Optimal. Leaf size=90 \[ \frac {3 a^5}{2 b^6 \left (a+b \sqrt [3]{x}\right )^2}-\frac {15 a^4}{b^6 \left (a+b \sqrt [3]{x}\right )}-\frac {30 a^3 \log \left (a+b \sqrt [3]{x}\right )}{b^6}+\frac {18 a^2 \sqrt [3]{x}}{b^5}-\frac {9 a x^{2/3}}{2 b^4}+\frac {x}{b^3} \]
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Rubi [A] time = 0.06, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac {3 a^5}{2 b^6 \left (a+b \sqrt [3]{x}\right )^2}-\frac {15 a^4}{b^6 \left (a+b \sqrt [3]{x}\right )}+\frac {18 a^2 \sqrt [3]{x}}{b^5}-\frac {30 a^3 \log \left (a+b \sqrt [3]{x}\right )}{b^6}-\frac {9 a x^{2/3}}{2 b^4}+\frac {x}{b^3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x}{\left (a+b \sqrt [3]{x}\right )^3} \, dx &=3 \operatorname {Subst}\left (\int \frac {x^5}{(a+b x)^3} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname {Subst}\left (\int \left (\frac {6 a^2}{b^5}-\frac {3 a x}{b^4}+\frac {x^2}{b^3}-\frac {a^5}{b^5 (a+b x)^3}+\frac {5 a^4}{b^5 (a+b x)^2}-\frac {10 a^3}{b^5 (a+b x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {3 a^5}{2 b^6 \left (a+b \sqrt [3]{x}\right )^2}-\frac {15 a^4}{b^6 \left (a+b \sqrt [3]{x}\right )}+\frac {18 a^2 \sqrt [3]{x}}{b^5}-\frac {9 a x^{2/3}}{2 b^4}+\frac {x}{b^3}-\frac {30 a^3 \log \left (a+b \sqrt [3]{x}\right )}{b^6}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 83, normalized size = 0.92 \[ \frac {\frac {3 a^5}{\left (a+b \sqrt [3]{x}\right )^2}-\frac {30 a^4}{a+b \sqrt [3]{x}}-60 a^3 \log \left (a+b \sqrt [3]{x}\right )+36 a^2 b \sqrt [3]{x}-9 a b^2 x^{2/3}+2 b^3 x}{2 b^6} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.75, size = 159, normalized size = 1.77 \[ \frac {2 \, b^{9} x^{3} + 4 \, a^{3} b^{6} x^{2} - 34 \, a^{6} b^{3} x - 27 \, a^{9} - 60 \, {\left (a^{3} b^{6} x^{2} + 2 \, a^{6} b^{3} x + a^{9}\right )} \log \left (b x^{\frac {1}{3}} + a\right ) - 3 \, {\left (3 \, a b^{8} x^{2} + 16 \, a^{4} b^{5} x + 10 \, a^{7} b^{2}\right )} x^{\frac {2}{3}} + 3 \, {\left (12 \, a^{2} b^{7} x^{2} + 35 \, a^{5} b^{4} x + 20 \, a^{8} b\right )} x^{\frac {1}{3}}}{2 \, {\left (b^{12} x^{2} + 2 \, a^{3} b^{9} x + a^{6} b^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 79, normalized size = 0.88 \[ -\frac {30 \, a^{3} \log \left ({\left | b x^{\frac {1}{3}} + a \right |}\right )}{b^{6}} - \frac {3 \, {\left (10 \, a^{4} b x^{\frac {1}{3}} + 9 \, a^{5}\right )}}{2 \, {\left (b x^{\frac {1}{3}} + a\right )}^{2} b^{6}} + \frac {2 \, b^{6} x - 9 \, a b^{5} x^{\frac {2}{3}} + 36 \, a^{2} b^{4} x^{\frac {1}{3}}}{2 \, b^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 77, normalized size = 0.86 \[ \frac {3 a^{5}}{2 \left (b \,x^{\frac {1}{3}}+a \right )^{2} b^{6}}-\frac {15 a^{4}}{\left (b \,x^{\frac {1}{3}}+a \right ) b^{6}}-\frac {30 a^{3} \ln \left (b \,x^{\frac {1}{3}}+a \right )}{b^{6}}+\frac {x}{b^{3}}-\frac {9 a \,x^{\frac {2}{3}}}{2 b^{4}}+\frac {18 a^{2} x^{\frac {1}{3}}}{b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.63, size = 94, normalized size = 1.04 \[ -\frac {30 \, a^{3} \log \left (b x^{\frac {1}{3}} + a\right )}{b^{6}} + \frac {{\left (b x^{\frac {1}{3}} + a\right )}^{3}}{b^{6}} - \frac {15 \, {\left (b x^{\frac {1}{3}} + a\right )}^{2} a}{2 \, b^{6}} + \frac {30 \, {\left (b x^{\frac {1}{3}} + a\right )} a^{2}}{b^{6}} - \frac {15 \, a^{4}}{{\left (b x^{\frac {1}{3}} + a\right )} b^{6}} + \frac {3 \, a^{5}}{2 \, {\left (b x^{\frac {1}{3}} + a\right )}^{2} b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 87, normalized size = 0.97 \[ \frac {x}{b^3}-\frac {\frac {27\,a^5}{2\,b}+15\,a^4\,x^{1/3}}{a^2\,b^5+b^7\,x^{2/3}+2\,a\,b^6\,x^{1/3}}-\frac {9\,a\,x^{2/3}}{2\,b^4}-\frac {30\,a^3\,\ln \left (a+b\,x^{1/3}\right )}{b^6}+\frac {18\,a^2\,x^{1/3}}{b^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.87, size = 362, normalized size = 4.02 \[ \begin {cases} - \frac {60 a^{5} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{2 a^{2} b^{6} + 4 a b^{7} \sqrt [3]{x} + 2 b^{8} x^{\frac {2}{3}}} - \frac {90 a^{5}}{2 a^{2} b^{6} + 4 a b^{7} \sqrt [3]{x} + 2 b^{8} x^{\frac {2}{3}}} - \frac {120 a^{4} b \sqrt [3]{x} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{2 a^{2} b^{6} + 4 a b^{7} \sqrt [3]{x} + 2 b^{8} x^{\frac {2}{3}}} - \frac {120 a^{4} b \sqrt [3]{x}}{2 a^{2} b^{6} + 4 a b^{7} \sqrt [3]{x} + 2 b^{8} x^{\frac {2}{3}}} - \frac {60 a^{3} b^{2} x^{\frac {2}{3}} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{2 a^{2} b^{6} + 4 a b^{7} \sqrt [3]{x} + 2 b^{8} x^{\frac {2}{3}}} + \frac {20 a^{2} b^{3} x}{2 a^{2} b^{6} + 4 a b^{7} \sqrt [3]{x} + 2 b^{8} x^{\frac {2}{3}}} - \frac {5 a b^{4} x^{\frac {4}{3}}}{2 a^{2} b^{6} + 4 a b^{7} \sqrt [3]{x} + 2 b^{8} x^{\frac {2}{3}}} + \frac {2 b^{5} x^{\frac {5}{3}}}{2 a^{2} b^{6} + 4 a b^{7} \sqrt [3]{x} + 2 b^{8} x^{\frac {2}{3}}} & \text {for}\: b \neq 0 \\\frac {x^{2}}{2 a^{3}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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