac {(d e-c f)^2 (3 a f (3 d e+c f)-2 b e (5 d e+c f)) x \sqrt {a+b x^2}}{8 e^2 f^3 (b e-a f) \left (e+f x^2\right )}-\frac {d^2 (6 b d e-6 b c f-a d f) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b} f^4}+\frac {(d e-c f) \left (24 b^2 d^2 e^4-4 a b e f \left (10 d^2 e^2+c d e f+c^2 f^2\right )+3 a^2 f^2 \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{8 e^{5/2} f^4 (b e-a f)^{3/2}} \] Output:
1/2*d^3*x*(b*x^2+a)^(1/2)/f^3-1/4*(-c*f+d*e)^3*x*(b*x^2+a)^(1/2)/e/f^3/(f* x^2+e)^2-1/8*(-c*f+d*e)^2*(3*a*f*(c*f+3*d*e)-2*b*e*(c*f+5*d*e))*x*(b*x^2+a )^(1/2)/e^2/f^3/(-a*f+b*e)/(f*x^2+e)-1/2*d^2*(-a*d*f-6*b*c*f+6*b*d*e)*arct anh(b^(1/2)*x/(b*x^2+a)^(1/2))/b^(1/2)/f^4+1/8*(-c*f+d*e)*(24*b^2*d^2*e^4- 4*a*b*e*f*(c^2*f^2+c*d*e*f+10*d^2*e^2)+3*a^2*f^2*(c^2*f^2+2*c*d*e*f+5*d^2* e^2))*arctanh((-a*f+b*e)^(1/2)*x/e^(1/2)/(b*x^2+a)^(1/2))/e^(5/2)/f^4/(-a* f+b*e)^(3/2)
Time = 11.12 (sec) , antiderivative size = 285, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^3}{\left (e+f x^2\right )^3} \, dx=\frac {f x \sqrt {a+b x^2} \left (4 d^3-\frac {2 (d e-c f)^3}{e \left (e+f x^2\right )^2}+\frac {(d e-c f)^2 (-3 a f (3 d e+c f)+2 b e (5 d e+c f))}{e^2 (b e-a f) \left (e+f x^2\right )}\right )-\frac {(d e-c f) \left (24 b^2 d^2 e^4-4 a b e f \left (10 d^2 e^2+c d e f+c^2 f^2\right )+3 a^2 f^2 \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {-b e+a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{e^{5/2} (-b e+a f)^{3/2}}-\frac {4 d^2 (6 b d e-6 b c f-a d f) \log \left (b x+\sqrt {b} \sqrt {a+b x^2}\right )}{\sqrt {b}}}{8 f^4} \] Input:
Integrate[(Sqrt[a + b*x^2]*(c + d*x^2)^3)/(e + f*x^2)^3,x]
Output:
(f*x*Sqrt[a + b*x^2]*(4*d^3 - (2*(d*e - c*f)^3)/(e*(e + f*x^2)^2) + ((d*e - c*f)^2*(-3*a*f*(3*d*e + c*f) + 2*b*e*(5*d*e + c*f)))/(e^2*(b*e - a*f)*(e + f*x^2))) - ((d*e - c*f)*(24*b^2*d^2*e^4 - 4*a*b*e*f*(10*d^2*e^2 + c*d*e *f + c^2*f^2) + 3*a^2*f^2*(5*d^2*e^2 + 2*c*d*e*f + c^2*f^2))*ArcTan[(Sqrt[ -(b*e) + a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])])/(e^(5/2)*(-(b*e) + a*f)^(3/2) ) - (4*d^2*(6*b*d*e - 6*b*c*f - a*d*f)*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]]) /Sqrt[b])/(8*f^4)
Leaf count is larger than twice the leaf count of optimal. \(1024\) vs. \(2(322)=644\).
