Optimal. Leaf size=449 \[ \frac{b^2 n^2 \left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 e^3}-\frac{9 b^2 d n^2 \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{4 e^3}+\frac{9 a b^2 d^2 n^2 x^{2/3}}{e^2}-\frac{9 b d^2 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 e^3}+\frac{3 d^2 \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^3}-\frac{b n \left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 e^3}+\frac{9 b d n \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 e^3}+\frac{\left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^3}-\frac{3 d \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^3}+\frac{9 b^3 d^2 n^2 \left (d+e x^{2/3}\right ) \log \left (c \left (d+e x^{2/3}\right )^n\right )}{e^3}-\frac{9 b^3 d^2 n^3 x^{2/3}}{e^2}-\frac{b^3 n^3 \left (d+e x^{2/3}\right )^3}{9 e^3}+\frac{9 b^3 d n^3 \left (d+e x^{2/3}\right )^2}{8 e^3} \]
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Rubi [A] time = 0.458871, antiderivative size = 449, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {2454, 2401, 2389, 2296, 2295, 2390, 2305, 2304} \[ \frac{b^2 n^2 \left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 e^3}-\frac{9 b^2 d n^2 \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{4 e^3}+\frac{9 a b^2 d^2 n^2 x^{2/3}}{e^2}-\frac{9 b d^2 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 e^3}+\frac{3 d^2 \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^3}-\frac{b n \left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 e^3}+\frac{9 b d n \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 e^3}+\frac{\left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^3}-\frac{3 d \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^3}+\frac{9 b^3 d^2 n^2 \left (d+e x^{2/3}\right ) \log \left (c \left (d+e x^{2/3}\right )^n\right )}{e^3}-\frac{9 b^3 d^2 n^3 x^{2/3}}{e^2}-\frac{b^3 n^3 \left (d+e x^{2/3}\right )^3}{9 e^3}+\frac{9 b^3 d n^3 \left (d+e x^{2/3}\right )^2}{8 e^3} \]
Antiderivative was successfully verified.
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Rule 2454
Rule 2401
Rule 2389
Rule 2296
Rule 2295
Rule 2390
Rule 2305
Rule 2304
Rubi steps
\begin{align*} \int x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx &=\frac{3}{2} \operatorname{Subst}\left (\int x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,x^{2/3}\right )\\ &=\frac{3}{2} \operatorname{Subst}\left (\int \left (\frac{d^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^2}-\frac{2 d (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^2}+\frac{(d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^2}\right ) \, dx,x,x^{2/3}\right )\\ &=\frac{3 \operatorname{Subst}\left (\int (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,x^{2/3}\right )}{2 e^2}-\frac{(3 d) \operatorname{Subst}\left (\int (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,x^{2/3}\right )}{e^2}+\frac{\left (3 d^2\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,x^{2/3}\right )}{2 e^2}\\ &=\frac{3 \operatorname{Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e x^{2/3}\right )}{2 e^3}-\frac{(3 d) \operatorname{Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e x^{2/3}\right )}{e^3}+\frac{\left (3 d^2\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e x^{2/3}\right )}{2 e^3}\\ &=\frac{3 d^2 \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^3}-\frac{3 d \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^3}+\frac{\left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^3}-\frac{(3 b n) \operatorname{Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x^{2/3}\right )}{2 e^3}+\frac{(9 b d n) \operatorname{Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x^{2/3}\right )}{2 e^3}-\frac{\left (9 b d^2 n\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x^{2/3}\right )}{2 e^3}\\ &=-\frac{9 b d^2 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 e^3}+\frac{9 b d n \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 e^3}-\frac{b n \left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 e^3}+\frac{3 d^2 \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^3}-\frac{3 d \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^3}+\frac{\left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^3}+\frac{\left (b^2 n^2\right ) \operatorname{Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x^{2/3}\right )}{e^3}-\frac{\left (9 b^2 d n^2\right ) \operatorname{Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x^{2/3}\right )}{2 e^3}+\frac{\left (9 b^2 d^2 n^2\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x^{2/3}\right )}{e^3}\\ &=\frac{9 b^3 d n^3 \left (d+e x^{2/3}\right )^2}{8 e^3}-\frac{b^3 n^3 \left (d+e x^{2/3}\right )^3}{9 e^3}+\frac{9 a b^2 d^2 n^2 x^{2/3}}{e^2}-\frac{9 b^2 d n^2 \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{4 e^3}+\frac{b^2 n^2 \left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 e^3}-\frac{9 b d^2 