Optimal. Leaf size=370 \[ -\frac{\left (-9 a^2 d^2+6 a b c d+b^2 c^2\right ) \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{18 a^{4/3} c^3}-\frac{\left (-9 a^2 d^2+6 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{4/3} c^3}+\frac{\log (x) \left (-9 a^2 d^2+6 a b c d+b^2 c^2\right )}{18 a^{4/3} c^3}+\frac{d^{4/3} (b c-a d)^{2/3} \log \left (c+d x^3\right )}{6 c^3}-\frac{d^{4/3} (b c-a d)^{2/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c^3}-\frac{d^{4/3} (b c-a d)^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt{3}}\right )}{\sqrt{3} c^3}+\frac{\left (a+b x^3\right )^{2/3} (6 a d+b c)}{18 a c^2 x^3}-\frac{\left (a+b x^3\right )^{5/3}}{6 a c x^6} \]
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Rubi [A] time = 0.496307, antiderivative size = 370, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {446, 103, 149, 156, 55, 617, 204, 31, 56} \[ -\frac{\left (-9 a^2 d^2+6 a b c d+b^2 c^2\right ) \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{18 a^{4/3} c^3}-\frac{\left (-9 a^2 d^2+6 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{4/3} c^3}+\frac{\log (x) \left (-9 a^2 d^2+6 a b c d+b^2 c^2\right )}{18 a^{4/3} c^3}+\frac{d^{4/3} (b c-a d)^{2/3} \log \left (c+d x^3\right )}{6 c^3}-\frac{d^{4/3} (b c-a d)^{2/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c^3}-\frac{d^{4/3} (b c-a d)^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt{3}}\right )}{\sqrt{3} c^3}+\frac{\left (a+b x^3\right )^{2/3} (6 a d+b c)}{18 a c^2 x^3}-\frac{\left (a+b x^3\right )^{5/3}}{6 a c x^6} \]
Antiderivative was successfully verified.
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Rule 446
Rule 103
Rule 149
Rule 156
Rule 55
Rule 617
Rule 204
Rule 31
Rule 56
Rubi steps
\begin{align*} \int \frac{\left (a+b x^3\right )^{2/3}}{x^7 \left (c+d x^3\right )} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{(a+b x)^{2/3}}{x^3 (c+d x)} \, dx,x,x^3\right )\\ &=-\frac{\left (a+b x^3\right )^{5/3}}{6 a c x^6}-\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^{2/3} \left (\frac{1}{3} (b c+6 a d)+\frac{b d x}{3}\right )}{x^2 (c+d x)} \, dx,x,x^3\right )}{6 a c}\\ &=\frac{(b c+6 a d) \left (a+b x^3\right )^{2/3}}{18 a c^2 x^3}-\frac{\left (a+b x^3\right )^{5/3}}{6 a c x^6}-\frac{\operatorname{Subst}\left (\int \frac{\frac{2}{9} \left (b^2 c^2+6 a b c d-9 a^2 d^2\right )+\frac{2}{9} b d (b c-3 a d) x}{x \sqrt [3]{a+b x} (c+d x)} \, dx,x,x^3\right )}{6 a c^2}\\ &=\frac{(b c+6 a d) \left (a+b x^3\right )^{2/3}}{18 a c^2 x^3}-\frac{\left (a+b x^3\right )^{5/3}}{6 a c x^6}+\frac{\left (d^2 (b c-a d)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a+b x} (c+d x)} \, dx,x,x^3\right )}{3 c^3}-\frac{\left (b^2 c^2+6 a b c d-9 a^2 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt [3]{a+b x}} \, dx,x,x^3\right )}{27 a c^3}\\ &=\frac{(b c+6 a d) \left (a+b x^3\right )^{2/3}}{18 a c^2 x^3}-\frac{\left (a+b x^3\right )^{5/3}}{6 a c x^6}+\frac{\left (b^2 c^2+6 a b c d-9 a^2 d^2\right ) \log (x)}{18 a^{4/3} c^3}+\frac{d^{4/3} (b c-a d)^{2/3} \log \left (c+d x^3\right )}{6 c^3}-\frac{\left (d^{4/3} (b c-a d)^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt [3]{b c-a d}}{\sqrt [3]{d}}+x} \, dx,x,\sqrt [3]{a+b x^3}\right )}{2 c^3}+\frac{(d (b c-a d)) \operatorname{Subst}\left (\int \frac{1}{\frac{(b c-a d)^{2/3}}{d^{2/3}}-\frac{\sqrt [3]{b c-a d} x}{\sqrt [3]{d}}+x^2} \, dx,x,\sqrt [3]{a+b x^3}\right )}{2 c^3}+\frac{\left (b^2 c^2+6 a b c d-9 a^2 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x^3}\right )}{18 a^{4/3} c^3}-\frac{\left (b^2 c^2+6 a b c d-9 a^2 d^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 )}{18 a c^3}\\ &=\frac{(b c+6 a d) \left (a+b x^3\right )^{2/3}}{18 a c^2 x^3}-\frac{\left (a+b x^3\right )^{5/3}}{6 a c x^6}+\frac{\left (b^2 c^2+6 a b c d-9 a^2 d^2\right ) \log (x)}{18 a^{4/3} c^3}+\frac{d^{4/3} (b c-a d)^{2/3} \log \left (c+d x^3\right )}{6 c^3}-\frac{\left (b^2 c^2+6 a b c d-9 a^2 d^2\right ) \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{18 a^{4/3} c^3}-\frac{d^{4/3} (b c-a d)^{2/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c^3}+\frac{\left (d^{4/3} (b c-a