Optimal. Leaf size=161 \[ -\frac{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{9 x^9 \left (a+b x^3\right )}-\frac{a^2 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^6 \left (a+b x^3\right )}-\frac{a b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x^3 \left (a+b x^3\right )}+\frac{b^3 \log (x) \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0464384, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {1355, 266, 43} \[ -\frac{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{9 x^9 \left (a+b x^3\right )}-\frac{a^2 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^6 \left (a+b x^3\right )}-\frac{a b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x^3 \left (a+b x^3\right )}+\frac{b^3 \log (x) \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1355
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^{10}} \, dx &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \int \frac{\left (a b+b^2 x^3\right )^3}{x^{10}} \, dx}{b^2 \left (a b+b^2 x^3\right )}\\ &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \operatorname{Subst}\left (\int \frac{\left (a b+b^2 x\right )^3}{x^4} \, dx,x,x^3\right )}{3 b^2 \left (a b+b^2 x^3\right )}\\ &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \operatorname{Subst}\left (\int \left (\frac{a^3 b^3}{x^4}+\frac{3 a^2 b^4}{x^3}+\frac{3 a b^5}{x^2}+\frac{b^6}{x}\right ) \, dx,x,x^3\right )}{3 b^2 \left (a b+b^2 x^3\right )}\\ &=-\frac{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{9 x^9 \left (a+b x^3\right )}-\frac{a^2 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^6 \left (a+b x^3\right )}-\frac{a b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x^3 \left (a+b x^3\right )}+\frac{b^3 \sqrt{a^2+2 a b x^3+b^2 x^6} \log (x)}{a+b x^3}\\ \end{align*}
Mathematica [A] time = 0.0220902, size = 63, normalized size = 0.39 \[ -\frac{\sqrt{\left (a+b x^3\right )^2} \left (a \left (2 a^2+9 a b x^3+18 b^2 x^6\right )-18 b^3 x^9 \log (x)\right )}{18 x^9 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.011, size = 60, normalized size = 0.4 \begin{align*}{\frac{18\,{b}^{3}\ln \left ( x \right ){x}^{9}-18\,a{b}^{2}{x}^{6}-9\,{a}^{2}b{x}^{3}-2\,{a}^{3}}{18\, \left ( b{x}^{3}+a \right ) ^{3}{x}^{9}} \left ( \left ( b{x}^{3}+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 = 1.7255, size = 90, normalized size = 0.56 \begin{align*} \frac{18 \, b^{3} x^{9} \log \left (x\right ) - 18 \, a b^{2} x^{6} - 9 \, a^{2} b x^{3} - 2 \, a^{3}}{18 \, x^{9}} \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^{3}\right )^{2}\right )^{\frac{3}{2}}}{x^{10}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.11498, size = 115, normalized size = 0.71 \begin{align*} b^{3} \log \left ({\left | x \right |}\right ) \mathrm{sgn}\left (b x^{3} + a\right ) - \frac{11 \, b^{3} x^{9} \mathrm{sgn}\left (b x^{3} + a\right ) + 18 \, a b^{2} x^{6} \mathrm{sgn}\left (b x^{3} + a\right ) + 9 \, a^{2} b x^{3} \mathrm{sgn}\left (b x^{3} + a\right ) + 2 \, a^{3} \mathrm{sgn}\left (b x^{3} + a\right )}{18 \, x^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]