Optimal. Leaf size=340 \[ \frac{(b c-3 a d) \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{6 a^{2/3} c^2}-\frac{(b c-3 a d) \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{2/3} c^2}-\frac{\log (x) (b c-3 a d)}{6 a^{2/3} c^2}+\frac{d^{2/3} \sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 c^2}-\frac{d^{2/3} \sqrt [3]{b c-a d} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c^2}+\frac{d^{2/3} \sqrt [3]{b c-a d} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt{3}}\right )}{\sqrt{3} c^2}+\frac{d \sqrt [3]{a+b x^3}}{c^2}+\frac{\sqrt [3]{a+b x^3} (b c-3 a d)}{3 a c^2}-\frac{\left (a+b x^3\right )^{4/3}}{3 a c x^3} \]
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Rubi [A] time = 0.383712, antiderivative size = 340, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {446, 103, 156, 50, 57, 617, 204, 31, 58} \[ \frac{(b c-3 a d) \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{6 a^{2/3} c^2}-\frac{(b c-3 a d) \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{2/3} c^2}-\frac{\log (x) (b c-3 a d)}{6 a^{2/3} c^2}+\frac{d^{2/3} \sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 c^2}-\frac{d^{2/3} \sqrt [3]{b c-a d} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c^2}+\frac{d^{2/3} \sqrt [3]{b c-a d} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt{3}}\right )}{\sqrt{3} c^2}+\frac{d \sqrt [3]{a+b x^3}}{c^2}+\frac{\sqrt [3]{a+b x^3} (b c-3 a d)}{3 a c^2}-\frac{\left (a+b x^3\right )^{4/3}}{3 a c x^3} \]
Antiderivative was successfully verified.
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Rule 446
Rule 103
Rule 156
Rule 50
Rule 57
Rule 617
Rule 204
Rule 31
Rule 58
Rubi steps
\begin{align*} \int \frac{\sqrt [3]{a+b x^3}}{x^4 \left (c+d x^3\right )} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{\sqrt [3]{a+b x}}{x^2 (c+d x)} \, dx,x,x^3\right )\\ &=-\frac{\left (a+b x^3\right )^{4/3}}{3 a c x^3}-\frac{\operatorname{Subst}\left (\int \frac{\sqrt [3]{a+b x} \left (\frac{1}{3} (-b c+3 a d)-\frac{b d x}{3}\right )}{x (c+d x)} \, dx,x,x^3\right )}{3 a c}\\ &=-\frac{\left (a+b x^3\right )^{4/3}}{3 a c x^3}+\frac{d^2 \operatorname{Subst}\left (\int \frac{\sqrt [3]{a+b x}}{c+d x} \, dx,x,x^3\right )}{3 c^2}+\frac{(b c-3 a d) \operatorname{Subst}\left (\int \frac{\sqrt [3]{a+b x}}{x} \, dx,x,x^3\right )}{9 a c^2}\\ &=\frac{d \sqrt [3]{a+b x^3}}{c^2}+\frac{(b c-3 a d) \sqrt [3]{a+b x^3}}{3 a c^2}-\frac{\left (a+b x^3\right )^{4/3}}{3 a c x^3}+\frac{(b c-3 a d) \operatorname{Subst}\left (\int \frac{1}{x (a+b x)^{2/3}} \, dx,x,x^3\right )}{9 c^2}-\frac{(d (b c-a d)) \operatorname{Subst}\left (\int \frac{1}{(a+b x)^{2/3} (c+d x)} \, dx,x,x^3\right )}{3 c^2}\\ &=\frac{d \sqrt [3]{a+b x^3}}{c^2}+\frac{(b c-3 a d) \sqrt [3]{a+b x^3}}{3 a c^2}-\frac{\left (a+b x^3\right )^{4/3}}{3 a c x^3}-\frac{(b c-3 a d) \log (x)}{6 a^{2/3} c^2}+\frac{d^{2/3} \sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 c^2}-\frac{(b c-3 a d) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x^3}\right )}{6 a^{2/3} c^2}-\frac{(b c-3 a d) \operatorname{Subst}\left (\int \frac{1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x^3}\right )}{6 \sqrt [3]{a} c^2}-\frac{\left (d^{2/3} \sqrt [3]{b c-a d}\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^2}-\frac{\left (\sqrt [3]{d} (b c-a d)^{2/3}\right ) \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^2}\\ &=\frac{d \sqrt [3]{a+b x^3}}{c^2}+\frac{(b c-3 a d) \sqrt [3]{a+b x^3}}{3 a c^2}-\frac{\left (a+b x^3\right )^{4/3}}{3 a c x^3}-\frac{(b c-3 a d) \log (x)}{6 a^{2/3} c^2}+\frac{d^{2/3} \sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 c^2}+\frac{(b c-3 a d) \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{6 a^{2/3} c^2}-\frac{d^{2/3} \sqrt [3]{b c-a d} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c^2}+\frac{(b c-3 a d) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}\right )}{3 a^{2/3} c^2}-\frac{\left (d^{2/3} \sqrt [3]{b c-a d}\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^2}\\ &=\frac{d \sqrt [3]{a+b x^3}}{c^2}+\frac{(b c-3 a d) \sqrt [3]{a+b x^3}}{3 a c^2}-\frac{\left (a+b x^3\right )^{4/3}}{3 a c x^3}-\frac{(b c-3 a d) \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{3 \sqrt{3} a^{2/3} c^2}+\frac{d^{2/3} \sqrt [3]{b c-a d} \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^2}-\frac{(b c-3 a d) \log (x)}{6 a^{2/3} c^2}+\frac{d^{2/3} \sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 c^2}+\frac{(b c-3 a d) \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{6 a^{2/3} c^2}-\frac{d^{2/3} \sqrt [3]{b c-a d} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c^2}\\ \end{align*}
