\(\int \frac {(c+d x^2)^2 (e+f x^2)^2}{(a+b x^2)^{3/2}} \, dx\) [344]

Optimal result

Integrand size = 30, antiderivative size = 272 \[ \int \frac {\left (c+d x^2\right )^2 \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {(b c-a d)^2 (b e-a f)^2 x}{a b^4 \sqrt {a+b x^2}}+\frac {\left (19 a^2 d^2 f^2-28 a b d f (d e+c f)+8 b^2 \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) x \sqrt {a+b x^2}}{16 b^4}-\frac {d f (11 a d f-12 b (d e+c f)) x^3 \sqrt {a+b x^2}}{24 b^3}+\frac {d^2 f^2 x^5 \sqrt {a+b x^2}}{6 b^2}-\frac {\left (35 a^3 d^2 f^2-32 b^3 c e (d e+c f)-60 a^2 b d f (d e+c f)+24 a b^2 \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{16 b^{9/2}} \] Output:

(-a*d+b*c)^2*(-a*f+b*e)^2*x/a/b^4/(b*x^2+a)^(1/2)+1/16*(19*a^2*d^2*f^2-28* 
a*b*d*f*(c*f+d*e)+8*b^2*(c^2*f^2+4*c*d*e*f+d^2*e^2))*x*(b*x^2+a)^(1/2)/b^4 
-1/24*d*f*(11*a*d*f-12*b*(c*f+d*e))*x^3*(b*x^2+a)^(1/2)/b^3+1/6*d^2*f^2*x^ 
5*(b*x^2+a)^(1/2)/b^2-1/16*(35*a^3*d^2*f^2-32*b^3*c*e*(c*f+d*e)-60*a^2*b*d 
*f*(c*f+d*e)+24*a*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 = 1.04 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.10 \[ \int \frac {\left (c+d x^2\right )^2 \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {\frac {\sqrt {b} x \left (48 b^4 c^2 e^2+105 a^4 d^2 f^2+5 a^3 b d f \left (-36 d e-36 c f+7 d f x^2\right )+2 a^2 b^2 \left (36 c^2 f^2+6 c d f \left (24 e-5 f x^2\right )+d^2 \left (36 e^2-30 e f x^2-7 f^2 x^4\right )\right )+8 a b^3 \left (3 c^2 f \left (-4 e+f x^2\right )+d^2 x^2 \left (3 e^2+3 e f x^2+f^2 x^4\right )+3 c d \left (-4 e^2+4 e f x^2+f^2 x^4\right )\right )\right )}{a \sqrt {a+b x^2}}+3 \left (35 a^3 d^2 f^2-32 b^3 c e (d e+c f)-60 a^2 b d f (d e+c f)+24 a b^2 \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{48 b^{9/2}} \] Input:

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

Output:

((Sqrt[b]*x*(48*b^4*c^2*e^2 + 105*a^4*d^2*f^2 + 5*a^3*b*d*f*(-36*d*e - 36* 
c*f + 7*d*f*x^2) + 2*a^2*b^2*(36*c^2*f^2 + 6*c*d*f*(24*e - 5*f*x^2) + d^2* 
(36*e^2 - 30*e*f*x^2 - 7*f^2*x^4)) + 8*a*b^3*(3*c^2*f*(-4*e + f*x^2) + d^2 
*x^2*(3*e^2 + 3*e*f*x^2 + f^2*x^4) + 3*c*d*(-4*e^2 + 4*e*f*x^2 + f^2*x^4)) 
))/(a*Sqrt[a + b*x^2]) + 3*(35*a^3*d^2*f^2 - 32*b^3*c*e*(d*e + c*f) - 60*a 
^2*b*d*f*(d*e + c*f) + 24*a*b^2*(d^2*e^2 + 4*c*d*e*f + c^2*f^2))*Log[-(Sqr 
t[b]*x) + Sqrt[a + b*x^2]])/(48*b^(9/2))
 

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 486, normalized size of antiderivative = 1.79, 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.

