Optimal. Leaf size=131 \[ \frac{12 b^2 n^2 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-24 a b^3 n^3 x-\frac{4 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^4}{e}-\frac{24 b^4 n^3 (d+e x) \log \left (c (d+e x)^n\right )}{e}+24 b^4 n^4 x \]
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Rubi [A] time = 0.0738035, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {2389, 2296, 2295} \[ \frac{12 b^2 n^2 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-24 a b^3 n^3 x-\frac{4 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^4}{e}-\frac{24 b^4 n^3 (d+e x) \log \left (c (d+e x)^n\right )}{e}+24 b^4 n^4 x \]
Antiderivative was successfully verified.
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Rule 2389
Rule 2296
Rule 2295
Rubi steps
\begin{align*} \int \left (a+b \log \left (c (d+e x)^n\right )\right )^4 \, dx &=\frac{\operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^4 \, dx,x,d+e x\right )}{e}\\ &=\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^4}{e}-\frac{(4 b n) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e x\right )}{e}\\ &=-\frac{4 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^4}{e}+\frac{\left (12 b^2 n^2\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e}\\ &=\frac{12 b^2 n^2 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{4 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^4}{e}-\frac{\left (24 b^3 n^3\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e}\\ &=-24 a b^3 n^3 x+\frac{12 b^2 n^2 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{4 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^4}{e}-\frac{\left (24 b^4 n^3\right ) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e}\\ &=-24 a b^3 n^3 x+24 b^4 n^4 x-\frac{24 b^4 n^3 (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac{12 b^2 n^2 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{4 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^4}{e}\\ \end{align*}
Mathematica [A] time = 0.0551883, size = 112, normalized size = 0.85 \[ \frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^4-4 b n \left ((d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3-3 b n \left ((d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2-2 b n \left (e x (a-b n)+b (d+e x) \log \left (c (d+e x)^n\right )\right )\right )\right )}{e} \]
Antiderivative was successfully verified.
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Maple [C] time = 1.475, size = 15871, normalized size = 121.2 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.20283, size = 675, normalized size = 5.15 \begin{align*} b^{4} x \log \left ({\left (e x + d\right )}^{n} c\right )^{4} + 4 \, a b^{3} x \log \left ({\left (e x + d\right )}^{n} c\right )^{3} - 4 \, a^{3} b e n{\left (\frac{x}{e} - \frac{d \log \left (e x + d\right )}{e^{2}}\right )} + 6 \, a^{2} b^{2} x \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + 4 \, a^{3} b x \log \left ({\left (e x + d\right )}^{n} c\right ) - 6 \,{\left (2 \, e n{\left (\frac{x}{e} - \frac{d \log \left (e x + d\right )}{e^{2}}\right )} \log \left ({\left (e x + d\right )}^{n} c\right ) + \frac{{\left (d \log \left (e x + d\right )^{2} - 2 \, e x + 2 \, d \log \left (e x + d\right )\right )} n^{2}}{e}\right )} a^{2} b^{2} - 4 \,{\left (3 \, e n{\left (\frac{x}{e} - \frac{d \log \left (e x + d\right )}{e^{2}}\right )} \log \left ({\left (e x + d\right )}^{n} c\right )^{2} - e n{\left (\frac{{\left (d \log \left (e x + d\right )^{3} + 3 \, d \log \left (e x + d\right )^{2} - 6 \, e x + 6 \, d \log \left (e x + d\right )\right )} n^{2}}{e^{2}} - \frac{3 \,{\left (d \log \left (e x + d\right )^{2} - 2 \, e x + 2 \, d \log \left (e x + d\right )\right )} n \log \left ({\left (e x + d\right )}^{n} c\right )}{e^{2}}\right )}\right )} a b^{3} -{\left (4 \, e n{\left (\frac{x}{e} - \frac{d \log \left (e x + d\right )}{e^{2}}\right )} \log \left ({\left (e x + d\right )}^{n} c\right )^{3} +{\left (e n{\left (\frac{{\left (d \log \left (e x + d\right )^{4} + 4 \, d \log \left (e x + d\right )^{3} + 12 \, d \log \left (e x + d\right )^{2} - 24 \, e x + 24 \, d \log \left (e x + d\right )\right )} n^{2}}{e^{3}} - \frac{4 \,{\left (d \log \left (e x + d\right )^{3} + 3 \, d \log \left (e x + d\right )^{2} - 6 \, e x + 6 \, d \log \left (e x + d\right )\right )} n \log \left ({\left (e x + d\right )}^{n} c\right )}{e^{3}}\right )} + \frac{6 \,{\left (d \log \left (e x + d\right )^{2} - 2 \, e x + 2 \, d \log \left (e x + d\right )\right )} n \log \left ({\left (e x + d\right )}^{n} c\right )^{2}}{e^{2}}\right )} e n\right )} b^{4} + a^{4} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.0372, size = 1312, normalized size = 10.02 \begin{align*} \frac{b^{4} e x \log \left (c\right )^{4} +{\left (b^{4} e n^{4} x + b^{4} d n^{4}\right )} \log \left (e x + d\right )^{4} - 4 \,{\left (b^{4} e n - a b^{3} e\right )} x \log \left (c\right )^{3} - 4 \,{\left (b^{4} d n^{4} - a b^{3} d n^{3} +{\left (b^{4} e n^{4} - a b^{3} e n^{3}\right )} x -{\left (b^{4} e n^{3} x + b^{4} d n^{3}\right )} \log \left (c\right )\right )} \log \left (e x + d\right )^{3} + 6 \,{\left (2 \, b^{4} e n^{2} - 2 \, a b^{3} e n + a^{2} b^{2} e\right )} x \log \left (c\right )^{2} + 6 \,{\left (2 \, b^{4} d n^{4} - 2 \, a b^{3} d n^{3} + a^{2} b^{2} d n^{2} +{\left (b^{4} e n^{2} x + b^{4} d n^{2}\right )} \log \left (c\right )^{2} +{\left (2 \, b^{4} e n^{4} - 2 \, a b^{3} e n^{3} + a^{2} b^{2} e n^{2}\right )} x - 2 \,{\left (b^{4} d n^{3} - a b^{3} d n^{2} +{\left (b^{4} e n^{3} - a b^{3} e n^{2}\right )} x\right )} \log \left (c\right )\right )} \log \left (e x + d\right )^{2} - 4 \,{\left (6 \, b^{4} e n^{3} - 6 \, a b^{3} e n^{2} + 3 \, a^{2} b^{2} e n - a^{3} b e\right )} x \log \left (c\right ) +{\left (24 \, b^{4} e n^{4} - 24 \, a b^{3} e n^{3} + 12 \, a^{2} b^{2} e n^{2} - 4 \, a^{3} b e n + a^{4} e\right )} x - 4 \,{\left (6 \, b^{4} d n^{4} - 6 \, a b^{3} d n^{3} + 3 \, a^{2} b^{2} d n^{2} - a^{3} b d n -{\left (b^{4} e n x + b^{4} d n\right )} \log \left (c\right )^{3} + 3 \,{\left (b^{4} d n^{2} - a b^{3} d n +{\left (b^{4} e n^{2} - a b^{3} e n\right )} x\right )} \log \left (c\right )^{2} +{\left (6 \, b^{4} e n^{4} - 6 \, a b^{3} e n^{3} + 3 \, a^{2} b^{2} e n^{2} - a^{3} b e n\right )} x - 3 \,{\left (2 \, b^{4} d n^{3} - 2 \, a b^{3} d n^{2} + a^{2} b^{2} d n +{\left (2 \, b^{4} e n^{3} - 2 \, a b^{3} e n^{2} + a^{2} b^{2} e n\right )} x\right )} \log \left (c\right )\right )} \log \left (e x + d\right )}{e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 9.23738, size = 1059, normalized size = 8.08 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.27707, size = 1050, normalized size = 8.02 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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