\(\int \frac {(c+d x^2)^2}{(a+b x^2)^{3/2} (e+f x^2)^3} \, dx\) [347]

Optimal result

Integrand size = 30, antiderivative size = 427 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^3} \, dx=\frac {d (4 b d e-4 b c f+a d f) x}{4 a e f^2 (b e-a f) \sqrt {a+b x^2}}+\frac {b \left (a^2 f^2 \left (5 d^2 e^2+6 c d e f-3 c^2 f^2\right )-8 b^2 e^2 \left (d^2 e^2-c d e f-c^2 f^2\right )+2 a b e f \left (9 d^2 e^2-22 c d e f+5 c^2 f^2\right )\right ) x}{8 a e^2 f^2 (b e-a f)^3 \sqrt {a+b x^2}}-\frac {f x \left (c+d x^2\right )^2}{4 e (b e-a f) \sqrt {a+b x^2} \left (e+f x^2\right )^2}-\frac {(d e-c f) (4 b e (d e-2 c f)+a f (d e+3 c f)) x}{8 e^2 f (b e-a f)^2 \sqrt {a+b x^2} \left (e+f x^2\right )}+\frac {\left (8 b^2 c e^2 (2 d e-3 c f)-4 a b e \left (3 d^2 e^2-4 c d e f-3 c^2 f^2\right )-a^2 f \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{8 e^{5/2} (b e-a f)^{7/2}} \] Output:

1/4*d*(a*d*f-4*b*c*f+4*b*d*e)*x/a/e/f^2/(-a*f+b*e)/(b*x^2+a)^(1/2)+1/8*b*( 
a^2*f^2*(-3*c^2*f^2+6*c*d*e*f+5*d^2*e^2)-8*b^2*e^2*(-c^2*f^2-c*d*e*f+d^2*e 
^2)+2*a*b*e*f*(5*c^2*f^2-22*c*d*e*f+9*d^2*e^2))*x/a/e^2/f^2/(-a*f+b*e)^3/( 
b*x^2+a)^(1/2)-1/4*f*x*(d*x^2+c)^2/e/(-a*f+b*e)/(b*x^2+a)^(1/2)/(f*x^2+e)^ 
2-1/8*(-c*f+d*e)*(4*b*e*(-2*c*f+d*e)+a*f*(3*c*f+d*e))*x/e^2/f/(-a*f+b*e)^2 
/(b*x^2+a)^(1/2)/(f*x^2+e)+1/8*(8*b^2*c*e^2*(-3*c*f+2*d*e)-4*a*b*e*(-3*c^2 
*f^2-4*c*d*e*f+3*d^2*e^2)-a^2*f*(3*c^2*f^2+2*c*d*e*f+3*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)
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 20.66 (sec) , antiderivative size = 2542, normalized size of antiderivative = 5.95 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^3} \, dx=\text {Result too large to show} \] Input:

Integrate[(c + d*x^2)^2/((a + b*x^2)^(3/2)*(e + f*x^2)^3),x]
 

Output:

(d^2*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*ArcTanh[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)*Hypergeometric2F1[2, 5/2, 7/2, ((b*e - a*f)*x^2)/(e*(a + 
b*x^2))]))/(5*a*e^2*f^2*(((b*e - a*f)*x^2)/(e*(a + b*x^2)))^(3/2)*Sqrt[a + 
 b*x^2]*(1 + (b*x^2)/a)) - (d*(d*e - c*f)*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*ArcT 
anh[Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^2))]])/e^2 - (945*(b*e - a*f)*x^2*A 
rcTanh[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, 5/2}, {1, 9/2}, ((b*e - a*f)*x^2)/...
 

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 517, normalized size of antiderivative = 1.21, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.367, Rules used = {425, 402, 25, 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.

\(\displaystyle \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^3} \, dx\)

