3.946 \(\int \frac{(a+b x^2+c x^4)^{3/2}}{x^{13}} \, dx\)

Optimal. Leaf size=216 \[ -\frac{\left (7 b^2-4 a c\right ) \left (2 a+b x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{384 a^3 x^8}+\frac{\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (2 a+b x^2\right ) \sqrt{a+b x^2+c x^4}}{1024 a^4 x^4}-\frac{\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{2048 a^{9/2}}+\frac{7 b \left (a+b x^2+c x^4\right )^{5/2}}{120 a^2 x^{10}}-\frac{\left (a+b x^2+c x^4\right )^{5/2}}{12 a x^{12}} \]

[Out]

((b^2 - 4*a*c)*(7*b^2 - 4*a*c)*(2*a + b*x^2)*Sqrt[a + b*x^2 + c*x^4])/(1024*a^4*x^4) - ((7*b^2 - 4*a*c)*(2*a +
b*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(384*a^3*x^8) - (a + b*x^2 + c*x^4)^(5/2)/(12*a*x^12) + (7*b*(a + b*x^2 + c
*x^4)^(5/2))/(120*a^2*x^10) - ((b^2 - 4*a*c)^2*(7*b^2 - 4*a*c)*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2
+ c*x^4])])/(2048*a^(9/2))

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Rubi [A]  time = 0.217046, antiderivative size = 216, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {1114, 744, 806, 720, 724, 206} \[ -\frac{\left (7 b^2-4 a c\right ) \left (2 a+b x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{384 a^3 x^8}+\frac{\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (2 a+b x^2\right ) \sqrt{a+b x^2+c x^4}}{1024 a^4 x^4}-\frac{\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{2048 a^{9/2}}+\frac{7 b \left (a+b x^2+c x^4\right )^{5/2}}{120 a^2 x^{10}}-\frac{\left (a+b x^2+c x^4\right )^{5/2}}{12 a x^{12}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)^(3/2)/x^13,x]

[Out]

((b^2 - 4*a*c)*(7*b^2 - 4*a*c)*(2*a + b*x^2)*Sqrt[a + b*x^2 + c*x^4])/(1024*a^4*x^4) - ((7*b^2 - 4*a*c)*(2*a +
b*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(384*a^3*x^8) - (a + b*x^2 + c*x^4)^(5/2)/(12*a*x^12) + (7*b*(a + b*x^2 + c
*x^4)^(5/2))/(120*a^2*x^10) - ((b^2 - 4*a*c)^2*(7*b^2 - 4*a*c)*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2
+ c*x^4])])/(2048*a^(9/2))

Rule 1114

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*(a +
b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[(m - 1)/2]

Rule 744

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m + 1)
*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)),
Int[(d + e*x)^(m + 1)*Simp[c*d*(m + 1) - b*e*(m + p + 2) - c*e*(m + 2*p + 3)*x, x]*(a + b*x + c*x^2)^p, x], x]
/; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e
, 0] && NeQ[m, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]) || (SumSimplerQ[m, 1] && IntegerQ
[p]) || ILtQ[Simplify[m + 2*p + 3], 0])

Rule 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)), x] - Dist[(b
*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x],
x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[Sim
plify[m + 2*p + 3], 0]

Rule 720

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*
(d*b - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^p)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[(p*(b^2 -
4*a*c))/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[
{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m +
2*p + 2, 0] && GtQ[p, 0]