Time = 1.59 (sec) , antiderivative size = 1024, normalized size of antiderivative = 3.18, number of steps used = 26, number of rules used = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {425, 425, 420, 299, 224, 219, 398, 224, 219, 291, 221, 425, 398, 224, 219, 291, 221, 402, 27, 291, 221, 402, 27, 291, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^3}{\left (e+f x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 425 |
| \(\displaystyle \frac {b \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\) |
| \(\Big \downarrow \) 425 |
| \(\displaystyle \frac {b \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 420 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a}}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {d \left (\frac {(2 b c-a d) \int \frac {1}{\sqrt {b x^2+a}}dx}{2 b}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {d \left (\frac {(2 b c-a d) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 b}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 398 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \int \frac {1}{\sqrt {b x^2+a}}dx}{f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}\right )}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}\right )}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}\right )}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}\right )}{f}\right )}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 425 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 398 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {d \int \frac {1}{\sqrt {b x^2+a}}dx}{f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {d \int \frac {1}{\sqrt {b x^2+a}}dx}{f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {d \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {d \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {d \left (\frac {d \sqrt {b x^2+a} x}{2 b}+\frac {(2 b c-a d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{2 b^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\int \frac {2 b c e-a (d e+c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\int \frac {2 b c e-a (d e+c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\int \frac {2 b c e-a (d e+c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\int \frac {2 b (d e-c f) x^2+4 b c e-a d e-3 a c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {d \left (\frac {d \sqrt {b x^2+a} x}{2 b}+\frac {(2 b c-a d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{2 b^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\int \frac {2 b (d e-c f) x^2+4 b c e-a d e-3 a c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {d \left (\frac {d \sqrt {b x^2+a} x}{2 b}+\frac {(2 b c-a d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{2 b^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\int \frac {2 b (d e-c f) x^2+4 b c e-a d e-3 a c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {d \left (\frac {d \sqrt {b x^2+a} x}{2 b}+\frac {(2 b c-a d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{2 b^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\int \frac {2 b (d e-c f) x^2+4 b c e-a d e-3 a c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {d \left (\frac {d \sqrt {b x^2+a} x}{2 b}+\frac {(2 b c-a d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{2 b^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\int \frac {f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}}{4 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {d \left (\frac {d \sqrt {b x^2+a} x}{2 b}+\frac {(2 b c-a d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{2 b^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\left (f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2\right ) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}}{4 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {d \left (\frac {d \sqrt {b x^2+a} x}{2 b}+\frac {(2 b c-a d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{2 b^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\left (f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2\right ) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}}{4 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {d \left (\frac {d \sqrt {b x^2+a} x}{2 b}+\frac {(2 b c-a d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{2 b^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\left (f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}\) |
Input:
Int[(Sqrt[a + b*x^2]*(c + d*x^2)^3)/(e + f*x^2)^3,x]
Output:
(b*((d*((d*((d*x*Sqrt[a + b*x^2])/(2*b) + ((2*b*c - a*d)*ArcTanh[(Sqrt[b]* x)/Sqrt[a + b*x^2]])/(2*b^(3/2))))/f - ((d*e - c*f)*((d*ArcTanh[(Sqrt[b]*x )/Sqrt[a + b*x^2]])/(Sqrt[b]*f) - ((d*e - c*f)*ArcTanh[(Sqrt[b*e - a*f]*x) /(Sqrt[e]*Sqrt[a + b*x^2])])/(Sqrt[e]*f*Sqrt[b*e - a*f])))/f))/f - ((d*e - c*f)*((d*((d*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(Sqrt[b]*f) - ((d*e - c*f)*ArcTanh[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])])/(Sqrt[e]*f*Sq rt[b*e - a*f])))/f - ((d*e - c*f)*(((d*e - c*f)*x*Sqrt[a + b*x^2])/(2*e*(b *e - a*f)*(e + f*x^2)) + ((2*b*c*e - a*(d*e + c*f))*ArcTanh[(Sqrt[b*e - a* f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])])/(2*e^(3/2)*(b*e - a*f)^(3/2))))/f))/f))/ f - ((b*e - a*f)*((d*((d*((d*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(Sqrt[b ]*f) - ((d*e - c*f)*ArcTanh[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])] )/(Sqrt[e]*f*Sqrt[b*e - a*f])))/f - ((d*e - c*f)*(((d*e - c*f)*x*Sqrt[a + b*x^2])/(2*e*(b*e - a*f)*(e + f*x^2)) + ((2*b*c*e - a*(d*e + c*f))*ArcTanh [(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])])/(2*e^(3/2)*(b*e - a*f)^(3 /2))))/f))/f - ((d*e - c*f)*((d*(((d*e - c*f)*x*Sqrt[a + b*x^2])/(2*e*(b*e - a*f)*(e + f*x^2)) + ((2*b*c*e - a*(d*e + c*f))*ArcTanh[(Sqrt[b*e - a*f] *x)/(Sqrt[e]*Sqrt[a + b*x^2])])/(2*e^(3/2)*(b*e - a*f)^(3/2))))/f - ((d*e - c*f)*(((d*e - c*f)*x*Sqrt[a + b*x^2])/(4*e*(b*e - a*f)*(e + f*x^2)^2) + (((2*b*e*(d*e - 3*c*f) + a*f*(d*e + 3*c*f))*x*Sqrt[a + b*x^2])/(2*e*(b*e - a*f)*(e + f*x^2)) + ((8*b^2*c*e^2 - 4*a*b*e*(d*e + 2*c*f) + a^2*f*(d*e...
