Integrand size = 30, antiderivative size = 395 \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^3} \, dx=\frac {d^3 x}{a f^3 \sqrt {a+b x^2}}+\frac {b (d e-c f) \left (2 a b e f \left (13 d^2 e^2+16 c d e f-5 c^2 f^2\right )-3 a^2 f^2 \left (11 d^2 e^2-2 c d e f-c^2 f^2\right )-8 b^2 e^2 \left (d^2 e^2+c d e f+c^2 f^2\right )\right ) x}{8 a e^2 f^3 (b e-a f)^3 \sqrt {a+b x^2}}+\frac {(d e-c f)^3 x}{4 e f^2 (b e-a f) \sqrt {a+b x^2} \left (e+f x^2\right )^2}+\frac {(d e-c f)^2 (3 a f (3 d e+c f)-4 b e (d e+2 c f)) x}{8 e^2 f^2 (b e-a f)^2 \sqrt {a+b x^2} \left (e+f x^2\right )}+\frac {3 (d e-c f) \left (8 b^2 c^2 e^2-4 a b c e (3 d e+c f)+a^2 \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{8 e^{5/2} (b e-a f)^{7/2}} \] Output:
d^3*x/a/f^3/(b*x^2+a)^(1/2)+1/8*b*(-c*f+d*e)*(2*a*b*e*f*(-5*c^2*f^2+16*c*d *e*f+13*d^2*e^2)-3*a^2*f^2*(-c^2*f^2-2*c*d*e*f+11*d^2*e^2)-8*b^2*e^2*(c^2* f^2+c*d*e*f+d^2*e^2))*x/a/e^2/f^3/(-a*f+b*e)^3/(b*x^2+a)^(1/2)+1/4*(-c*f+d *e)^3*x/e/f^2/(-a*f+b*e)/(b*x^2+a)^(1/2)/(f*x^2+e)^2+1/8*(-c*f+d*e)^2*(3*a *f*(c*f+3*d*e)-4*b*e*(2*c*f+d*e))*x/e^2/f^2/(-a*f+b*e)^2/(b*x^2+a)^(1/2)/( f*x^2+e)+3/8*(-c*f+d*e)*(8*b^2*c^2*e^2-4*a*b*c*e*(c*f+3*d*e)+a^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)/(-a*f+b*e)^(7/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 20.03 (sec) , antiderivative size = 2523, normalized size of antiderivative = 6.39 \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^3} \, dx=\text {Result too large to show} \] Input:
Integrate[(c + d*x^2)^3/((a + b*x^2)^(3/2)*(e + f*x^2)^3),x]
Output:
(d^3*x)/(a*f^3*Sqrt[a + b*x^2]) - (3*d^2*(d*e - c*f)*x*(-15*e*Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^2))] - 10*f*x^2*Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^ 2))] + 15*e*ArcTanh[Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^2))]] + 10*f*x^2*Ar cTanh[Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^2))]] + 2*e*(((b*e - a*f)*x^2)/(e *(a + b*x^2)))^(5/2)*Hypergeometric2F1[2, 5/2, 7/2, ((b*e - a*f)*x^2)/(e*( a + b*x^2))] + 2*f*x^2*(((b*e - a*f)*x^2)/(e*(a + b*x^2)))^(5/2)*Hypergeom etric2F1[2, 5/2, 7/2, ((b*e - a*f)*x^2)/(e*(a + b*x^2))]))/(5*e^2*f^3*(((b *e - a*f)*x^2)/(e*(a + b*x^2)))^(3/2)*(a + b*x^2)^(3/2)) + (d*(d*e - c*f)^ 2*x*(-2625*Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^2))] - (5250*f*x^2*Sqrt[((b* e - a*f)*x^2)/(e*(a + b*x^2))])/e - (2310*f^2*x^4*Sqrt[((b*e - a*f)*x^2)/( e*(a + b*x^2))])/e^2 + 70*(((b*e - a*f)*x^2)/(e*(a + b*x^2)))^(3/2) + (560 *f*x^2*(((b*e - a*f)*x^2)/(e*(a + b*x^2)))^(3/2))/e + (280*f^2*x^4*(((b*e - a*f)*x^2)/(e*(a + b*x^2)))^(3/2))/e^2 + 2625*ArcTanh[Sqrt[((b*e - a*f)*x ^2)/(e*(a + b*x^2))]] + (5250*f*x^2*ArcTanh[Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^2))]])/e + (2310*f^2*x^4*ArcTanh[Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^2 ))]])/e^2 - (945*(b*e - a*f)*x^2*ArcTanh[Sqrt[((b*e - a*f)*x^2)/(e*(a + b* x^2))]])/(e*(a + b*x^2)) + (2310*f*(-(b*e) + a*f)*x^4*ArcTanh[Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^2))]])/(e^2*(a + b*x^2)) + (1050*f^2*(-(b*e) + a*f) *x^6*ArcTanh[Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^2))]])/(e^3*(a + b*x^2)) + 24*(((b*e - a*f)*x^2)/(e*(a + b*x^2)))^(7/2)*HypergeometricPFQ[{2, 2, ...
