Optimal. Leaf size=77 \[ a x+b x \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )+\frac {b d^3 n \log \left (d+e \sqrt [3]{x}\right )}{e^3}-\frac {b d^2 n \sqrt [3]{x}}{e^2}+\frac {b d n x^{2/3}}{2 e}-\frac {b n x}{3} \]
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Rubi [A] time = 0.05, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2448, 266, 43} \[ a x+b x \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )-\frac {b d^2 n \sqrt [3]{x}}{e^2}+\frac {b d^3 n \log \left (d+e \sqrt [3]{x}\right )}{e^3}+\frac {b d n x^{2/3}}{2 e}-\frac {b n x}{3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 2448
Rubi steps
\begin {align*} \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \, dx &=a x+b \int \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right ) \, dx\\ &=a x+b x \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )-\frac {1}{3} (b e n) \int \frac {\sqrt [3]{x}}{d+e \sqrt [3]{x}} \, dx\\ &=a x+b x \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )-(b e n) \operatorname {Subst}\left (\int \frac {x^3}{d+e x} \, dx,x,\sqrt [3]{x}\right )\\ &=a x+b x \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )-(b e n) \operatorname {Subst}\left (\int \left (\frac {d^2}{e^3}-\frac {d x}{e^2}+\frac {x^2}{e}-\frac {d^3}{e^3 (d+e x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {b d^2 n \sqrt [3]{x}}{e^2}+\frac {b d n x^{2/3}}{2 e}+a x-\frac {b n x}{3}+\frac {b d^3 n \log \left (d+e \sqrt [3]{x}\right )}{e^3}+b x \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 77, normalized size = 1.00 \[ a x+b x \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )+\frac {b d^3 n \log \left (d+e \sqrt [3]{x}\right )}{e^3}-\frac {b d^2 n \sqrt [3]{x}}{e^2}+\frac {b d n x^{2/3}}{2 e}-\frac {b n x}{3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 77, normalized size = 1.00 \[ \frac {6 \, b e^{3} x \log \relax (c) + 3 \, b d e^{2} n x^{\frac {2}{3}} - 6 \, b d^{2} e n x^{\frac {1}{3}} - 2 \, {\left (b e^{3} n - 3 \, a e^{3}\right )} x + 6 \, {\left (b e^{3} n x + b d^{3} n\right )} \log \left (e x^{\frac {1}{3}} + d\right )}{6 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 135, normalized size = 1.75 \[ \frac {1}{6} \, {\left (6 \, x e \log \relax (c) + {\left (6 \, {\left (x^{\frac {1}{3}} e + d\right )}^{3} e^{\left (-2\right )} \log \left (x^{\frac {1}{3}} e + d\right ) - 18 \, {\left (x^{\frac {1}{3}} e + d\right )}^{2} d e^{\left (-2\right )} \log \left (x^{\frac {1}{3}} e + d\right ) + 18 \, {\left (x^{\frac {1}{3}} e + d\right )} d^{2} e^{\left (-2\right )} \log \left (x^{\frac {1}{3}} e + d\right ) - 2 \, {\left (x^{\frac {1}{3}} e + d\right )}^{3} e^{\left (-2\right )} + 9 \, {\left (x^{\frac {1}{3}} e + d\right )}^{2} d e^{\left (-2\right )} - 18 \, {\left (x^{\frac {1}{3}} e + d\right )} d^{2} e^{\left (-2\right )}\right )} n\right )} b e^{\left (-1\right )} + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 66, normalized size = 0.86 \[ \frac {b \,d^{3} n \ln \left (e \,x^{\frac {1}{3}}+d \right )}{e^{3}}-\frac {b n x}{3}+b x \ln \left (c \left (e \,x^{\frac {1}{3}}+d \right )^{n}\right )+\frac {b d n \,x^{\frac {2}{3}}}{2 e}-\frac {b \,d^{2} n \,x^{\frac {1}{3}}}{e^{2}}+a x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.81, size = 70, normalized size = 0.91 \[ \frac {1}{6} \, {\left (e n {\left (\frac {6 \, d^{3} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{4}} - \frac {2 \, e^{2} x - 3 \, d e x^{\frac {2}{3}} + 6 \, d^{2} x^{\frac {1}{3}}}{e^{3}}\right )} + 6 \, x \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right )\right )} b + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.34, size = 65, normalized size = 0.84 \[ a\,x+b\,x\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )-\frac {b\,n\,x}{3}+\frac {b\,d\,n\,x^{2/3}}{2\,e}+\frac {b\,d^3\,n\,\ln \left (d+e\,x^{1/3}\right )}{e^3}-\frac {b\,d^2\,n\,x^{1/3}}{e^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.71, size = 82, normalized size = 1.06 \[ a x + b \left (- \frac {e n \left (- \frac {3 d^{3} \left (\begin {cases} \frac {\sqrt [3]{x}}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e \sqrt [3]{x} \right )}}{e} & \text {otherwise} \end {cases}\right )}{e^{3}} + \frac {3 d^{2} \sqrt [3]{x}}{e^{3}} - \frac {3 d x^{\frac {2}{3}}}{2 e^{2}} + \frac {x}{e}\right )}{3} + x \log {\left (c \left (d + e \sqrt [3]{x}\right )^{n} \right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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