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 0]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]) , x_Symbol] :> Simp[f/b Int[1/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/ b Int[1/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f} , x]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
Int[(((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_)^2)^(r_))/((a_) + (b_.)*( x_)^2), x_Symbol] :> Simp[d/b Int[(c + d*x^2)^(q - 1)*(e + f*x^2)^r, x], x] + Simp[(b*c - a*d)/b Int[(c + d*x^2)^(q - 1)*((e + f*x^2)^r/(a + b*x^2 )), x], x] /; FreeQ[{a, b, c, d, e, f, r}, x] && GtQ[q, 1]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_ )^2)^(r_), x_Symbol] :> Simp[d/b Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1)*(e + f*x^2)^r, x], x] + Simp[(b*c - a*d)/b Int[(a + b*x^2)^p*(c + d*x ^2)^(q - 1)*(e + f*x^2)^r, x], x] /; FreeQ[{a, b, c, d, e, f, r}, x] && ILt Q[p, 0] && GtQ[q, 0]
Time = 1.18 (sec) , antiderivative size = 419, normalized size of antiderivative = 1.30
| method | result | size |
| pseudoelliptic | \(-\frac {3 \left (\left (c f -d e \right ) \left (f \,x^{2}+e \right )^{2} \left (\frac {f^{2} a^{2} \left (c^{2} f^{2}+2 c d e f +5 d^{2} e^{2}\right ) \sqrt {b}}{8}+\left (e^{3} b \,d^{2}-\frac {1}{6} a \,c^{2} f^{3}-\frac {1}{6} a c d e \,f^{2}-\frac {5}{3} a \,d^{2} e^{2} f \right ) b^{\frac {3}{2}} e \right ) \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )+\frac {\sqrt {\left (a f -b e \right ) e}\, \left (-d^{2} \left (a f -b e \right ) \left (f \,x^{2}+e \right )^{2} e^{2} \left (-6 b d e +f \left (a d +6 b c \right )\right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )+\left (-\frac {5 a \left (\frac {11 e^{4} d^{3}}{5}-\frac {9 d^{2} f \left (-\frac {17 x^{2} d}{9}+c \right ) e^{3}}{5}-\frac {3 \left (-\frac {4}{3} d^{2} x^{4}+5 c d \,x^{2}+c^{2}\right ) d \,f^{2} e^{2}}{5}+c^{2} f^{3} \left (\frac {3 x^{2} d}{5}+c \right ) e +\frac {3 c^{3} f^{4} x^{2}}{5}\right ) f \sqrt {b}}{4}+\left (3 e^{4} d^{3}-3 d^{2} \left (-\frac {3 x^{2} d}{2}+c \right ) f \,e^{3}-\frac {9 d^{2} x^{2} f^{2} \left (-\frac {2 x^{2} d}{9}+c \right ) e^{2}}{2}+c^{2} f^{3} \left (\frac {3 x^{2} d}{2}+c \right ) e +\frac {c^{3} f^{4} x^{2}}{2}\right ) b^{\frac {3}{2}} e \right ) \sqrt {b \,x^{2}+a}\, x f \right )}{6}\right )}{\sqrt {\left (a f -b e \right ) e}\, \sqrt {b}\, f^{4} \left (f \,x^{2}+e \right )^{2} \left (a f -b e \right ) e^{2}}\) | \(419\) |
