Integrand size = 30, antiderivative size = 265 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^3}{\left (e+f x^2\right )^2} \, dx=-\frac {d^2 (8 b d e-12 b c f-a d f) x \sqrt {a+b x^2}}{8 b f^3}+\frac {d^3 x^3 \sqrt {a+b x^2}}{4 f^2}-\frac {(d e-c f)^3 x \sqrt {a+b x^2}}{2 e f^3 \left (e+f x^2\right )}-\frac {d \left (a^2 d^2 f^2+4 a b d f (2 d e-3 c f)-24 b^2 (d e-c f)^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{3/2} f^4}-\frac {(d e-c f)^2 \left (6 b d e^2-a f (5 d e+c f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{2 e^{3/2} f^4 \sqrt {b e-a f}} \] Output:
-1/8*d^2*(-a*d*f-12*b*c*f+8*b*d*e)*x*(b*x^2+a)^(1/2)/b/f^3+1/4*d^3*x^3*(b* x^2+a)^(1/2)/f^2-1/2*(-c*f+d*e)^3*x*(b*x^2+a)^(1/2)/e/f^3/(f*x^2+e)-1/8*d* (a^2*d^2*f^2+4*a*b*d*f*(-3*c*f+2*d*e)-24*b^2*(-c*f+d*e)^2)*arctanh(b^(1/2) *x/(b*x^2+a)^(1/2))/b^(3/2)/f^4-1/2*(-c*f+d*e)^2*(6*b*d*e^2-a*f*(c*f+5*d*e ))*arctanh((-a*f+b*e)^(1/2)*x/e^(1/2)/(b*x^2+a)^(1/2))/e^(3/2)/f^4/(-a*f+b *e)^(1/2)
Time = 1.87 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^3}{\left (e+f x^2\right )^2} \, dx=\frac {\frac {f x \sqrt {a+b x^2} \left (a d^3 e f \left (e+f x^2\right )-2 b \left (6 c^2 d e f^2-2 c^3 f^3-6 c d^2 e f \left (2 e+f x^2\right )+d^3 e \left (6 e^2+3 e f x^2-f^2 x^4\right )\right )\right )}{b e \left (e+f x^2\right )}+\frac {4 (d e-c f)^2 \left (6 b d e^2-a f (5 d e+c f)\right ) \arctan \left (\frac {-f x \sqrt {a+b x^2}+\sqrt {b} \left (e+f x^2\right )}{\sqrt {e} \sqrt {-b e+a f}}\right )}{e^{3/2} \sqrt {-b e+a f}}-\frac {d \left (-a^2 d^2 f^2+24 b^2 (d e-c f)^2+4 a b d f (-2 d e+3 c f)\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{b^{3/2}}}{8 f^4} \] Input:
Integrate[(Sqrt[a + b*x^2]*(c + d*x^2)^3)/(e + f*x^2)^2,x]
Output:
((f*x*Sqrt[a + b*x^2]*(a*d^3*e*f*(e + f*x^2) - 2*b*(6*c^2*d*e*f^2 - 2*c^3* f^3 - 6*c*d^2*e*f*(2*e + f*x^2) + d^3*e*(6*e^2 + 3*e*f*x^2 - f^2*x^4))))/( b*e*(e + f*x^2)) + (4*(d*e - c*f)^2*(6*b*d*e^2 - a*f*(5*d*e + c*f))*ArcTan [(-(f*x*Sqrt[a + b*x^2]) + Sqrt[b]*(e + f*x^2))/(Sqrt[e]*Sqrt[-(b*e) + a*f ])])/(e^(3/2)*Sqrt[-(b*e) + a*f]) - (d*(-(a^2*d^2*f^2) + 24*b^2*(d*e - c*f )^2 + 4*a*b*d*f*(-2*d*e + 3*c*f))*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/b^( 3/2))/(8*f^4)
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.
