Optimal. Leaf size=94 \[ \frac {3 b^6 \log \left (a \sqrt [3]{x}+b\right )}{a^7}-\frac {3 b^5 \sqrt [3]{x}}{a^6}+\frac {3 b^4 x^{2/3}}{2 a^5}-\frac {b^3 x}{a^4}+\frac {3 b^2 x^{4/3}}{4 a^3}-\frac {3 b x^{5/3}}{5 a^2}+\frac {x^2}{2 a} \]
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Rubi [A] time = 0.05, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {263, 266, 43} \[ \frac {3 b^4 x^{2/3}}{2 a^5}+\frac {3 b^2 x^{4/3}}{4 a^3}-\frac {3 b^5 \sqrt [3]{x}}{a^6}-\frac {b^3 x}{a^4}+\frac {3 b^6 \log \left (a \sqrt [3]{x}+b\right )}{a^7}-\frac {3 b x^{5/3}}{5 a^2}+\frac {x^2}{2 a} \]
Antiderivative was successfully verified.
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Rule 43
Rule 263
Rule 266
Rubi steps
\begin {align*} \int \frac {x}{a+\frac {b}{\sqrt [3]{x}}} \, dx &=\int \frac {x^{4/3}}{b+a \sqrt [3]{x}} \, dx\\ &=3 \operatorname {Subst}\left (\int \frac {x^6}{b+a x} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname {Subst}\left (\int \left (-\frac {b^5}{a^6}+\frac {b^4 x}{a^5}-\frac {b^3 x^2}{a^4}+\frac {b^2 x^3}{a^3}-\frac {b x^4}{a^2}+\frac {x^5}{a}+\frac {b^6}{a^6 (b+a x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {3 b^5 \sqrt [3]{x}}{a^6}+\frac {3 b^4 x^{2/3}}{2 a^5}-\frac {b^3 x}{a^4}+\frac {3 b^2 x^{4/3}}{4 a^3}-\frac {3 b x^{5/3}}{5 a^2}+\frac {x^2}{2 a}+\frac {3 b^6 \log \left (b+a \sqrt [3]{x}\right )}{a^7}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 95, normalized size = 1.01 \[ \frac {10 a^6 x^2-12 a^5 b x^{5/3}+15 a^4 b^2 x^{4/3}-20 a^3 b^3 x+30 a^2 b^4 x^{2/3}+60 b^6 \log \left (a+\frac {b}{\sqrt [3]{x}}\right )-60 a b^5 \sqrt [3]{x}+20 b^6 \log (x)}{20 a^7} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 77, normalized size = 0.82 \[ \frac {10 \, a^{6} x^{2} - 20 \, a^{3} b^{3} x + 60 \, b^{6} \log \left (a x^{\frac {1}{3}} + b\right ) - 6 \, {\left (2 \, a^{5} b x - 5 \, a^{2} b^{4}\right )} x^{\frac {2}{3}} + 15 \, {\left (a^{4} b^{2} x - 4 \, a b^{5}\right )} x^{\frac {1}{3}}}{20 \, a^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 78, normalized size = 0.83 \[ \frac {3 \, b^{6} \log \left ({\left | a x^{\frac {1}{3}} + b \right |}\right )}{a^{7}} + \frac {10 \, a^{5} x^{2} - 12 \, a^{4} b x^{\frac {5}{3}} + 15 \, a^{3} b^{2} x^{\frac {4}{3}} - 20 \, a^{2} b^{3} x + 30 \, a b^{4} x^{\frac {2}{3}} - 60 \, b^{5} x^{\frac {1}{3}}}{20 \, a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 77, normalized size = 0.82 \[ \frac {x^{2}}{2 a}-\frac {3 b \,x^{\frac {5}{3}}}{5 a^{2}}+\frac {3 b^{2} x^{\frac {4}{3}}}{4 a^{3}}-\frac {b^{3} x}{a^{4}}+\frac {3 b^{6} \ln \left (a \,x^{\frac {1}{3}}+b \right )}{a^{7}}+\frac {3 b^{4} x^{\frac {2}{3}}}{2 a^{5}}-\frac {3 b^{5} x^{\frac {1}{3}}}{a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 88, normalized size = 0.94 \[ \frac {3 \, b^{6} \log \left (a + \frac {b}{x^{\frac {1}{3}}}\right )}{a^{7}} + \frac {b^{6} \log \relax (x)}{a^{7}} + \frac {{\left (10 \, a^{5} - \frac {12 \, a^{4} b}{x^{\frac {1}{3}}} + \frac {15 \, a^{3} b^{2}}{x^{\frac {2}{3}}} - \frac {20 \, a^{2} b^{3}}{x} + \frac {30 \, a b^{4}}{x^{\frac {4}{3}}} - \frac {60 \, b^{5}}{x^{\frac {5}{3}}}\right )} x^{2}}{20 \, a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 76, normalized size = 0.81 \[ \frac {x^2}{2\,a}-\frac {b^3\,x}{a^4}-\frac {3\,b\,x^{5/3}}{5\,a^2}+\frac {3\,b^6\,\ln \left (b+a\,x^{1/3}\right )}{a^7}+\frac {3\,b^2\,x^{4/3}}{4\,a^3}+\frac {3\,b^4\,x^{2/3}}{2\,a^5}-\frac {3\,b^5\,x^{1/3}}{a^6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.26, size = 100, normalized size = 1.06 \[ \begin {cases} \frac {x^{2}}{2 a} - \frac {3 b x^{\frac {5}{3}}}{5 a^{2}} + \frac {3 b^{2} x^{\frac {4}{3}}}{4 a^{3}} - \frac {b^{3} x}{a^{4}} + \frac {3 b^{4} x^{\frac {2}{3}}}{2 a^{5}} - \frac {3 b^{5} \sqrt [3]{x}}{a^{6}} + \frac {3 b^{6} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{a^{7}} & \text {for}\: a \neq 0 \\\frac {3 x^{\frac {7}{3}}}{7 b} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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