Optimal. Leaf size=87 \[ -\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{x}+\frac{b e^2 n}{d^2 \sqrt [3]{x}}-\frac{b e^3 n \log \left (d+e \sqrt [3]{x}\right )}{d^3}+\frac{b e^3 n \log (x)}{3 d^3}-\frac{b e n}{2 d x^{2/3}} \]
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Rubi [A] time = 0.0679198, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {2454, 2395, 44} \[ -\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{x}+\frac{b e^2 n}{d^2 \sqrt [3]{x}}-\frac{b e^3 n \log \left (d+e \sqrt [3]{x}\right )}{d^3}+\frac{b e^3 n \log (x)}{3 d^3}-\frac{b e n}{2 d x^{2/3}} \]
Antiderivative was successfully verified.
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Rule 2454
Rule 2395
Rule 44
Rubi steps
\begin{align*} \int \frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{x^2} \, dx &=3 \operatorname{Subst}\left (\int \frac{a+b \log \left (c (d+e x)^n\right )}{x^4} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{x}+(b e n) \operatorname{Subst}\left (\int \frac{1}{x^3 (d+e x)} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{x}+(b e n) \operatorname{Subst}\left (\int \left (\frac{1}{d x^3}-\frac{e}{d^2 x^2}+\frac{e^2}{d^3 x}-\frac{e^3}{d^3 (d+e x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{b e n}{2 d x^{2/3}}+\frac{b e^2 n}{d^2 \sqrt [3]{x}}-\frac{b e^3 n \log \left (d+e \sqrt [3]{x}\right )}{d^3}-\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{x}+\frac{b e^3 n \log (x)}{3 d^3}\\ \end{align*}
Mathematica [A] time = 0.0331193, size = 84, normalized size = 0.97 \[ -\frac{a}{x}-\frac{b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{x}+b e n \left (-\frac{e^2 \log \left (d+e \sqrt [3]{x}\right )}{d^3}+\frac{e^2 \log (x)}{3 d^3}+\frac{e}{d^2 \sqrt [3]{x}}-\frac{1}{2 d x^{2/3}}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.098, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ( a+b\ln \left ( c \left ( d+e\sqrt [3]{x} \right ) ^{n} \right ) \right ) }\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02714, size = 101, normalized size = 1.16 \begin{align*} -\frac{1}{6} \, b e n{\left (\frac{6 \, e^{2} \log \left (e x^{\frac{1}{3}} + d\right )}{d^{3}} - \frac{2 \, e^{2} \log \left (x\right )}{d^{3}} - \frac{3 \,{\left (2 \, e x^{\frac{1}{3}} - d\right )}}{d^{2} x^{\frac{2}{3}}}\right )} - \frac{b \log \left ({\left (e x^{\frac{1}{3}} + d\right )}^{n} c\right )}{x} - \frac{a}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.8584, size = 208, normalized size = 2.39 \begin{align*} \frac{2 \, b e^{3} n x \log \left (x^{\frac{1}{3}}\right ) + 2 \, b d e^{2} n x^{\frac{2}{3}} - b d^{2} e n x^{\frac{1}{3}} - 2 \, b d^{3} \log \left (c\right ) - 2 \, a d^{3} - 2 \,{\left (b e^{3} n x + b d^{3} n\right )} \log \left (e x^{\frac{1}{3}} + d\right )}{2 \, d^{3} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.3379, size = 378, normalized size = 4.34 \begin{align*} -\frac{{\left (2 \,{\left (x^{\frac{1}{3}} e + d\right )}^{3} b n e^{4} \log \left (x^{\frac{1}{3}} e + d\right ) - 6 \,{\left (x^{\frac{1}{3}} e + d\right )}^{2} b d n e^{4} \log \left (x^{\frac{1}{3}} e + d\right ) + 6 \,{\left (x^{\frac{1}{3}} e + d\right )} b d^{2} n e^{4} \log \left (x^{\frac{1}{3}} e + d\right ) - 2 \,{\left (x^{\frac{1}{3}} e + d\right )}^{3} b n e^{4} \log \left (x^{\frac{1}{3}} e\right ) + 6 \,{\left (x^{\frac{1}{3}} e + d\right )}^{2} b d n e^{4} \log \left (x^{\frac{1}{3}} e\right ) - 6 \,{\left (x^{\frac{1}{3}} e + d\right )} b d^{2} n e^{4} \log \left (x^{\frac{1}{3}} e\right ) + 2 \, b d^{3} n e^{4} \log \left (x^{\frac{1}{3}} e\right ) - 2 \,{\left (x^{\frac{1}{3}} e + d\right )}^{2} b d n e^{4} + 5 \,{\left (x^{\frac{1}{3}} e + d\right )} b d^{2} n e^{4} - 3 \, b d^{3} n e^{4} + 2 \, b d^{3} e^{4} \log \left (c\right ) + 2 \, a d^{3} e^{4}\right )} e^{\left (-1\right )}}{2 \,{\left ({\left (x^{\frac{1}{3}} e + d\right )}^{3} d^{3} - 3 \,{\left (x^{\frac{1}{3}} e + d\right )}^{2} d^{4} + 3 \,{\left (x^{\frac{1}{3}} e + d\right )} d^{5} - d^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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