Leaf count is larger than twice the leaf count of optimal. \(836\) vs. \(2(395)=790\).
Time = 1.13 (sec) , antiderivative size = 836, normalized size of antiderivative = 2.12, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {425, 425, 402, 25, 27, 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.
| \(\displaystyle \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 425 |
| \(\displaystyle \frac {d \int \frac {\left (d x^2+c\right )^2}{\left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^3}dx}{f}\) |
| \(\Big \downarrow \) 425 |
| \(\displaystyle \frac {d \left (\frac {d \int \frac {d x^2+c}{\left (b x^2+a\right )^{3/2} \left (f x^2+e\right )}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \int \frac {d x^2+c}{\left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {x (b c-a d)}{a \sqrt {a+b x^2} (b e-a f)}-\frac {\int -\frac {a (d e-c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{a (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}-\frac {\int -\frac {2 (b c-a d) f x^2+a (d e-c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{a (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}-\frac {\int -\frac {2 (b c-a d) f x^2+a (d e-c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{a (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\int -\frac {4 (b c-a d) f x^2+a (d e-c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{a (b e-a f)}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {\int \frac {a (d e-c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{a (b e-a f)}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\int \frac {2 (b c-a d) f x^2+a (d e-c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{a (b e-a f)}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\int \frac {2 (b c-a d) f x^2+a (d e-c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{a (b e-a f)}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\int \frac {4 (b c-a d) f x^2+a (d e-c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{a (b e-a f)}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{b e-a f}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\int \frac {2 (b c-a d) f x^2+a (d e-c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{a (b e-a f)}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\int \frac {2 (b c-a d) f x^2+a (d e-c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{a (b e-a f)}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\int \frac {4 (b c-a d) f x^2+a (d e-c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{a (b e-a f)}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {(d e-c f) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{b e-a f}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\int \frac {2 (b c-a d) f x^2+a (d e-c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{a (b e-a f)}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\int \frac {2 (b c-a d) f x^2+a (d e-c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{a (b e-a f)}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\int \frac {4 (b c-a d) f x^2+a (d e-c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{a (b e-a f)}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} (b e-a f)^{3/2}}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\int \frac {2 (b c-a d) f x^2+a (d e-c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{a (b e-a f)}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\int \frac {2 (b c-a d) f x^2+a (d e-c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{a (b e-a f)}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\int \frac {4 (b c-a d) f x^2+a (d e-c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{a (b e-a f)}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} (b e-a f)^{3/2}}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\frac {\int \frac {a (2 b e (d e-2 c f)+a f (d e+c f))}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}+\frac {f x \sqrt {a+b x^2} (a c f-3 a d e+2 b c e)}{2 e \left (e+f x^2\right ) (b e-a f)}}{a (b e-a