Optimal. Leaf size=104 \[ \frac{11 \sqrt [3]{x-1}}{27 x^2}+\frac{11 \sqrt [3]{x-1}}{36 x^3}+\frac{\sqrt [3]{x-1}}{4 x^4}+\frac{55 \sqrt [3]{x-1}}{81 x}+\frac{55}{81} \log \left (\sqrt [3]{x-1}+1\right )-\frac{55 \log (x)}{243}-\frac{110 \tan ^{-1}\left (\frac{1-2 \sqrt [3]{x-1}}{\sqrt{3}}\right )}{81 \sqrt{3}} \]
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Rubi [A] time = 0.0400664, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454, Rules used = {51, 58, 618, 204, 31} \[ \frac{11 \sqrt [3]{x-1}}{27 x^2}+\frac{11 \sqrt [3]{x-1}}{36 x^3}+\frac{\sqrt [3]{x-1}}{4 x^4}+\frac{55 \sqrt [3]{x-1}}{81 x}+\frac{55}{81} \log \left (\sqrt [3]{x-1}+1\right )-\frac{55 \log (x)}{243}-\frac{110 \tan ^{-1}\left (\frac{1-2 \sqrt [3]{x-1}}{\sqrt{3}}\right )}{81 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 58
Rule 618
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{(-1+x)^{2/3} x^5} \, dx &=\frac{\sqrt [3]{-1+x}}{4 x^4}+\frac{11}{12} \int \frac{1}{(-1+x)^{2/3} x^4} \, dx\\ &=\frac{\sqrt [3]{-1+x}}{4 x^4}+\frac{11 \sqrt [3]{-1+x}}{36 x^3}+\frac{22}{27} \int \frac{1}{(-1+x)^{2/3} x^3} \, dx\\ &=\frac{\sqrt [3]{-1+x}}{4 x^4}+\frac{11 \sqrt [3]{-1+x}}{36 x^3}+\frac{11 \sqrt [3]{-1+x}}{27 x^2}+\frac{55}{81} \int \frac{1}{(-1+x)^{2/3} x^2} \, dx\\ &=\frac{\sqrt [3]{-1+x}}{4 x^4}+\frac{11 \sqrt [3]{-1+x}}{36 x^3}+\frac{11 \sqrt [3]{-1+x}}{27 x^2}+\frac{55 \sqrt [3]{-1+x}}{81 x}+\frac{110}{243} \int \frac{1}{(-1+x)^{2/3} x} \, dx\\ &=\frac{\sqrt [3]{-1+x}}{4 x^4}+\frac{11 \sqrt [3]{-1+x}}{36 x^3}+\frac{11 \sqrt [3]{-1+x}}{27 x^2}+\frac{55 \sqrt [3]{-1+x}}{81 x}-\frac{55 \log (x)}{243}+\frac{55}{81} \operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,\sqrt [3]{-1+x}\right )+\frac{55}{81} \operatorname{Subst}\left (\int \frac{1}{1-x+x^2} \, dx,x,\sqrt [3]{-1+x}\right )\\ &=\frac{\sqrt [3]{-1+x}}{4 x^4}+\frac{11 \sqrt [3]{-1+x}}{36 x^3}+\frac{11 \sqrt [3]{-1+x}}{27 x^2}+\frac{55 \sqrt [3]{-1+x}}{81 x}+\frac{55}{81} \log \left (1+\sqrt [3]{-1+x}\right )-\frac{55 \log (x)}{243}-\frac{110}{81} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,-1+2 \sqrt [3]{-1+x}\right )\\ &=\frac{\sqrt [3]{-1+x}}{4 x^4}+\frac{11 \sqrt [3]{-1+x}}{36 x^3}+\frac{11 \sqrt [3]{-1+x}}{27 x^2}+\frac{55 \sqrt [3]{-1+x}}{81 x}-\frac{110 \tan ^{-1}\left (\frac{1-2 \sqrt [3]{-1+x}}{\sqrt{3}}\right )}{81 \sqrt{3}}+\frac{55}{81} \log \left (1+\sqrt [3]{-1+x}\right )-\frac{55 \log (x)}{243}\\ \end{align*}
