Optimal. Leaf size=100 \[ \frac {3 b^3}{2 a^4 \left (a+\frac {b}{\sqrt [3]{x}}\right )^2}+\frac {12 b^3}{a^5 \left (a+\frac {b}{\sqrt [3]{x}}\right )}+\frac {18 b^2 \sqrt [3]{x}}{a^5}-\frac {9 b x^{2/3}}{2 a^4}+\frac {x}{a^3}-\frac {30 b^3 \log \left (a+\frac {b}{\sqrt [3]{x}}\right )}{a^6}-\frac {10 b^3 \log (x)}{a^6} \]
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Rubi [A]
time = 0.05, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {196, 46}
\begin {gather*} -\frac {30 b^3 \log \left (a+\frac {b}{\sqrt [3]{x}}\right )}{a^6}-\frac {10 b^3 \log (x)}{a^6}+\frac {12 b^3}{a^5 \left (a+\frac {b}{\sqrt [3]{x}}\right )}+\frac {18 b^2 \sqrt [3]{x}}{a^5}+\frac {3 b^3}{2 a^4 \left (a+\frac {b}{\sqrt [3]{x}}\right )^2}-\frac {9 b x^{2/3}}{2 a^4}+\frac {x}{a^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 196
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3} \, dx &=-\left (3 \text {Subst}\left (\int \frac {1}{x^4 (a+b x)^3} \, dx,x,\frac {1}{\sqrt [3]{x}}\right )\right )\\ &=-\left (3 \text {Subst}\left (\int \left (\frac {1}{a^3 x^4}-\frac {3 b}{a^4 x^3}+\frac {6 b^2}{a^5 x^2}-\frac {10 b^3}{a^6 x}+\frac {b^4}{a^4 (a+b x)^3}+\frac {4 b^4}{a^5 (a+b x)^2}+\frac {10 b^4}{a^6 (a+b x)}\right ) \, dx,x,\frac {1}{\sqrt [3]{x}}\right )\right )\\ &=\frac {3 b^3}{2 a^4 \left (a+\frac {b}{\sqrt [3]{x}}\right )^2}+\frac {12 b^3}{a^5 \left (a+\frac {b}{\sqrt [3]{x}}\right )}+\frac {18 b^2 \sqrt [3]{x}}{a^5}-\frac {9 b x^{2/3}}{2 a^4}+\frac {x}{a^3}-\frac {30 b^3 \log \left (a+\frac {b}{\sqrt [3]{x}}\right )}{a^6}-\frac {10 b^3 \log (x)}{a^6}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 97, normalized size = 0.97 \begin {gather*} \frac {-27 b^5+6 a b^4 \sqrt [3]{x}+63 a^2 b^3 x^{2/3}+20 a^3 b^2 x-5 a^4 b x^{4/3}+2 a^5 x^{5/3}}{2 a^6 \left (b+a \sqrt [3]{x}\right )^2}-\frac {30 b^3 \log \left (b+a \sqrt [3]{x}\right )}{a^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.21, size = 79, normalized size = 0.79
method | result | size |
derivativedivides | \(\frac {a^{2} x -\frac {9 a b \,x^{\frac {2}{3}}}{2}+18 b^{2} x^{\frac {1}{3}}}{a^{5}}+\frac {3 b^{5}}{2 a^{6} \left (b +a \,x^{\frac {1}{3}}\right )^{2}}-\frac {15 b^{4}}{a^{6} \left (b +a \,x^{\frac {1}{3}}\right )}-\frac {30 b^{3} \ln \left (b +a \,x^{\frac {1}{3}}\right )}{a^{6}}\) | \(79\) |
default | \(\frac {a^{2} x -\frac {9 a b \,x^{\frac {2}{3}}}{2}+18 b^{2} x^{\frac {1}{3}}}{a^{5}}+\frac {3 b^{5}}{2 a^{6} \left (b +a \,x^{\frac {1}{3}}\right )^{2}}-\frac {15 b^{4}}{a^{6} \left (b +a \,x^{\frac {1}{3}}\right )}-\frac {30 b^{3} \ln \left (b +a \,x^{\frac {1}{3}}\right )}{a^{6}}\) | \(79\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 101, normalized size = 1.01 \begin {gather*} \frac {2 \, a^{4} - \frac {5 \, a^{3} b}{x^{\frac {1}{3}}} + \frac {20 \, a^{2} b^{2}}{x^{\frac {2}{3}}} + \frac {90 \, a b^{3}}{x} + \frac {60 \, b^{4}}{x^{\frac {4}{3}}}}{2 \, {\left (\frac {a^{7}}{x} + \frac {2 \, a^{6} b}{x^{\frac {4}{3}}} + \frac {a^{5} b^{2}}{x^{\frac {5}{3}}}\right )}} - \frac {30 \, b^{3} \log \left (a + \frac {b}{x^{\frac {1}{3}}}\right )}{a^{6}} - \frac {10 \, b^{3} \log \left (x\right )}{a^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 159, normalized size = 1.59 \begin {gather*} \frac {2 \, a^{9} x^{3} + 4 \, a^{6} b^{3} x^{2} - 34 \, a^{3} b^{6} x - 27 \, b^{9} - 60 \, {\left (a^{6} b^{3} x^{2} + 2 \, a^{3} b^{6} x + b^{9}\right )} \log \left (a x^{\frac {1}{3}} + b\right ) - 3 \, {\left (3 \, a^{8} b x^{2} + 16 \, a^{5} b^{4} x + 10 \, a^{2} b^{7}\right )} x^{\frac {2}{3}} + 3 \, {\left (12 \, a^{7} b^{2} x^{2} + 35 \, a^{4} b^{5} x + 20 \, a b^{8}\right )} x^{\frac {1}{3}}}{2 \, {\left (a^{12} x^{2} + 2 \, a^{9} b^{3} x + a^{6} b^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 362 vs.
\(2 (99) = 198\).
time = 0.28, size = 362, normalized size = 3.62 \begin {gather*} \begin {cases} \frac {2 a^{5} x^{\frac {5}{3}}}{2 a^{8} x^{\frac {2}{3}} + 4 a^{7} b \sqrt [3]{x} + 2 a^{6} b^{2}} - \frac {5 a^{4} b x^{\frac {4}{3}}}{2 a^{8} x^{\frac {2}{3}} + 4 a^{7} b \sqrt [3]{x} + 2 a^{6} b^{2}} + \frac {20 a^{3} b^{2} x}{2 a^{8} x^{\frac {2}{3}} + 4 a^{7} b \sqrt [3]{x} + 2 a^{6} b^{2}} - \frac {60 a^{2} b^{3} x^{\frac {2}{3}} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{2 a^{8} x^{\frac {2}{3}} + 4 a^{7} b \sqrt [3]{x} + 2 a^{6} b^{2}} - \frac {120 a b^{4} \sqrt [3]{x} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{2 a^{8} x^{\frac {2}{3}} + 4 a^{7} b \sqrt [3]{x} + 2 a^{6} b^{2}} - \frac {120 a b^{4} \sqrt [3]{x}}{2 a^{8} x^{\frac {2}{3}} + 4 a^{7} b \sqrt [3]{x} + 2 a^{6} b^{2}} - \frac {60 b^{5} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{2 a^{8} x^{\frac {2}{3}} + 4 a^{7} b \sqrt [3]{x} + 2 a^{6} b^{2}} - \frac {90 b^{5}}{2 a^{8} x^{\frac {2}{3}} + 4 a^{7} b \sqrt [3]{x} + 2 a^{6} b^{2}} & \text {for}\: a \neq 0 \\\frac {x^{2}}{2 b^{3}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.63, size = 79, normalized size = 0.79 \begin {gather*} -\frac {30 \, b^{3} \log \left ({\left | a x^{\frac {1}{3}} + b \right |}\right )}{a^{6}} - \frac {3 \, {\left (10 \, a b^{4} x^{\frac {1}{3}} + 9 \, b^{5}\right )}}{2 \, {\left (a x^{\frac {1}{3}} + b\right )}^{2} a^{6}} + \frac {2 \, a^{6} x - 9 \, a^{5} b x^{\frac {2}{3}} + 36 \, a^{4} b^{2} x^{\frac {1}{3}}}{2 \, a^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 87, normalized size = 0.87 \begin {gather*} \frac {x}{a^3}-\frac {\frac {27\,b^5}{2\,a}+15\,b^4\,x^{1/3}}{a^5\,b^2+a^7\,x^{2/3}+2\,a^6\,b\,x^{1/3}}-\frac {9\,b\,x^{2/3}}{2\,a^4}-\frac {30\,b^3\,\ln \left (b+a\,x^{1/3}\right )}{a^6}+\frac {18\,b^2\,x^{1/3}}{a^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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