f)}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\frac {\int \frac {a (2 b e (d e-2 c f)+a f (d e+c f))}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}+\frac {f x \sqrt {a+b x^2} (a c f-3 a d e+2 b c e)}{2 e \left (e+f x^2\right ) (b e-a f)}}{a (b e-a f)}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\frac {\int \frac {2 b f (4 b c e-5 a d e+a c f) x^2+a (4 b e (d e-2 c f)+a f (d e+3 c f))}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}+\frac {f x \sqrt {a+b x^2} (a c f-5 a d e+4 b c e)}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} (b e-a f)^{3/2}}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\frac {a (a f (c f+d e)+2 b e (d e-2 c f)) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}+\frac {f x \sqrt {a+b x^2} (a c f-3 a d e+2 b c e)}{2 e \left (e+f x^2\right ) (b e-a f)}}{a (b e-a f)}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\frac {a (a f (c f+d e)+2 b e (d e-2 c f)) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}+\frac {f x \sqrt {a+b x^2} (a c f-3 a d e+2 b c e)}{2 e \left (e+f x^2\right ) (b e-a f)}}{a (b e-a f)}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\frac {\int \frac {2 b f (4 b c e-5 a d e+a c f) x^2+a (4 b e (d e-2 c f)+a f (d e+3 c f))}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}+\frac {f x \sqrt {a+b x^2} (a c f-5 a d e+4 b c e)}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} (b e-a f)^{3/2}}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\frac {a (a f (c f+d e)+2 b e (d e-2 c f)) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}+\frac {f x \sqrt {a+b x^2} (a c f-3 a d e+2 b c e)}{2 e \left (e+f x^2\right ) (b e-a f)}}{a (b e-a f)}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\frac {a (a f (c f+d e)+2 b e (d e-2 c f)) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}+\frac {f x \sqrt {a+b x^2} (a c f-3 a d e+2 b c e)}{2 e \left (e+f x^2\right ) (b e-a f)}}{a (b e-a f)}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\frac {\int \frac {2 b f (4 b c e-5 a d e+a c f) x^2+a (4 b e (d e-2 c f)+a f (d e+3 c f))}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}+\frac {f x \sqrt {a+b x^2} (a c f-5 a d e+4 b c e)}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} (b e-a f)^{3/2}}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (a f (c f+d e)+2 b e (d e-2 c f))}{2 e^{3/2} (b e-a f)^{3/2}}+\frac {f x \sqrt {a+b x^2} (a c f-3 a d e+2 b c e)}{2 e \left (e+f x^2\right ) (b e-a f)}}{a (b e-a f)}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (a f (c f+d e)+2 b e (d e-2 c f))}{2 e^{3/2} (b e-a f)^{3/2}}+\frac {f x \sqrt {a+b x^2} (a c f-3 a d e+2 b c e)}{2 e \left (e+f x^2\right ) (b e-a f)}}{a (b e-a f)}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\frac {\int \frac {2 b f (4 b c e-5 a d e+a c f) x^2+a (4 b e (d e-2 c f)+a f (d e+3 c f))}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}+\frac {f x \sqrt {a+b x^2} (a c f-5 a d e+4 b c e)}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {(b c-a d) x}{a (b e-a f) \sqrt {b x^2+a}}+\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} (b e-a f)^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {(b c-a d) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )}+\frac {\frac {f (2 b c e-3 a d e+a c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {a (2 b e (d e-2 c f)+a f (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{a (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(b c-a d) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )}+\frac {\frac {f (2 b c e-3 a d e+a c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {a (2 b e (d e-2 c f)+a f (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{a (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(b c-a d) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}+\frac {\frac {f (4 b c e-5 a d e+a c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {f \left (-f (d e+3 c f) a^2-2 b e (7 d e-5 c f) a+8 b^2 c e^2\right ) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\int \frac {a \left (8 b^2 (d e-3 c f) e^2+4 a b f (2 d e+3 c f) e-a^2 f^2 (d e+3 c f)\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}}{4 e (b e-a f)}}{a (b e-a f)}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {(b c-a d) x}{a (b e-a f) \sqrt {b x^2+a}}+\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} (b e-a f)^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {(b c-a d) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )}+\frac {\frac {f (2 b c e-3 a d e+a c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {a (2 b e (d e-2 c f)+a f (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{a (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(b c-a d) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )}+\frac {\frac {f (2 b c e-3 a d e+a c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {a (2 b e (d e-2 c f)+a f (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{a (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(b c-a d) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}+\frac {\frac {f (4 b c e-5 a d e+a c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {f \left (-f (d e+3 c f) a^2-2 b e (7 d e-5 c f) a+8 b^2 c e^2\right ) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {a \left (8 b^2 (d e-3 c f) e^2+4 a b f (2 d e+3 c f) e-a^2 f^2 (d e+3 c f)\right ) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}}{4 e (b e-a f)}}{a (b e-a f)}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {(b c-a d) x}{a (b e-a f) \sqrt {b x^2+a}}+\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} (b e-a f)^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {(b c-a d) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )}+\frac {\frac {f (2 b c e-3 a d e+a c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {a (2 b e (d e-2 c f)+a f (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{a (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(b c-a d) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )}+\frac {\frac {f (2 b c e-3 a d e+a c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {a (2 b e (d e-2 c f)+a f (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{a (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(b c-a d) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}+\frac {\frac {f (4 b c e-5 a d e+a c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {f \left (-f (d e+3 c f) a^2-2 b e (7 d e-5 c f) a+8 b^2 c e^2\right ) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {a \left (8 b^2 (d e-3 c f) e^2+4 a b f (2 d e+3 c f) e-a^2 f^2 (d e+3 c f)\right ) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}}{4 e (b e-a f)}}{a (b e-a f)}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {(b c-a d) x}{a (b e-a f) \sqrt {b x^2+a}}+\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} (b e-a f)^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {(b c-a d) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )}+\frac {\frac {f (2 b c e-3 a d e+a c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {a (2 b e (d e-2 c f)+a f (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{a (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(b c-a d) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )}+\frac {\frac {f (2 b c e-3 a d e+a c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {a (2 b e (d e-2 c f)+a f (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{a (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(b c-a d) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}+\frac {\frac {f (4 b c e-5 a d e+a c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {f \left (-f (d e+3 c f) a^2-2 b e (7 d e-5 c f) a+8 b^2 c e^2\right ) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {a \left (8 b^2 (d e-3 c f) e^2+4 a b f (2 d e+3 c f) e-a^2 f^2 (d e+3 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}}{a (b e-a f)}\right )}{f}\right )}{f}\) |
Input:
Int[(c + d*x^2)^3/((a + b*x^2)^(3/2)*(e + f*x^2)^3),x]
Output:
