Optimal. Leaf size=131 \[ -24 a b^3 n^3 x+24 b^4 n^4 x-\frac {24 b^4 n^3 (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac {12 b^2 n^2 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac {4 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^4}{e} \]
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Rubi [A]
time = 0.05, 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 = {2436, 2333,
2332} \begin {gather*} -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 \end {gather*}
Antiderivative was successfully verified.
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Rule 2332
Rule 2333
Rule 2436
Rubi steps
\begin {align*} \int \left (a+b \log \left (c (d+e x)^n\right )\right )^4 \, dx &=\frac {\text {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) \text {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 ) \text {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 ) \text {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 ) \text {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 = 0.02, size = 112, normalized size = 0.85 \begin {gather*} \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 (a-b n) x+b (d+e x) \log \left (c (d+e x)^n\right )\right )\right )\right )}{e} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.28, size = 15871, normalized size = 121.15
method | result | size |
risch | \(\text {Expression too large to display}\) | \(15871\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 528 vs.
\(2 (135) = 270\).
time = 0.30, size = 528, normalized size = 4.03 \begin {gather*} b^{4} x \log \left ({\left (x e + d\right )}^{n} c\right )^{4} + 4 \, a b^{3} x \log \left ({\left (x e + d\right )}^{n} c\right )^{3} + 4 \, {\left (d e^{\left (-2\right )} \log \left (x e + d\right ) - x e^{\left (-1\right )}\right )} a^{3} b n e + 6 \, a^{2} b^{2} x \log \left ({\left (x e + d\right )}^{n} c\right )^{2} + 4 \, a^{3} b x \log \left ({\left (x e + d\right )}^{n} c\right ) - 6 \, {\left ({\left (d \log \left (x e + d\right )^{2} - 2 \, x e + 2 \, d \log \left (x e + d\right )\right )} n^{2} e^{\left (-1\right )} - 2 \, {\left (d e^{\left (-2\right )} \log \left (x e + d\right ) - x e^{\left (-1\right )}\right )} n e \log \left ({\left (x e + d\right )}^{n} c\right )\right )} a^{2} b^{2} + 4 \, {\left (3 \, {\left (d e^{\left (-2\right )} \log \left (x e + d\right ) - x e^{\left (-1\right )}\right )} n e \log \left ({\left (x e + d\right )}^{n} c\right )^{2} + {\left ({\left (d \log \left (x e + d\right )^{3} + 3 \, d \log \left (x e + d\right )^{2} - 6 \, x e + 6 \, d \log \left (x e + d\right )\right )} n^{2} e^{\left (-2\right )} - 3 \, {\left (d \log \left (x e + d\right )^{2} - 2 \, x e + 2 \, d \log \left (x e + d\right )\right )} n e^{\left (-2\right )} \log \left ({\left (x e + d\right )}^{n} c\right )\right )} n e\right )} a b^{3} + {\left (4 \, {\left (d e^{\left (-2\right )} \log \left (x e + d\right ) - x e^{\left (-1\right )}\right )} n e \log \left ({\left (x e + d\right )}^{n} c\right )^{3} - {\left (6 \, {\left (d \log \left (x e + d\right )^{2} - 2 \, x e + 2 \, d \log \left (x e + d\right )\right )} n e^{\left (-2\right )} \log \left ({\left (x e + d\right )}^{n} c\right )^{2} + {\left ({\left (d \log \left (x e + d\right )^{4} + 4 \, d \log \left (x e + d\right )^{3} + 12 \, d \log \left (x e + d\right )^{2} - 24 \, x e + 24 \, d \log \left (x e + d\right )\right )} n^{2} e^{\left (-3\right )} - 4 \, {\left (d \log \left (x e + d\right )^{3} + 3 \, d \log \left (x e + d\right )^{2} - 6 \, x e + 6 \, d \log \left (x e + d\right )\right )} n e^{\left (-3\right )} \log \left ({\left (x e + d\right )}^{n} c\right )\right )} n e\right )} n e\right )} b^{4} + a^{4} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 611 vs.
