Optimal. Leaf size=153 \[ \frac{a^{2/3} \log \left (a-b x^3\right )}{3 \sqrt [3]{2} b d}-\frac{a^{2/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} b d}-\frac{2^{2/3} a^{2/3} \tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b d}-\frac{\left (a+b x^3\right )^{2/3}}{2 b d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.125755, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {444, 50, 55, 617, 204, 31} \[ \frac{a^{2/3} \log \left (a-b x^3\right )}{3 \sqrt [3]{2} b d}-\frac{a^{2/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} b d}-\frac{2^{2/3} a^{2/3} \tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b d}-\frac{\left (a+b x^3\right )^{2/3}}{2 b d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 444
Rule 50
Rule 55
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{x^2 \left (a+b x^3\right )^{2/3}}{a d-b d x^3} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{(a+b x)^{2/3}}{a d-b d x} \, dx,x,x^3\right )\\ &=-\frac{\left (a+b x^3\right )^{2/3}}{2 b d}+\frac{1}{3} (2 a) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a+b x} (a d-b d x)} \, dx,x,x^3\right )\\ &=-\frac{\left (a+b x^3\right )^{2/3}}{2 b d}+\frac{a^{2/3} \log \left (a-b x^3\right )}{3 \sqrt [3]{2} b d}+\frac{a^{2/3} \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{2} \sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} b d}-\frac{a \operatorname{Subst}\left (\int \frac{1}{2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x^3}\right )}{b d}\\ &=-\frac{\left (a+b x^3\right )^{2/3}}{2 b d}+\frac{a^{2/3} \log \left (a-b x^3\right )}{3 \sqrt [3]{2} b d}-\frac{a^{2/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} b d}+\frac{\left (2^{2/3} a^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}\right )}{b d}\\ &=-\frac{\left (a+b x^3\right )^{2/3}}{2 b d}-\frac{2^{2/3} a^{2/3} \tan ^{-1}\left (\frac{1+\frac{2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt{3} b d}+\frac{a^{2/3} \log \left (a-b x^3\right )}{3 \sqrt [3]{2} b d}-\frac{a^{2/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} b d}\\ \end{align*}
Mathematica [A] time = 0.0713689, size = 130, normalized size = 0.85 \[ \frac{2^{2/3} a^{2/3} \log \left (a-b x^3\right )-3 \left (2^{2/3} a^{2/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )+\left (a+b x^3\right )^{2/3}\right )-2\ 2^{2/3} \sqrt{3} a^{2/3} \tan ^{-1}\left (\frac{\frac{2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}+1}{\sqrt{3}}\right )}{6 b d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.04, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2}}{-bd{x}^{3}+ad} \left ( b{x}^{3}+a \right ) ^{{\frac{2}{3}}}}\, dx \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 = 1.56344, size = 454, normalized size = 2.97 \begin{align*} -\frac{2 \cdot 4^{\frac{1}{3}} \sqrt{3} \left (-a^{2}\right )^{\frac{1}{3}} \arctan \left (\frac{4^{\frac{1}{3}} \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-a^{2}\right )^{\frac{1}{3}} - \sqrt{3} a}{3 \, a}\right ) + 4^{\frac{1}{3}} \left (-a^{2}\right )^{\frac{1}{3}} \log \left (4^{\frac{2}{3}}{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-a^{2}\right )^{\frac{2}{3}} + 2 \,{\left (b x^{3} + a\right )}^{\frac{2}{3}} a - 2 \cdot 4^{\frac{1}{3}} \left (-a^{2}\right )^{\frac{1}{3}} a\right ) - 2 \cdot 4^{\frac{1}{3}} \left (-a^{2}\right )^{\frac{1}{3}} \log \left (-4^{\frac{2}{3}} \left (-a^{2}\right )^{\frac{2}{3}} + 2 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}} a\right ) + 3 \,{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{6 \, b d} \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*} - \frac{\int \frac{x^{2} \left (a + b x^{3}\right )^{\frac{2}{3}}}{- a + b x^{3}}\, dx}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]