Optimal. Leaf size=438 \[ \frac{2 b^2 n^2 \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3 e^3}-\frac{9 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{2 e^3}+\frac{18 a b^2 d^2 n^2 \sqrt [3]{x}}{e^2}-\frac{9 b d^2 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{e^3}+\frac{3 d^2 \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^3}-\frac{b n \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{e^3}+\frac{9 b d n \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 e^3}+\frac{\left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^3}-\frac{3 d \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^3}+\frac{18 b^3 d^2 n^2 \left (d+e \sqrt [3]{x}\right ) \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{e^3}-\frac{18 b^3 d^2 n^3 \sqrt [3]{x}}{e^2}-\frac{2 b^3 n^3 \left (d+e \sqrt [3]{x}\right )^3}{9 e^3}+\frac{9 b^3 d n^3 \left (d+e \sqrt [3]{x}\right )^2}{4 e^3} \]
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Rubi [A] time = 0.443928, antiderivative size = 438, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {2451, 2401, 2389, 2296, 2295, 2390, 2305, 2304} \[ \frac{2 b^2 n^2 \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3 e^3}-\frac{9 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{2 e^3}+\frac{18 a b^2 d^2 n^2 \sqrt [3]{x}}{e^2}-\frac{9 b d^2 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{e^3}+\frac{3 d^2 \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^3}-\frac{b n \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{e^3}+\frac{9 b d n \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 e^3}+\frac{\left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^3}-\frac{3 d \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^3}+\frac{18 b^3 d^2 n^2 \left (d+e \sqrt [3]{x}\right ) \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{e^3}-\frac{18 b^3 d^2 n^3 \sqrt [3]{x}}{e^2}-\frac{2 b^3 n^3 \left (d+e \sqrt [3]{x}\right )^3}{9 e^3}+\frac{9 b^3 d n^3 \left (d+e \sqrt [3]{x}\right )^2}{4 e^3} \]
Antiderivative was successfully verified.
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Rule 2451
Rule 2401
Rule 2389
Rule 2296
Rule 2295
Rule 2390
Rule 2305
Rule 2304
Rubi steps
\begin{align*} \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx &=3 \operatorname{Subst}\left (\int x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt [3]{x}\right )\\ &=3 \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,\sqrt [3]{x}\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,\sqrt [3]{x}\right )}{e^2}-\frac{(6 d) \operatorname{Subst}\left (\int (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt [3]{x}\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,\sqrt [3]{x}\right )}{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 \sqrt [3]{x}\right )}{e^3}-\frac{(6 d) \operatorname{Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt [3]{x}\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 \sqrt [3]{x}\right )}{e^3}\\ &=\frac{3 d^2 \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^3}-\frac{3 d \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^3}+\frac{\left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{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 \sqrt [3]{x}\right )}{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 \sqrt [3]{x}\right )}{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 \sqrt [3]{x}\right )}{e^3}\\ &=-\frac{9 b d^2 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{e^3}+\frac{9 b d n \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 e^3}-\frac{b n \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{e^3}+\frac{3 d^2 \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^3}-\frac{3 d \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^3}+\frac{\left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^3}+\frac{\left (2 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 \sqrt [3]{x}\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 \sqrt [3]{x}\right )}{e^3}+\frac{\left (18 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 \sqrt [3]{x}\right )}{e^3}\\ &=\frac{9 b^3 d n^3 \left (d+e \sqrt [3]{x}\right )^2}{4 e^3}-\frac{2 b^3 n^3 \left (d+e \sqrt [3]{x}\right )^3}{9 e^3}+\frac{18 a b^2 d^2 n^2 \sqrt [3]{x}}{e^2}-\frac{9 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{2 e^3}+\frac{2 b^2 n^2 \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3 e^3}-\frac{9 b d^2 