Optimal. Leaf size=136 \[ -\frac {3 b^9 \log \left (a \sqrt [3]{x}+b\right )}{a^{10}}+\frac {3 b^8 \sqrt [3]{x}}{a^9}-\frac {3 b^7 x^{2/3}}{2 a^8}+\frac {b^6 x}{a^7}-\frac {3 b^5 x^{4/3}}{4 a^6}+\frac {3 b^4 x^{5/3}}{5 a^5}-\frac {b^3 x^2}{2 a^4}+\frac {3 b^2 x^{7/3}}{7 a^3}-\frac {3 b x^{8/3}}{8 a^2}+\frac {x^3}{3 a} \]
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Rubi [A] time = 0.09, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {263, 266, 43} \[ -\frac {3 b^7 x^{2/3}}{2 a^8}-\frac {3 b^5 x^{4/3}}{4 a^6}+\frac {3 b^4 x^{5/3}}{5 a^5}-\frac {b^3 x^2}{2 a^4}+\frac {3 b^2 x^{7/3}}{7 a^3}+\frac {3 b^8 \sqrt [3]{x}}{a^9}+\frac {b^6 x}{a^7}-\frac {3 b^9 \log \left (a \sqrt [3]{x}+b\right )}{a^{10}}-\frac {3 b x^{8/3}}{8 a^2}+\frac {x^3}{3 a} \]
Antiderivative was successfully verified.
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Rule 43
Rule 263
Rule 266
Rubi steps
\begin {align*} \int \frac {x^2}{a+\frac {b}{\sqrt [3]{x}}} \, dx &=\int \frac {x^{7/3}}{b+a \sqrt [3]{x}} \, dx\\ &=3 \operatorname {Subst}\left (\int \frac {x^9}{b+a x} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname {Subst}\left (\int \left (\frac {b^8}{a^9}-\frac {b^7 x}{a^8}+\frac {b^6 x^2}{a^7}-\frac {b^5 x^3}{a^6}+\frac {b^4 x^4}{a^5}-\frac {b^3 x^5}{a^4}+\frac {b^2 x^6}{a^3}-\frac {b x^7}{a^2}+\frac {x^8}{a}-\frac {b^9}{a^9 (b+a x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {3 b^8 \sqrt [3]{x}}{a^9}-\frac {3 b^7 x^{2/3}}{2 a^8}+\frac {b^6 x}{a^7}-\frac {3 b^5 x^{4/3}}{4 a^6}+\frac {3 b^4 x^{5/3}}{5 a^5}-\frac {b^3 x^2}{2 a^4}+\frac {3 b^2 x^{7/3}}{7 a^3}-\frac {3 b x^{8/3}}{8 a^2}+\frac {x^3}{3 a}-\frac {3 b^9 \log \left (b+a \sqrt [3]{x}\right )}{a^{10}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 146, normalized size = 1.07 \[ -\frac {3 b^9 \log \left (a+\frac {b}{\sqrt [3]{x}}\right )}{a^{10}}-\frac {b^9 \log (x)}{a^{10}}+\frac {3 b^8 \sqrt [3]{x}}{a^9}-\frac {3 b^7 x^{2/3}}{2 a^8}+\frac {b^6 x}{a^7}-\frac {3 b^5 x^{4/3}}{4 a^6}+\frac {3 b^4 x^{5/3}}{5 a^5}-\frac {b^3 x^2}{2 a^4}+\frac {3 b^2 x^{7/3}}{7 a^3}-\frac {3 b x^{8/3}}{8 a^2}+\frac {x^3}{3 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.03, size = 111, normalized size = 0.82 \[ \frac {280 \, a^{9} x^{3} - 420 \, a^{6} b^{3} x^{2} + 840 \, a^{3} b^{6} x - 2520 \, b^{9} \log \left (a x^{\frac {1}{3}} + b\right ) - 63 \, {\left (5 \, a^{8} b x^{2} - 8 \, a^{5} b^{4} x + 20 \, a^{2} b^{7}\right )} x^{\frac {2}{3}} + 90 \, {\left (4 \, a^{7} b^{2} x^{2} - 7 \, a^{4} b^{5} x + 28 \, a b^{8}\right )} x^{\frac {1}{3}}}{840 \, a^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 