d)^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}\right )}{c^3}+\frac{\left (b^2 c^2+6 a b c d-9 a^2 d^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^{4/3} c^3}\\ &=\frac{(b c+6 a d) \left (a+b x^3\right )^{2/3}}{18 a c^2 x^3}-\frac{\left (a+b x^3\right )^{5/3}}{6 a c x^6}-\frac{\left (b^2 c^2+6 a b c d-9 a^2 d^2\right ) \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{9 \sqrt{3} a^{4/3} c^3}-\frac{d^{4/3} (b c-a d)^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt{3}}\right )}{\sqrt{3} c^3}+\frac{\left (b^2 c^2+6 a b c d-9 a^2 d^2\right ) \log (x)}{18 a^{4/3} c^3}+\frac{d^{4/3} (b c-a d)^{2/3} \log \left (c+d x^3\right )}{6 c^3}-\frac{\left (b^2 c^2+6 a b c d-9 a^2 d^2\right ) \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{18 a^{4/3} c^3}-\frac{d^{4/3} (b c-a d)^{2/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c^3}\\ \end{align*}
Mathematica [C] time = 0.272646, size = 240, normalized size = 0.65 \[ \frac{3 x^6 \log (x) \left (-9 a^2 d^2+6 a b c d+b^2 c^2\right )-3 x^6 \left (-9 a^2 d^2+6 a b c d+b^2 c^2\right ) \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )-2 \sqrt{3} x^6 \left (-9 a^2 d^2+6 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}+1}{\sqrt{3}}\right )+27 a^{4/3} d^2 x^6 \left (a+b x^3\right )^{2/3} \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{d \left (b x^3+a\right )}{a d-b c}\right )+3 \sqrt [3]{a} c \left (a+b x^3\right )^{2/3} \left (-3 a c+6 a d x^3-2 b c x^3\right )}{54 a^{4/3} c^3 x^6} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.066, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{7} \left ( d{x}^{3}+c \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 (d x^{3} + c\right )} x^{7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 9.31567, size = 2691, normalized size = 7.27 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x^{3}\right )^{\frac{2}{3}}}{x^{7} \left (c + d x^{3}\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.64391, size = 698, normalized size = 1.89 \begin{align*} -\frac{1}{54} \,{\left (\frac{18 \,{\left (b c d^{2} \left (-\frac{b c - a d}{d}\right )^{\frac{1}{3}} - a d^{3} \left (-\frac{b c - a d}{d}\right )^{\frac{1}{3}}\right )} \left (-\frac{b c - a d}{d}\right )^{\frac{1}{3}} \log \left ({\left |{\left (b x^{3} + a\right )}^{\frac{1}{3}} - \left (-\frac{b c - a d}{d}\right )^{\frac{1}{3}} \right |}\right )}{b^{4} c^{4} - a b^{3} c^{3} d} + \frac{18 \, \sqrt{3}{\left (-b c d^{2} + a d^{3}\right )}^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}} + \left (-\frac{b c - a d}{d}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{b c - a d}{d}\right )^{\frac{1}{3}}}\right )}{b^{3} c^{3}} - \frac{9 \,{\left (-b c d^{2} + a d^{3}\right )}^{\frac{2}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac{2}{3}} +{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-\frac{b c - a d}{d}\right )^{\frac{1}{3}} + \left (-\frac{b c - a d}{d}\right )^{\frac{2}{3}}\right )}{b^{3} c^{3}} + \frac{2 \, \sqrt{3}{\left (a^{\frac{2}{3}} b^{2} c^{2} + 6 \, a^{\frac{5}{3}} b c d - 9 \, a^{\frac{8}{3}} d^{2}\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}} + a^{\frac{1}{3}}\right )}}{3 \, a^{\frac{1}{3}}}\right )}{a^{2} b^{3} c^{3}} + \frac{2 \,{\left (a^{\frac{1}{3}} b^{2} c^{2} + 6 \, a^{\frac{4}{3}} b c d - 9 \, a^{\frac{7}{3}} d^{2}\right )} \log \left ({\left |{\left (b x^{3} + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}} \right |}\right )}{a^{\frac{5}{3}} b^{3} c^{3}} - \frac{{\left (a^{\frac{2}{3}} b^{2} c^{2} + 6 \, a^{\frac{5}{3}} b c d - 9 \, a^{\frac{8}{3}} d^{2}\right )} \log \left ({\left (b x^{3} + a\right )}^{\frac{2}{3}} +{\left (b x^{3} + a\right )}^{\frac{1}{3}} a^{\frac{1}{3}} + a^{\frac{2}{3}}\right )}{a^{2} b^{3} c^{3}} + \frac{3 \,{\left (2 \,{\left (b x^{3} + a\right )}^{\frac{5}{3}} b c +{\left (b x^{3} + a\right )}^{\frac{2}{3}} a b c - 6 \,{\left (b x^{3} + a\right )}^{\frac{5}{3}} a d + 6 \,{\left (b x^{3} + a\right )}^{\frac{2}{3}} a^{2} d\right )}}{a b^{4} c^{2} x^{6}}\right )} b^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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