Mathematica [A] time = 0.92963, size = 366, normalized size = 1.08 \[ \frac{\frac{(b c-3 a d) \left (3 \sqrt [3]{a+b x^3}-\frac{1}{2} \sqrt [3]{a} \left (\log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )-2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}+1}{\sqrt{3}}\right )\right )\right )}{3 c}+\frac{a d^{2/3} \left (\sqrt [3]{b c-a d} \log \left (-\sqrt [3]{d} \sqrt [3]{a+b x^3} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+d^{2/3} \left (a+b x^3\right )^{2/3}\right )-2 \sqrt [3]{b c-a d} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )-2 \sqrt{3} \sqrt [3]{b c-a d} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}-1}{\sqrt{3}}\right )+6 \sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c}-\frac{\left (a+b x^3\right )^{4/3}}{x^3}}{3 a c} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.059, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4} \left ( d{x}^{3}+c \right ) }\sqrt [3]{b{x}^{3}+a}}\, 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{1}{3}}}{{\left (d x^{3} + c\right )} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.69548, size = 1044, normalized size = 3.07 \begin{align*} -\frac{6 \, \sqrt{3}{\left (-b c d^{2} + a d^{3}\right )}^{\frac{1}{3}} a^{2} x^{3} \arctan \left (-\frac{2 \, \sqrt{3}{\left (-b c d^{2} + a d^{3}\right )}^{\frac{2}{3}}{\left (b x^{3} + a\right )}^{\frac{1}{3}} - \sqrt{3}{\left (b c d - a d^{2}\right )}}{3 \,{\left (b c d - a d^{2}\right )}}\right ) + 3 \,{\left (-b c d^{2} + a d^{3}\right )}^{\frac{1}{3}} a^{2} x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac{2}{3}} d^{2} +{\left (-b c d^{2} + a d^{3}\right )}^{\frac{1}{3}}{\left (b x^{3} + a\right )}^{\frac{1}{3}} d +{\left (-b c d^{2} + a d^{3}\right )}^{\frac{2}{3}}\right ) - 6 \,{\left (-b c d^{2} + a d^{3}\right )}^{\frac{1}{3}} a^{2} x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac{1}{3}} d -{\left (-b c d^{2} + a d^{3}\right )}^{\frac{1}{3}}\right ) + 2 \, \sqrt{3}{\left (a b c - 3 \, a^{2} d\right )} x^{3} \sqrt{-\left (-a^{2}\right )^{\frac{1}{3}}} \arctan \left (-\frac{{\left (\sqrt{3} \left (-a^{2}\right )^{\frac{1}{3}} a - 2 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-a^{2}\right )^{\frac{2}{3}}\right )} \sqrt{-\left (-a^{2}\right )^{\frac{1}{3}}}}{3 \, a^{2}}\right ) + \left (-a^{2}\right )^{\frac{2}{3}}{\left (b c - 3 \, a d\right )} x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac{2}{3}} a - \left (-a^{2}\right )^{\frac{1}{3}} a +{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-a^{2}\right )^{\frac{2}{3}}\right ) - 2 \, \left (-a^{2}\right )^{\frac{2}{3}}{\left (b c - 3 \, a d\right )} x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac{1}{3}} a - \left (-a^{2}\right )^{\frac{2}{3}}\right ) + 6 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}} a^{2} c}{18 \, a^{2} c^{2} x^{3}} \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{\sqrt [3]{a + b x^{3}}}{x^{4} \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 = 2.60832, size = 525, normalized size = 1.54 \begin{align*} \frac{1}{18} \,{\left (\frac{6 \,{\left (b c d - a d^{2}\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^{3} c^{3} - a b^{2} c^{2} d} - \frac{6 \, \sqrt{3}{\left (-b c d^{2} + a d^{3}\right )}^{\frac{1}{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^{2} c^{2}} - \frac{2 \, \sqrt{3}{\left (a^{\frac{1}{3}} b c - 3 \, a^{\frac{4}{3}} d\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 b^{2} c^{2}} - \frac{3 \,{\left (-b c d^{2} + a d^{3}\right )}^{\frac{1}{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^{2} c^{2}} + \frac{2 \,{\left (b c - 3 \, a d\right )} \log \left ({\left |{\left (b x^{3} + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}} \right |}\right )}{a^{\frac{2}{3}} b^{2} c^{2}} - \frac{{\left (a^{\frac{1}{3}} b c - 3 \, a^{\frac{4}{3}} d\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 b^{2} c^{2}} - \frac{6 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{b^{2} c x^{3}}\right )} b^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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