111, normalized size = 0.82 \[ -\frac {3 \, b^{9} \log \left ({\left | a x^{\frac {1}{3}} + b \right |}\right )}{a^{10}} + \frac {280 \, a^{8} x^{3} - 315 \, a^{7} b x^{\frac {8}{3}} + 360 \, a^{6} b^{2} x^{\frac {7}{3}} - 420 \, a^{5} b^{3} x^{2} + 504 \, a^{4} b^{4} x^{\frac {5}{3}} - 630 \, a^{3} b^{5} x^{\frac {4}{3}} + 840 \, a^{2} b^{6} x - 1260 \, a b^{7} x^{\frac {2}{3}} + 2520 \, b^{8} x^{\frac {1}{3}}}{840 \, a^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 109, normalized size = 0.80 \[ \frac {x^{3}}{3 a}-\frac {3 b \,x^{\frac {8}{3}}}{8 a^{2}}+\frac {3 b^{2} x^{\frac {7}{3}}}{7 a^{3}}-\frac {b^{3} x^{2}}{2 a^{4}}+\frac {3 b^{4} x^{\frac {5}{3}}}{5 a^{5}}-\frac {3 b^{5} x^{\frac {4}{3}}}{4 a^{6}}+\frac {b^{6} x}{a^{7}}-\frac {3 b^{9} \ln \left (a \,x^{\frac {1}{3}}+b \right )}{a^{10}}-\frac {3 b^{7} x^{\frac {2}{3}}}{2 a^{8}}+\frac {3 b^{8} x^{\frac {1}{3}}}{a^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 122, normalized size = 0.90 \[ -\frac {3 \, b^{9} \log \left (a + \frac {b}{x^{\frac {1}{3}}}\right )}{a^{10}} - \frac {b^{9} \log \relax (x)}{a^{10}} + \frac {{\left (280 \, a^{8} - \frac {315 \, a^{7} b}{x^{\frac {1}{3}}} + \frac {360 \, a^{6} b^{2}}{x^{\frac {2}{3}}} - \frac {420 \, a^{5} b^{3}}{x} + \frac {504 \, a^{4} b^{4}}{x^{\frac {4}{3}}} - \frac {630 \, a^{3} b^{5}}{x^{\frac {5}{3}}} + \frac {840 \, a^{2} b^{6}}{x^{2}} - \frac {1260 \, a b^{7}}{x^{\frac {7}{3}}} + \frac {2520 \, b^{8}}{x^{\frac {8}{3}}}\right )} x^{3}}{840 \, a^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 108, normalized size = 0.79 \[ \frac {x^3}{3\,a}-\frac {3\,b\,x^{8/3}}{8\,a^2}+\frac {b^6\,x}{a^7}-\frac {3\,b^9\,\ln \left (b+a\,x^{1/3}\right )}{a^{10}}-\frac {b^3\,x^2}{2\,a^4}+\frac {3\,b^2\,x^{7/3}}{7\,a^3}+\frac {3\,b^4\,x^{5/3}}{5\,a^5}-\frac {3\,b^5\,x^{4/3}}{4\,a^6}-\frac {3\,b^7\,x^{2/3}}{2\,a^8}+\frac {3\,b^8\,x^{1/3}}{a^9} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.14, size = 143, normalized size = 1.05 \[ \begin {cases} \frac {x^{3}}{3 a} - \frac {3 b x^{\frac {8}{3}}}{8 a^{2}} + \frac {3 b^{2} x^{\frac {7}{3}}}{7 a^{3}} - \frac {b^{3} x^{2}}{2 a^{4}} + \frac {3 b^{4} x^{\frac {5}{3}}}{5 a^{5}} - \frac {3 b^{5} x^{\frac {4}{3}}}{4 a^{6}} + \frac {b^{6} x}{a^{7}} - \frac {3 b^{7} x^{\frac {2}{3}}}{2 a^{8}} + \frac {3 b^{8} \sqrt [3]{x}}{a^{9}} - \frac {3 b^{9} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{a^{10}} & \text {for}\: a \neq 0 \\\frac {3 x^{\frac {10}{3}}}{10 b} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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