(d*((d*(((b*c - a*d)*x)/(a*(b*e - a*f)*Sqrt[a + b*x^2]) + ((d*e - c*f)*Arc Tanh[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])])/(Sqrt[e]*(b*e - a*f)^ (3/2))))/f - ((d*e - c*f)*(((b*c - a*d)*x)/(a*(b*e - a*f)*Sqrt[a + b*x^2]* (e + f*x^2)) + ((f*(2*b*c*e - 3*a*d*e + a*c*f)*x*Sqrt[a + b*x^2])/(2*e*(b* e - a*f)*(e + f*x^2)) + (a*(2*b*e*(d*e - 2*c*f) + a*f*(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)))/(a*(b*e - a*f))))/f))/f - ((d*e - c*f)*((d*(((b*c - a*d)*x)/(a*(b*e - a*f)*Sqrt[a + b*x^2]*(e + f*x^2)) + ((f*(2*b*c*e - 3*a*d*e + a*c*f)*x*Sq rt[a + b*x^2])/(2*e*(b*e - a*f)*(e + f*x^2)) + (a*(2*b*e*(d*e - 2*c*f) + a *f*(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)))/(a*(b*e - a*f))))/f - ((d*e - c*f)*(((b*c - a *d)*x)/(a*(b*e - a*f)*Sqrt[a + b*x^2]*(e + f*x^2)^2) + ((f*(4*b*c*e - 5*a* d*e + a*c*f)*x*Sqrt[a + b*x^2])/(4*e*(b*e - a*f)*(e + f*x^2)^2) + ((f*(8*b ^2*c*e^2 - 2*a*b*e*(7*d*e - 5*c*f) - a^2*f*(d*e + 3*c*f))*x*Sqrt[a + b*x^2 ])/(2*e*(b*e - a*f)*(e + f*x^2)) + (a*(8*b^2*e^2*(d*e - 3*c*f) - a^2*f^2*( d*e + 3*c*f) + 4*a*b*e*f*(2*d*e + 3*c*f))*ArcTanh[(Sqrt[b*e - a*f]*x)/(Sqr t[e]*Sqrt[a + b*x^2])])/(2*e^(3/2)*(b*e - a*f)^(3/2)))/(4*e*(b*e - a*f)))/ (a*(b*e - a*f))))/f))/f
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[(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]*((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)^(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[((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 = 1.21 (sec) , antiderivative size = 460, normalized size of antiderivative = 1.16
| method | result | size |
| pseudoelliptic | \(\frac {-\frac {3 \left (\left (5 a^{2} d^{2}-12 a b c d +8 b^{2} c^{2}\right ) e^{2}+2 a c f \left (a d -2 b c \right ) e +a^{2} c^{2} f^{2}\right ) a \sqrt {b \,x^{2}+a}\, \left (c f -d e \right ) \left (f \,x^{2}+e \right )^{2} \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )}{8}+\frac {5 \sqrt {\left (a f -b e \right ) e}\, \left (\left (3 a^{3} d^{3}-\frac {36 \left (-\frac {5 x^{2} d}{36}+c \right ) d^{2} b \,a^{2}}{5}+\frac {24 d \,b^{2} \left (-\frac {1}{12} d^{2} x^{4}-\frac {1}{2} c d \,x^{2}+c^{2}\right ) a}{5}-\frac {8 b^{3} c^{3}}{5}\right ) e^{4}-\frac {9 \left (d^{2} \left (-\frac {25 x^{2} d}{9}+c \right ) a^{3}-\frac {8 d \left (\frac {3}{8} d^{2} x^{4}-\frac {21}{8} c d \,x^{2}+c^{2}\right ) b \,a^{2}}{3}-8 \left (-\frac {x^{2} d}{12}+c \right ) c d \,b^{2} x^{2} a +\frac {16 b^{3} c^{3} x^{2}}{9}\right ) f \,e^{3}}{5}-\frac {3 f^{2} \left (d \left (-\frac {8}{3} d^{2} x^{4}+5 c d \,x^{2}+c^{2}\right ) a^{3}+4 c \left (\frac {13}{4} d^{2} x^{4}-\frac {5}{4} c d \,x^{2}+c^{2}\right ) b \,a^{2}+4 \left (-\frac {7 x^{2} d}{2}+c \right ) c^{2} b^{2} x^{2} a +\frac {8 c^{3} b^{3} x^{4}}{3}\right ) e^{2}}{5}+a \,c^{2} \left (b \,x^{2}+a \right ) f^{3} \left (a \left (\frac {3 x^{2} d}{5}+c \right )-2 x^{2} b c \right ) e +\frac {3 a^{2} c^{3} f^{4} x^{2} \left (b \,x^{2}+a \right )}{5}\right ) x}{8}}{\left (f \,x^{2}+e \right )^{2} e^{2} \left (a f -b e \right )^{3} \sqrt {\left (a f -b e \right ) e}\, \sqrt {b \,x^{2}+a}\, a}\) | \(460\) |
| default | \(\text {Expression too large to display}\) | \(4273\) |
Input:
int((d*x^2+c)^3/(b*x^2+a)^(3/2)/(f*x^2+e)^3,x,method=_RETURNVERBOSE)
Output:
5/8*(-3/5*((5*a^2*d^2-12*a*b*c*d+8*b^2*c^2)*e^2+2*a*c*f*(a*d-2*b*c)*e+a^2* c^2*f^2)*a*(b*x^2+a)^(1/2)*(c*f-d*e)*(f*x^2+e)^2*arctan(e*(b*x^2+a)^(1/2)/ x/((a*f-b*e)*e)^(1/2))+((a*f-b*e)*e)^(1/2)*((3*a^3*d^3-36/5*(-5/36*x^2*d+c )*d^2*b*a^2+24/5*d*b^2*(-1/12*d^2*x^4-1/2*c*d*x^2+c^2)*a-8/5*b^3*c^3)*e^4- 9/5*(d^2*(-25/9*x^2*d+c)*a^3-8/3*d*(3/8*d^2*x^4-21/8*c*d*x^2+c^2)*b*a^2-8* (-1/12*x^2*d+c)*c*d*b^2*x^2*a+16/9*b^3*c^3*x^2)*f*e^3-3/5*f^2*(d*(-8/3*d^2 *x^4+5*c*d*x^2+c^2)*a^3+4*c*(13/4*d^2*x^4-5/4*c*d*x^2+c^2)*b*a^2+4*(-7/2*x ^2*d+c)*c^2*b^2*x^2*a+8/3*c^3*b^3*x^4)*e^2+a*c^2*(b*x^2+a)*f^3*(a*(3/5*x^2 *d+c)-2*x^2*b*c)*e+3/5*a^2*c^3*f^4*x^2*(b*x^2+a))*x)/((a*f-b*e)*e)^(1/2)/( b*x^2+a)^(1/2)/(f*x^2+e)^2/e^2/(a*f-b*e)^3/a
Leaf count of result is larger than twice the leaf count of optimal. 1491 vs. \(2 (369) = 738\).