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

\(\Big \downarrow \) 433

\(\displaystyle \int \left (\frac {x^4 \left (c^2 f^2+4 c d e f+d^2 e^2\right )}{\left (a+b x^2\right )^{3/2}}+\frac {c^2 e^2}{\left (a+b x^2\right )^{3/2}}+\frac {2 c e x^2 (c f+d e)}{\left (a+b x^2\right )^{3/2}}+\frac {2 d f x^6 (c f+d e)}{\left (a+b x^2\right )^{3/2}}+\frac {d^2 f^2 x^8}{\left (a+b x^2\right )^{3/2}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {35 a^3 d^2 f^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{16 b^{9/2}}+\frac {15 a^2 d f \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (c f+d e)}{4 b^{7/2}}+\frac {35 a^2 d^2 f^2 x \sqrt {a+b x^2}}{16 b^4}-\frac {3 a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (c^2 f^2+4 c d e f+d^2 e^2\right )}{2 b^{5/2}}+\frac {2 c e \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (c f+d e)}{b^{3/2}}-\frac {15 a d f x \sqrt {a+b x^2} (c f+d e)}{4 b^3}-\frac {35 a d^2 f^2 x^3 \sqrt {a+b x^2}}{24 b^3}+\frac {3 x \sqrt {a+b x^2} \left (c^2 f^2+4 c d e f+d^2 e^2\right )}{2 b^2}+\frac {5 d f x^3 \sqrt {a+b x^2} (c f+d e)}{2 b^2}+\frac {7 d^2 f^2 x^5 \sqrt {a+b x^2}}{6 b^2}-\frac {x^3 \left (c^2 f^2+4 c d e f+d^2 e^2\right )}{b \sqrt {a+b x^2}}+\frac {c^2 e^2 x}{a \sqrt {a+b x^2}}-\frac {2 c e x (c f+d e)}{b \sqrt {a+b x^2}}-\frac {2 d f x^5 (c f+d e)}{b \sqrt {a+b x^2}}-\frac {d^2 f^2 x^7}{b \sqrt {a+b x^2}}\)

Input:

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

Output:

(c^2*e^2*x)/(a*Sqrt[a + b*x^2]) - (2*c*e*(d*e + c*f)*x)/(b*Sqrt[a + b*x^2] 
) - ((d^2*e^2 + 4*c*d*e*f + c^2*f^2)*x^3)/(b*Sqrt[a + b*x^2]) - (2*d*f*(d* 
e + c*f)*x^5)/(b*Sqrt[a + b*x^2]) - (d^2*f^2*x^7)/(b*Sqrt[a + b*x^2]) + (3 
5*a^2*d^2*f^2*x*Sqrt[a + b*x^2])/(16*b^4) - (15*a*d*f*(d*e + c*f)*x*Sqrt[a 
 + b*x^2])/(4*b^3) + (3*(d^2*e^2 + 4*c*d*e*f + c^2*f^2)*x*Sqrt[a + b*x^2]) 
/(2*b^2) - (35*a*d^2*f^2*x^3*Sqrt[a + b*x^2])/(24*b^3) + (5*d*f*(d*e + c*f 
)*x^3*Sqrt[a + b*x^2])/(2*b^2) + (7*d^2*f^2*x^5*Sqrt[a + b*x^2])/(6*b^2) - 
 (35*a^3*d^2*f^2*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(16*b^(9/2)) + (2*c 
*e*(d*e + c*f)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/b^(3/2) + (15*a^2*d*f 
*(d*e + c*f)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(4*b^(7/2)) - (3*a*(d^2 
*e^2 + 4*c*d*e*f + c^2*f^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(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 = 0.78 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.05

