Optimal. Leaf size=357 \[ -\frac{3 a d+4 b c}{3 a^2 c^2 \sqrt [3]{a+b x^3}}-\frac{(3 a d+4 b c) \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{6 a^{7/3} c^2}-\frac{(3 a d+4 b c) \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{7/3} c^2}+\frac{\log (x) (3 a d+4 b c)}{6 a^{7/3} c^2}-\frac{d^2}{c^2 \sqrt [3]{a+b x^3} (b c-a d)}-\frac{d^{7/3} \log \left (c+d x^3\right )}{6 c^2 (b c-a d)^{4/3}}+\frac{d^{7/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c^2 (b c-a d)^{4/3}}+\frac{d^{7/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt{3}}\right )}{\sqrt{3} c^2 (b c-a d)^{4/3}}-\frac{1}{3 a c x^3 \sqrt [3]{a+b x^3}} \]
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Rubi [A] time = 0.393891, antiderivative size = 357, 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, 51, 55, 617, 204, 31, 56} \[ -\frac{3 a d+4 b c}{3 a^2 c^2 \sqrt [3]{a+b x^3}}-\frac{(3 a d+4 b c) \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{6 a^{7/3} c^2}-\frac{(3 a d+4 b c) \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{7/3} c^2}+\frac{\log (x) (3 a d+4 b c)}{6 a^{7/3} c^2}-\frac{d^2}{c^2 \sqrt [3]{a+b x^3} (b c-a d)}-\frac{d^{7/3} \log \left (c+d x^3\right )}{6 c^2 (b c-a d)^{4/3}}+\frac{d^{7/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c^2 (b c-a d)^{4/3}}+\frac{d^{7/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt{3}}\right )}{\sqrt{3} c^2 (b c-a d)^{4/3}}-\frac{1}{3 a c x^3 \sqrt [3]{a+b x^3}} \]
Antiderivative was successfully verified.
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Rule 446
Rule 103
Rule 156
Rule 51
Rule 55
Rule 617
Rule 204
Rule 31
Rule 56
Rubi steps
\begin{align*} \int \frac{1}{x^4 \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{x^2 (a+b x)^{4/3} (c+d x)} \, dx,x,x^3\right )\\ &=-\frac{1}{3 a c x^3 \sqrt [3]{a+b x^3}}-\frac{\operatorname{Subst}\left (\int \frac{\frac{1}{3} (4 b c+3 a d)+\frac{4 b d x}{3}}{x (a+b x)^{4/3} (c+d x)} \, dx,x,x^3\right )}{3 a c}\\ &=-\frac{1}{3 a c x^3 \sqrt [3]{a+b x^3}}+\frac{d^2 \operatorname{Subst}\left (\int \frac{1}{(a+b x)^{4/3} (c+d x)} \, dx,x,x^3\right )}{3 c^2}-\frac{(4 b c+3 a d) \operatorname{Subst}\left (\int \frac{1}{x (a+b x)^{4/3}} \, dx,x,x^3\right )}{9 a c^2}\\ &=-\frac{d^2}{c^2 (b c-a d) \sqrt [3]{a+b x^3}}-\frac{4 b c+3 a d}{3 a^2 c^2 \sqrt [3]{a+b x^3}}-\frac{1}{3 a c x^3 \sqrt [3]{a+b x^3}}-\frac{d^3 \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a+b x} (c+d x)} \, dx,x,x^3\right )}{3 c^2 (b c-a d)}-\frac{(4 b c+3 a d) \operatorname{Subst}\left (\int \frac{1}{x \sqrt [3]{a+b x}} \, dx,x,x^3\right )}{9 a^2 c^2}\\ &=-\frac{d^2}{c^2 (b c-a d) \sqrt [3]{a+b x^3}}-\frac{4 b c+3 a d}{3 a^2 c^2 \sqrt [3]{a+b x^3}}-\frac{1}{3 a c x^3 \sqrt [3]{a+b x^3}}+\frac{(4 b c+3 a d) \log (x)}{6 a^{7/3} c^2}-\frac{d^{7/3} \log \left (c+d x^3\right )}{6 c^2 (b c-a d)^{4/3}}+\frac{d^{7/3} \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 (b c-a d)^{4/3}}-\frac{d^2 \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 (b c-a d)}+\frac{(4 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^{7/3} c^2}-\frac{(4 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 a^2 c^2}\\ &=-\frac{d^2}{c^2 (b c-a d) \sqrt [3]{a+b x^3}}-\frac{4 b c+3 a d}{3 a^2 c^2 \sqrt [3]{a+b x^3}}-\frac{1}{3 a c x^3 \sqrt [3]{a+b x^3}}+\frac{(4 b c+3 a d) \log (x)}{6 a^{7/3} c^2}-\frac{d^{7/3} \log \left (c+d x^3\right )}{6 c^2 (b