\(\Big \downarrow \) 425

\(\displaystyle \frac {d \int \frac {d x^2+c}{\left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^3}dx}{f}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {d \left (\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}-\frac {\int -\frac {2 (b c-a d) f x^2+a (d e-c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{a (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\int -\frac {4 (b c-a d) f x^2+a (d e-c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{a (b e-a f)}\right )}{f}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {d \left (\frac {\int \frac {2 (b c-a d) f x^2+a (d e-c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{a (b e-a f)}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\int \frac {4 (b c-a d) f x^2+a (d e-c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{a (b e-a f)}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{f}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {d \left (\frac {\frac {\int \frac {a (2 b e (d e-2 c f)+a f (d e+c f))}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}+\frac {f x \sqrt {a+b x^2} (a c f-3 a d e+2 b c e)}{2 e \left (e+f x^2\right ) (b e-a f)}}{a (b e-a f)}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\frac {\int \frac {2 b f (4 b c e-5 a d e+a c f) x^2+a (4 b e (d e-2 c f)+a f (d e+3 c f))}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}+\frac {f x \sqrt {a+b x^2} (a c f-5 a d e+4 b c e)}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {d \left (\frac {\frac {a (a f (c f+d e)+2 b e (d e-2 c f)) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}+\frac {f x \sqrt {a+b x^2} (a c f-3 a d e+2 b c e)}{2 e \left (e+f x^2\right ) (b e-a f)}}{a (b e-a f)}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\frac {\int \frac {2 b f (4 b c e-5 a d e+a c f) x^2+a (4 b e (d e-2 c f)+a f (d e+3 c f))}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}+\frac {f x \sqrt {a+b x^2} (a c f-5 a d e+4 b c e)}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{f}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {d \left (\frac {\frac {a (a f (c f+d e)+2 b e (d e-2 c f)) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}+\frac {f x \sqrt {a+b x^2} (a c f-3 a d e+2 b c e)}{2 e \left (e+f x^2\right ) (b e-a f)}}{a (b e-a f)}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\frac {\int \frac {2 b f (4 b c e-5 a d e+a c f) x^2+a (4 b e (d e-2 c f)+a f (d e+3 c f))}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}+\frac {f x \sqrt {a+b x^2} (a c f-5 a d e+4 b c e)}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{f}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {d \left (\frac {\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (a f (c f+d e)+2 b e (d e-2 c f))}{2 e^{3/2} (b e-a f)^{3/2}}+\frac {f x \sqrt {a+b x^2} (a c f-3 a d e+2 b c e)}{2 e \left (e+f x^2\right ) (b e-a f)}}{a (b e-a f)}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\frac {\int \frac {2 b f (4 b c e-5 a d e+a c f) x^2+a (4 b e (d e-2 c f)+a f (d e+3 c f))}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}+\frac {f x \sqrt {a+b x^2} (a c f-5 a d e+4 b c e)}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{f}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {d \left (\frac {\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (a f (c f+d e)+2 b e (d e-2 c f))}{2 e^{3/2} (b e-a f)^{3/2}}+\frac {f x \sqrt {a+b x^2} (a c f-3 a d e+2 b c e)}{2 e \left (e+f x^2\right ) (b e-a f)}}{a (b e-a f)}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\frac {\frac {\int \frac {a \left (8 b^2 (d e-3 c f) e^2+4 a b f (2 d e+3 c f) e-a^2 f^2 (d e+3 c f)\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}+\frac {f x \sqrt {a+b x^2} \left (a^2 (-f) (3 c f+d e)-2 a b e (7 d e-5 c f)+8 b^2 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}+\frac {f x \sqrt {a+b x^2} (a c f-5 a d e+4 b c e)}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {d \left (\frac {\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (a f (c f+d e)+2 b e (d e-2 c f))}{2 e^{3/2} (b e-a f)^{3/2}}+\frac {f x \sqrt {a+b x^2} (a c f-3 a d e+2 b c e)}{2 e \left (e+f x^2\right ) (b e-a f)}}{a (b e-a f)}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\frac {\frac {a \left (-a^2 f^2 (3 c f+d e)+4 a b e f (3 c f+2 d e)+8 b^2 e^2 (d e-3 c f)\right ) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}+\frac {f x \sqrt {a+b x^2} \left (a^2 (-f) (3 c f+d e)-2 a b e (7 d e-5 c f)+8 b^2 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}+\frac {f x \sqrt {a+b x^2} (a c f-5 a d e+4 b c e)}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{f}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {d \left (\frac {\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (a f (c f+d e)+2 b e (d e-2 c f))}{2 e^{3/2} (b e-a f)^{3/2}}+\frac {f x \sqrt {a+b x^2} (a c f-3 a d e+2 b c e)}{2 e \left (e+f x^2\right ) (b e-a f)}}{a (b e-a f)}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\frac {\frac {a \left (-a^2 f^2 (3 c f+d e)+4 a b e f (3 c f+2 d e)+8 b^2 e^2 (d e-3 c f)\right ) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}+\frac {f x \sqrt {a+b x^2} \left (a^2 (-f) (3 c f+d e)-2 a b e (7 d e-5 c f)+8 b^2 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}+\frac {f x \sqrt {a+b x^2} (a c f-5 a d e+4 b c e)}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{f}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {d \left (\frac {\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (a f (c f+d e)+2 b e (d e-2 c f))}{2 e^{3/2} (b e-a f)^{3/2}}+\frac {f x \sqrt {a+b x^2} (a c f-3 a d e+2 b c e)}{2 e \left (e+f x^2\right ) (b e-a f)}}{a (b e-a f)}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\frac {\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (-a^2 f^2 (3 c f+d e)+4 a b e f (3 c f+2 d e)+8 b^2 e^2 (d e-3 c f)\right )}{2 e^{3/2} (b e-a f)^{3/2}}+\frac {f x \sqrt {a+b x^2} \left (a^2 (-f) (3 c f+d e)-2 a b e (7 d e-5 c f)+8 b^2 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}+\frac {f x \sqrt {a+b x^2} (a c f-5 a d e+4 b c e)}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{f}\)

