\(\int \frac {(a+b x^2)^2 (c+d x^2)^{3/2}}{x^6} \, dx\) [855]

Optimal result

Integrand size = 24, antiderivative size = 144 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x^6} \, dx=-\frac {b (3 b c+4 a d) \sqrt {c+d x^2}}{3 x}+\frac {b d (3 b c+4 a d) x \sqrt {c+d x^2}}{6 c}-\frac {a^2 \left (c+d x^2\right )^{5/2}}{5 c x^5}-\frac {2 a b \left (c+d x^2\right )^{5/2}}{3 c x^3}+\frac {1}{2} b \sqrt {d} (3 b c+4 a d) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right ) \] Output:

-1/3*b*(4*a*d+3*b*c)*(d*x^2+c)^(1/2)/x+1/6*b*d*(4*a*d+3*b*c)*x*(d*x^2+c)^( 
1/2)/c-1/5*a^2*(d*x^2+c)^(5/2)/c/x^5-2/3*a*b*(d*x^2+c)^(5/2)/c/x^3+1/2*b*d 
^(1/2)*(4*a*d+3*b*c)*arctanh(d^(1/2)*x/(d*x^2+c)^(1/2))
 

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x^6} \, dx=-\frac {\sqrt {c+d x^2} \left (15 b^2 c x^4 \left (2 c-d x^2\right )+6 a^2 \left (c+d x^2\right )^2+20 a b c x^2 \left (c+4 d x^2\right )\right )}{30 c x^5}-\frac {1}{2} b \sqrt {d} (3 b c+4 a d) \log \left (-\sqrt {d} x+\sqrt {c+d x^2}\right ) \] Input:

Integrate[((a + b*x^2)^2*(c + d*x^2)^(3/2))/x^6,x]
 

Output:

-1/30*(Sqrt[c + d*x^2]*(15*b^2*c*x^4*(2*c - d*x^2) + 6*a^2*(c + d*x^2)^2 + 
 20*a*b*c*x^2*(c + 4*d*x^2)))/(c*x^5) - (b*Sqrt[d]*(3*b*c + 4*a*d)*Log[-(S 
qrt[d]*x) + Sqrt[c + d*x^2]])/2
 

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.92, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {365, 27, 359, 247, 211, 224, 219}

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.

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

\(\Big \downarrow \) 365

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 359

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

\(\Big \downarrow \) 247

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

Input:

Int[((a + b*x^2)^2*(c + d*x^2)^(3/2))/x^6,x]
 

Output:

-1/5*(a^2*(c + d*x^2)^(5/2))/(c*x^5) + b*((-2*a*(c + d*x^2)^(5/2))/(3*c*x^ 
3) + ((3*b*c + 4*a*d)*(-((c + d*x^2)^(3/2)/x) + 3*d*((x*Sqrt[c + d*x^2])/2 
 + (c*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(2*Sqrt[d]))))/(3*c))
 

Defintions of rubi rules used

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 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

rule 219

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])
 

rule 224

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]
 

rule 247

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ 
(m + 1)*((a + b*x^2)^p/(c*(m + 1))), x] - Simp[2*b*(p/(c^2*(m + 1)))   Int[ 
(c*x)^(m + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 
0] && LtQ[m, -1] &&  !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomialQ[a, b, c, 2, 
m, p, x]
 

rule 359

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x 
_Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + 
Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(a*e^2*(m + 1))   Int[(e*x)^(m + 2)* 
(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] 
&& LtQ[m, -1] &&  !ILtQ[p, -1]
 

rule 365

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^2, x 
_Symbol] :> Simp[c^2*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] 
- Simp[1/(a*e^2*(m + 1))   Int[(e*x)^(m + 2)*(a + b*x^2)^p*Simp[2*b*c^2*(p 
+ 1) + c*(b*c - 2*a*d)*(m + 1) - a*d^2*(m + 1)*x^2, x], x], x] /; FreeQ[{a, 
 b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1]
 

Maple [A] (verified)