c-a d)^{4/3}}-\frac{(4 b c+3 a d) \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{6 a^{7/3} c^2}+\frac{d^{7/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c^2 (b c-a d)^{4/3}}-\frac{d^{7/3} \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 (b c-a d)^{4/3}}+\frac{(4 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^{7/3} c^2}\\ &=-\frac{d^2}{c^2 (b c-a d) \sqrt [3]{a+b x^3}}-\frac{4 b c+3 a d}{3 a^2 c^2 \sqrt [3]{a+b x^3}}-\frac{1}{3 a c x^3 \sqrt [3]{a+b x^3}}-\frac{(4 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^{7/3} c^2}+\frac{d^{7/3} \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^2 (b c-a d)^{4/3}}+\frac{(4 b c+3 a d) \log (x)}{6 a^{7/3} c^2}-\frac{d^{7/3} \log \left (c+d x^3\right )}{6 c^2 (b c-a d)^{4/3}}-\frac{(4 b c+3 a d) \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{6 a^{7/3} c^2}+\frac{d^{7/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c^2 (b c-a d)^{4/3}}\\ \end{align*}
Mathematica [C] time = 0.0475005, size = 117, normalized size = 0.33 \[ \frac{3 a^2 d^2 x^3 \, _2F_1\left (-\frac{1}{3},1;\frac{2}{3};\frac{d \left (b x^3+a\right )}{a d-b c}\right )+(b c-a d) \left (x^3 (3 a d+4 b c) \, _2F_1\left (-\frac{1}{3},1;\frac{2}{3};\frac{b x^3}{a}+1\right )+a c\right )}{3 a^2 c^2 x^3 \sqrt [3]{a+b x^3} (a d-b c)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.065, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4} \left ( d{x}^{3}+c \right ) } \left ( b{x}^{3}+a \right ) ^{-{\frac{4}{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{1}{{\left (b x^{3} + a\right )}^{\frac{4}{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 [B] time = 6.19937, size = 3062, normalized size = 8.58 \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{1}{x^{4} \left (a + b x^{3}\right )^{\frac{4}{3}} \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 = 2.89965, size = 699, normalized size = 1.96 \begin{align*} \frac{1}{18} \,{\left (\frac{6 \, d^{3} \left (-\frac{b c - a d}{d}\right )^{\frac{2}{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} - 2 \, a b^{3} c^{3} d + a^{2} b^{2} c^{2} d^{2}} + \frac{18 \,{\left (-b c d^{2} + a d^{3}\right )}^{\frac{2}{3}} d \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 )}{\sqrt{3} b^{4} c^{4} - 2 \, \sqrt{3} a b^{3} c^{3} d + \sqrt{3} a^{2} b^{2} c^{2} d^{2}} - \frac{3 \,{\left (-b c d^{2} + a d^{3}\right )}^{\frac{2}{3}} d \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^{4} c^{4} - 2 \, a b^{3} c^{3} d + a^{2} b^{2} c^{2} d^{2}} - \frac{6 \,{\left (4 \,{\left (b x^{3} + a\right )} b c - 3 \, a b c -{\left (b x^{3} + a\right )} a d\right )}}{{\left (a^{2} b^{2} c^{2} - a^{3} b c d\right )}{\left ({\left (b x^{3} + a\right )}^{\frac{4}{3}} -{\left (b x^{3} + a\right )}^{\frac{1}{3}} a\right )}} - \frac{2 \, \sqrt{3}{\left (4 \, a^{\frac{2}{3}} b c + 3 \, a^{\frac{5}{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^{3} b^{2} c^{2}} - \frac{2 \,{\left (4 \, a^{\frac{1}{3}} b c + 3 \, a^{\frac{4}{3}} d\right )} \log \left ({\left |{\left (b x^{3} + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}} \right |}\right )}{a^{\frac{8}{3}} b^{2} c^{2}} + \frac{{\left (4 \, a^{\frac{2}{3}} b c + 3 \, a^{\frac{5}{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^{3} b^{2} c^{2}}\right )} b^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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