method	result	size
pseudoelliptic	\(\frac {-\frac {35 a \left (a^{3} d^{2} f^{2}-\frac {12 a^{2} b d f \left (c f +d e \right )}{7}+\frac {24 a \,b^{2} \left (c^{2} f^{2}+4 c d e f +d^{2} e^{2}\right )}{35}-\frac {32 b^{3} c e \left (c f +d e \right )}{35}\right ) \sqrt {b \,x^{2}+a}\, \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )}{16}+\left (\frac {35 a^{4} d^{2} f^{2}}{16}-\frac {15 \left (-\frac {7}{36} d f \,x^{2}+c f +d e \right ) d b f \,a^{3}}{4}+\frac {3 b^{2} \left (-\frac {7 d^{2} f^{2} x^{4}}{36}-\frac {5 d f \left (c f +d e \right ) x^{2}}{6}+c^{2} f^{2}+4 c d e f +d^{2} e^{2}\right ) a^{2}}{2}-2 \left (-\frac {d^{2} f^{2} x^{6}}{12}-\frac {d f \left (c f +d e \right ) x^{4}}{4}+\left (-\frac {1}{4} c^{2} f^{2}-c d e f -\frac {1}{4} d^{2} e^{2}\right ) x^{2}+c e \left (c f +d e \right )\right ) b^{3} a +b^{4} c^{2} e^{2}\right ) x \sqrt {b}}{\sqrt {b \,x^{2}+a}\, b^{\frac {9}{2}} a}\)	\(286\)
default	\(\frac {c^{2} e^{2} x}{a \sqrt {b \,x^{2}+a}}+2 d f \left (c f +d e \right ) \left (\frac {x^{5}}{4 b \sqrt {b \,x^{2}+a}}-\frac {5 a \left (\frac {x^{3}}{2 b \sqrt {b \,x^{2}+a}}-\frac {3 a \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )}{2 b}\right )}{4 b}\right )+2 c e \left (c f +d e \right ) \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )+\left (c^{2} f^{2}+4 c d e f +d^{2} e^{2}\right ) \left (\frac {x^{3}}{2 b \sqrt {b \,x^{2}+a}}-\frac {3 a \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )}{2 b}\right )+d^{2} f^{2} \left (\frac {x^{7}}{6 b \sqrt {b \,x^{2}+a}}-\frac {7 a \left (\frac {x^{5}}{4 b \sqrt {b \,x^{2}+a}}-\frac {5 a \left (\frac {x^{3}}{2 b \sqrt {b \,x^{2}+a}}-\frac {3 a \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )}{2 b}\right )}{4 b}\right )}{6 b}\right )\)	\(361\)
risch	\(\frac {x \left (8 f^{2} x^{4} b^{2} d^{2}-22 a b \,d^{2} f^{2} x^{2}+24 b^{2} c d \,f^{2} x^{2}+24 b^{2} d^{2} e f \,x^{2}+57 a^{2} d^{2} f^{2}-84 a b c d \,f^{2}-84 a b \,d^{2} e f +24 b^{2} c^{2} f^{2}+96 b^{2} c d e f +24 b^{2} d^{2} e^{2}\right ) \sqrt {b \,x^{2}+a}}{48 b^{4}}-\frac {b \left (35 a^{3} d^{2} f^{2}-60 a^{2} b c d \,f^{2}-60 a^{2} b \,d^{2} e f +24 a \,b^{2} c^{2} f^{2}+96 a \,b^{2} c d e f +24 a \,b^{2} d^{2} e^{2}-32 b^{3} c^{2} e f -32 b^{3} c d \,e^{2}\right ) \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )+\frac {19 a^{3} d^{2} f^{2} x}{\sqrt {b \,x^{2}+a}}-\frac {16 b^{4} c^{2} e^{2} x}{a \sqrt {b \,x^{2}+a}}+\frac {8 a \,b^{2} c^{2} f^{2} x}{\sqrt {b \,x^{2}+a}}+\frac {8 a \,b^{2} d^{2} e^{2} x}{\sqrt {b \,x^{2}+a}}-\frac {28 a^{2} b c d \,f^{2} x}{\sqrt {b \,x^{2}+a}}-\frac {28 a^{2} b \,d^{2} e f x}{\sqrt {b \,x^{2}+a}}+\frac {32 a \,b^{2} c d e f x}{\sqrt {b \,x^{2}+a}}}{16 b^{4}}\)	\(414\)

Input:

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

Output:

1/(b*x^2+a)^(1/2)*(-35/16*a*(a^3*d^2*f^2-12/7*a^2*b*d*f*(c*f+d*e)+24/35*a* 
b^2*(c^2*f^2+4*c*d*e*f+d^2*e^2)-32/35*b^3*c*e*(c*f+d*e))*(b*x^2+a)^(1/2)*a 
rctanh((b*x^2+a)^(1/2)/x/b^(1/2))+(35/16*a^4*d^2*f^2-15/4*(-7/36*d*f*x^2+c 
*f+d*e)*d*b*f*a^3+3/2*b^2*(-7/36*d^2*f^2*x^4-5/6*d*f*(c*f+d*e)*x^2+c^2*f^2 
+4*c*d*e*f+d^2*e^2)*a^2-2*(-1/12*d^2*f^2*x^6-1/4*d*f*(c*f+d*e)*x^4+(-1/4*c 
^2*f^2-c*d*e*f-1/4*d^2*e^2)*x^2+c*e*(c*f+d*e))*b^3*a+b^4*c^2*e^2)*x*b^(1/2 
))/b^(9/2)/a
 

Fricas [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 994, normalized size of antiderivative = 3.65 \[ \int \frac {\left (c+d x^2\right )^2 \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

[-1/96*(3*(8*(4*a^2*b^3*c*d - 3*a^3*b^2*d^2)*e^2 + 4*(8*a^2*b^3*c^2 - 24*a 
^3*b^2*c*d + 15*a^4*b*d^2)*e*f - (24*a^3*b^2*c^2 - 60*a^4*b*c*d + 35*a^5*d 
^2)*f^2 + (8*(4*a*b^4*c*d - 3*a^2*b^3*d^2)*e^2 + 4*(8*a*b^4*c^2 - 24*a^2*b 
^3*c*d + 15*a^3*b^2*d^2)*e*f - (24*a^2*b^3*c^2 - 60*a^3*b^2*c*d + 35*a^4*b 
*d^2)*f^2)*x^2)*sqrt(b)*log(-2*b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - 
2*(8*a*b^4*d^2*f^2*x^7 + 2*(12*a*b^4*d^2*e*f + (12*a*b^4*c*d - 7*a^2*b^3*d 
^2)*f^2)*x^5 + (24*a*b^4*d^2*e^2 + 12*(8*a*b^4*c*d - 5*a^2*b^3*d^2)*e*f + 
(24*a*b^4*c^2 - 60*a^2*b^3*c*d + 35*a^3*b^2*d^2)*f^2)*x^3 + 3*(8*(2*b^5*c^ 
2 - 4*a*b^4*c*d + 3*a^2*b^3*d^2)*e^2 - 4*(8*a*b^4*c^2 - 24*a^2*b^3*c*d + 1 
5*a^3*b^2*d^2)*e*f + (24*a^2*b^3*c^2 - 60*a^3*b^2*c*d + 35*a^4*b*d^2)*f^2) 
*x)*sqrt(b*x^2 + a))/(a*b^6*x^2 + a^2*b^5), -1/48*(3*(8*(4*a^2*b^3*c*d - 3 
*a^3*b^2*d^2)*e^2 + 4*(8*a^2*b^3*c^2 - 24*a^3*b^2*c*d + 15*a^4*b*d^2)*e*f 
- (24*a^3*b^2*c^2 - 60*a^4*b*c*d + 35*a^5*d^2)*f^2 + (8*(4*a*b^4*c*d - 3*a 
^2*b^3*d^2)*e^2 + 4*(8*a*b^4*c^2 - 24*a^2*b^3*c*d + 15*a^3*b^2*d^2)*e*f - 
(24*a^2*b^3*c^2 - 60*a^3*b^2*c*d + 35*a^4*b*d^2)*f^2)*x^2)*sqrt(-b)*arctan 
(sqrt(-b)*x/sqrt(b*x^2 + a)) - (8*a*b^4*d^2*f^2*x^7 + 2*(12*a*b^4*d^2*e*f 
