\(\int \frac {(c+d x^2)^2 (e+f x^2)^2}{(a+b x^2)^{5/2}} \, dx\) [365]

Optimal result

Integrand size = 30, antiderivative size = 249 \[ \int \frac {\left (c+d x^2\right )^2 \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {(b c-a d)^2 (b e-a f)^2 x}{3 a b^4 \left (a+b x^2\right )^{3/2}}+\frac {2 (b c-a d) (b e-a f) \left (b^2 c e-5 a^2 d f+2 a b (d e+c f)\right ) x}{3 a^2 b^4 \sqrt {a+b x^2}}-\frac {d f (11 a d f-8 b (d e+c f)) x \sqrt {a+b x^2}}{8 b^4}+\frac {d^2 f^2 x^3 \sqrt {a+b x^2}}{4 b^3}+\frac {\left (35 a^2 d^2 f^2-40 a b d f (d e+c f)+8 b^2 \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{9/2}} \] Output:

1/3*(-a*d+b*c)^2*(-a*f+b*e)^2*x/a/b^4/(b*x^2+a)^(3/2)+2/3*(-a*d+b*c)*(-a*f 
+b*e)*(b^2*c*e-5*a^2*d*f+2*a*b*(c*f+d*e))*x/a^2/b^4/(b*x^2+a)^(1/2)-1/8*d* 
f*(11*a*d*f-8*b*(c*f+d*e))*x*(b*x^2+a)^(1/2)/b^4+1/4*d^2*f^2*x^3*(b*x^2+a) 
^(1/2)/b^3+1/8*(35*a^2*d^2*f^2-40*a*b*d*f*(c*f+d*e)+8*b^2*(c^2*f^2+4*c*d*e 
*f+d^2*e^2))*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/b^(9/2)
 

Mathematica [A] (verified)

Time = 0.94 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.20 \[ \int \frac {\left (c+d x^2\right )^2 \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {x \left (-105 a^5 d^2 f^2+16 b^5 c^2 e^2 x^2+8 a b^4 c e \left (3 c e+2 d e x^2+2 c f x^2\right )+20 a^4 b d f \left (6 d e+6 c f-7 d f x^2\right )+2 a^2 b^3 x^2 \left (-16 c^2 f^2+4 c d f \left (-16 e+3 f x^2\right )+d^2 \left (-16 e^2+12 e f x^2+3 f^2 x^4\right )\right )-a^3 b^2 \left (24 c^2 f^2+32 c d f \left (3 e-5 f x^2\right )+d^2 \left (24 e^2-160 e f x^2+21 f^2 x^4\right )\right )\right )}{24 a^2 b^4 \left (a+b x^2\right )^{3/2}}-\frac {\left (35 a^2 d^2 f^2-40 a b d f (d e+c f)+8 b^2 \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{8 b^{9/2}} \] Input:

Integrate[((c + d*x^2)^2*(e + f*x^2)^2)/(a + b*x^2)^(5/2),x]
 

Output:

(x*(-105*a^5*d^2*f^2 + 16*b^5*c^2*e^2*x^2 + 8*a*b^4*c*e*(3*c*e + 2*d*e*x^2 
 + 2*c*f*x^2) + 20*a^4*b*d*f*(6*d*e + 6*c*f - 7*d*f*x^2) + 2*a^2*b^3*x^2*( 
-16*c^2*f^2 + 4*c*d*f*(-16*e + 3*f*x^2) + d^2*(-16*e^2 + 12*e*f*x^2 + 3*f^ 
2*x^4)) - a^3*b^2*(24*c^2*f^2 + 32*c*d*f*(3*e - 5*f*x^2) + d^2*(24*e^2 - 1 
60*e*f*x^2 + 21*f^2*x^4))))/(24*a^2*b^4*(a + b*x^2)^(3/2)) - ((35*a^2*d^2* 
f^2 - 40*a*b*d*f*(d*e + c*f) + 8*b^2*(d^2*e^2 + 4*c*d*e*f + c^2*f^2))*Log[ 
-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/(8*b^(9/2))
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 473, normalized size of antiderivative = 1.90, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {433, 2009}

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.