Time = 0.51 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.83

method	result	size
risch	\(-\frac {\sqrt {x^{2} d +c}\, \left (-15 b^{2} c d \,x^{6}+6 a^{2} d^{2} x^{4}+80 a b c d \,x^{4}+30 b^{2} c^{2} x^{4}+12 a^{2} c d \,x^{2}+20 a b \,c^{2} x^{2}+6 a^{2} c^{2}\right )}{30 x^{5} c}+\frac {\left (4 a d +3 b c \right ) b \sqrt {d}\, \ln \left (\sqrt {d}\, x +\sqrt {x^{2} d +c}\right )}{2}\)	\(120\)
pseudoelliptic	\(-\frac {-10 c d \left (a d +\frac {3 b c}{4}\right ) b \,x^{5} \operatorname {arctanh}\left (\frac {\sqrt {x^{2} d +c}}{x \sqrt {d}}\right )+\left (2 c \,x^{2} \left (-\frac {5}{4} b^{2} x^{4}+\frac {20}{3} a b \,x^{2}+a^{2}\right ) d^{\frac {3}{2}}+d^{\frac {5}{2}} a^{2} x^{4}+c^{2} \sqrt {d}\, \left (5 b^{2} x^{4}+\frac {10}{3} a b \,x^{2}+a^{2}\right )\right ) \sqrt {x^{2} d +c}}{5 \sqrt {d}\, x^{5} c}\)	\(121\)
default	\(-\frac {a^{2} \left (x^{2} d +c \right )^{\frac {5}{2}}}{5 c \,x^{5}}+b^{2} \left (-\frac {\left (x^{2} d +c \right )^{\frac {5}{2}}}{c x}+\frac {4 d \left (\frac {x \left (x^{2} d +c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {x^{2} d +c}}{2}+\frac {c \ln \left (\sqrt {d}\, x +\sqrt {x^{2} d +c}\right )}{2 \sqrt {d}}\right )}{4}\right )}{c}\right )+2 a b \left (-\frac {\left (x^{2} d +c \right )^{\frac {5}{2}}}{3 c \,x^{3}}+\frac {2 d \left (-\frac {\left (x^{2} d +c \right )^{\frac {5}{2}}}{c x}+\frac {4 d \left (\frac {x \left (x^{2} d +c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {x^{2} d +c}}{2}+\frac {c \ln \left (\sqrt {d}\, x +\sqrt {x^{2} d +c}\right )}{2 \sqrt {d}}\right )}{4}\right )}{c}\right )}{3 c}\right )\)	\(204\)

Input:

int((b*x^2+a)^2*(d*x^2+c)^(3/2)/x^6,x,method=_RETURNVERBOSE)
 

Output:

-1/30*(d*x^2+c)^(1/2)*(-15*b^2*c*d*x^6+6*a^2*d^2*x^4+80*a*b*c*d*x^4+30*b^2 
*c^2*x^4+12*a^2*c*d*x^2+20*a*b*c^2*x^2+6*a^2*c^2)/x^5/c+1/2*(4*a*d+3*b*c)* 
b*d^(1/2)*ln(d^(1/2)*x+(d*x^2+c)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.85 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x^6} \, dx=\left [\frac {15 \, {\left (3 \, b^{2} c^{2} + 4 \, a b c d\right )} \sqrt {d} x^{5} \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) + 2 \, {\left (15 \, b^{2} c d x^{6} - 2 \, {\left (15 \, b^{2} c^{2} + 40 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{4} - 6 \, a^{2} c^{2} - 4 \, {\left (5 \, a b c^{2} + 3 \, a^{2} c d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{60 \, c x^{5}}, -\frac {15 \, {\left (3 \, b^{2} c^{2} + 4 \, a b c d\right )} \sqrt {-d} x^{5} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) - {\left (15 \, b^{2} c d x^{6} - 2 \, {\left (15 \, b^{2} c^{2} + 40 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{4} - 6 \, a^{2} c^{2} - 4 \, {\left (5 \, a b c^{2} + 3 \, a^{2} c d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{30 \, c x^{5}}\right ] \] Input:

integrate((b*x^2+a)^2*(d*x^2+c)^(3/2)/x^6,x, algorithm="fricas")
 