+ (12*a*b^4*c*d - 7*a^2*b^3*d^2)*f^2)*x^5 + (24*a*b^4*d^2*e^2 + 12*(8*a*b^ 
4*c*d - 5*a^2*b^3*d^2)*e*f + (24*a*b^4*c^2 - 60*a^2*b^3*c*d + 35*a^3*b^2*d 
^2)*f^2)*x^3 + 3*(8*(2*b^5*c^2 - 4*a*b^4*c*d + 3*a^2*b^3*d^2)*e^2 - 4*(8*a 
*b^4*c^2 - 24*a^2*b^3*c*d + 15*a^3*b^2*d^2)*e*f + (24*a^2*b^3*c^2 - 60*...
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.58 \[ \int \frac {\left (c+d x^2\right )^2 \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {d^{2} f^{2} x^{7}}{6 \, \sqrt {b x^{2} + a} b} - \frac {7 \, a d^{2} f^{2} x^{5}}{24 \, \sqrt {b x^{2} + a} b^{2}} + \frac {35 \, a^{2} d^{2} f^{2} x^{3}}{48 \, \sqrt {b x^{2} + a} b^{3}} + \frac {{\left (d^{2} e f + c d f^{2}\right )} x^{5}}{2 \, \sqrt {b x^{2} + a} b} + \frac {c^{2} e^{2} x}{\sqrt {b x^{2} + a} a} + \frac {35 \, a^{3} d^{2} f^{2} x}{16 \, \sqrt {b x^{2} + a} b^{4}} - \frac {35 \, a^{3} d^{2} f^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{16 \, b^{\frac {9}{2}}} - \frac {5 \, {\left (d^{2} e f + c d f^{2}\right )} a x^{3}}{4 \, \sqrt {b x^{2} + a} b^{2}} + \frac {{\left (d^{2} e^{2} + 4 \, c d e f + c^{2} f^{2}\right )} x^{3}}{2 \, \sqrt {b x^{2} + a} b} - \frac {15 \, {\left (d^{2} e f + c d f^{2}\right )} a^{2} x}{4 \, \sqrt {b x^{2} + a} b^{3}} + \frac {3 \, {\left (d^{2} e^{2} + 4 \, c d e f + c^{2} f^{2}\right )} a x}{2 \, \sqrt {b x^{2} + a} b^{2}} - \frac {2 \, {\left (c d e^{2} + c^{2} e f\right )} x}{\sqrt {b x^{2} + a} b} + \frac {15 \, {\left (d^{2} e f + c d f^{2}\right )} a^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{4 \, b^{\frac {7}{2}}} - \frac {3 \, {\left (d^{2} e^{2} + 4 \, c d e f + c^{2} f^{2}\right )} a \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {5}{2}}} + \frac {2 \, {\left (c d e^{2} + c^{2} e f\right )} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {3}{2}}} \] Input:

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

Output:

1/6*d^2*f^2*x^7/(sqrt(b*x^2 + a)*b) - 7/24*a*d^2*f^2*x^5/(sqrt(b*x^2 + a)* 
b^2) + 35/48*a^2*d^2*f^2*x^3/(sqrt(b*x^2 + a)*b^3) + 1/2*(d^2*e*f + c*d*f^ 
2)*x^5/(sqrt(b*x^2 + a)*b) + c^2*e^2*x/(sqrt(b*x^2 + a)*a) + 35/16*a^3*d^2 
*f^2*x/(sqrt(b*x^2 + a)*b^4) - 35/16*a^3*d^2*f^2*arcsinh(b*x/sqrt(a*b))/b^ 
(9/2) - 5/4*(d^2*e*f + c*d*f^2)*a*x^3/(sqrt(b*x^2 + a)*b^2) + 1/2*(d^2*e^2 
 + 4*c*d*e*f + c^2*f^2)*x^3/(sqrt(b*x^2 + a)*b) - 15/4*(d^2*e*f + c*d*f^2) 
*a^2*x/(sqrt(b*x^2 + a)*b^3) + 3/2*(d^2*e^2 + 4*c*d*e*f + c^2*f^2)*a*x/(sq 
rt(b*x^2 + a)*b^2) - 2*(c*d*e^2 + c^2*e*f)*x/(sqrt(b*x^2 + a)*b) + 15/4*(d 
^2*e*f + c*d*f^2)*a^2*arcsinh(b*x/sqrt(a*b))/b^(7/2) - 3/2*(d^2*e^2 + 4*c* 
d*e*f + c^2*f^2)*a*arcsinh(b*x/sqrt(a*b))/b^(5/2) + 2*(c*d*e^2 + c^2*e*f)* 
arcsinh(b*x/sqrt(a*b))/b^(3/2)
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.46 \[ \int \frac {\left (c+d x^2\right )^2 \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {{\left ({\left (2 \, {\left (\frac {4 \, d^{2} f^{2} x^{2}}{b} + \frac {12 \, a b^{6} d^{2} e f + 12 \, a b^{6} c d f^{2} - 7 \, a^{2} b^{5} d^{2} f^{2}}{a b^{7}}\right )} x^{2} + \frac {24 \, a b^{6} d^{2} e^{2} + 96 \, a b^{6} c d e f - 60 \, a^{2} b^{5} d^{2} e f + 24 \, a b^{6} c^{2} f^{2} - 60 \, a^{2} b^{5} c d f^{2} + 35 \, a^{3} b^{4} d^{2} f^{2}}{a b^{7}}\right )} x^{2} + \frac {3 \, {\left (16 \, b^{7} c^{2} e^{2} - 32 \, a b^{6} c d e^{2} + 24 \, a^{2} b^{5} d^{2} e^{2} - 32 \, a b^{6} c^{2} e f + 96 \, a^{2} b^{5} c d e f - 60 \, a^{3} b^{4} d^{2} e f + 24 \, a^{2} b^{5} c^{2} f^{2} - 60 \, a^{3} b^{4} c d f^{2} + 35 \, a^{4} b^{3} d^{2} f^{2}\right )}}{a b^{7}}\right )} x}{48 \, \sqrt {b x^{2} + a}} - \frac {{\left (32 \, b^{3} c d e^{2} - 24 \, a b^{2} d^{2} e^{2} + 32 \, b^{3} c^{2} e f - 96 \, a b^{2} c d e f + 60 \, a^{2} b d^{2} e f - 24 \, a b^{2} c^{2} f^{2} + 60 \, a^{2} b c d f^{2} - 35 \, a^{3} d^{2} f^{2}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{16 \, b^{\frac {9}{2}}} \] Input:

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

Output:

1/48*((2*(4*d^2*f^2*x^2/b + (12*a*b^6*d^2*e*f + 12*a*b^6*c*d*f^2 - 7*a^2*b 
^5*d^2*f^2)/(a*b^7))*x^2 + (24*a*b^6*d^2*e^2 + 96*a*b^6*c*d*e*f - 60*a^2*b 
^5*d^2*e*f + 24*a*b^6*c^2*f^2 - 60*a^2*b^5*c*d*f^2 + 35*a^3*b^4*d^2*f^2)/( 
a*b^7))*x^2 + 3*(16*b^7*c^2*e^2 - 32*a*b^6*c*d*e^2 + 24*a^2*b^5*d^2*e^2 - 
32*a*b^6*c^2*e*f + 96*a^2*b^5*c*d*e*f - 60*a^3*b^4*d^2*e*f + 24*a^2*b^5*c^ 
2*f^2 - 60*a^3*b^4*c*d*f^2 + 35*a^4*b^3*d^2*f^2)/(a*b^7))*x/sqrt(b*x^2 + a 
) - 1/16*(32*b^3*c*d*e^2 - 24*a*b^2*d^2*e^2 + 32*b^3*c^2*e*f - 96*a*b^2*c* 
d*e*f + 60*a^2*b*d^2*e*f - 24*a*b^2*c^2*f^2 + 60*a^2*b*c*d*f^2 - 35*a^3*d^ 
2*f^2)*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(9/2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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