| risch | \(\text {Expression too large to display}\) | \(2428\) |
| default | \(\text {Expression too large to display}\) | \(4373\) |
Input:
int((b*x^2+a)^(1/2)*(d*x^2+c)^3/(f*x^2+e)^3,x,method=_RETURNVERBOSE)
Output:
-3*((c*f-d*e)*(f*x^2+e)^2*(1/8*f^2*a^2*(c^2*f^2+2*c*d*e*f+5*d^2*e^2)*b^(1/ 2)+(e^3*b*d^2-1/6*a*c^2*f^3-1/6*a*c*d*e*f^2-5/3*a*d^2*e^2*f)*b^(3/2)*e)*ar ctan(e*(b*x^2+a)^(1/2)/x/((a*f-b*e)*e)^(1/2))+1/6*((a*f-b*e)*e)^(1/2)*(-d^ 2*(a*f-b*e)*(f*x^2+e)^2*e^2*(-6*b*d*e+f*(a*d+6*b*c))*arctanh((b*x^2+a)^(1/ 2)/x/b^(1/2))+(-5/4*a*(11/5*e^4*d^3-9/5*d^2*f*(-17/9*x^2*d+c)*e^3-3/5*(-4/ 3*d^2*x^4+5*c*d*x^2+c^2)*d*f^2*e^2+c^2*f^3*(3/5*x^2*d+c)*e+3/5*c^3*f^4*x^2 )*f*b^(1/2)+(3*e^4*d^3-3*d^2*(-3/2*x^2*d+c)*f*e^3-9/2*d^2*x^2*f^2*(-2/9*x^ 2*d+c)*e^2+c^2*f^3*(3/2*x^2*d+c)*e+1/2*c^3*f^4*x^2)*b^(3/2)*e)*(b*x^2+a)^( 1/2)*x*f))/((a*f-b*e)*e)^(1/2)/b^(1/2)/f^4/(f*x^2+e)^2/(a*f-b*e)/e^2
Leaf count of result is larger than twice the leaf count of optimal. 1268 vs. \(2 (290) = 580\).
Time = 46.91 (sec) , antiderivative size = 5166, normalized size of antiderivative = 16.04 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^3}{\left (e+f x^2\right )^3} \, dx=\text {Too large to display} \] Input:
integrate((b*x^2+a)^(1/2)*(d*x^2+c)^3/(f*x^2+e)^3,x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^3}{\left (e+f x^2\right )^3} \, dx=\int \frac {\sqrt {a + b x^{2}} \left (c + d x^{2}\right )^{3}}{\left (e + f x^{2}\right )^{3}}\, dx \] Input:
integrate((b*x**2+a)**(1/2)*(d*x**2+c)**3/(f*x**2+e)**3,x)
Output:
Integral(sqrt(a + b*x**2)*(c + d*x**2)**3/(e + f*x**2)**3, x)
\[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^3}{\left (e+f x^2\right )^3} \, dx=\int { \frac {\sqrt {b x^{2} + a} {\left (d x^{2} + c\right )}^{3}}{{\left (f x^{2} + e\right )}^{3}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)*(d*x^2+c)^3/(f*x^2+e)^3,x, algorithm="maxima")
Output:
integrate(sqrt(b*x^2 + a)*(d*x^2 + c)^3/(f*x^2 + e)^3, x)
Leaf count of result is larger than twice the leaf count of optimal. 1875 vs. \(2 (290) = 580\).