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\left (a+b x^2+c x^4\right )^{3/2}}{x^{13}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\left (a+b x+c x^2\right )^{3/2}}{x^7} \, dx,x,x^2\right )\\ &=-\frac{\left (a+b x^2+c x^4\right )^{5/2}}{12 a x^{12}}-\frac{\operatorname{Subst}\left (\int \frac{\left (\frac{7 b}{2}+c x\right ) \left (a+b x+c x^2\right )^{3/2}}{x^6} \, dx,x,x^2\right )}{12 a}\\ &=-\frac{\left (a+b x^2+c x^4\right )^{5/2}}{12 a x^{12}}+\frac{7 b \left (a+b x^2+c x^4\right )^{5/2}}{120 a^2 x^{10}}+\frac{\left (7 b^2-4 a c\right ) \operatorname{Subst}\left (\int \frac{\left (a+b x+c x^2\right )^{3/2}}{x^5} \, dx,x,x^2\right )}{48 a^2}\\ &=-\frac{\left (7 b^2-4 a c\right ) \left (2 a+b x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{384 a^3 x^8}-\frac{\left (a+b x^2+c x^4\right )^{5/2}}{12 a x^{12}}+\frac{7 b \left (a+b x^2+c x^4\right )^{5/2}}{120 a^2 x^{10}}-\frac{\left (\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right )\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x+c x^2}}{x^3} \, dx,x,x^2\right )}{256 a^3}\\ &=\frac{\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (2 a+b x^2\right ) \sqrt{a+b x^2+c x^4}}{1024 a^4 x^4}-\frac{\left (7 b^2-4 a c\right ) \left (2 a+b x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{384 a^3 x^8}-\frac{\left (a+b x^2+c x^4\right )^{5/2}}{12 a x^{12}}+\frac{7 b \left (a+b x^2+c x^4\right )^{5/2}}{120 a^2 x^{10}}+\frac{\left (\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{2048 a^4}\\ &=\frac{\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (2 a+b x^2\right ) \sqrt{a+b x^2+c x^4}}{1024 a^4 x^4}-\frac{\left (7 b^2-4 a c\right ) \left (2 a+b x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{384 a^3 x^8}-\frac{\left (a+b x^2+c x^4\right )^{5/2}}{12 a x^{12}}+\frac{7 b \left (a+b x^2+c x^4\right )^{5/2}}{120 a^2 x^{10}}-\frac{\left (\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 a-x^2} \, dx,x,\frac{2 a+b x^2}{\sqrt{a+b x^2+c x^4}}\right )}{1024 a^4}\\ &=\frac{\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (2 a+b x^2\right ) \sqrt{a+b x^2+c x^4}}{1024 a^4 x^4}-\frac{\left (7 b^2-4 a c\right ) \left (2 a+b x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{384 a^3 x^8}-\frac{\left (a+b x^2+c x^4\right )^{5/2}}{12 a x^{12}}+\frac{7 b \left (a+b x^2+c x^4\right )^{5/2}}{120 a^2 x^{10}}-\frac{\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{2048 a^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.210329, size = 206, normalized size = 0.95 \[ -\frac{\frac{\left (\frac{7 b^2}{2}-2 a c\right ) \left (16 a^{3/2} \left (2 a+b x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}-3 x^4 \left (b^2-4 a c\right ) \left (2 \sqrt{a} \left (2 a+b x^2\right ) \sqrt{a+b x^2+c x^4}-x^4 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )\right )\right )}{256 a^{7/2} x^8}-\frac{7 b \left (a+b x^2+c x^4\right )^{5/2}}{10 a x^{10}}+\frac{\left (a+b x^2+c x^4\right )^{5/2}}{x^{12}}}{12 a} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)^(3/2)/x^13,x]

[Out]

-((a + b*x^2 + c*x^4)^(5/2)/x^12 - (7*b*(a + b*x^2 + c*x^4)^(5/2))/(10*a*x^10) + (((7*b^2)/2 - 2*a*c)*(16*a^(3
/2)*(2*a + b*x^2)*(a + b*x^2 + c*x^4)^(3/2) - 3*(b^2 - 4*a*c)*x^4*(2*Sqrt[a]*(2*a + b*x^2)*Sqrt[a + b*x^2 + c*
x^4] - (b^2 - 4*a*c)*x^4*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])])))/(256*a^(7/2)*x^8))/(12*
a)

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Maple [B]  time = 0.184, size = 457, normalized size = 2.1 \begin{align*} -{\frac{{b}^{2}}{320\,a{x}^{8}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{7\,{b}^{3}}{1920\,{a}^{2}{x}^{6}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{7\,{b}^{4}}{1536\,{a}^{3}{x}^{4}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{7\,{b}^{5}}{1024\,{a}^{4}{x}^{2}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{7\,{b}^{6}}{2048}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{9}{2}}}}+{\frac{27\,b{c}^{2}}{320\,{a}^{2}{x}^{2}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{9\,{b}^{2}c}{320\,{a}^{2}{x}^{4}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{19\,{b}^{3}c}{384\,{x}^{2}{a}^{3}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{3\,bc}{160\,a{x}^{6}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{13\,b}{120\,{x}^{10}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{7\,c}{48\,{x}^{8}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{15\,c{b}^{4}}{512}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{7}{2}}}}-{\frac{9\,{b}^{2}{c}^{2}}{128}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}-{\frac{a}{12\,{x}^{12}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{{c}^{2}}{32\,a{x}^{4}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{{c}^{3}}{32}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2+a)^(3/2)/x^13,x)

[Out]