\(\displaystyle \int \frac {\left (c+d x^2\right )^2 \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 433

\(\displaystyle \int \left (\frac {x^4 \left (c^2 f^2+4 c d e f+d^2 e^2\right )}{\left (a+b x^2\right )^{5/2}}+\frac {c^2 e^2}{\left (a+b x^2\right )^{5/2}}+\frac {2 c e x^2 (c f+d e)}{\left (a+b x^2\right )^{5/2}}+\frac {2 d f x^6 (c f+d e)}{\left (a+b x^2\right )^{5/2}}+\frac {d^2 f^2 x^8}{\left (a+b x^2\right )^{5/2}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {35 a^2 d^2 f^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{9/2}}+\frac {2 c^2 e^2 x}{3 a^2 \sqrt {a+b x^2}}+\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (c^2 f^2+4 c d e f+d^2 e^2\right )}{b^{5/2}}-\frac {5 a d f \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (c f+d e)}{b^{7/2}}-\frac {35 a d^2 f^2 x \sqrt {a+b x^2}}{8 b^4}+\frac {5 d f x \sqrt {a+b x^2} (c f+d e)}{b^3}+\frac {35 d^2 f^2 x^3 \sqrt {a+b x^2}}{12 b^3}-\frac {x \left (c^2 f^2+4 c d e f+d^2 e^2\right )}{b^2 \sqrt {a+b x^2}}-\frac {10 d f x^3 (c f+d e)}{3 b^2 \sqrt {a+b x^2}}-\frac {7 d^2 f^2 x^5}{3 b^2 \sqrt {a+b x^2}}-\frac {x^3 \left (c^2 f^2+4 c d e f+d^2 e^2\right )}{3 b \left (a+b x^2\right )^{3/2}}+\frac {c^2 e^2 x}{3 a \left (a+b x^2\right )^{3/2}}-\frac {2 d f x^5 (c f+d e)}{3 b \left (a+b x^2\right )^{3/2}}+\frac {2 c e x^3 (c f+d e)}{3 a \left (a+b x^2\right )^{3/2}}-\frac {d^2 f^2 x^7}{3 b \left (a+b x^2\right )^{3/2}}\)

Input:

Int[((c + d*x^2)^2*(e + f*x^2)^2)/(a + b*x^2)^(5/2),x]
 

Output:

(c^2*e^2*x)/(3*a*(a + b*x^2)^(3/2)) + (2*c*e*(d*e + c*f)*x^3)/(3*a*(a + b* 
x^2)^(3/2)) - ((d^2*e^2 + 4*c*d*e*f + c^2*f^2)*x^3)/(3*b*(a + b*x^2)^(3/2) 
) - (2*d*f*(d*e + c*f)*x^5)/(3*b*(a + b*x^2)^(3/2)) - (d^2*f^2*x^7)/(3*b*( 
a + b*x^2)^(3/2)) + (2*c^2*e^2*x)/(3*a^2*Sqrt[a + b*x^2]) - ((d^2*e^2 + 4* 
c*d*e*f + c^2*f^2)*x)/(b^2*Sqrt[a + b*x^2]) - (10*d*f*(d*e + c*f)*x^3)/(3* 
b^2*Sqrt[a + b*x^2]) - (7*d^2*f^2*x^5)/(3*b^2*Sqrt[a + b*x^2]) - (35*a*d^2 
*f^2*x*Sqrt[a + b*x^2])/(8*b^4) + (5*d*f*(d*e + c*f)*x*Sqrt[a + b*x^2])/b^ 
3 + (35*d^2*f^2*x^3*Sqrt[a + b*x^2])/(12*b^3) + (35*a^2*d^2*f^2*ArcTanh[(S 
qrt[b]*x)/Sqrt[a + b*x^2]])/(8*b^(9/2)) - (5*a*d*f*(d*e + c*f)*ArcTanh[(Sq 
rt[b]*x)/Sqrt[a + b*x^2]])/b^(7/2) + ((d^2*e^2 + 4*c*d*e*f + c^2*f^2)*ArcT 
anh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/b^(5/2)
 

