Optimal. Leaf size=284 \[ \frac{b^2 \log \left (a-b x^3\right )}{3 \sqrt [3]{2} a^{7/3} d}+\frac{7 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{9 a^{7/3} d}-\frac{b^2 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} a^{7/3} d}+\frac{14 b^2 \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{7/3} d}-\frac{2^{2/3} b^2 \tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{7/3} d}-\frac{7 b^2 \log (x)}{9 a^{7/3} d}-\frac{5 b \left (a+b x^3\right )^{2/3}}{18 a^2 d x^3}-\frac{\left (a+b x^3\right )^{5/3}}{6 a^2 d x^6} \]
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Rubi [A] time = 0.279263, antiderivative size = 284, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {446, 103, 149, 156, 55, 617, 204, 31} \[ \frac{b^2 \log \left (a-b x^3\right )}{3 \sqrt [3]{2} a^{7/3} d}+\frac{7 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{9 a^{7/3} d}-\frac{b^2 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} a^{7/3} d}+\frac{14 b^2 \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{7/3} d}-\frac{2^{2/3} b^2 \tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{7/3} d}-\frac{7 b^2 \log (x)}{9 a^{7/3} d}-\frac{5 b \left (a+b x^3\right )^{2/3}}{18 a^2 d x^3}-\frac{\left (a+b x^3\right )^{5/3}}{6 a^2 d x^6} \]
Antiderivative was successfully verified.
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Rule 446
Rule 103
Rule 149
Rule 156
Rule 55
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{\left (a+b x^3\right )^{2/3}}{x^7 \left (a d-b d x^3\right )} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{(a+b x)^{2/3}}{x^3 (a d-b d x)} \, dx,x,x^3\right )\\ &=-\frac{\left (a+b x^3\right )^{5/3}}{6 a^2 d x^6}-\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^{2/3} \left (-\frac{5}{3} a b d-\frac{1}{3} b^2 d x\right )}{x^2 (a d-b d x)} \, dx,x,x^3\right )}{6 a^2 d}\\ &=-\frac{5 b \left (a+b x^3\right )^{2/3}}{18 a^2 d x^3}-\frac{\left (a+b x^3\right )^{5/3}}{6 a^2 d x^6}-\frac{\operatorname{Subst}\left (\int \frac{-\frac{28}{9} a^2 b^2 d^2-\frac{8}{9} a b^3 d^2 x}{x \sqrt [3]{a+b x} (a d-b d x)} \, dx,x,x^3\right )}{6 a^3 d^2}\\ &=-\frac{5 b \left (a+b x^3\right )^{2/3}}{18 a^2 d x^3}-\frac{\left (a+b x^3\right )^{5/3}}{6 a^2 d x^6}+\frac{\left (2 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a+b x} (a d-b d x)} \, dx,x,x^3\right )}{3 a^2}+\frac{\left (14 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt [3]{a+b x}} \, dx,x,x^3\right )}{27 a^2 d}\\ &=-\frac{5 b \left (a+b x^3\right )^{2/3}}{18 a^2 d x^3}-\frac{\left (a+b x^3\right )^{5/3}}{6 a^2 d x^6}-\frac{7 b^2 \log (x)}{9 a^{7/3} d}+\frac{b^2 \log \left (a-b x^3\right )}{3 \sqrt [3]{2} a^{7/3} d}-\frac{\left (7 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x^3}\right )}{9 a^{7/3} d}+\frac{b^2 \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} a^{7/3} d}+\frac{\left (7 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x^3}\right )}{9 a^2 d}-\frac{b^2 \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 )}{a^2 d}\\ &=-\frac{5 b \left (a+b x^3\right )^{2/3}}{18 a^2 d x^3}-\frac{\left (a+b x^3\right )^{5/3}}{6 a^2 d x^6}-\frac{7 b^2 \log (x)}{9 a^{7/3} d}+\frac{b^2 \log \left (a-b x^3\right )}{3 \sqrt [3]{2} a^{7/3} d}+\frac{7 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{9 a^{7/3} d}-\frac{b^2 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} a^{7/3} d}-\frac{\left (14 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}\right )}{9 a^{7/3} d}+\frac{\left (2^{2/3} b^2\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 )}{a^{7/3} d}\\ &=-\frac{5 b \left (a+b x^3\right )^{2/3}}{18 a^2 d x^3}-\frac{\left (a+b x^3\right )^{5/3}}{6 a^2 d x^6}+\frac{14 b^2 \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{9 \sqrt{3} a^{7/3} d}-\frac{2^{2/3} b^2 \tan ^{-1}\left (\frac{1+\frac{2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt{3} a^{7/3} d}-\frac{7 b^2 \log (x)}{9 a^{7/3} d}+\frac{b^2 \log \left (a-b x^3\right )}{3 \sqrt [3]{2} a^{7/3} d}+\frac{7 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{9 a^{7/3} d}-\frac{b^2 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} a^{7/3} d}\\ \end{align*}
Mathematica [A] time = 0.14117, size = 247, normalized size = 0.87 \[ \frac{28 \sqrt{3} b^2 x^6 \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}+1}{\sqrt{3}}\right )-3 \left (3 a^{4/3} \left (a+b x^3\right )^{2/3}-3\ 2^{2/3} b^2 x^6 \log \left (a-b x^3\right )-14 b^2 x^6 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )+9\ 2^{2/3} b^2 x^6 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )+6\ 2^{2/3} \sqrt{3} b^2 x^6 \tan ^{-1}\left (\frac{\frac{2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}+1}{\sqrt{3}}\right )+8 \sqrt [3]{a} b x^3 \left (a+b x^3\right )^{2/3}+14 b^2 x^6 \log (x)\right )}{54 a^{7/3} d x^6} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.063, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{7} \left ( -bd{x}^{3}+ad \right ) } \left ( b{x}^{3}+a \right ) ^{{\frac{2}{3}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{{\left (b d x^{3} - a d\right )} x^{7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72148, size = 1816, normalized size = 6.39 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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