Optimal. Leaf size=70 \[ a x+b x \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )-\frac{b e^2 n \sqrt [3]{x}}{d^2}+\frac{b e^3 n \log \left (d \sqrt [3]{x}+e\right )}{d^3}+\frac{b e n x^{2/3}}{2 d} \]
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Rubi [A] time = 0.0501666, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2448, 263, 190, 43} \[ a x+b x \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )-\frac{b e^2 n \sqrt [3]{x}}{d^2}+\frac{b e^3 n \log \left (d \sqrt [3]{x}+e\right )}{d^3}+\frac{b e n x^{2/3}}{2 d} \]
Antiderivative was successfully verified.
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Rule 2448
Rule 263
Rule 190
Rule 43
Rubi steps
\begin{align*} \int \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right ) \, dx &=a x+b \int \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right ) \, dx\\ &=a x+b x \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )+\frac{1}{3} (b e n) \int \frac{1}{\left (d+\frac{e}{\sqrt [3]{x}}\right ) \sqrt [3]{x}} \, dx\\ &=a x+b x \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )+\frac{1}{3} (b e n) \int \frac{1}{e+d \sqrt [3]{x}} \, dx\\ &=a x+b x \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )+(b e n) \operatorname{Subst}\left (\int \frac{x^2}{e+d x} \, dx,x,\sqrt [3]{x}\right )\\ &=a x+b x \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )+(b e n) \operatorname{Subst}\left (\int \left (-\frac{e}{d^2}+\frac{x}{d}+\frac{e^2}{d^2 (e+d x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{b e^2 n \sqrt [3]{x}}{d^2}+\frac{b e n x^{2/3}}{2 d}+a x+b x \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )+\frac{b e^3 n \log \left (e+d \sqrt [3]{x}\right )}{d^3}\\ \end{align*}
Mathematica [A] time = 0.0470079, size = 79, normalized size = 1.13 \[ a x+b x \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )-b e n \left (-\frac{e^2 \log \left (d+\frac{e}{\sqrt [3]{x}}\right )}{d^3}-\frac{e^2 \log (x)}{3 d^3}+\frac{e \sqrt [3]{x}}{d^2}-\frac{x^{2/3}}{2 d}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.102, size = 115, normalized size = 1.6 \begin{align*} ax+xb\ln \left ( c \left ({ \left ( e+d\sqrt [3]{x} \right ){\frac{1}{\sqrt [3]{x}}}} \right ) ^{n} \right ) +{\frac{b{e}^{3}n\ln \left ({d}^{3}x+{e}^{3} \right ) }{3\,{d}^{3}}}+{\frac{enb}{2\,d}{x}^{{\frac{2}{3}}}}-{\frac{b{e}^{3}n}{3\,{d}^{3}}\ln \left ({d}^{2}{x}^{{\frac{2}{3}}}-ed\sqrt [3]{x}+{e}^{2} \right ) }+{\frac{2\,b{e}^{3}n}{3\,{d}^{3}}\ln \left ( e+d\sqrt [3]{x} \right ) }-{\frac{b{e}^{2}n}{{d}^{2}}\sqrt [3]{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03024, size = 80, normalized size = 1.14 \begin{align*} \frac{1}{2} \,{\left (e n{\left (\frac{2 \, e^{2} \log \left (d x^{\frac{1}{3}} + e\right )}{d^{3}} + \frac{d x^{\frac{2}{3}} - 2 \, e x^{\frac{1}{3}}}{d^{2}}\right )} + 2 \, x \log \left (c{\left (d + \frac{e}{x^{\frac{1}{3}}}\right )}^{n}\right )\right )} b + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.15552, size = 269, normalized size = 3.84 \begin{align*} \frac{2 \, b d^{3} x \log \left (c\right ) - 2 \, b d^{3} n \log \left (x^{\frac{1}{3}}\right ) + b d^{2} e n x^{\frac{2}{3}} - 2 \, b d e^{2} n x^{\frac{1}{3}} + 2 \, a d^{3} x + 2 \,{\left (b d^{3} + b e^{3}\right )} n \log \left (d x^{\frac{1}{3}} + e\right ) + 2 \,{\left (b d^{3} n x - b d^{3} n\right )} \log \left (\frac{d x + e x^{\frac{2}{3}}}{x}\right )}{2 \, d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 15.0529, size = 92, normalized size = 1.31 \begin{align*} a x + b \left (\frac{e n \left (\frac{3 x^{\frac{2}{3}}}{2 d} - \frac{3 e \sqrt [3]{x}}{d^{2}} + \frac{3 e^{3} \left (\begin{cases} \frac{1}{d \sqrt [3]{x}} & \text{for}\: e = 0 \\\frac{\log{\left (d + \frac{e}{\sqrt [3]{x}} \right )}}{e} & \text{otherwise} \end{cases}\right )}{d^{3}} - \frac{3 e^{2} \log{\left (\frac{1}{\sqrt [3]{x}} \right )}}{d^{3}}\right )}{3} + x \log{\left (c \left (d + \frac{e}{\sqrt [3]{x}}\right )^{n} \right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.38545, size = 89, normalized size = 1.27 \begin{align*} \frac{1}{2} \,{\left ({\left ({\left (\frac{d x^{\frac{2}{3}} - 2 \, x^{\frac{1}{3}} e}{d^{2}} + \frac{2 \, e^{2} \log \left ({\left | d x^{\frac{1}{3}} + e \right |}\right )}{d^{3}}\right )} e + 2 \, x \log \left (d + \frac{e}{x^{\frac{1}{3}}}\right )\right )} n + 2 \, x \log \left (c\right )\right )} b + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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