Optimal. Leaf size=54 \[ -\frac {3 a^2}{2 b^3 \left (a+b \sqrt [3]{x}\right )^2}+\frac {6 a}{b^3 \left (a+b \sqrt [3]{x}\right )}+\frac {3 \log \left (a+b \sqrt [3]{x}\right )}{b^3} \]
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Rubi [A] time = 0.03, antiderivative size = 54, 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 = {190, 43} \[ -\frac {3 a^2}{2 b^3 \left (a+b \sqrt [3]{x}\right )^2}+\frac {6 a}{b^3 \left (a+b \sqrt [3]{x}\right )}+\frac {3 \log \left (a+b \sqrt [3]{x}\right )}{b^3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 190
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b \sqrt [3]{x}\right )^3} \, dx &=3 \operatorname {Subst}\left (\int \frac {x^2}{(a+b x)^3} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname {Subst}\left (\int \left (\frac {a^2}{b^2 (a+b x)^3}-\frac {2 a}{b^2 (a+b x)^2}+\frac {1}{b^2 (a+b x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {3 a^2}{2 b^3 \left (a+b \sqrt [3]{x}\right )^2}+\frac {6 a}{b^3 \left (a+b \sqrt [3]{x}\right )}+\frac {3 \log \left (a+b \sqrt [3]{x}\right )}{b^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 45, normalized size = 0.83 \[ \frac {3 \left (\frac {a \left (3 a+4 b \sqrt [3]{x}\right )}{\left (a+b \sqrt [3]{x}\right )^2}+2 \log \left (a+b \sqrt [3]{x}\right )\right )}{2 b^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.55, size = 113, normalized size = 2.09 \[ \frac {3 \, {\left (6 \, a^{3} b^{3} x + 3 \, a^{6} + 2 \, {\left (b^{6} x^{2} + 2 \, a^{3} b^{3} x + a^{6}\right )} \log \left (b x^{\frac {1}{3}} + a\right ) + {\left (4 \, a b^{5} x + a^{4} b^{2}\right )} x^{\frac {2}{3}} - {\left (5 \, a^{2} b^{4} x + 2 \, a^{5} b\right )} x^{\frac {1}{3}}\right )}}{2 \, {\left (b^{9} x^{2} + 2 \, a^{3} b^{6} x + a^{6} b^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 44, normalized size = 0.81 \[ \frac {3 \, \log \left ({\left | b x^{\frac {1}{3}} + a \right |}\right )}{b^{3}} + \frac {3 \, {\left (4 \, a x^{\frac {1}{3}} + \frac {3 \, a^{2}}{b}\right )}}{2 \, {\left (b x^{\frac {1}{3}} + a\right )}^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 237, normalized size = 4.39 \[ -\frac {9 a^{6}}{2 \left (b^{3} x +a^{3}\right )^{2} b^{3}}+\frac {2 a x}{\left (b^{2} x^{\frac {2}{3}}-a b \,x^{\frac {1}{3}}+a^{2}\right )^{2}}-\frac {13 a^{2} x^{\frac {2}{3}}}{2 \left (b^{2} x^{\frac {2}{3}}-a b \,x^{\frac {1}{3}}+a^{2}\right )^{2} b}+\frac {5 a^{3} x^{\frac {1}{3}}}{\left (b^{2} x^{\frac {2}{3}}-a b \,x^{\frac {1}{3}}+a^{2}\right )^{2} b^{2}}-\frac {3 a^{4}}{\left (b^{2} x^{\frac {2}{3}}-a b \,x^{\frac {1}{3}}+a^{2}\right )^{2} b^{3}}+\frac {9 a^{3}}{\left (b^{3} x +a^{3}\right ) b^{3}}-\frac {a^{2}}{\left (b \,x^{\frac {1}{3}}+a \right )^{2} b^{3}}+\frac {4 a}{\left (b \,x^{\frac {1}{3}}+a \right ) b^{3}}+\frac {2 \ln \left (b \,x^{\frac {1}{3}}+a \right )}{b^{3}}-\frac {\ln \left (b^{2} x^{\frac {2}{3}}-a b \,x^{\frac {1}{3}}+a^{2}\right )}{b^{3}}+\frac {\ln \left (b^{3} x +a^{3}\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 46, normalized size = 0.85 \[ \frac {3 \, \log \left (b x^{\frac {1}{3}} + a\right )}{b^{3}} + \frac {6 \, a}{{\left (b x^{\frac {1}{3}} + a\right )} b^{3}} - \frac {3 \, a^{2}}{2 \, {\left (b x^{\frac {1}{3}} + a\right )}^{2} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.14, size = 53, normalized size = 0.98 \[ \frac {\frac {9\,a^2}{2\,b^3}+\frac {6\,a\,x^{1/3}}{b^2}}{a^2+b^2\,x^{2/3}+2\,a\,b\,x^{1/3}}+\frac {3\,\ln \left (a+b\,x^{1/3}\right )}{b^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.67, size = 228, normalized size = 4.22 \[ \begin {cases} \frac {6 a^{2} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{2 a^{2} b^{3} + 4 a b^{4} \sqrt [3]{x} + 2 b^{5} x^{\frac {2}{3}}} + \frac {9 a^{2}}{2 a^{2} b^{3} + 4 a b^{4} \sqrt [3]{x} + 2 b^{5} x^{\frac {2}{3}}} + \frac {12 a b \sqrt [3]{x} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{2 a^{2} b^{3} + 4 a b^{4} \sqrt [3]{x} + 2 b^{5} x^{\frac {2}{3}}} + \frac {12 a b \sqrt [3]{x}}{2 a^{2} b^{3} + 4 a b^{4} \sqrt [3]{x} + 2 b^{5} x^{\frac {2}{3}}} + \frac {6 b^{2} x^{\frac {2}{3}} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{2 a^{2} b^{3} + 4 a b^{4} \sqrt [3]{x} + 2 b^{5} x^{\frac {2}{3}}} & \text {for}\: b \neq 0 \\\frac {x}{a^{3}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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