Optimal. Leaf size=41 \[ -\frac{\left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 a x^8} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0394895, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {1111, 646, 37} \[ -\frac{\left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 a x^8} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1111
Rule 646
Rule 37
Rubi steps
\begin{align*} \int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{x^9} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^5} \, dx,x,x^2\right )\\ &=\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \operatorname{Subst}\left (\int \frac{\left (a b+b^2 x\right )^3}{x^5} \, dx,x,x^2\right )}{2 b^2 \left (a b+b^2 x^2\right )}\\ &=-\frac{\left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 a x^8}\\ \end{align*}
Mathematica [A] time = 0.0148303, size = 59, normalized size = 1.44 \[ -\frac{\sqrt{\left (a+b x^2\right )^2} \left (4 a^2 b x^2+a^3+6 a b^2 x^4+4 b^3 x^6\right )}{8 x^8 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.176, size = 56, normalized size = 1.4 \begin{align*} -{\frac{4\,{b}^{3}{x}^{6}+6\,a{x}^{4}{b}^{2}+4\,{a}^{2}b{x}^{2}+{a}^{3}}{8\,{x}^{8} \left ( b{x}^{2}+a \right ) ^{3}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.42766, size = 76, normalized size = 1.85 \begin{align*} -\frac{4 \, b^{3} x^{6} + 6 \, a b^{2} x^{4} + 4 \, a^{2} b x^{2} + a^{3}}{8 \, x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac{3}{2}}}{x^{9}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.13841, size = 92, normalized size = 2.24 \begin{align*} -\frac{4 \, b^{3} x^{6} \mathrm{sgn}\left (b x^{2} + a\right ) + 6 \, a b^{2} x^{4} \mathrm{sgn}\left (b x^{2} + a\right ) + 4 \, a^{2} b x^{2} \mathrm{sgn}\left (b x^{2} + a\right ) + a^{3} \mathrm{sgn}\left (b x^{2} + a\right )}{8 \, x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]