Input:

Int[(c + d*x^2)^2/((a + b*x^2)^(3/2)*(e + f*x^2)^3),x]
 

Output:

(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*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)/(S 
qrt[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))*ArcT 
anh[(Sqrt[b*e - a*f]*x)/(Sqrt[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
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 221

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]
 

rule 291

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]
 

rule 402

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]
 

rule 425

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]
 

Maple [A] (verified)

Time = 1.14 (sec) , antiderivative size = 408, normalized size of antiderivative = 0.96

method	result	size
pseudoelliptic	\(\frac {-\frac {3 a \sqrt {b \,x^{2}+a}\, \left (4 \left (a b \,d^{2}-\frac {4}{3} b^{2} c d \right ) e^{3}+f \left (a^{2} d^{2}-\frac {16}{3} a b c d +8 b^{2} c^{2}\right ) e^{2}+\frac {2 a c \,f^{2} \left (a d -6 b c \right ) e}{3}+a^{2} c^{2} f^{3}\right ) \left (f \,x^{2}+e \right )^{2} \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )}{8}+\frac {5 \left (-\frac {12 \left (a^{2} d^{2}-\frac {4 \left (-\frac {x^{2} d}{4}+c \right ) d b a}{3}+\frac {2 b^{2} c^{2}}{3}\right ) b \,e^{4}}{5}-\frac {3 f \left (a^{3} d^{2}-\frac {16 d \left (-\frac {21 x^{2} d}{16}+c \right ) b \,a^{2}}{3}-16 d \left (-\frac {x^{2} d}{24}+c \right ) b^{2} x^{2} a +\frac {16 x^{2} b^{3} c^{2}}{3}\right ) e^{3}}{5}-\frac {2 \left (d \left (\frac {5 x^{2} d}{2}+c \right ) a^{3}+6 \left (\frac {13}{12} d^{2} x^{4}-\frac {5}{6} c d \,x^{2}+c^{2}\right ) b \,a^{2}+6 \left (-\frac {7 x^{2} d}{3}+c \right ) c \,b^{2} x^{2} a +4 c^{2} b^{3} x^{4}\right ) f^{2} e^{2}}{5}+a c \left (b \,x^{2}+a \right ) \left (a \left (\frac {2 x^{2} d}{5}+c \right )-2 x^{2} b c \right ) f^{3} e +\frac {3 a^{2} c^{2} f^{4} x^{2} \left (b \,x^{2}+a \right )}{5}\right ) \sqrt {\left (a f -b e \right ) e}\, x}{8}}{\left (f \,x^{2}+e \right )^{2} e^{2} \left (a f -b e \right )^{3} \sqrt {\left (a f -b e \right ) e}\, \sqrt {b \,x^{2}+a}\, a}\)	\(408\)
default	\(\text {Expression too large to display}\)	\(4175\)

Input:

int((d*x^2+c)^2/(b*x^2+a)^(3/2)/(f*x^2+e)^3,x,method=_RETURNVERBOSE)
 

Output:

5/8/((a*f-b*e)*e)^(1/2)/(b*x^2+a)^(1/2)*(-3/5*a*(b*x^2+a)^(1/2)*(4*(a*b*d^ 
2-4/3*b^2*c*d)*e^3+f*(a^2*d^2-16/3*a*b*c*d+8*b^2*c^2)*e^2+2/3*a*c*f^2*(a*d 
-6*b*c)*e+a^2*c^2*f^3)*(f*x^2+e)^2*arctan(e*(b*x^2+a)^(1/2)/x/((a*f-b*e)*e 
)^(1/2))+(-12/5*(a^2*d^2-4/3*(-1/4*x^2*d+c)*d*b*a+2/3*b^2*c^2)*b*e^4-3/5*f 
*(a^3*d^2-16/3*d*(-21/16*x^2*d+c)*b*a^2-16*d*(-1/24*x^2*d+c)*b^2*x^2*a+16/ 
3*x^2*b^3*c^2)*e^3-2/5*(d*(5/2*x^2*d+c)*a^3+6*(13/12*d^2*x^4-5/6*c*d*x^2+c 
^2)*b*a^2+6*(-7/3*x^2*d+c)*c*b^2*x^2*a+4*c^2*b^3*x^4)*f^2*e^2+a*c*(b*x^2+a 
)*(a*(2/5*x^2*d+c)-2*x^2*b*c)*f^3*e+3/5*a^2*c^2*f^4*x^2*(b*x^2+a))*((a*f-b 
*e)*e)^(1/2)*x)/(f*x^2+e)^2/e^2/(a*f-b*e)^3/a
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1288 vs. \(2 (400) = 800\).