| \(\displaystyle \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^3}{\left (e+f x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 425 |
| \(\displaystyle \frac {b \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 420 |
| \(\displaystyle \frac {b \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a}}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 318 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\int \frac {3 d (2 b c-a d) x^2+c (4 b c-a d)}{\sqrt {b x^2+a}}dx}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\frac {\left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right ) \int \frac {1}{\sqrt {b x^2+a}}dx}{2 b}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\frac {\left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right ) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 b}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 420 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a}}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(2 b c-a d) \int \frac {1}{\sqrt {b x^2+a}}dx}{2 b}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(2 b c-a d) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 b}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 398 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \int \frac {1}{\sqrt {b x^2+a}}dx}{f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 425 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 420 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a}}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {(2 b c-a d) \int \frac {1}{\sqrt {b x^2+a}}dx}{2 b}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {(2 b c-a d) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 b}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 398 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \int \frac {1}{\sqrt {b x^2+a}}dx}{f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}\right )}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}\right )}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}\right )}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}\right )}{f}\right )}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 425 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 398 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {d \int \frac {1}{\sqrt {b x^2+a}}dx}{f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {d \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}\) |
Input:
Int[(Sqrt[a + b*x^2]*(c + d*x^2)^3)/(e + f*x^2)^2,x]
Output:
$Aborted
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[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[d*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(b*(2*(p + q) + 1))), x] + S imp[1/(b*(2*(p + q) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b *c*(2*(p + q) + 1) - a*d) + d*(b*c*(2*(p + 2*q - 1) + 1) - a*d*(2*(q - 1) + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && G tQ[q, 1] && NeQ[2*(p + q) + 1, 0] && !IGtQ[p, 1] && IntBinomialQ[a, b, c, d, 2, p, q, x]
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[(((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 = 1.08 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.11
| method | result | size |
| pseudoelliptic | \(\frac {6 b^{\frac {5}{2}} \left (b d \,e^{2}-\frac {1}{6} a c \,f^{2}-\frac {5}{6} a d e f \right ) \left (c f -d e \right )^{2} \left (f \,x^{2}+e \right ) \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )+\left (-\frac {d b \left (f \,x^{2}+e \right ) e \left (-24 b^{2} d^{2} e^{2}+8 b d f \left (a d +6 b c \right ) e +f^{2} \left (a^{2} d^{2}-12 a b c d -24 b^{2} c^{2}\right )\right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )}{4}+b^{\frac {3}{2}} \sqrt {b \,x^{2}+a}\, x f \left (-3 b \,d^{3} e^{3}+\frac {d^{2} \left (\left (-6 x^{2} d +24 c \right ) b +a d \right ) f \,e^{2}}{4}+\frac {d \,f^{2} \left (\left (2 d^{2} x^{4}+12 c d \,x^{2}-12 c^{2}\right ) b +a \,d^{2} x^{2}\right ) e}{4}+b \,c^{3} f^{3}\right )\right ) \sqrt {\left (a f -b e \right ) e}}{2 \sqrt {\left (a f -b e \right ) e}\, b^{\frac {5}{2}} f^{4} e \left (f \,x^{2}+e \right )}\) | \(295\) |
| risch | \(\text {Expression too large to display}\) | \(1297\) |
| default | \(\text {Expression too large to display}\) | \(2241\) |
Input:
int((b*x^2+a)^(1/2)*(d*x^2+c)^3/(f*x^2+e)^2,x,method=_RETURNVERBOSE)
Output:
1/2/((a*f-b*e)*e)^(1/2)*(6*b^(5/2)*(b*d*e^2-1/6*a*c*f^2-5/6*a*d*e*f)*(c*f- d*e)^2*(f*x^2+e)*arctan(e*(b*x^2+a)^(1/2)/x/((a*f-b*e)*e)^(1/2))+(-1/4*d*b *(f*x^2+e)*e*(-24*b^2*d^2*e^2+8*b*d*f*(a*d+6*b*c)*e+f^2*(a^2*d^2-12*a*b*c* d-24*b^2*c^2))*arctanh((b*x^2+a)^(1/2)/x/b^(1/2))+b^(3/2)*(b*x^2+a)^(1/2)* x*f*(-3*b*d^3*e^3+1/4*d^2*((-6*d*x^2+24*c)*b+a*d)*f*e^2+1/4*d*f^2*((2*d^2* x^4+12*c*d*x^2-12*c^2)*b+a*d^2*x^2)*e+b*c^3*f^3))*((a*f-b*e)*e)^(1/2))/b^( 5/2)/f^4/e/(f*x^2+e)
Leaf count of result is larger than twice the leaf count of optimal. 827 vs. \(2 (233) = 466\).