Defintions of rubi rules used

rule 433

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_ 
)^2)^(r_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*x^2)^p*(c + d*x^2) 
^q*(e + f*x^2)^r, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, f, p, 
 q, r}, x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 1.09 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.13

method	result	size
pseudoelliptic	\(\frac {\frac {35 a^{2} \left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (a^{2} d^{2} f^{2}-\frac {8 a b d f \left (c f +d e \right )}{7}+\frac {8 b^{2} \left (c^{2} f^{2}+4 c d e f +d^{2} e^{2}\right )}{35}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )}{8}+\frac {2 \left (-\frac {3 a^{3} \left (\left (\frac {7}{8} f^{2} x^{4}-\frac {20}{3} e f \,x^{2}+e^{2}\right ) d^{2}+4 c \left (-\frac {5 f \,x^{2}}{3}+e \right ) f d +c^{2} f^{2}\right ) b^{\frac {5}{2}}}{2}-2 a^{2} x^{2} \left (\left (-\frac {3}{16} f^{2} x^{4}-\frac {3}{4} e f \,x^{2}+e^{2}\right ) d^{2}+4 c \left (-\frac {3 f \,x^{2}}{16}+e \right ) f d +c^{2} f^{2}\right ) b^{\frac {7}{2}}+\frac {15 a^{4} d \left (\left (-\frac {7 f \,x^{2}}{6}+e \right ) d +c f \right ) f \,b^{\frac {3}{2}}}{2}-\frac {105 a^{5} d^{2} f^{2} \sqrt {b}}{16}+\left (\left (d e \,x^{2}+\frac {3 c \left (\frac {2 f \,x^{2}}{3}+e \right )}{2}\right ) a +b c e \,x^{2}\right ) c \,b^{\frac {9}{2}} e \right ) x}{3}}{b^{\frac {9}{2}} \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{2}}\)	\(282\)
default	\(c^{2} e^{2} \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )+2 d f \left (c f +d e \right ) \left (\frac {x^{5}}{2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {5 a \left (-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}\right )}{2 b}\right )+2 c e \left (c f +d e \right ) \left (-\frac {x}{2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {a \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )}{2 b}\right )+\left (c^{2} f^{2}+4 c d e f +d^{2} e^{2}\right ) \left (-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}\right )+d^{2} f^{2} \left (\frac {x^{7}}{4 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {7 a \left (\frac {x^{5}}{2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {5 a \left (-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}\right )}{2 b}\right )}{4 b}\right )\)	\(390\)
risch	\(\text {Expression too large to display}\)	\(1019\)

Input:

int((d*x^2+c)^2*(f*x^2+e)^2/(b*x^2+a)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

2/3/(b*x^2+a)^(3/2)*(105/16*a^2*(b*x^2+a)^(3/2)*(a^2*d^2*f^2-8/7*a*b*d*f*( 
c*f+d*e)+8/35*b^2*(c^2*f^2+4*c*d*e*f+d^2*e^2))*arctanh((b*x^2+a)^(1/2)/x/b 
^(1/2))+(-3/2*a^3*((7/8*f^2*x^4-20/3*e*f*x^2+e^2)*d^2+4*c*(-5/3*f*x^2+e)*f 
*d+c^2*f^2)*b^(5/2)-2*a^2*x^2*((-3/16*f^2*x^4-3/4*e*f*x^2+e^2)*d^2+4*c*(-3 
/16*f*x^2+e)*f*d+c^2*f^2)*b^(7/2)+15/2*a^4*d*((-7/6*f*x^2+e)*d+c*f)*f*b^(3 
/2)-105/16*a^5*d^2*f^2*b^(1/2)+((d*e*x^2+3/2*c*(2/3*f*x^2+e))*a+b*c*e*x^2) 
*c*b^(9/2)*e)*x)/b^(9/2)/a^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 566 vs. \(2 (225) = 450\).

Time = 0.39 (sec) , antiderivative size = 1140, normalized size of antiderivative = 4.58 \[ \int \frac {\left (c+d x^2\right )^2 \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate((d*x^2+c)^2*(f*x^2+e)^2/(b*x^2+a)^(5/2),x, algorithm="fricas")
 

Output:

[1/48*(3*(8*a^4*b^2*d^2*e^2 + (8*a^2*b^4*d^2*e^2 + 8*(4*a^2*b^4*c*d - 5*a^ 
3*b^3*d^2)*e*f + (8*a^2*b^4*c^2 - 40*a^3*b^3*c*d + 35*a^4*b^2*d^2)*f^2)*x^ 
4 + 8*(4*a^4*b^2*c*d - 5*a^5*b*d^2)*e*f + (8*a^4*b^2*c^2 - 40*a^5*b*c*d + 
35*a^6*d^2)*f^2 + 2*(8*a^3*b^3*d^2*e^2 + 8*(4*a^3*b^3*c*d - 5*a^4*b^2*d^2) 
*e*f + (8*a^3*b^3*c^2 - 40*a^4*b^2*c*d + 35*a^5*b*d^2)*f^2)*x^2)*sqrt(b)*l 
og(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2*(6*a^2*b^4*d^2*f^2*x^7 
+ 3*(8*a^2*b^4*d^2*e*f + (8*a^2*b^4*c*d - 7*a^3*b^3*d^2)*f^2)*x^5 + 4*(4*( 
b^6*c^2 + a*b^5*c*d - 2*a^2*b^4*d^2)*e^2 + 4*(a*b^5*c^2 - 8*a^2*b^4*c*d + 
10*a^3*b^3*d^2)*e*f - (8*a^2*b^4*c^2 - 40*a^3*b^3*c*d + 35*a^4*b^2*d^2)*f^ 
2)*x^3 + 3*(8*(a*b^5*c^2 - a^3*b^3*d^2)*e^2 - 8*(4*a^3*b^3*c*d - 5*a^4*b^2 
*d^2)*e*f - (8*a^3*b^3*c^2 - 40*a^4*b^2*c*d + 35*a^5*b*d^2)*f^2)*x)*sqrt(b 
*x^2 + a))/(a^2*b^7*x^4 + 2*a^3*b^6*x^2 + a^4*b^5), -1/24*(3*(8*a^4*b^2*d^ 
2*e^2 + (8*a^2*b^4*d^2*e^2 + 8*(4*a^2*b^4*c*d - 5*a^3*b^3*d^2)*e*f + (8*a^ 
2*b^4*c^2 - 40*a^3*b^3*c*d + 35*a^4*b^2*d^2)*f^2)*x^4 + 8*(4*a^4*b^2*c*d - 
 5*a^5*b*d^2)*e*f + (8*a^4*b^2*c^2 - 40*a^5*b*c*d + 35*a^6*d^2)*f^2 + 2*(8 
*a^3*b^3*d^2*e^2 + 8*(4*a^3*b^3*c*d - 5*a^4*b^2*d^2)*e*f + (8*a^3*b^3*c^2 
- 40*a^4*b^2*c*d + 35*a^5*b*d^2)*f^2)*x^2)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt 
(b*x^2 + a)) - (6*a^2*b^4*d^2*f^2*x^7 + 3*(8*a^2*b^4*d^2*e*f + (8*a^2*b^4* 
c*d - 7*a^3*b^3*d^2)*f^2)*x^5 + 4*(4*(b^6*c^2 + a*b^5*c*d - 2*a^2*b^4*d^2) 
*e^2 + 4*(a*b^5*c^2 - 8*a^2*b^4*c*d + 10*a^3*b^3*d^2)*e*f - (8*a^2*b^4*...
 

Sympy [F]

\[ \int \frac {\left (c+d x^2\right )^2 \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{5/2}} \, dx=\int \frac {\left (c + d x^{2}\right )^{2} \left (e + f x^{2}\right )^{2}}{\left (a + b x^{2}\right )^{\frac {5}{2}}}\, dx \] Input:

integrate((d*x**2+c)**2*(f*x**2+e)**2/(b*x**2+a)**(5/2),x)
                                                                                    
                                                                                    
 

Output:

Integral((c + d*x**2)**2*(e + f*x**2)**2/(a + b*x**2)**(5/2), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 510 vs. \(2 (225) = 450\).