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 e^3}+\frac{9 b d n \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 e^3}-\frac{b n \left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 e^3}+\frac{3 d^2 \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^3}-\frac{3 d \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^3}+\frac{\left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^3}+\frac{\left (9 b^3 d^2 n^2\right ) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x^{2/3}\right )}{e^3}\\ &=\frac{9 b^3 d n^3 \left (d+e x^{2/3}\right )^2}{8 e^3}-\frac{b^3 n^3 \left (d+e x^{2/3}\right )^3}{9 e^3}+\frac{9 a b^2 d^2 n^2 x^{2/3}}{e^2}-\frac{9 b^3 d^2 n^3 x^{2/3}}{e^2}+\frac{9 b^3 d^2 n^2 \left (d+e x^{2/3}\right ) \log \left (c \left (d+e x^{2/3}\right )^n\right )}{e^3}-\frac{9 b^2 d n^2 \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{4 e^3}+\frac{b^2 n^2 \left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 e^3}-\frac{9 b d^2 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 e^3}+\frac{9 b d n \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 e^3}-\frac{b n \left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 e^3}+\frac{3 d^2 \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^3}-\frac{3 d \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^3}+\frac{\left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^3}\\ \end{align*}
Mathematica [A] time = 0.407071, size = 428, normalized size = 0.95 \[ \frac{6 b \left (18 a^2 \left (d^3+e^3 x^2\right )-6 a b n \left (6 d^2 e x^{2/3}+11 d^3-3 d e^2 x^{4/3}+2 e^3 x^2\right )+b^2 n^2 \left (66 d^2 e x^{2/3}+66 d^3-15 d e^2 x^{4/3}+4 e^3 x^2\right )\right ) \log \left (c \left (d+e x^{2/3}\right )^n\right )-108 a^2 b d^2 e n x^{2/3}-198 a^2 b d^3 n+54 a^2 b d e^2 n x^{4/3}-36 a^2 b e^3 n x^2+36 a^3 d^3+36 a^3 e^3 x^2+18 b^2 \left (6 a \left (d^3+e^3 x^2\right )-b n \left (6 d^2 e x^{2/3}+11 d^3-3 d e^2 x^{4/3}+2 e^3 x^2\right )\right ) \log ^2\left (c \left (d+e x^{2/3}\right )^n\right )+396 a b^2 d^2 e n^2 x^{2/3}-90 a b^2 d e^2 n^2 x^{4/3}+24 a b^2 e^3 n^2 x^2+36 b^3 \left (d^3+e^3 x^2\right ) \log ^3\left (c \left (d+e x^{2/3}\right )^n\right )-510 b^3 d^2 e n^3 x^{2/3}+114 b^3 d^3 n^3 \log \left (d+e x^{2/3}\right )+57 b^3 d e^2 n^3 x^{4/3}-8 b^3 e^3 n^3 x^2}{72 e^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.347, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b\ln \left ( c \left ( d+e{x}^{{\frac{2}{3}}} \right ) ^{n} \right ) \right ) ^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11061, size = 653, normalized size = 1.45 \begin{align*} \frac{1}{2} \, b^{3} x^{2} \log \left ({\left (e x^{\frac{2}{3}} + d\right )}^{n} c\right )^{3} + \frac{3}{2} \, a b^{2} x^{2} \log \left ({\left (e x^{\frac{2}{3}} + d\right )}^{n} c\right )^{2} + \frac{1}{4} \, a^{2} b e n{\left (\frac{6 \, d^{3} \log \left (e x^{\frac{2}{3}} + d\right )}{e^{4}} - \frac{2 \, e^{2} x^{2} - 3 \, d e x^{\frac{4}{3}} + 6 \, d^{2} x^{\frac{2}{3}}}{e^{3}}\right )} + \frac{3}{2} \, a^{2} b x^{2} \log \left ({\left (e x^{\frac{2}{3}} + d\right )}^{n} c\right ) + \frac{1}{2} \, a^{3} x^{2} + \frac{1}{12} \,{\left (6 \, e n{\left (\frac{6 \, d^{3} \log \left (e x^{\frac{2}{3}} + d\right )}{e^{4}} - \frac{2 \, e^{2} x^{2} - 3 \, d e x^{\frac{4}{3}} + 6 \, d^{2} x^{\frac{2}{3}}}{e^{3}}\right )} \log \left ({\left (e x^{\frac{2}{3}} + d\right )}^{n} c\right ) + \frac{{\left (4 \, e^{3} x^{2} - 18 \, d^{3} \log \left (e x^{\frac{2}{3}} + d\right )^{2} - 15 \, d e^{2} x^{\frac{4}{3}} - 66 \, d^{3} \log \left (e x^{\frac{2}{3}} + d\right ) + 66 \, d^{2} e x^{\frac{2}{3}}\right )} n^{2}}{e^{3}}\right )} a b^{2} + \frac{1}{72} \,{\left (18 \, e n{\left (\frac{6 \, d^{3} \log \left (e x^{\frac{2}{3}} + d\right )}{e^{4}} - \frac{2 \, e^{2} x^{2} - 3 \, d e x^{\frac{4}{3}} + 6 \, d^{2} x^{\frac{2}{3}}}{e^{3}}\right )} \log \left ({\left (e x^{\frac{2}{3}} + d\right )}^{n} c\right )^{2} + e n{\left (\frac{{\left (36 \, d^{3} \log \left (e x^{\frac{2}{3}} + d\right )^{3} - 8 \, e^{3} x^{2} + 198 \, d^{3} \log \left (e x^{\frac{2}{3}} + d\right )^{2} + 57 \, d e^{2} x^{\frac{4}{3}} + 510 \, d^{3} \log \left (e x^{\frac{2}{3}} + d\right ) - 510 \, d^{2} e x^{\frac{2}{3}}\right )} n^{2}}{e^{4}} + \frac{6 \,{\left (4 \, e^{3} x^{2} - 18 \, d^{3} \log \left (e x^{\frac{2}{3}} + d\right )^{2} - 15 \, d e^{2} x^{\frac{4}{3}} - 66 \, d^{3} \log \left (e x^{\frac{2}{3}} + d\right ) + 66 \, d^{2} e x^{\frac{2}{3}}\right )} n \log \left ({\left (e x^{\frac{2}{3}} + d\right )}^{n} c\right )}{e^{4}}\right )}\right )} b^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.3055, size = 1590, normalized size = 3.54 \begin{align*} \text{result too large to display} \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 = 2.30135, size = 1050, normalized size = 2.34 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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