Output:

[1/60*(15*(3*b^2*c^2 + 4*a*b*c*d)*sqrt(d)*x^5*log(-2*d*x^2 - 2*sqrt(d*x^2 
+ c)*sqrt(d)*x - c) + 2*(15*b^2*c*d*x^6 - 2*(15*b^2*c^2 + 40*a*b*c*d + 3*a 
^2*d^2)*x^4 - 6*a^2*c^2 - 4*(5*a*b*c^2 + 3*a^2*c*d)*x^2)*sqrt(d*x^2 + c))/ 
(c*x^5), -1/30*(15*(3*b^2*c^2 + 4*a*b*c*d)*sqrt(-d)*x^5*arctan(sqrt(-d)*x/ 
sqrt(d*x^2 + c)) - (15*b^2*c*d*x^6 - 2*(15*b^2*c^2 + 40*a*b*c*d + 3*a^2*d^ 
2)*x^4 - 6*a^2*c^2 - 4*(5*a*b*c^2 + 3*a^2*c*d)*x^2)*sqrt(d*x^2 + c))/(c*x^ 
5)]
 

Sympy [A] (verification not implemented)

Time = 2.79 (sec) , antiderivative size = 347, normalized size of antiderivative = 2.41 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x^6} \, dx=- \frac {a^{2} c \sqrt {d} \sqrt {\frac {c}{d x^{2}} + 1}}{5 x^{4}} - \frac {2 a^{2} d^{\frac {3}{2}} \sqrt {\frac {c}{d x^{2}} + 1}}{5 x^{2}} - \frac {a^{2} d^{\frac {5}{2}} \sqrt {\frac {c}{d x^{2}} + 1}}{5 c} - \frac {2 a b \sqrt {c} d}{x \sqrt {1 + \frac {d x^{2}}{c}}} - \frac {2 a b c \sqrt {d} \sqrt {\frac {c}{d x^{2}} + 1}}{3 x^{2}} - \frac {2 a b d^{\frac {3}{2}} \sqrt {\frac {c}{d x^{2}} + 1}}{3} + 2 a b d^{\frac {3}{2}} \operatorname {asinh}{\left (\frac {\sqrt {d} x}{\sqrt {c}} \right )} - \frac {2 a b d^{2} x}{\sqrt {c} \sqrt {1 + \frac {d x^{2}}{c}}} - \frac {b^{2} c^{\frac {3}{2}}}{x \sqrt {1 + \frac {d x^{2}}{c}}} - \frac {b^{2} \sqrt {c} d x}{\sqrt {1 + \frac {d x^{2}}{c}}} + b^{2} c \sqrt {d} \operatorname {asinh}{\left (\frac {\sqrt {d} x}{\sqrt {c}} \right )} + b^{2} d \left (\begin {cases} \frac {c \left (\begin {cases} \frac {\log {\left (2 \sqrt {d} \sqrt {c + d x^{2}} + 2 d x \right )}}{\sqrt {d}} & \text {for}\: c \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {d x^{2}}} & \text {otherwise} \end {cases}\right )}{2} + \frac {x \sqrt {c + d x^{2}}}{2} & \text {for}\: d \neq 0 \\\sqrt {c} x & \text {otherwise} \end {cases}\right ) \] Input:

integrate((b*x**2+a)**2*(d*x**2+c)**(3/2)/x**6,x)
 

Output:

-a**2*c*sqrt(d)*sqrt(c/(d*x**2) + 1)/(5*x**4) - 2*a**2*d**(3/2)*sqrt(c/(d* 
x**2) + 1)/(5*x**2) - a**2*d**(5/2)*sqrt(c/(d*x**2) + 1)/(5*c) - 2*a*b*sqr 
t(c)*d/(x*sqrt(1 + d*x**2/c)) - 2*a*b*c*sqrt(d)*sqrt(c/(d*x**2) + 1)/(3*x* 
*2) - 2*a*b*d**(3/2)*sqrt(c/(d*x**2) + 1)/3 + 2*a*b*d**(3/2)*asinh(sqrt(d) 
*x/sqrt(c)) - 2*a*b*d**2*x/(sqrt(c)*sqrt(1 + d*x**2/c)) - b**2*c**(3/2)/(x 
*sqrt(1 + d*x**2/c)) - b**2*sqrt(c)*d*x/sqrt(1 + d*x**2/c) + b**2*c*sqrt(d 
)*asinh(sqrt(d)*x/sqrt(c)) + b**2*d*Piecewise((c*Piecewise((log(2*sqrt(d)* 
sqrt(c + d*x**2) + 2*d*x)/sqrt(d), Ne(c, 0)), (x*log(x)/sqrt(d*x**2), True 
))/2 + x*sqrt(c + d*x**2)/2, Ne(d, 0)), (sqrt(c)*x, True))
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x^6} \, dx=\frac {3}{2} \, \sqrt {d x^{2} + c} b^{2} d x + \frac {2 \, \sqrt {d x^{2} + c} a b d^{2} x}{c} + \frac {3}{2} \, b^{2} c \sqrt {d} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right ) + 2 \, a b d^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right ) - \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2}}{x} - \frac {4 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b d}{3 \, c x} - \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a b}{3 \, c x^{3}} - \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2}}{5 \, c x^{5}} \] Input:

integrate((b*x^2+a)^2*(d*x^2+c)^(3/2)/x^6,x, algorithm="maxima")
 

Output:

3/2*sqrt(d*x^2 + c)*b^2*d*x + 2*sqrt(d*x^2 + c)*a*b*d^2*x/c + 3/2*b^2*c*sq 
rt(d)*arcsinh(d*x/sqrt(c*d)) + 2*a*b*d^(3/2)*arcsinh(d*x/sqrt(c*d)) - (d*x 
^2 + c)^(3/2)*b^2/x - 4/3*(d*x^2 + c)^(3/2)*a*b*d/(c*x) - 2/3*(d*x^2 + c)^ 
(5/2)*a*b/(c*x^3) - 1/5*(d*x^2 + c)^(5/2)*a^2/(c*x^5)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 407 vs. \(2 (120) = 240\).

Time = 0.16 (sec) , antiderivative size = 407, normalized size of antiderivative = 2.83 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x^6} \, dx=\frac {1}{2} \, \sqrt {d x^{2} + c} b^{2} d x - \frac {1}{4} \, {\left (3 \, b^{2} c \sqrt {d} + 4 \, a b d^{\frac {3}{2}}\right )} \log \left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2}\right ) + \frac {2 \, {\left (15 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} b^{2} c^{2} \sqrt {d} + 60 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} a b c d^{\frac {3}{2}} + 15 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} a^{2} d^{\frac {5}{2}} - 60 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} b^{2} c^{3} \sqrt {d} - 180 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} a b c^{2} d^{\frac {3}{2}} + 90 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b^{2} c^{4} \sqrt {d} + 220 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a b c^{3} d^{\frac {3}{2}} + 30 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a^{2} c^{2} d^{\frac {5}{2}} - 60 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b^{2} c^{5} \sqrt {d} - 140 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b c^{4} d^{\frac {3}{2}} + 15 \, b^{2} c^{6} \sqrt {d} + 40 \, a b c^{5} d^{\frac {3}{2}} + 3 \, a^{2} c^{4} d^{\frac {5}{2}}\right )}}{15 \, {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} - c\right )}^{5}} \] Input:

integrate((b*x^2+a)^2*(d*x^2+c)^(3/2)/x^6,x, algorithm="giac")
 

Output:

1/2*sqrt(d*x^2 + c)*b^2*d*x - 1/4*(3*b^2*c*sqrt(d) + 4*a*b*d^(3/2))*log((s 
qrt(d)*x - sqrt(d*x^2 + c))^2) + 2/15*(15*(sqrt(d)*x - sqrt(d*x^2 + c))^8* 
b^2*c^2*sqrt(d) + 60*(sqrt(d)*x - sqrt(d*x^2 + c))^8*a*b*c*d^(3/2) + 15*(s 
qrt(d)*x - sqrt(d*x^2 + c))^8*a^2*d^(5/2) - 60*(sqrt(d)*x - sqrt(d*x^2 + c 
))^6*b^2*c^3*sqrt(d) - 180*(sqrt(d)*x - sqrt(d*x^2 + c))^6*a*b*c^2*d^(3/2) 
 + 90*(sqrt(d)*x - sqrt(d*x^2 + c))^4*b^2*c^4*sqrt(d) + 220*(sqrt(d)*x - s 
qrt(d*x^2 + c))^4*a*b*c^3*d^(3/2) + 30*(sqrt(d)*x - sqrt(d*x^2 + c))^4*a^2 
*c^2*d^(5/2) - 60*(sqrt(d)*x - sqrt(d*x^2 + c))^2*b^2*c^5*sqrt(d) - 140*(s 
qrt(d)*x - sqrt(d*x^2 + c))^2*a*b*c^4*d^(3/2) + 15*b^2*c^6*sqrt(d) + 40*a* 
b*c^5*d^(3/2) + 3*a^2*c^4*d^(5/2))/((sqrt(d)*x - sqrt(d*x^2 + c))^2 - c)^5
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x^6} \, dx=\int \frac {{\left (b\,x^2+a\right )}^2\,{\left (d\,x^2+c\right )}^{3/2}}{x^6} \,d x \] Input:

int(((a + b*x^2)^2*(c + d*x^2)^(3/2))/x^6,x)
                                                                                    
                                                                                    
 

Output:

int(((a + b*x^2)^2*(c + d*x^2)^(3/2))/x^6, x)
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.62 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x^6} \, dx=\frac {-24 \sqrt {d \,x^{2}+c}\, a^{2} c^{2}-48 \sqrt {d \,x^{2}+c}\, a^{2} c d \,x^{2}-24 \sqrt {d \,x^{2}+c}\, a^{2} d^{2} x^{4}-80 \sqrt {d \,x^{2}+c}\, a b \,c^{2} x^{2}-320 \sqrt {d \,x^{2}+c}\, a b c d \,x^{4}-120 \sqrt {d \,x^{2}+c}\, b^{2} c^{2} x^{4}+60 \sqrt {d \,x^{2}+c}\, b^{2} c d \,x^{6}+240 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d \,x^{2}+c}+\sqrt {d}\, x}{\sqrt {c}}\right ) a b c d \,x^{5}+180 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d \,x^{2}+c}+\sqrt {d}\, x}{\sqrt {c}}\right ) b^{2} c^{2} x^{5}-24 \sqrt {d}\, a^{2} d^{2} x^{5}+128 \sqrt {d}\, a b c d \,x^{5}+99 \sqrt {d}\, b^{2} c^{2} x^{5}}{120 c \,x^{5}} \] Input:

int((b*x^2+a)^2*(d*x^2+c)^(3/2)/x^6,x)
 

Output:

( - 24*sqrt(c + d*x**2)*a**2*c**2 - 48*sqrt(c + d*x**2)*a**2*c*d*x**2 - 24 
*sqrt(c + d*x**2)*a**2*d**2*x**4 - 80*sqrt(c + d*x**2)*a*b*c**2*x**2 - 320 
*sqrt(c + d*x**2)*a*b*c*d*x**4 - 120*sqrt(c + d*x**2)*b**2*c**2*x**4 + 60* 
sqrt(c + d*x**2)*b**2*c*d*x**6 + 240*sqrt(d)*log((sqrt(c + d*x**2) + sqrt( 
d)*x)/sqrt(c))*a*b*c*d*x**5 + 180*sqrt(d)*log((sqrt(c + d*x**2) + sqrt(d)* 
x)/sqrt(c))*b**2*c**2*x**5 - 24*sqrt(d)*a**2*d**2*x**5 + 128*sqrt(d)*a*b*c 
*d*x**5 + 99*sqrt(d)*b**2*c**2*x**5)/(120*c*x**5)