Optimal. Leaf size=131 \[ -24 a b^3 n^3 x+\frac {12 b^2 n^2 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac {4 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^4}{e}-\frac {24 b^4 n^3 (d+e x) \log \left (c (d+e x)^n\right )}{e}+24 b^4 n^4 x \]
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Rubi [A] time = 0.07, antiderivative size = 131, normalized size of antiderivative = 1.00, 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 2295
Rule 2296
Rule 2389
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*}
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Mathematica [A] time = 0.03, 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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fricas [B] time = 0.49, size = 614, normalized size = 4.69 \[ \frac {b^{4} e x \log \relax (c)^{4} + {\left (b^{4} e n^{4} x + b^{4} d n^{4}\right )} \log \left (e x + d\right )^{4} - 4 \, {\left (b^{4} e n - a b^{3} e\right )} x \log \relax (c)^{3} - 4 \, {\left (b^{4} d n^{4} - a b^{3} d n^{3} + {\left (b^{4} e n^{4} - a b^{3} e n^{3}\right )} x - {\left (b^{4} e n^{3} x + b^{4} d n^{3}\right )} \log \relax (c)\right )} \log \left (e x + d\right )^{3} + 6 \, {\left (2 \, b^{4} e n^{2} - 2 \, a b^{3} e n + a^{2} b^{2} e\right )} x \log \relax (c)^{2} + 6 \, {\left (2 \, b^{4} d n^{4} - 2 \, a b^{3} d n^{3} + a^{2} b^{2} d n^{2} + {\left (b^{4} e n^{2} x + b^{4} d n^{2}\right )} \log \relax (c)^{2} + {\left (2 \, b^{4} e n^{4} - 2 \, a b^{3} e n^{3} + a^{2} b^{2} e n^{2}\right )} x - 2 \, {\left (b^{4} d n^{3} - a b^{3} d n^{2} + {\left (b^{4} e n^{3} - a b^{3} e n^{2}\right )} x\right )} \log \relax (c)\right )} \log \left (e x + d\right )^{2} - 4 \, {\left (6 \, b^{4} e n^{3} - 6 \, a b^{3} e n^{2} + 3 \, a^{2} b^{2} e n - a^{3} b e\right )} x \log \relax (c) + {\left (24 \, b^{4} e n^{4} - 24 \, a b^{3} e n^{3} + 12 \, a^{2} b^{2} e n^{2} - 4 \, a^{3} b e n + a^{4} e\right )} x - 4 \, {\left (6 \, b^{4} d n^{4} - 6 \, a b^{3} d n^{3} + 3 \, a^{2} b^{2} d n^{2} - a^{3} b d n - {\left (b^{4} e n x + b^{4} d n\right )} \log \relax (c)^{3} + 3 \, {\left (b^{4} d n^{2} - a b^{3} d n + {\left (b^{4} e n^{2} - a b^{3} e n\right )} x\right )} \log \relax (c)^{2} + {\left (6 \, b^{4} e n^{4} - 6 \, a b^{3} e n^{3} + 3 \, a^{2} b^{2} e n^{2} - a^{3} b e n\right )} x - 3 \, {\left (2 \, b^{4} d n^{3} - 2 \, a b^{3} d n^{2} + a^{2} b^{2} d n + {\left (2 \, b^{4} e n^{3} - 2 \, a b^{3} e n^{2} + a^{2} b^{2} e n\right )} x\right )} \log \relax (c)\right )} \log \left (e x + d\right )}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 778, normalized size = 5.94 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.42, size = 15871, normalized size = 121.15 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.29, size = 500, normalized size = 3.82 \[ 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 \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.00, size = 275, normalized size = 2.10 \[ {\ln \left (c\,{\left (d+e\,x\right )}^n\right )}^2\,\left (\frac {6\,\left (d\,a^2\,b^2-2\,d\,a\,b^3\,n+2\,d\,b^4\,n^2\right )}{e}+6\,b^2\,x\,\left (a^2-2\,a\,b\,n+2\,b^2\,n^2\right )\right )+x\,\left (a^4-4\,a^3\,b\,n+12\,a^2\,b^2\,n^2-24\,a\,b^3\,n^3+24\,b^4\,n^4\right )+{\ln \left (c\,{\left (d+e\,x\right )}^n\right )}^4\,\left (b^4\,x+\frac {b^4\,d}{e}\right )+{\ln \left (c\,{\left (d+e\,x\right )}^n\right )}^3\,\left (\frac {4\,\left (a\,b^3\,d-b^4\,d\,n\right )}{e}+4\,b^3\,x\,\left (a-b\,n\right )\right )-\frac {\ln \left (d+e\,x\right )\,\left (-4\,d\,a^3\,b\,n+12\,d\,a^2\,b^2\,n^2-24\,d\,a\,b^3\,n^3+24\,d\,b^4\,n^4\right )}{e}+4\,b\,x\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\,\left (a^3-3\,a^2\,b\,n+6\,a\,b^2\,n^2-6\,b^3\,n^3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.92, size = 1059, normalized size = 8.08 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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