Time = 0.19 (sec) , antiderivative size = 1875, normalized size of antiderivative = 5.82 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^3}{\left (e+f x^2\right )^3} \, dx=\text {Too large to display} \] Input:
integrate((b*x^2+a)^(1/2)*(d*x^2+c)^3/(f*x^2+e)^3,x, algorithm="giac")
Output:
1/2*sqrt(b*x^2 + a)*d^3*x/f^3 - 1/8*(24*b^(5/2)*d^3*e^5 - 24*b^(5/2)*c*d^2 *e^4*f - 40*a*b^(3/2)*d^3*e^4*f + 36*a*b^(3/2)*c*d^2*e^3*f^2 + 15*a^2*sqrt (b)*d^3*e^3*f^2 - 9*a^2*sqrt(b)*c*d^2*e^2*f^3 + 4*a*b^(3/2)*c^3*e*f^4 - 3* a^2*sqrt(b)*c^2*d*e*f^4 - 3*a^2*sqrt(b)*c^3*f^5)*arctan(1/2*((sqrt(b)*x - sqrt(b*x^2 + a))^2*f + 2*b*e - a*f)/sqrt(-b^2*e^2 + a*b*e*f))/((b*e^3*f^4 - a*e^2*f^5)*sqrt(-b^2*e^2 + a*b*e*f)) + 1/4*(24*(sqrt(b)*x - sqrt(b*x^2 + a))^6*b^(5/2)*d^3*e^5*f - 48*(sqrt(b)*x - sqrt(b*x^2 + a))^6*b^(5/2)*c*d^ 2*e^4*f^2 - 32*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a*b^(3/2)*d^3*e^4*f^2 + 24* (sqrt(b)*x - sqrt(b*x^2 + a))^6*b^(5/2)*c^2*d*e^3*f^3 + 60*(sqrt(b)*x - sq rt(b*x^2 + a))^6*a*b^(3/2)*c*d^2*e^3*f^3 + 9*(sqrt(b)*x - sqrt(b*x^2 + a)) ^6*a^2*sqrt(b)*d^3*e^3*f^3 - 24*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a*b^(3/2)* c^2*d*e^2*f^4 - 15*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^2*sqrt(b)*c*d^2*e^2*f ^4 - 4*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a*b^(3/2)*c^3*e*f^5 + 3*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^2*sqrt(b)*c^2*d*e*f^5 + 3*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^2*sqrt(b)*c^3*f^6 + 80*(sqrt(b)*x - sqrt(b*x^2 + a))^4*b^(7/2)*d^ 3*e^6 - 144*(sqrt(b)*x - sqrt(b*x^2 + a))^4*b^(7/2)*c*d^2*e^5*f - 152*(sqr t(b)*x - sqrt(b*x^2 + a))^4*a*b^(5/2)*d^3*e^5*f + 48*(sqrt(b)*x - sqrt(b*x ^2 + a))^4*b^(7/2)*c^2*d*e^4*f^2 + 264*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a*b ^(5/2)*c*d^2*e^4*f^2 + 102*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^2*b^(3/2)*d^3 *e^4*f^2 + 16*(sqrt(b)*x - sqrt(b*x^2 + a))^4*b^(7/2)*c^3*e^3*f^3 - 72*...
Timed out. \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^3}{\left (e+f x^2\right )^3} \, dx=\int \frac {\sqrt {b\,x^2+a}\,{\left (d\,x^2+c\right )}^3}{{\left (f\,x^2+e\right )}^3} \,d x \] Input:
int(((a + b*x^2)^(1/2)*(c + d*x^2)^3)/(e + f*x^2)^3,x)
Output:
int(((a + b*x^2)^(1/2)*(c + d*x^2)^3)/(e + f*x^2)^3, x)
Time = 0.58 (sec) , antiderivative size = 8152, normalized size of antiderivative = 25.32 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^3}{\left (e+f x^2\right )^3} \, dx =\text {Too large to display} \] Input:
int((b*x^2+a)^(1/2)*(d*x^2+c)^3/(f*x^2+e)^3,x)
Output:
( - 12*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b* x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**3*b*c**3*e**2*f**6 - 24*s qrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**3*b*c**3*e*f**7*x**2 - 12*sqrt(e) *sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f )*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**3*b*c**3*f**8*x**4 - 12*sqrt(e)*sqrt(a* f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b )*x)/(sqrt(e)*sqrt(b)))*a**3*b*c**2*d*e**3*f**5 - 24*sqrt(e)*sqrt(a*f - b* e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/( sqrt(e)*sqrt(b)))*a**3*b*c**2*d*e**2*f**6*x**2 - 12*sqrt(e)*sqrt(a*f - b*e )*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(s qrt(e)*sqrt(b)))*a**3*b*c**2*d*e*f**7*x**4 - 36*sqrt(e)*sqrt(a*f - b*e)*at an((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt( e)*sqrt(b)))*a**3*b*c*d**2*e**4*f**4 - 72*sqrt(e)*sqrt(a*f - b*e)*atan((sq rt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqr t(b)))*a**3*b*c*d**2*e**3*f**5*x**2 - 36*sqrt(e)*sqrt(a*f - b*e)*atan((sqr t(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt (b)))*a**3*b*c*d**2*e**2*f**6*x**4 + 60*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt (a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt( b)))*a**3*b*d**3*e**5*f**3 + 120*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f...