Time = 9.58 (sec) , antiderivative size = 2616, normalized size of antiderivative = 6.13 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((d*x^2+c)^2/(b*x^2+a)^(3/2)/(f*x^2+e)^3,x, algorithm="fricas")
 

Output:

[-1/32*((3*a^4*c^2*e^2*f^3 + (3*a^3*b*c^2*f^5 - 4*(4*a*b^3*c*d - 3*a^2*b^2 
*d^2)*e^3*f^2 + (24*a*b^3*c^2 - 16*a^2*b^2*c*d + 3*a^3*b*d^2)*e^2*f^3 - 2* 
(6*a^2*b^2*c^2 - a^3*b*c*d)*e*f^4)*x^6 - 4*(4*a^2*b^2*c*d - 3*a^3*b*d^2)*e 
^5 + (24*a^2*b^2*c^2 - 16*a^3*b*c*d + 3*a^4*d^2)*e^4*f - 2*(6*a^3*b*c^2 - 
a^4*c*d)*e^3*f^2 + (3*a^4*c^2*f^5 - 8*(4*a*b^3*c*d - 3*a^2*b^2*d^2)*e^4*f 
+ 6*(8*a*b^3*c^2 - 8*a^2*b^2*c*d + 3*a^3*b*d^2)*e^3*f^2 - 3*(4*a^3*b*c*d - 
 a^4*d^2)*e^2*f^3 - 2*(3*a^3*b*c^2 - a^4*c*d)*e*f^4)*x^4 + (6*a^4*c^2*e*f^ 
4 - 4*(4*a*b^3*c*d - 3*a^2*b^2*d^2)*e^5 + 3*(8*a*b^3*c^2 - 16*a^2*b^2*c*d 
+ 9*a^3*b*d^2)*e^4*f + 6*(6*a^2*b^2*c^2 - 5*a^3*b*c*d + a^4*d^2)*e^3*f^2 - 
 (21*a^3*b*c^2 - 4*a^4*c*d)*e^2*f^3)*x^2)*sqrt(b*e^2 - a*e*f)*log(((8*b^2* 
e^2 - 8*a*b*e*f + a^2*f^2)*x^4 + a^2*e^2 + 2*(4*a*b*e^2 - 3*a^2*e*f)*x^2 + 
 4*((2*b*e - a*f)*x^3 + a*e*x)*sqrt(b*e^2 - a*e*f)*sqrt(b*x^2 + a))/(f^2*x 
^4 + 2*e*f*x^2 + e^2)) - 4*((2*a*b^3*d^2*e^5*f + 3*a^3*b*c^2*e*f^5 + (8*b^ 
4*c^2 - 28*a*b^3*c*d + 11*a^2*b^2*d^2)*e^4*f^2 + (2*a*b^3*c^2 + 26*a^2*b^2 
*c*d - 13*a^3*b*d^2)*e^3*f^3 - (13*a^2*b^2*c^2 - 2*a^3*b*c*d)*e^2*f^4)*x^5 
 + (4*a*b^3*d^2*e^6 + 3*a^4*c^2*e*f^5 + (16*b^4*c^2 - 48*a*b^3*c*d + 17*a^ 
2*b^2*d^2)*e^5*f - 2*(2*a*b^3*c^2 - 19*a^2*b^2*c*d + 8*a^3*b*d^2)*e^4*f^2 
- (7*a^2*b^2*c^2 - 8*a^3*b*c*d + 5*a^4*d^2)*e^3*f^3 - 2*(4*a^3*b*c^2 - a^4 
*c*d)*e^2*f^4)*x^3 + (5*a^4*c^2*e^2*f^4 + 4*(2*b^4*c^2 - 4*a*b^3*c*d + 3*a 
^2*b^2*d^2)*e^6 - (8*a*b^3*c^2 + 9*a^3*b*d^2)*e^5*f + 3*(4*a^2*b^2*c^2 ...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate((d*x**2+c)**2/(b*x**2+a)**(3/2)/(f*x**2+e)**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^3} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{3}} \,d x } \] Input:

integrate((d*x^2+c)^2/(b*x^2+a)^(3/2)/(f*x^2+e)^3,x, algorithm="maxima")
 

Output:

integrate((d*x^2 + c)^2/((b*x^2 + a)^(3/2)*(f*x^2 + e)^3), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1451 vs. \(2 (400) = 800\).