Mathematica [C] time = 0.0048085, size = 22, normalized size = 0.21 \[ 3 \sqrt [3]{x-1} \, _2F_1\left (\frac{1}{3},5;\frac{4}{3};1-x\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.015, size = 158, normalized size = 1.5 \begin{align*} -{\frac{1}{324} \left ( 1+\sqrt [3]{-1+x} \right ) ^{-4}}-{\frac{5}{243} \left ( 1+\sqrt [3]{-1+x} \right ) ^{-3}}-{\frac{20}{243} \left ( 1+\sqrt [3]{-1+x} \right ) ^{-2}}-{\frac{25}{81} \left ( 1+\sqrt [3]{-1+x} \right ) ^{-1}}+{\frac{110}{243}\ln \left ( 1+\sqrt [3]{-1+x} \right ) }-{\frac{1}{243} \left ( -75\, \left ( -1+x \right ) ^{7/3}+190\, \left ( -1+x \right ) ^{2}-350\, \left ( -1+x \right ) ^{5/3}+{\frac{1157}{4} \left ( -1+x \right ) ^{{\frac{4}{3}}}}+{\frac{149}{4}}-138\,x-116\, \left ( -1+x \right ) ^{2/3}+137\,\sqrt [3]{-1+x} \right ) \left ( \left ( -1+x \right ) ^{{\frac{2}{3}}}-\sqrt [3]{-1+x}+1 \right ) ^{-4}}-{\frac{55}{243}\ln \left ( \left ( -1+x \right ) ^{{\frac{2}{3}}}-\sqrt [3]{-1+x}+1 \right ) }+{\frac{110\,\sqrt{3}}{243}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,\sqrt [3]{-1+x}-1 \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.41639, size = 142, normalized size = 1.37 \begin{align*} \frac{110}{243} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \,{\left (x - 1\right )}^{\frac{1}{3}} - 1\right )}\right ) + \frac{220 \,{\left (x - 1\right )}^{\frac{10}{3}} + 792 \,{\left (x - 1\right )}^{\frac{7}{3}} + 1023 \,{\left (x - 1\right )}^{\frac{4}{3}} + 532 \,{\left (x - 1\right )}^{\frac{1}{3}}}{324 \,{\left ({\left (x - 1\right )}^{4} + 4 \,{\left (x - 1\right )}^{3} + 6 \,{\left (x - 1\right )}^{2} + 4 \, x - 3\right )}} - \frac{55}{243} \, \log \left ({\left (x - 1\right )}^{\frac{2}{3}} -{\left (x - 1\right )}^{\frac{1}{3}} + 1\right ) + \frac{110}{243} \, \log \left ({\left (x - 1\right )}^{\frac{1}{3}} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.69599, size = 282, normalized size = 2.71 \begin{align*} \frac{440 \, \sqrt{3} x^{4} \arctan \left (\frac{2}{3} \, \sqrt{3}{\left (x - 1\right )}^{\frac{1}{3}} - \frac{1}{3} \, \sqrt{3}\right ) - 220 \, x^{4} \log \left ({\left (x - 1\right )}^{\frac{2}{3}} -{\left (x - 1\right )}^{\frac{1}{3}} + 1\right ) + 440 \, x^{4} \log \left ({\left (x - 1\right )}^{\frac{1}{3}} + 1\right ) + 3 \,{\left (220 \, x^{3} + 132 \, x^{2} + 99 \, x + 81\right )}{\left (x - 1\right )}^{\frac{1}{3}}}{972 \, x^{4}} \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 [A] time = 1.07167, size = 111, normalized size = 1.07 \begin{align*} \frac{110}{243} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \,{\left (x - 1\right )}^{\frac{1}{3}} - 1\right )}\right ) + \frac{220 \,{\left (x - 1\right )}^{\frac{10}{3}} + 792 \,{\left (x - 1\right )}^{\frac{7}{3}} + 1023 \,{\left (x - 1\right )}^{\frac{4}{3}} + 532 \,{\left (x - 1\right )}^{\frac{1}{3}}}{324 \, x^{4}} - \frac{55}{243} \, \log \left ({\left (x - 1\right )}^{\frac{2}{3}} -{\left (x - 1\right )}^{\frac{1}{3}} + 1\right ) + \frac{110}{243} \, \log \left ({\left (x - 1\right )}^{\frac{1}{3}} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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