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{e^3}+\frac{9 b d n \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 e^3}-\frac{b n \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{e^3}+\frac{3 d^2 \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^3}-\frac{3 d \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^3}+\frac{\left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^3}+\frac{\left (18 b^3 d^2 n^2\right ) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e \sqrt [3]{x}\right )}{e^3}\\ &=\frac{9 b^3 d n^3 \left (d+e \sqrt [3]{x}\right )^2}{4 e^3}-\frac{2 b^3 n^3 \left (d+e \sqrt [3]{x}\right )^3}{9 e^3}+\frac{18 a b^2 d^2 n^2 \sqrt [3]{x}}{e^2}-\frac{18 b^3 d^2 n^3 \sqrt [3]{x}}{e^2}+\frac{18 b^3 d^2 n^2 \left (d+e \sqrt [3]{x}\right ) \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{e^3}-\frac{9 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{2 e^3}+\frac{2 b^2 n^2 \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3 e^3}-\frac{9 b d^2 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{e^3}+\frac{9 b d n \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 e^3}-\frac{b n \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{e^3}+\frac{3 d^2 \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^3}-\frac{3 d \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^3}+\frac{\left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^3}\\ \end{align*}
Mathematica [A] time = 0.224464, size = 362, normalized size = 0.83 \[ \frac{6 b \left (d+e \sqrt [3]{x}\right ) \left (18 a^2 \left (d^2-d e \sqrt [3]{x}+e^2 x^{2/3}\right )-6 a b n \left (11 d^2-5 d e \sqrt [3]{x}+2 e^2 x^{2/3}\right )+b^2 n^2 \left (85 d^2-19 d e \sqrt [3]{x}+4 e^2 x^{2/3}\right )\right ) \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )-18 a^2 b n \left (6 d^2 e \sqrt [3]{x}+11 d^3-3 d e^2 x^{2/3}+2 e^3 x\right )+36 a^3 \left (d^3+e^3 x\right )+18 b^2 \left (6 a \left (d^3+e^3 x\right )-b n \left (6 d^2 e \sqrt [3]{x}+11 d^3-3 d e^2 x^{2/3}+2 e^3 x\right )\right ) \log ^2\left (c \left (d+e \sqrt [3]{x}\right )^n\right )-6 a b^2 n^2 \left (-66 d^2 e \sqrt [3]{x}+23 d^3+15 d e^2 x^{2/3}-4 e^3 x\right )+36 b^3 \left (d^3+e^3 x\right ) \log ^3\left (c \left (d+e \sqrt [3]{x}\right )^n\right )+b^3 e n^3 \sqrt [3]{x} \left (-510 d^2+57 d e \sqrt [3]{x}-8 e^2 x^{2/3}\right )}{36 e^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.095, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c \left ( d+e\sqrt [3]{x} \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 = 1.08376, size = 614, normalized size = 1.4 \begin{align*} \frac{1}{2} \,{\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 )} a^{2} b + \frac{1}{6} \,{\left (6 \, 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 )} \log \left ({\left (e x^{\frac{1}{3}} + d\right )}^{n} c\right ) + 18 \, x \log \left ({\left (e x^{\frac{1}{3}} + d\right )}^{n} c\right )^{2} - \frac{{\left (18 \, d^{3} \log \left (e x^{\frac{1}{3}} + d\right )^{2} - 4 \, e^{3} x + 66 \, d^{3} \log \left (e x^{\frac{1}{3}} + d\right ) + 15 \, d e^{2} x^{\frac{2}{3}} - 66 \, d^{2} e x^{\frac{1}{3}}\right )} n^{2}}{e^{3}}\right )} a b^{2} + \frac{1}{36} \,{\left (18 \, 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 )} \log \left ({\left (e x^{\frac{1}{3}} + d\right )}^{n} c\right )^{2} + 36 \, x \log \left ({\left (e x^{\frac{1}{3}} + d\right )}^{n} c\right )^{3} + e n{\left (\frac{{\left (36 \, d^{3} \log \left (e x^{\frac{1}{3}} + d\right )^{3} + 198 \, d^{3} \log \left (e x^{\frac{1}{3}} + d\right )^{2} - 8 \, e^{3} x + 510 \, d^{3} \log \left (e x^{\frac{1}{3}} + d\right ) + 57 \, d e^{2} x^{\frac{2}{3}} - 510 \, d^{2} e x^{\frac{1}{3}}\right )} n^{2}}{e^{4}} - \frac{6 \,{\left (18 \, d^{3} \log \left (e x^{\frac{1}{3}} + d\right )^{2} - 4 \, e^{3} x + 66 \, d^{3} \log \left (e x^{\frac{1}{3}} + d\right ) + 15 \, d e^{2} x^{\frac{2}{3}} - 66 \, d^{2} e x^{\frac{1}{3}}\right )} n \log \left ({\left (e x^{\frac{1}{3}} + d\right )}^{n} c\right )}{e^{4}}\right )}\right )} b^{3} + a^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.09555, size = 1544, normalized size = 3.53 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \log{\left (c \left (d + e \sqrt [3]{x}\right )^{n} \right )}\right )^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.33871, size = 1492, normalized size = 3.41 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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