Optimal. Leaf size=134 \[ -\frac{3 a^6}{2 b^7 x^{2/3}}-\frac{3 a^4}{4 b^5 x^{4/3}}+\frac{3 a^3}{5 b^4 x^{5/3}}-\frac{a^2}{2 b^3 x^2}+\frac{3 a^7}{b^8 \sqrt [3]{x}}+\frac{a^5}{b^6 x}-\frac{3 a^8 \log \left (a \sqrt [3]{x}+b\right )}{b^9}+\frac{a^8 \log (x)}{b^9}+\frac{3 a}{7 b^2 x^{7/3}}-\frac{3}{8 b x^{8/3}} \]
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Rubi [A] time = 0.0756118, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {263, 266, 44} \[ -\frac{3 a^6}{2 b^7 x^{2/3}}-\frac{3 a^4}{4 b^5 x^{4/3}}+\frac{3 a^3}{5 b^4 x^{5/3}}-\frac{a^2}{2 b^3 x^2}+\frac{3 a^7}{b^8 \sqrt [3]{x}}+\frac{a^5}{b^6 x}-\frac{3 a^8 \log \left (a \sqrt [3]{x}+b\right )}{b^9}+\frac{a^8 \log (x)}{b^9}+\frac{3 a}{7 b^2 x^{7/3}}-\frac{3}{8 b x^{8/3}} \]
Antiderivative was successfully verified.
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Rule 263
Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{\sqrt [3]{x}}\right ) x^4} \, dx &=\int \frac{1}{\left (b+a \sqrt [3]{x}\right ) x^{11/3}} \, dx\\ &=3 \operatorname{Subst}\left (\int \frac{1}{x^9 (b+a x)} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{1}{b x^9}-\frac{a}{b^2 x^8}+\frac{a^2}{b^3 x^7}-\frac{a^3}{b^4 x^6}+\frac{a^4}{b^5 x^5}-\frac{a^5}{b^6 x^4}+\frac{a^6}{b^7 x^3}-\frac{a^7}{b^8 x^2}+\frac{a^8}{b^9 x}-\frac{a^9}{b^9 (b+a x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{3}{8 b x^{8/3}}+\frac{3 a}{7 b^2 x^{7/3}}-\frac{a^2}{2 b^3 x^2}+\frac{3 a^3}{5 b^4 x^{5/3}}-\frac{3 a^4}{4 b^5 x^{4/3}}+\frac{a^5}{b^6 x}-\frac{3 a^6}{2 b^7 x^{2/3}}+\frac{3 a^7}{b^8 \sqrt [3]{x}}-\frac{3 a^8 \log \left (b+a \sqrt [3]{x}\right )}{b^9}+\frac{a^8 \log (x)}{b^9}\\ \end{align*}
Mathematica [A] time = 0.106573, size = 121, normalized size = 0.9 \[ \frac{\frac{b \left (280 a^5 b^2 x^{5/3}-210 a^4 b^3 x^{4/3}-140 a^2 b^5 x^{2/3}+168 a^3 b^4 x-420 a^6 b x^2+840 a^7 x^{7/3}+120 a b^6 \sqrt [3]{x}-105 b^7\right )}{x^{8/3}}-840 a^8 \log \left (a \sqrt [3]{x}+b\right )+280 a^8 \log (x)}{280 b^9} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 109, normalized size = 0.8 \begin{align*} -{\frac{3}{8\,b}{x}^{-{\frac{8}{3}}}}+{\frac{3\,a}{7\,{b}^{2}}{x}^{-{\frac{7}{3}}}}-{\frac{{a}^{2}}{2\,{b}^{3}{x}^{2}}}+{\frac{3\,{a}^{3}}{5\,{b}^{4}}{x}^{-{\frac{5}{3}}}}-{\frac{3\,{a}^{4}}{4\,{b}^{5}}{x}^{-{\frac{4}{3}}}}+{\frac{{a}^{5}}{{b}^{6}x}}-{\frac{3\,{a}^{6}}{2\,{b}^{7}}{x}^{-{\frac{2}{3}}}}+3\,{\frac{{a}^{7}}{{b}^{8}\sqrt [3]{x}}}-3\,{\frac{{a}^{8}\ln \left ( b+a\sqrt [3]{x} \right ) }{{b}^{9}}}+{\frac{{a}^{8}\ln \left ( x \right ) }{{b}^{9}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03907, size = 197, normalized size = 1.47 \begin{align*} -\frac{3 \, a^{8} \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{b^{9}} - \frac{3 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{8}}{8 \, b^{9}} + \frac{24 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{7} a}{7 \, b^{9}} - \frac{14 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{6} a^{2}}{b^{9}} + \frac{168 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{5} a^{3}}{5 \, b^{9}} - \frac{105 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{4} a^{4}}{2 \, b^{9}} + \frac{56 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{3} a^{5}}{b^{9}} - \frac{42 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{2} a^{6}}{b^{9}} + \frac{24 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )} a^{7}}{b^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56741, size = 292, normalized size = 2.18 \begin{align*} -\frac{840 \, a^{8} x^{3} \log \left (a x^{\frac{1}{3}} + b\right ) - 840 \, a^{8} x^{3} \log \left (x^{\frac{1}{3}}\right ) - 280 \, a^{5} b^{3} x^{2} + 140 \, a^{2} b^{6} x - 30 \,{\left (28 \, a^{7} b x^{2} - 7 \, a^{4} b^{4} x + 4 \, a b^{7}\right )} x^{\frac{2}{3}} + 21 \,{\left (20 \, a^{6} b^{2} x^{2} - 8 \, a^{3} b^{5} x + 5 \, b^{8}\right )} x^{\frac{1}{3}}}{280 \, b^{9} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 17.711, size = 158, normalized size = 1.18 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{x^{\frac{8}{3}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{1}{3 a x^{3}} & \text{for}\: b = 0 \\- \frac{3}{8 b x^{\frac{8}{3}}} & \text{for}\: a = 0 \\\frac{a^{8} \log{\left (x \right )}}{b^{9}} - \frac{3 a^{8} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{b^{9}} + \frac{3 a^{7}}{b^{8} \sqrt [3]{x}} - \frac{3 a^{6}}{2 b^{7} x^{\frac{2}{3}}} + \frac{a^{5}}{b^{6} x} - \frac{3 a^{4}}{4 b^{5} x^{\frac{4}{3}}} + \frac{3 a^{3}}{5 b^{4} x^{\frac{5}{3}}} - \frac{a^{2}}{2 b^{3} x^{2}} + \frac{3 a}{7 b^{2} x^{\frac{7}{3}}} - \frac{3}{8 b x^{\frac{8}{3}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19143, size = 153, normalized size = 1.14 \begin{align*} -\frac{3 \, a^{8} \log \left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{b^{9}} + \frac{a^{8} \log \left ({\left | x \right |}\right )}{b^{9}} + \frac{840 \, a^{7} b x^{\frac{7}{3}} - 420 \, a^{6} b^{2} x^{2} + 280 \, a^{5} b^{3} x^{\frac{5}{3}} - 210 \, a^{4} b^{4} x^{\frac{4}{3}} + 168 \, a^{3} b^{5} x - 140 \, a^{2} b^{6} x^{\frac{2}{3}} + 120 \, a b^{7} x^{\frac{1}{3}} - 105 \, b^{8}}{280 \, b^{9} x^{\frac{8}{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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