Optimal. Leaf size=42 \[ \frac {3 a^2 \log \left (a+b \sqrt [3]{x}\right )}{b^3}-\frac {3 a \sqrt [3]{x}}{b^2}+\frac {3 x^{2/3}}{2 b} \]
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Rubi [A] time = 0.02, antiderivative size = 42, 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} \[ \frac {3 a^2 \log \left (a+b \sqrt [3]{x}\right )}{b^3}-\frac {3 a \sqrt [3]{x}}{b^2}+\frac {3 x^{2/3}}{2 b} \]
Antiderivative was successfully verified.
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Rule 43
Rule 190
Rubi steps
\begin {align*} \int \frac {1}{a+b \sqrt [3]{x}} \, dx &=3 \operatorname {Subst}\left (\int \frac {x^2}{a+b x} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname {Subst}\left (\int \left (-\frac {a}{b^2}+\frac {x}{b}+\frac {a^2}{b^2 (a+b x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {3 a \sqrt [3]{x}}{b^2}+\frac {3 x^{2/3}}{2 b}+\frac {3 a^2 \log \left (a+b \sqrt [3]{x}\right )}{b^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 42, normalized size = 1.00 \[ \frac {3 a^2 \log \left (a+b \sqrt [3]{x}\right )}{b^3}-\frac {3 a \sqrt [3]{x}}{b^2}+\frac {3 x^{2/3}}{2 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 33, normalized size = 0.79 \[ \frac {3 \, {\left (2 \, a^{2} \log \left (b x^{\frac {1}{3}} + a\right ) + b^{2} x^{\frac {2}{3}} - 2 \, a b x^{\frac {1}{3}}\right )}}{2 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 35, normalized size = 0.83 \[ \frac {3 \, a^{2} \log \left ({\left | b x^{\frac {1}{3}} + a \right |}\right )}{b^{3}} + \frac {3 \, {\left (b x^{\frac {2}{3}} - 2 \, a x^{\frac {1}{3}}\right )}}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 79, normalized size = 1.88 \[ \frac {2 a^{2} \ln \left (b \,x^{\frac {1}{3}}+a \right )}{b^{3}}-\frac {a^{2} \ln \left (b^{2} x^{\frac {2}{3}}-a b \,x^{\frac {1}{3}}+a^{2}\right )}{b^{3}}+\frac {a^{2} \ln \left (b^{3} x +a^{3}\right )}{b^{3}}+\frac {3 x^{\frac {2}{3}}}{2 b}-\frac {3 a \,x^{\frac {1}{3}}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 44, normalized size = 1.05 \[ \frac {3 \, a^{2} \log \left (b x^{\frac {1}{3}} + a\right )}{b^{3}} + \frac {3 \, {\left (b x^{\frac {1}{3}} + a\right )}^{2}}{2 \, b^{3}} - \frac {6 \, {\left (b x^{\frac {1}{3}} + a\right )} a}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 34, normalized size = 0.81 \[ \frac {6\,a^2\,\ln \left (a+b\,x^{1/3}\right )+3\,b^2\,x^{2/3}-6\,a\,b\,x^{1/3}}{2\,b^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.19, size = 42, normalized size = 1.00 \[ \begin {cases} \frac {3 a^{2} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{b^{3}} - \frac {3 a \sqrt [3]{x}}{b^{2}} + \frac {3 x^{\frac {2}{3}}}{2 b} & \text {for}\: b \neq 0 \\\frac {x}{a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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