\(2 (135) = 270\).
time = 0.36, size = 611, normalized size = 4.66 \begin {gather*} {\left (b^{4} x e \log \left (c\right )^{4} - 4 \, {\left (b^{4} n - a b^{3}\right )} x e \log \left (c\right )^{3} + {\left (b^{4} n^{4} x e + b^{4} d n^{4}\right )} \log \left (x e + d\right )^{4} + 6 \, {\left (2 \, b^{4} n^{2} - 2 \, a b^{3} n + a^{2} b^{2}\right )} x e \log \left (c\right )^{2} - 4 \, {\left (b^{4} d n^{4} - a b^{3} d n^{3} + {\left (b^{4} n^{4} - a b^{3} n^{3}\right )} x e - {\left (b^{4} n^{3} x e + b^{4} d n^{3}\right )} \log \left (c\right )\right )} \log \left (x e + d\right )^{3} - 4 \, {\left (6 \, b^{4} n^{3} - 6 \, a b^{3} n^{2} + 3 \, a^{2} b^{2} n - a^{3} b\right )} x e \log \left (c\right ) + {\left (24 \, b^{4} n^{4} - 24 \, a b^{3} n^{3} + 12 \, a^{2} b^{2} n^{2} - 4 \, a^{3} b n + a^{4}\right )} x e + 6 \, {\left (2 \, b^{4} d n^{4} - 2 \, a b^{3} d n^{3} + a^{2} b^{2} d n^{2} + {\left (2 \, b^{4} n^{4} - 2 \, a b^{3} n^{3} + a^{2} b^{2} n^{2}\right )} x e + {\left (b^{4} n^{2} x e + b^{4} d n^{2}\right )} \log \left (c\right )^{2} - 2 \, {\left (b^{4} d n^{3} - a b^{3} d n^{2} + {\left (b^{4} n^{3} - a b^{3} n^{2}\right )} x e\right )} \log \left (c\right )\right )} \log \left (x e + d\right )^{2} - 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} n x e + b^{4} d n\right )} \log \left (c\right )^{3} + {\left (6 \, b^{4} n^{4} - 6 \, a b^{3} n^{3} + 3 \, a^{2} b^{2} n^{2} - a^{3} b n\right )} x e + 3 \, {\left (b^{4} d n^{2} - a b^{3} d n + {\left (b^{4} n^{2} - a b^{3} n\right )} x e\right )} \log \left (c\right )^{2} - 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} n^{3} - 2 \, a b^{3} n^{2} + a^{2} b^{2} n\right )} x e\right )} \log \left (c\right )\right )} \log \left (x e + d\right )\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 495 vs.
\(2 (126) = 252\).
time = 1.09, size = 495, normalized size = 3.78 \begin {gather*} \begin {cases} a^{4} x + \frac {4 a^{3} b d \log {\left (c \left (d + e x\right )^{n} \right )}}{e} - 4 a^{3} b n x + 4 a^{3} b x \log {\left (c \left (d + e x\right )^{n} \right )} - \frac {12 a^{2} b^{2} d n \log {\left (c \left (d + e x\right )^{n} \right )}}{e} + \frac {6 a^{2} b^{2} d \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{e} + 12 a^{2} b^{2} n^{2} x - 12 a^{2} b^{2} n x \log {\left (c \left (d + e x\right )^{n} \right )} + 6 a^{2} b^{2} x \log {\left (c \left (d + e x\right )^{n} \right )}^{2} + \frac {24 a b^{3} d n^{2} \log {\left (c \left (d + e x\right )^{n} \right )}}{e} - \frac {12 a b^{3} d n \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{e} + \frac {4 a b^{3} d \log {\left (c \left (d + e x\right )^{n} \right )}^{3}}{e} - 24 a b^{3} n^{3} x + 24 a b^{3} n^{2} x \log {\left (c \left (d + e x\right )^{n} \right )} - 12 a b^{3} n x \log {\left (c \left (d + e x\right )^{n} \right )}^{2} + 4 a b^{3} x \log {\left (c \left (d + e x\right )^{n} \right )}^{3} - \frac {24 b^{4} d n^{3} \log {\left (c \left (d + e x\right )^{n} \right )}}{e} + \frac {12 b^{4} d n^{2} \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{e} - \frac {4 b^{4} d n \log {\left (c \left (d + e x\right )^{n} \right )}^{3}}{e} + \frac {b^{4} d \log {\left (c \left (d + e x\right )^{n} \right )}^{4}}{e} + 24 b^{4} n^{4} x - 24 b^{4} n^{3} x \log {\left (c \left (d + e x\right )^{n} \right )} + 12 b^{4} n^{2} x \log {\left (c \left (d + e x\right )^{n} \right )}^{2} - 4 b^{4} n x \log {\left (c \left (d + e x\right )^{n} \right )}^{3} + b^{4} x \log {\left (c \left (d + e x\right )^{n} \right )}^{4} & \text {for}\: e \neq 0 \\x \left (a + b \log {\left (c d^{n} \right )}\right )^{4} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 778 vs.
