Optimal. Leaf size=103 \[ \frac {30 a^3 \log \left (a \sqrt [3]{x}+b\right )}{b^6}-\frac {10 a^3 \log (x)}{b^6}-\frac {12 a^3}{b^5 \left (a \sqrt [3]{x}+b\right )}-\frac {3 a^3}{2 b^4 \left (a \sqrt [3]{x}+b\right )^2}-\frac {18 a^2}{b^5 \sqrt [3]{x}}+\frac {9 a}{2 b^4 x^{2/3}}-\frac {1}{b^3 x} \]
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Rubi [A] time = 0.07, antiderivative size = 103, 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, 44} \[ -\frac {12 a^3}{b^5 \left (a \sqrt [3]{x}+b\right )}-\frac {3 a^3}{2 b^4 \left (a \sqrt [3]{x}+b\right )^2}-\frac {18 a^2}{b^5 \sqrt [3]{x}}+\frac {30 a^3 \log \left (a \sqrt [3]{x}+b\right )}{b^6}-\frac {10 a^3 \log (x)}{b^6}+\frac {9 a}{2 b^4 x^{2/3}}-\frac {1}{b^3 x} \]
Antiderivative was successfully verified.
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Rule 44
Rule 263
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x^3} \, dx &=\int \frac {1}{\left (b+a \sqrt [3]{x}\right )^3 x^2} \, dx\\ &=3 \operatorname {Subst}\left (\int \frac {1}{x^4 (b+a x)^3} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname {Subst}\left (\int \left (\frac {1}{b^3 x^4}-\frac {3 a}{b^4 x^3}+\frac {6 a^2}{b^5 x^2}-\frac {10 a^3}{b^6 x}+\frac {a^4}{b^4 (b+a x)^3}+\frac {4 a^4}{b^5 (b+a x)^2}+\frac {10 a^4}{b^6 (b+a x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {3 a^3}{2 b^4 \left (b+a \sqrt [3]{x}\right )^2}-\frac {12 a^3}{b^5 \left (b+a \sqrt [3]{x}\right )}-\frac {1}{b^3 x}+\frac {9 a}{2 b^4 x^{2/3}}-\frac {18 a^2}{b^5 \sqrt [3]{x}}+\frac {30 a^3 \log \left (b+a \sqrt [3]{x}\right )}{b^6}-\frac {10 a^3 \log (x)}{b^6}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 93, normalized size = 0.90 \[ -\frac {-60 a^3 \log \left (a \sqrt [3]{x}+b\right )+20 a^3 \log (x)+\frac {b \left (60 a^4 x^{4/3}+90 a^3 b x+20 a^2 b^2 x^{2/3}-5 a b^3 \sqrt [3]{x}+2 b^4\right )}{x \left (a \sqrt [3]{x}+b\right )^2}}{2 b^6} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.92, size = 191, normalized size = 1.85 \[ -\frac {20 \, a^{6} b^{3} x^{2} + 31 \, a^{3} b^{6} x + 2 \, b^{9} - 60 \, {\left (a^{9} x^{3} + 2 \, a^{6} b^{3} x^{2} + a^{3} b^{6} x\right )} \log \left (a x^{\frac {1}{3}} + b\right ) + 60 \, {\left (a^{9} x^{3} + 2 \, a^{6} b^{3} x^{2} + a^{3} b^{6} x\right )} \log \left (x^{\frac {1}{3}}\right ) + 3 \, {\left (20 \, a^{8} b x^{2} + 35 \, a^{5} b^{4} x + 12 \, a^{2} b^{7}\right )} x^{\frac {2}{3}} - 3 \, {\left (10 \, a^{7} b^{2} x^{2} + 16 \, a^{4} b^{5} x + 3 \, a b^{8}\right )} x^{\frac {1}{3}}}{2 \, {\left (a^{6} b^{6} x^{3} + 2 \, a^{3} b^{9} x^{2} + b^{12} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 90, normalized size = 0.87 \[ \frac {30 \, a^{3} \log \left ({\left | a x^{\frac {1}{3}} + b \right |}\right )}{b^{6}} - \frac {10 \, a^{3} \log \left ({\left | x \right |}\right )}{b^{6}} - \frac {60 \, a^{4} b x^{\frac {4}{3}} + 90 \, a^{3} b^{2} x + 20 \, a^{2} b^{3} x^{\frac {2}{3}} - 5 \, a b^{4} x^{\frac {1}{3}} + 2 \, b^{5}}{2 \, {\left (a x^{\frac {1}{3}} + b\right )}^{2} b^{6} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 90, normalized