Time = 15.93 (sec) , antiderivative size = 3401, normalized size of antiderivative = 12.83 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^3}{\left (e+f x^2\right )^2} \, dx=\text {Too large to display} \] Input:
integrate((b*x^2+a)^(1/2)*(d*x^2+c)^3/(f*x^2+e)^2,x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^3}{\left (e+f x^2\right )^2} \, dx=\int \frac {\sqrt {a + b x^{2}} \left (c + d x^{2}\right )^{3}}{\left (e + f x^{2}\right )^{2}}\, dx \] Input:
integrate((b*x**2+a)**(1/2)*(d*x**2+c)**3/(f*x**2+e)**2,x)
Output:
Integral(sqrt(a + b*x**2)*(c + d*x**2)**3/(e + f*x**2)**2, x)
\[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^3}{\left (e+f x^2\right )^2} \, dx=\int { \frac {\sqrt {b x^{2} + a} {\left (d x^{2} + c\right )}^{3}}{{\left (f x^{2} + e\right )}^{2}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)*(d*x^2+c)^3/(f*x^2+e)^2,x, algorithm="maxima")
Output:
integrate(sqrt(b*x^2 + a)*(d*x^2 + c)^3/(f*x^2 + e)^2, x)
Leaf count of result is larger than twice the leaf count of optimal. 720 vs. \(2 (233) = 466\).
Time = 0.17 (sec) , antiderivative size = 720, normalized size of antiderivative = 2.72 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^3}{\left (e+f x^2\right )^2} \, dx=\frac {1}{8} \, \sqrt {b x^{2} + a} {\left (\frac {2 \, d^{3} x^{2}}{f^{2}} - \frac {8 \, b^{2} d^{3} e f^{6} - 12 \, b^{2} c d^{2} f^{7} - a b d^{3} f^{7}}{b^{2} f^{9}}\right )} x - \frac {{\left (6 \, b^{\frac {3}{2}} d^{3} e^{4} - 12 \, b^{\frac {3}{2}} c d^{2} e^{3} f - 5 \, a \sqrt {b} d^{3} e^{3} f + 6 \, b^{\frac {3}{2}} c^{2} d e^{2} f^{2} + 9 \, a \sqrt {b} c d^{2} e^{2} f^{2} - 3 \, a \sqrt {b} c^{2} d e f^{3} - a \sqrt {b} c^{3} f^{4}\right )} \arctan \left (-\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} f + 2 \, b e - a f}{2 \, \sqrt {-b^{2} e^{2} + a b e f}}\right )}{2 \, \sqrt {-b^{2} e^{2} + a b e f} e f^{4}} - \frac {{\left (24 \, b^{2} d^{3} e^{2} - 48 \, b^{2} c d^{2} e f - 8 \, a b d^{3} e f + 24 \, b^{2} c^{2} d f^{2} + 12 \, a b c d^{2} f^{2} - a^{2} d^{3} f^{2}\right )} \log \left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2}\right )}{16 \, b^{\frac {3}{2}} f^{4}} - \frac {2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b^{2} d^{3} e^{4} - 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b^{2} c d^{2} e^{3} f - {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a b d^{3} e^{3} f + 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b^{2} c^{2} d e^{2} f^{2} + 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a b c d^{2} e^{2} f^{2} - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b^{2} c^{3} e f^{3} - 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a b c^{2} d e f^{3} + {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a b c^{3} f^{4} + a^{2} b d^{3} e^{3} f - 3 \, a^{2} b c d^{2} e^{2} f^{2} + 3 \, a^{2} b c^{2} d e f^{3} - a^{2} b c^{3} f^{4}}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} f + 4 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b e - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a f + a^{2} f\right )} \sqrt {b} e f^{4}} \] Input:
integrate((b*x^2+a)^(1/2)*(d*x^2+c)^3/(f*x^2+e)^2,x, algorithm="giac")
Output:
1/8*sqrt(b*x^2 + a)*(2*d^3*x^2/f^2 - (8*b^2*d^3*e*f^6 - 12*b^2*c*d^2*f^7 - a*b*d^3*f^7)/(b^2*f^9))*x - 1/2*(6*b^(3/2)*d^3*e^4 - 12*b^(3/2)*c*d^2*e^3 *f - 5*a*sqrt(b)*d^3*e^3*f + 6*b^(3/2)*c^2*d*e^2*f^2 + 9*a*sqrt(b)*c*d^2*e ^2*f^2 - 3*a*sqrt(b)*c^2*d*e*f^3 - a*sqrt(b)*c^3*f^4)*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))/(sqrt( -b^2*e^2 + a*b*e*f)*e*f^4) - 1/16*(24*b^2*d^3*e^2 - 48*b^2*c*d^2*e*f - 8*a *b*d^3*e*f + 24*b^2*c^2*d*f^2 + 12*a*b*c*d^2*f^2 - a^2*d^3*f^2)*log((sqrt( b)*x - sqrt(b*x^2 + a))^2)/(b^(3/2)*f^4) - (2*(sqrt(b)*x - sqrt(b*x^2 + a) )^2*b^2*d^3*e^4 - 6*(sqrt(b)*x - sqrt(b*x^2 + a))^2*b^2*c*d^2*e^3*f - (sqr t(b)*x - sqrt(b*x^2 + a))^2*a*b*d^3*e^3*f + 6*(sqrt(b)*x - sqrt(b*x^2 + a) )^2*b^2*c^2*d*e^2*f^2 + 3*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a*b*c*d^2*e^2*f^ 2 - 2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*b^2*c^3*e*f^3 - 3*(sqrt(b)*x - sqrt( b*x^2 + a))^2*a*b*c^2*d*e*f^3 + (sqrt(b)*x - sqrt(b*x^2 + a))^2*a*b*c^3*f^ 4 + a^2*b*d^3*e^3*f - 3*a^2*b*c*d^2*e^2*f^2 + 3*a^2*b*c^2*d*e*f^3 - a^2*b* c^3*f^4)/(((sqrt(b)*x - sqrt(b*x^2 + a))^4*f + 4*(sqrt(b)*x - sqrt(b*x^2 + a))^2*b*e - 2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a*f + a^2*f)*sqrt(b)*e*f^4)
Timed out. \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^3}{\left (e+f x^2\right )^2} \, dx=\int \frac {\sqrt {b\,x^2+a}\,{\left (d\,x^2+c\right )}^3}{{\left (f\,x^2+e\right )}^2} \,d x \] Input:
int(((a + b*x^2)^(1/2)*(c + d*x^2)^3)/(e + f*x^2)^2,x)
Output:
int(((a + b*x^2)^(1/2)*(c + d*x^2)^3)/(e + f*x^2)^2, x)
Time = 0.36 (sec) , antiderivative size = 2906, normalized size of antiderivative = 10.97 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^3}{\left (e+f x^2\right )^2} \, dx =\text {Too large to display} \] Input:
int((b*x^2+a)^(1/2)*(d*x^2+c)^3/(f*x^2+e)^2,x)
Output:
( - 4*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*b**2*c**3*e*f**4 - 4*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*b**2*c**3*f**5*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*b**2*c**2*d*e**2*f**3 - 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*b**2*c**2*d*e*f**4*x**2 + 36*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)/(sqr t(e)*sqrt(b)))*a*b**2*c*d**2*e**3*f**2 + 36*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)*s qrt(b)))*a*b**2*c*d**2*e**2*f**3*x**2 - 20*sqrt(e)*sqrt(a*f - b*e)*atan((s qrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sq rt(b)))*a*b**2*d**3*e**4*f - 20*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*b **2*d**3*e**3*f**2*x**2 + 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)))*b**3*c **2*d*e**3*f**2 + 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)))*b**3*c**2*d*e* *2*f**3*x**2 - 48*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(...