Time = 0.71 (sec) , antiderivative size = 1451, normalized size of antiderivative = 3.40 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((d*x^2+c)^2/(b*x^2+a)^(3/2)/(f*x^2+e)^3,x, algorithm="giac")
 

Output:

(b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*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)) - 1/8*(16*b^(5/2)*c*d*e^3 - 12*a*b^( 
3/2)*d^2*e^3 - 24*b^(5/2)*c^2*e^2*f + 16*a*b^(3/2)*c*d*e^2*f - 3*a^2*sqrt( 
b)*d^2*e^2*f + 12*a*b^(3/2)*c^2*e*f^2 - 2*a^2*sqrt(b)*c*d*e*f^2 - 3*a^2*sq 
rt(b)*c^2*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*(16*(sqrt(b)*x - sqrt(b*x^2 + 
 a))^6*b^(5/2)*c*d*e^3*f^2 - 12*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a*b^(3/2)* 
d^2*e^3*f^2 - 16*(sqrt(b)*x - sqrt(b*x^2 + a))^6*b^(5/2)*c^2*e^2*f^3 + 5*( 
sqrt(b)*x - sqrt(b*x^2 + a))^6*a^2*sqrt(b)*d^2*e^2*f^3 + 12*(sqrt(b)*x - s 
qrt(b*x^2 + a))^6*a*b^(3/2)*c^2*e*f^4 - 2*(sqrt(b)*x - sqrt(b*x^2 + a))^6* 
a^2*sqrt(b)*c*d*e*f^4 - 3*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^2*sqrt(b)*c^2* 
f^5 - 16*(sqrt(b)*x - sqrt(b*x^2 + a))^4*b^(7/2)*d^2*e^5 + 96*(sqrt(b)*x - 
 sqrt(b*x^2 + a))^4*b^(7/2)*c*d*e^4*f - 24*(sqrt(b)*x - sqrt(b*x^2 + a))^4 
*a*b^(5/2)*d^2*e^4*f - 80*(sqrt(b)*x - sqrt(b*x^2 + a))^4*b^(7/2)*c^2*e^3* 
f^2 - 80*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a*b^(5/2)*c*d*e^3*f^2 + 34*(sqrt( 
b)*x - sqrt(b*x^2 + a))^4*a^2*b^(3/2)*d^2*e^3*f^2 + 104*(sqrt(b)*x - sqrt( 
b*x^2 + a))^4*a*b^(5/2)*c^2*e^2*f^3 + 20*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a 
^2*b^(3/2)*c*d*e^2*f^3 - 15*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^3*sqrt(b)*d^ 
2*e^2*f^3 - 54*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^2*b^(3/2)*c^2*e*f^4 + ...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^3} \, dx=\int \frac {{\left (d\,x^2+c\right )}^2}{{\left (b\,x^2+a\right )}^{3/2}\,{\left (f\,x^2+e\right )}^3} \,d x \] Input:

int((c + d*x^2)^2/((a + b*x^2)^(3/2)*(e + f*x^2)^3),x)
 

Output:

int((c + d*x^2)^2/((a + b*x^2)^(3/2)*(e + f*x^2)^3), x)
 

Reduce [B] (verification not implemented)

Time = 0.55 (sec) , antiderivative size = 9235, normalized size of antiderivative = 21.63 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int((d*x^2+c)^2/(b*x^2+a)^(3/2)/(f*x^2+e)^3,x)
 

Output:

( - 9*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**5*c**2*e**2*f**5 - 18*sqrt 
(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqr 
t(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**5*c**2*e*f**6*x**2 - 9*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**5*c**2*f**7*x**4 - 6*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**5*c*d*e**3*f**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**5*c*d*e**2*f**5*x**2 - 6*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** 
5*c*d*e*f**6*x**4 - 9*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**5*d**2*e** 
4*f**3 - 18*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**5*d**2*e**3*f**4*x** 
2 - 9*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**5*d**2*e**2*f**5*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**4*b*c**2*e**3*f**4 + 111*sqrt(e 
)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sq...