Time = 20.43 (sec) , antiderivative size = 3022, normalized size of antiderivative = 7.65 \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^3} \, dx=\text {Too large to display} \] Input:
integrate((d*x^2+c)^3/(b*x^2+a)^(3/2)/(f*x^2+e)^3,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate((d*x**2+c)**3/(b*x**2+a)**(3/2)/(f*x**2+e)**3,x)
Output:
Timed out
\[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^3} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{3}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{3}} \,d x } \] Input:
integrate((d*x^2+c)^3/(b*x^2+a)^(3/2)/(f*x^2+e)^3,x, algorithm="maxima")
Output:
integrate((d*x^2 + c)^3/((b*x^2 + a)^(3/2)*(f*x^2 + e)^3), x)
Leaf count of result is larger than twice the leaf count of optimal. 1902 vs. \(2 (369) = 738\).
Time = 0.71 (sec) , antiderivative size = 1902, normalized size of antiderivative = 4.82 \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^3} \, dx=\text {Too large to display} \] Input:
integrate((d*x^2+c)^3/(b*x^2+a)^(3/2)/(f*x^2+e)^3,x, algorithm="giac")
Output:
(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*x/((a*b^3*e^3 - 3*a^2* b^2*e^2*f + 3*a^3*b*e*f^2 - a^4*f^3)*sqrt(b*x^2 + a)) - 3/8*(8*b^(5/2)*c^2 *d*e^3 - 12*a*b^(3/2)*c*d^2*e^3 + 5*a^2*sqrt(b)*d^3*e^3 - 8*b^(5/2)*c^3*e^ 2*f + 8*a*b^(3/2)*c^2*d*e^2*f - 3*a^2*sqrt(b)*c*d^2*e^2*f + 4*a*b^(3/2)*c^ 3*e*f^2 - a^2*sqrt(b)*c^2*d*e*f^2 - a^2*sqrt(b)*c^3*f^3)*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^ 3*e^5 - 3*a*b^2*e^4*f + 3*a^2*b*e^3*f^2 - a^3*e^2*f^3)*sqrt(-b^2*e^2 + a*b *e*f)) + 1/4*(8*(sqrt(b)*x - sqrt(b*x^2 + a))^6*b^(5/2)*d^3*e^5*f - 24*(sq rt(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 + 36*(sqrt(b)*x - sqrt(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 + 16*(sqrt(b)*x - sqrt(b*x^2 + a))^6*b^(5/2)*c^3*e^2*f^4 - 15*(sq rt(b)*x - sqrt(b*x^2 + a))^6*a^2*sqrt(b)*c*d^2*e^2*f^4 - 12*(sqrt(b)*x - s qrt(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 + 16*(sqrt(b)*x - sqrt(b*x^2 + a))^4*b^(7/2)*d^3*e^6 + 48*(sqrt(b)*x - sqrt(b*x^2 + a))^4*b^(7/2)*c*d^2*e^5*f - 88*(sqrt(b)*x - sqrt(b*x^2 + a ))^4*a*b^(5/2)*d^3*e^5*f - 144*(sqrt(b)*x - sqrt(b*x^2 + a))^4*b^(7/2)*c^2 *d*e^4*f^2 + 72*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a*b^(5/2)*c*d^2*e^4*f^2 + 78*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^2*b^(3/2)*d^3*e^4*f^2 + 80*(sqrt(b...
Timed out. \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^3} \, dx=\int \frac {{\left (d\,x^2+c\right )}^3}{{\left (b\,x^2+a\right )}^{3/2}\,{\left (f\,x^2+e\right )}^3} \,d x \] Input:
int((c + d*x^2)^3/((a + b*x^2)^(3/2)*(e + f*x^2)^3),x)
Output:
int((c + d*x^2)^3/((a + b*x^2)^(3/2)*(e + f*x^2)^3), x)
\[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^3} \, dx=\int \frac {\left (d \,x^{2}+c \right )^{3}}{\left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (f \,x^{2}+e \right )^{3}}d x \] Input:
int((d*x^2+c)^3/(b*x^2+a)^(3/2)/(f*x^2+e)^3,x)
Output:
int((d*x^2+c)^3/(b*x^2+a)^(3/2)/(f*x^2+e)^3,x)