-1/320/a*b^2/x^8*(c*x^4+b*x^2+a)^(1/2)+7/1920/a^2*b^3/x^6*(c*x^4+b*x^2+a)^(1/2)-7/1536/a^3*b^4/x^4*(c*x^4+b*x^
2+a)^(1/2)+7/1024/a^4*b^5/x^2*(c*x^4+b*x^2+a)^(1/2)-7/2048/a^(9/2)*b^6*ln((2*a+b*x^2+2*a^(1/2)*(c*x^4+b*x^2+a)
^(1/2))/x^2)+27/320*b*c^2/a^2/x^2*(c*x^4+b*x^2+a)^(1/2)+9/320*b^2*c/a^2/x^4*(c*x^4+b*x^2+a)^(1/2)-19/384*b^3*c
/a^3/x^2*(c*x^4+b*x^2+a)^(1/2)-3/160*b*c/a/x^6*(c*x^4+b*x^2+a)^(1/2)-13/120*b/x^10*(c*x^4+b*x^2+a)^(1/2)-7/48*
c/x^8*(c*x^4+b*x^2+a)^(1/2)+15/512*b^4*c/a^(7/2)*ln((2*a+b*x^2+2*a^(1/2)*(c*x^4+b*x^2+a)^(1/2))/x^2)-9/128*b^2
*c^2/a^(5/2)*ln((2*a+b*x^2+2*a^(1/2)*(c*x^4+b*x^2+a)^(1/2))/x^2)-1/12*a/x^12*(c*x^4+b*x^2+a)^(1/2)-1/32/a*c^2/
x^4*(c*x^4+b*x^2+a)^(1/2)+1/32/a^(3/2)*c^3*ln((2*a+b*x^2+2*a^(1/2)*(c*x^4+b*x^2+a)^(1/2))/x^2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^(3/2)/x^13,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 3.90069, size = 1108, normalized size = 5.13 \begin{align*} \left [-\frac{15 \,{\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt{a} x^{12} \log \left (-\frac{{\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a b x^{2} + 4 \, \sqrt{c x^{4} + b x^{2} + a}{\left (b x^{2} + 2 \, a\right )} \sqrt{a} + 8 \, a^{2}}{x^{4}}\right ) - 4 \,{\left ({\left (105 \, a b^{5} - 760 \, a^{2} b^{3} c + 1296 \, a^{3} b c^{2}\right )} x^{10} - 2 \,{\left (35 \, a^{2} b^{4} - 216 \, a^{3} b^{2} c + 240 \, a^{4} c^{2}\right )} x^{8} - 1664 \, a^{5} b x^{2} + 8 \,{\left (7 \, a^{3} b^{3} - 36 \, a^{4} b c\right )} x^{6} - 1280 \, a^{6} - 16 \,{\left (3 \, a^{4} b^{2} + 140 \, a^{5} c\right )} x^{4}\right )} \sqrt{c x^{4} + b x^{2} + a}}{61440 \, a^{5} x^{12}}, \frac{15 \,{\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt{-a} x^{12} \arctan \left (\frac{\sqrt{c x^{4} + b x^{2} + a}{\left (b x^{2} + 2 \, a\right )} \sqrt{-a}}{2 \,{\left (a c x^{4} + a b x^{2} + a^{2}\right )}}\right ) + 2 \,{\left ({\left (105 \, a b^{5} - 760 \, a^{2} b^{3} c + 1296 \, a^{3} b c^{2}\right )} x^{10} - 2 \,{\left (35 \, a^{2} b^{4} - 216 \, a^{3} b^{2} c + 240 \, a^{4} c^{2}\right )} x^{8} - 1664 \, a^{5} b x^{2} + 8 \,{\left (7 \, a^{3} b^{3} - 36 \, a^{4} b c\right )} x^{6} - 1280 \, a^{6} - 16 \,{\left (3 \, a^{4} b^{2} + 140 \, a^{5} c\right )} x^{4}\right )} \sqrt{c x^{4} + b x^{2} + a}}{30720 \, a^{5} x^{12}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^(3/2)/x^13,x, algorithm="fricas")

[Out]

[-1/61440*(15*(7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*sqrt(a)*x^12*log(-((b^2 + 4*a*c)*x^4 + 8*a*b
*x^2 + 4*sqrt(c*x^4 + b*x^2 + a)*(b*x^2 + 2*a)*sqrt(a) + 8*a^2)/x^4) - 4*((105*a*b^5 - 760*a^2*b^3*c + 1296*a^
3*b*c^2)*x^10 - 2*(35*a^2*b^4 - 216*a^3*b^2*c + 240*a^4*c^2)*x^8 - 1664*a^5*b*x^2 + 8*(7*a^3*b^3 - 36*a^4*b*c)
*x^6 - 1280*a^6 - 16*(3*a^4*b^2 + 140*a^5*c)*x^4)*sqrt(c*x^4 + b*x^2 + a))/(a^5*x^12), 1/30720*(15*(7*b^6 - 60
*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*sqrt(-a)*x^12*arctan(1/2*sqrt(c*x^4 + b*x^2 + a)*(b*x^2 + 2*a)*sqrt(-
a)/(a*c*x^4 + a*b*x^2 + a^2)) + 2*((105*a*b^5 - 760*a^2*b^3*c + 1296*a^3*b*c^2)*x^10 - 2*(35*a^2*b^4 - 216*a^3
*b^2*c + 240*a^4*c^2)*x^8 - 1664*a^5*b*x^2 + 8*(7*a^3*b^3 - 36*a^4*b*c)*x^6 - 1280*a^6 - 16*(3*a^4*b^2 + 140*a
^5*c)*x^4)*sqrt(c*x^4 + b*x^2 + a))/(a^5*x^12)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{x^{13}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2+a)**(3/2)/x**13,x)

[Out]

Integral((a + b*x**2 + c*x**4)**(3/2)/x**13, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c x^{4} + b x^{2} + a\right )}^{\frac{3}{2}}}{x^{13}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^(3/2)/x^13,x, algorithm="giac")

[Out]

integrate((c*x^4 + b*x^2 + a)^(3/2)/x^13, x)