\(2 (135) = 270\).
time = 4.12, size = 778, normalized size = 5.94 \begin {gather*} {\left (x e + d\right )} b^{4} n^{4} e^{\left (-1\right )} \log \left (x e + d\right )^{4} - 4 \, {\left (x e + d\right )} b^{4} n^{4} e^{\left (-1\right )} \log \left (x e + d\right )^{3} + 4 \, {\left (x e + d\right )} b^{4} n^{3} e^{\left (-1\right )} \log \left (x e + d\right )^{3} \log \left (c\right ) + 12 \, {\left (x e + d\right )} b^{4} n^{4} e^{\left (-1\right )} \log \left (x e + d\right )^{2} + 4 \, {\left (x e + d\right )} a b^{3} n^{3} e^{\left (-1\right )} \log \left (x e + d\right )^{3} - 12 \, {\left (x e + d\right )} b^{4} n^{3} e^{\left (-1\right )} \log \left (x e + d\right )^{2} \log \left (c\right ) + 6 \, {\left (x e + d\right )} b^{4} n^{2} e^{\left (-1\right )} \log \left (x e + d\right )^{2} \log \left (c\right )^{2} - 24 \, {\left (x e + d\right )} b^{4} n^{4} e^{\left (-1\right )} \log \left (x e + d\right ) - 12 \, {\left (x e + d\right )} a b^{3} n^{3} e^{\left (-1\right )} \log \left (x e + d\right )^{2} + 24 \, {\left (x e + d\right )} b^{4} n^{3} e^{\left (-1\right )} \log \left (x e + d\right ) \log \left (c\right ) + 12 \, {\left (x e + d\right )} a b^{3} n^{2} e^{\left (-1\right )} \log \left (x e + d\right )^{2} \log \left (c\right ) - 12 \, {\left (x e + d\right )} b^{4} n^{2} e^{\left (-1\right )} \log \left (x e + d\right ) \log \left (c\right )^{2} + 4 \, {\left (x e + d\right )} b^{4} n e^{\left (-1\right )} \log \left (x e + d\right ) \log \left (c\right )^{3} + 24 \, {\left (x e + d\right )} b^{4} n^{4} e^{\left (-1\right )} + 24 \, {\left (x e + d\right )} a b^{3} n^{3} e^{\left (-1\right )} \log \left (x e + d\right ) + 6 \, {\left (x e + d\right )} a^{2} b^{2} n^{2} e^{\left (-1\right )} \log \left (x e + d\right )^{2} - 24 \, {\left (x e + d\right )} b^{4} n^{3} e^{\left (-1\right )} \log \left (c\right ) - 24 \, {\left (x e + d\right )} a b^{3} n^{2} e^{\left (-1\right )} \log \left (x e + d\right ) \log \left (c\right ) + 12 \, {\left (x e + d\right )} b^{4} n^{2} e^{\left (-1\right )} \log \left (c\right )^{2} + 12 \, {\left (x e + d\right )} a b^{3} n e^{\left (-1\right )} \log \left (x e + d\right ) \log \left (c\right )^{2} - 4 \, {\left (x e + d\right )} b^{4} n e^{\left (-1\right )} \log \left (c\right )^{3} + {\left (x e + d\right )} b^{4} e^{\left (-1\right )} \log \left (c\right )^{4} - 24 \, {\left (x e + d\right )} a b^{3} n^{3} e^{\left (-1\right )} - 12 \, {\left (x e + d\right )} a^{2} b^{2} n^{2} e^{\left (-1\right )} \log \left (x e + d\right ) + 24 \, {\left (x e + d\right )} a b^{3} n^{2} e^{\left (-1\right )} \log \left (c\right ) + 12 \, {\left (x e + d\right )} a^{2} b^{2} n e^{\left (-1\right )} \log \left (x e + d\right ) \log \left (c\right ) - 12 \, {\left (x e + d\right )} a b^{3} n e^{\left (-1\right )} \log \left (c\right )^{2} + 4 \, {\left (x e + d\right )} a b^{3} e^{\left (-1\right )} \log \left (c\right )^{3} + 12 \, {\left (x e + d\right )} a^{2} b^{2} n^{2} e^{\left (-1\right )} + 4 \, {\left (x e + d\right )} a^{3} b n e^{\left (-1\right )} \log \left (x e + d\right ) - 12 \, {\left (x e + d\right )} a^{2} b^{2} n e^{\left (-1\right )} \log \left (c\right ) + 6 \, {\left (x e + d\right )} a^{2} b^{2} e^{\left (-1\right )} \log \left (c\right )^{2} - 4 \, {\left (x e + d\right )} a^{3} b n e^{\left (-1\right )} + 4 \, {\left (x e + d\right )} a^{3} b e^{\left (-1\right )} \log \left (c\right ) + {\left (x e + d\right )} a^{4} e^{\left (-1\right )} \end {gather*}
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 \begin {gather*} {\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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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