size = 0.87 \[ -\frac {3 a^{3}}{2 \left (a \,x^{\frac {1}{3}}+b \right )^{2} b^{4}}-\frac {12 a^{3}}{\left (a \,x^{\frac {1}{3}}+b \right ) b^{5}}-\frac {10 a^{3} \ln \relax (x )}{b^{6}}+\frac {30 a^{3} \ln \left (a \,x^{\frac {1}{3}}+b \right )}{b^{6}}-\frac {18 a^{2}}{b^{5} x^{\frac {1}{3}}}+\frac {9 a}{2 b^{4} x^{\frac {2}{3}}}-\frac {1}{b^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 95, normalized size = 0.92 \[ \frac {30 \, a^{3} \log \left (a + \frac {b}{x^{\frac {1}{3}}}\right )}{b^{6}} - \frac {{\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{3}}{b^{6}} + \frac {15 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{2} a}{2 \, b^{6}} - \frac {30 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )} a^{2}}{b^{6}} + \frac {15 \, a^{4}}{{\left (a + \frac {b}{x^{\frac {1}{3}}}\right )} b^{6}} - \frac {3 \, a^{5}}{2 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{2} b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 89, normalized size = 0.86 \[ \frac {60\,a^3\,\mathrm {atanh}\left (\frac {2\,a\,x^{1/3}}{b}+1\right )}{b^6}-\frac {\frac {1}{b}-\frac {5\,a\,x^{1/3}}{2\,b^2}+\frac {45\,a^3\,x}{b^4}+\frac {10\,a^2\,x^{2/3}}{b^3}+\frac {30\,a^4\,x^{4/3}}{b^5}}{b^2\,x+a^2\,x^{5/3}+2\,a\,b\,x^{4/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 11.10, size = 561, normalized size = 5.45 \[ \begin {cases} \frac {\tilde {\infty }}{x} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {1}{b^{3} x} & \text {for}\: a = 0 \\- \frac {1}{2 a^{3} x^{2}} & \text {for}\: b = 0 \\- \frac {20 a^{5} x^{\frac {10}{3}} \log {\relax (x )}}{2 a^{2} b^{6} x^{\frac {10}{3}} + 4 a b^{7} x^{3} + 2 b^{8} x^{\frac {8}{3}}} + \frac {60 a^{5} x^{\frac {10}{3}} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{2 a^{2} b^{6} x^{\frac {10}{3}} + 4 a b^{7} x^{3} + 2 b^{8} x^{\frac {8}{3}}} - \frac {40 a^{4} b x^{3} \log {\relax (x )}}{2 a^{2} b^{6} x^{\frac {10}{3}} + 4 a b^{7} x^{3} + 2 b^{8} x^{\frac {8}{3}}} + \frac {120 a^{4} b x^{3} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{2 a^{2} b^{6} x^{\frac {10}{3}} + 4 a b^{7} x^{3} + 2 b^{8} x^{\frac {8}{3}}} - \frac {60 a^{4} b x^{3}}{2 a^{2} b^{6} x^{\frac {10}{3}} + 4 a b^{7} x^{3} + 2 b^{8} x^{\frac {8}{3}}} - \frac {20 a^{3} b^{2} x^{\frac {8}{3}} \log {\relax (x )}}{2 a^{2} b^{6} x^{\frac {10}{3}} + 4 a b^{7} x^{3} + 2 b^{8} x^{\frac {8}{3}}} + \frac {60 a^{3} b^{2} x^{\frac {8}{3}} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{2 a^{2} b^{6} x^{\frac {10}{3}} + 4 a b^{7} x^{3} + 2 b^{8} x^{\frac {8}{3}}} - \frac {90 a^{3} b^{2} x^{\frac {8}{3}}}{2 a^{2} b^{6} x^{\frac {10}{3}} + 4 a b^{7} x^{3} + 2 b^{8} x^{\frac {8}{3}}} - \frac {20 a^{2} b^{3} x^{\frac {7}{3}}}{2 a^{2} b^{6} x^{\frac {10}{3}} + 4 a b^{7} x^{3} + 2 b^{8} x^{\frac {8}{3}}} + \frac {5 a b^{4} x^{2}}{2 a^{2} b^{6} x^{\frac {10}{3}} + 4 a b^{7} x^{3} + 2 b^{8} x^{\frac {8}{3}}} - \frac {2 b^{5} x^{\frac {5}{3}}}{2 a^{2} b^{6} x^{\frac {10}{3}} + 4 a b^{7} x^{3} + 2 b^{8} x^{\frac {8}{3}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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