Optimal. Leaf size=149 \[ \frac{3 b^7}{2 a^8 x^{2/3}}+\frac{3 b^5}{4 a^6 x^{4/3}}-\frac{3 b^4}{5 a^5 x^{5/3}}+\frac{b^3}{2 a^4 x^2}-\frac{3 b^2}{7 a^3 x^{7/3}}-\frac{3 b^8}{a^9 \sqrt [3]{x}}-\frac{b^6}{a^7 x}+\frac{3 b^9 \log \left (a+b \sqrt [3]{x}\right )}{a^{10}}-\frac{b^9 \log (x)}{a^{10}}+\frac{3 b}{8 a^2 x^{8/3}}-\frac{1}{3 a x^3} \]
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Rubi [A] time = 0.0785077, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 44} \[ \frac{3 b^7}{2 a^8 x^{2/3}}+\frac{3 b^5}{4 a^6 x^{4/3}}-\frac{3 b^4}{5 a^5 x^{5/3}}+\frac{b^3}{2 a^4 x^2}-\frac{3 b^2}{7 a^3 x^{7/3}}-\frac{3 b^8}{a^9 \sqrt [3]{x}}-\frac{b^6}{a^7 x}+\frac{3 b^9 \log \left (a+b \sqrt [3]{x}\right )}{a^{10}}-\frac{b^9 \log (x)}{a^{10}}+\frac{3 b}{8 a^2 x^{8/3}}-\frac{1}{3 a x^3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b \sqrt [3]{x}\right ) x^4} \, dx &=3 \operatorname{Subst}\left (\int \frac{1}{x^{10} (a+b x)} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{1}{a x^{10}}-\frac{b}{a^2 x^9}+\frac{b^2}{a^3 x^8}-\frac{b^3}{a^4 x^7}+\frac{b^4}{a^5 x^6}-\frac{b^5}{a^6 x^5}+\frac{b^6}{a^7 x^4}-\frac{b^7}{a^8 x^3}+\frac{b^8}{a^9 x^2}-\frac{b^9}{a^{10} x}+\frac{b^{10}}{a^{10} (a+b x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{1}{3 a x^3}+\frac{3 b}{8 a^2 x^{8/3}}-\frac{3 b^2}{7 a^3 x^{7/3}}+\frac{b^3}{2 a^4 x^2}-\frac{3 b^4}{5 a^5 x^{5/3}}+\frac{3 b^5}{4 a^6 x^{4/3}}-\frac{b^6}{a^7 x}+\frac{3 b^7}{2 a^8 x^{2/3}}-\frac{3 b^8}{a^9 \sqrt [3]{x}}+\frac{3 b^9 \log \left (a+b \sqrt [3]{x}\right )}{a^{10}}-\frac{b^9 \log (x)}{a^{10}}\\ \end{align*}
Mathematica [A] time = 0.0614976, size = 138, normalized size = 0.93 \[ -\frac{360 a^7 b^2 x^{2/3}+504 a^5 b^4 x^{4/3}-630 a^4 b^5 x^{5/3}+840 a^3 b^6 x^2-1260 a^2 b^7 x^{7/3}-420 a^6 b^3 x-315 a^8 b \sqrt [3]{x}+280 a^9+2520 a b^8 x^{8/3}-2520 b^9 x^3 \log \left (a+b \sqrt [3]{x}\right )+840 b^9 x^3 \log (x)}{840 a^{10} x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 122, normalized size = 0.8 \begin{align*} -{\frac{1}{3\,a{x}^{3}}}+{\frac{3\,b}{8\,{a}^{2}}{x}^{-{\frac{8}{3}}}}-{\frac{3\,{b}^{2}}{7\,{a}^{3}}{x}^{-{\frac{7}{3}}}}+{\frac{{b}^{3}}{2\,{a}^{4}{x}^{2}}}-{\frac{3\,{b}^{4}}{5\,{a}^{5}}{x}^{-{\frac{5}{3}}}}+{\frac{3\,{b}^{5}}{4\,{a}^{6}}{x}^{-{\frac{4}{3}}}}-{\frac{{b}^{6}}{{a}^{7}x}}+{\frac{3\,{b}^{7}}{2\,{a}^{8}}{x}^{-{\frac{2}{3}}}}-3\,{\frac{{b}^{8}}{{a}^{9}\sqrt [3]{x}}}+3\,{\frac{{b}^{9}\ln \left ( a+b\sqrt [3]{x} \right ) }{{a}^{10}}}-{\frac{{b}^{9}\ln \left ( x \right ) }{{a}^{10}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.968268, size = 162, normalized size = 1.09 \begin{align*} \frac{3 \, b^{9} \log \left (b x^{\frac{1}{3}} + a\right )}{a^{10}} - \frac{b^{9} \log \left (x\right )}{a^{10}} - \frac{2520 \, b^{8} x^{\frac{8}{3}} - 1260 \, a b^{7} x^{\frac{7}{3}} + 840 \, a^{2} b^{6} x^{2} - 630 \, a^{3} b^{5} x^{\frac{5}{3}} + 504 \, a^{4} b^{4} x^{\frac{4}{3}} - 420 \, a^{5} b^{3} x + 360 \, a^{6} b^{2} x^{\frac{2}{3}} - 315 \, a^{7} b x^{\frac{1}{3}} + 280 \, a^{8}}{840 \, a^{9} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54651, size = 313, normalized size = 2.1 \begin{align*} \frac{2520 \, b^{9} x^{3} \log \left (b x^{\frac{1}{3}} + a\right ) - 2520 \, b^{9} x^{3} \log \left (x^{\frac{1}{3}}\right ) - 840 \, a^{3} b^{6} x^{2} + 420 \, a^{6} b^{3} x - 280 \, a^{9} - 90 \,{\left (28 \, a b^{8} x^{2} - 7 \, a^{4} b^{5} x + 4 \, a^{7} b^{2}\right )} x^{\frac{2}{3}} + 63 \,{\left (20 \, a^{2} b^{7} x^{2} - 8 \, a^{5} b^{4} x + 5 \, a^{8} b\right )} x^{\frac{1}{3}}}{840 \, a^{10} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 24.0175, size = 172, normalized size = 1.15 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{x^{\frac{10}{3}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{3}{10 b x^{\frac{10}{3}}} & \text{for}\: a = 0 \\- \frac{1}{3 a x^{3}} & \text{for}\: b = 0 \\- \frac{1}{3 a x^{3}} + \frac{3 b}{8 a^{2} x^{\frac{8}{3}}} - \frac{3 b^{2}}{7 a^{3} x^{\frac{7}{3}}} + \frac{b^{3}}{2 a^{4} x^{2}} - \frac{3 b^{4}}{5 a^{5} x^{\frac{5}{3}}} + \frac{3 b^{5}}{4 a^{6} x^{\frac{4}{3}}} - \frac{b^{6}}{a^{7} x} + \frac{3 b^{7}}{2 a^{8} x^{\frac{2}{3}}} - \frac{3 b^{8}}{a^{9} \sqrt [3]{x}} - \frac{b^{9} \log{\left (x \right )}}{a^{10}} + \frac{3 b^{9} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{a^{10}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19857, size = 169, normalized size = 1.13 \begin{align*} \frac{3 \, b^{9} \log \left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{a^{10}} - \frac{b^{9} \log \left ({\left | x \right |}\right )}{a^{10}} - \frac{2520 \, a b^{8} x^{\frac{8}{3}} - 1260 \, a^{2} b^{7} x^{\frac{7}{3}} + 840 \, a^{3} b^{6} x^{2} - 630 \, a^{4} b^{5} x^{\frac{5}{3}} + 504 \, a^{5} b^{4} x^{\frac{4}{3}} - 420 \, a^{6} b^{3} x + 360 \, a^{7} b^{2} x^{\frac{2}{3}} - 315 \, a^{8} b x^{\frac{1}{3}} + 280 \, a^{9}}{840 \, a^{10} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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