Time = 0.05 (sec) , antiderivative size = 510, normalized size of antiderivative = 2.05 \[ \int \frac {\left (c+d x^2\right )^2 \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {d^{2} f^{2} x^{7}}{4 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b} - \frac {7 \, a d^{2} f^{2} x^{5}}{8 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}} - \frac {35 \, a^{2} d^{2} f^{2} x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, a}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}}\right )}}{24 \, b^{2}} + \frac {{\left (d^{2} e f + c d f^{2}\right )} x^{5}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} - \frac {1}{3} \, {\left (d^{2} e^{2} + 4 \, c d e f + c^{2} f^{2}\right )} x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, a}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}}\right )} + \frac {5 \, {\left (d^{2} e f + c d f^{2}\right )} a x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, a}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}}\right )}}{3 \, b} + \frac {2 \, c^{2} e^{2} x}{3 \, \sqrt {b x^{2} + a} a^{2}} + \frac {c^{2} e^{2} x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a} - \frac {35 \, a^{2} d^{2} f^{2} x}{24 \, \sqrt {b x^{2} + a} b^{4}} + \frac {35 \, a^{2} d^{2} f^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, b^{\frac {9}{2}}} + \frac {5 \, {\left (d^{2} e f + c d f^{2}\right )} a x}{3 \, \sqrt {b x^{2} + a} b^{3}} - \frac {{\left (d^{2} e^{2} + 4 \, c d e f + c^{2} f^{2}\right )} x}{3 \, \sqrt {b x^{2} + a} b^{2}} - \frac {2 \, {\left (c d e^{2} + c^{2} e f\right )} x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, {\left (c d e^{2} + c^{2} e f\right )} x}{3 \, \sqrt {b x^{2} + a} a b} - \frac {5 \, {\left (d^{2} e f + c d f^{2}\right )} a \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {7}{2}}} + \frac {{\left (d^{2} e^{2} + 4 \, c d e f + c^{2} f^{2}\right )} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {5}{2}}} \] Input:

integrate((d*x^2+c)^2*(f*x^2+e)^2/(b*x^2+a)^(5/2),x, algorithm="maxima")
 

Output:

1/4*d^2*f^2*x^7/((b*x^2 + a)^(3/2)*b) - 7/8*a*d^2*f^2*x^5/((b*x^2 + a)^(3/ 
2)*b^2) - 35/24*a^2*d^2*f^2*x*(3*x^2/((b*x^2 + a)^(3/2)*b) + 2*a/((b*x^2 + 
 a)^(3/2)*b^2))/b^2 + (d^2*e*f + c*d*f^2)*x^5/((b*x^2 + a)^(3/2)*b) - 1/3* 
(d^2*e^2 + 4*c*d*e*f + c^2*f^2)*x*(3*x^2/((b*x^2 + a)^(3/2)*b) + 2*a/((b*x 
^2 + a)^(3/2)*b^2)) + 5/3*(d^2*e*f + c*d*f^2)*a*x*(3*x^2/((b*x^2 + a)^(3/2 
)*b) + 2*a/((b*x^2 + a)^(3/2)*b^2))/b + 2/3*c^2*e^2*x/(sqrt(b*x^2 + a)*a^2 
) + 1/3*c^2*e^2*x/((b*x^2 + a)^(3/2)*a) - 35/24*a^2*d^2*f^2*x/(sqrt(b*x^2 
+ a)*b^4) + 35/8*a^2*d^2*f^2*arcsinh(b*x/sqrt(a*b))/b^(9/2) + 5/3*(d^2*e*f 
 + c*d*f^2)*a*x/(sqrt(b*x^2 + a)*b^3) - 1/3*(d^2*e^2 + 4*c*d*e*f + c^2*f^2 
)*x/(sqrt(b*x^2 + a)*b^2) - 2/3*(c*d*e^2 + c^2*e*f)*x/((b*x^2 + a)^(3/2)*b 
) + 2/3*(c*d*e^2 + c^2*e*f)*x/(sqrt(b*x^2 + a)*a*b) - 5*(d^2*e*f + c*d*f^2 
)*a*arcsinh(b*x/sqrt(a*b))/b^(7/2) + (d^2*e^2 + 4*c*d*e*f + c^2*f^2)*arcsi 
nh(b*x/sqrt(a*b))/b^(5/2)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.58 \[ \int \frac {\left (c+d x^2\right )^2 \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {{\left ({\left (3 \, {\left (\frac {2 \, d^{2} f^{2} x^{2}}{b} + \frac {8 \, a^{2} b^{6} d^{2} e f + 8 \, a^{2} b^{6} c d f^{2} - 7 \, a^{3} b^{5} d^{2} f^{2}}{a^{2} b^{7}}\right )} x^{2} + \frac {4 \, {\left (4 \, b^{8} c^{2} e^{2} + 4 \, a b^{7} c d e^{2} - 8 \, a^{2} b^{6} d^{2} e^{2} + 4 \, a b^{7} c^{2} e f - 32 \, a^{2} b^{6} c d e f + 40 \, a^{3} b^{5} d^{2} e f - 8 \, a^{2} b^{6} c^{2} f^{2} + 40 \, a^{3} b^{5} c d f^{2} - 35 \, a^{4} b^{4} d^{2} f^{2}\right )}}{a^{2} b^{7}}\right )} x^{2} + \frac {3 \, {\left (8 \, a b^{7} c^{2} e^{2} - 8 \, a^{3} b^{5} d^{2} e^{2} - 32 \, a^{3} b^{5} c d e f + 40 \, a^{4} b^{4} d^{2} e f - 8 \, a^{3} b^{5} c^{2} f^{2} + 40 \, a^{4} b^{4} c d f^{2} - 35 \, a^{5} b^{3} d^{2} f^{2}\right )}}{a^{2} b^{7}}\right )} x}{24 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}}} - \frac {{\left (8 \, b^{2} d^{2} e^{2} + 32 \, b^{2} c d e f - 40 \, a b d^{2} e f + 8 \, b^{2} c^{2} f^{2} - 40 \, a b c d f^{2} + 35 \, a^{2} d^{2} f^{2}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{8 \, b^{\frac {9}{2}}} \] Input:

integrate((d*x^2+c)^2*(f*x^2+e)^2/(b*x^2+a)^(5/2),x, algorithm="giac")
 

Output:

1/24*((3*(2*d^2*f^2*x^2/b + (8*a^2*b^6*d^2*e*f + 8*a^2*b^6*c*d*f^2 - 7*a^3 
*b^5*d^2*f^2)/(a^2*b^7))*x^2 + 4*(4*b^8*c^2*e^2 + 4*a*b^7*c*d*e^2 - 8*a^2* 
b^6*d^2*e^2 + 4*a*b^7*c^2*e*f - 32*a^2*b^6*c*d*e*f + 40*a^3*b^5*d^2*e*f - 
8*a^2*b^6*c^2*f^2 + 40*a^3*b^5*c*d*f^2 - 35*a^4*b^4*d^2*f^2)/(a^2*b^7))*x^ 
2 + 3*(8*a*b^7*c^2*e^2 - 8*a^3*b^5*d^2*e^2 - 32*a^3*b^5*c*d*e*f + 40*a^4*b 
^4*d^2*e*f - 8*a^3*b^5*c^2*f^2 + 40*a^4*b^4*c*d*f^2 - 35*a^5*b^3*d^2*f^2)/ 
(a^2*b^7))*x/(b*x^2 + a)^(3/2) - 1/8*(8*b^2*d^2*e^2 + 32*b^2*c*d*e*f - 40* 
a*b*d^2*e*f + 8*b^2*c^2*f^2 - 40*a*b*c*d*f^2 + 35*a^2*d^2*f^2)*log(abs(-sq 
rt(b)*x + sqrt(b*x^2 + a)))/b^(9/2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c+d x^2\right )^2 \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{5/2}} \, dx=\int \frac {{\left (d\,x^2+c\right )}^2\,{\left (f\,x^2+e\right )}^2}{{\left (b\,x^2+a\right )}^{5/2}} \,d x \] Input:

int(((c + d*x^2)^2*(e + f*x^2)^2)/(a + b*x^2)^(5/2),x)
 

Output:

int(((c + d*x^2)^2*(e + f*x^2)^2)/(a + b*x^2)^(5/2), x)
 

Reduce [F]

\[ \int \frac {\left (c+d x^2\right )^2 \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{5/2}} \, dx=\int \frac {\left (d \,x^{2}+c \right )^{2} \left (f \,x^{2}+e \right )^{2}}{\left (b \,x^{2}+a \right )^{\frac {5}{2}}}d x \] Input:

int((d*x^2+c)^2*(f*x^2+e)^2/(b*x^2+a)^(5/2),x)
 

Output:

int((d*x^2+c)^2*(f*x^2+e)^2/(b*x^2+a)^(5/2),x)