Optimal. Leaf size=99 \[ 6 a b^2 n^2 x-6 b^3 n^3 x+\frac {6 b^3 n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac {3 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e} \]
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Rubi [A]
time = 0.04, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps
used = 5, 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*} 6 a b^2 n^2 x-\frac {3 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac {6 b^3 n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-6 b^3 n^3 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 )^3 \, dx &=\frac {\text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e x\right )}{e}\\ &=\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac {(3 b n) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e}\\ &=-\frac {3 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac {\left (6 b^2 n^2\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e}\\ &=6 a b^2 n^2 x-\frac {3 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac {\left (6 b^3 n^2\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e}\\ &=6 a b^2 n^2 x-6 b^3 n^3 x+\frac {6 b^3 n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac {3 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 85, normalized size = 0.86 \begin {gather*} \frac {(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 )}{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 = 0.55, size = 4872, normalized size = 49.21
method | result | size |
risch | \(\text {Expression too large to display}\) | \(4872\) |
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. 294 vs.
\(2 (102) = 204\).
time = 0.30, size = 294, normalized size = 2.97 \begin {gather*} b^{3} x \log \left ({\left (x e + d\right )}^{n} c\right )^{3} + 3 \, {\left (d e^{\left (-2\right )} \log \left (x e + d\right ) - x e^{\left (-1\right )}\right )} a^{2} b n e + 3 \, a b^{2} x \log \left ({\left (x e + d\right )}^{n} c\right )^{2} + 3 \, a^{2} b x \log \left ({\left (x e + d\right )}^{n} c\right ) - 3 \, {\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 b^{2} + {\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 )} b^{3} + a^{3} 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. 326 vs.
\(2 (102) = 204\).
time = 0.34, size = 326, normalized size = 3.29 \begin {gather*} {\left (b^{3} x e \log \left (c\right )^{3} - 3 \, {\left (b^{3} n - a b^{2}\right )} x e \log \left (c\right )^{2} + {\left (b^{3} n^{3} x e + b^{3} d n^{3}\right )} \log \left (x e + d\right )^{3} + 3 \, {\left (2 \, b^{3} n^{2} - 2 \, a b^{2} n + a^{2} b\right )} x e \log \left (c\right ) - {\left (6 \, b^{3} n^{3} - 6 \, a b^{2} n^{2} + 3 \, a^{2} b n - a^{3}\right )} x e - 3 \, {\left (b^{3} d n^{3} - a b^{2} d n^{2} + {\left (b^{3} n^{3} - a b^{2} n^{2}\right )} x e - {\left (b^{3} n^{2} x e + b^{3} d n^{2}\right )} \log \left (c\right )\right )} \log \left (x e + d\right )^{2} + 3 \, {\left (2 \, b^{3} d n^{3} - 2 \, a b^{2} d n^{2} + a^{2} b d n + {\left (2 \, b^{3} n^{3} - 2 \, a b^{2} n^{2} + a^{2} b n\right )} x e + {\left (b^{3} n x e + b^{3} d n\right )} \log \left (c\right )^{2} - 2 \, {\left (b^{3} d n^{2} - a b^{2} d n + {\left (b^{3} n^{2} - a 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. 294 vs.
\(2 (95) = 190\).
time = 0.55, size = 294, normalized size = 2.97 \begin {gather*} \begin {cases} a^{3} x + \frac {3 a^{2} b d \log {\left (c \left (d + e x\right )^{n} \right )}}{e} - 3 a^{2} b n x + 3 a^{2} b x \log {\left (c \left (d + e x\right )^{n} \right )} - \frac {6 a b^{2} d n \log {\left (c \left (d + e x\right )^{n} \right )}}{e} + \frac {3 a b^{2} d \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{e} + 6 a b^{2} n^{2} x - 6 a b^{2} n x \log {\left (c \left (d + e x\right )^{n} \right )} + 3 a b^{2} x \log {\left (c \left (d + e x\right )^{n} \right )}^{2} + \frac {6 b^{3} d n^{2} \log {\left (c \left (d + e x\right )^{n} \right )}}{e} - \frac {3 b^{3} d n \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{e} + \frac {b^{3} d \log {\left (c \left (d + e x\right )^{n} \right )}^{3}}{e} - 6 b^{3} n^{3} x + 6 b^{3} n^{2} x \log {\left (c \left (d + e x\right )^{n} \right )} - 3 b^{3} n x \log {\left (c \left (d + e x\right )^{n} \right )}^{2} + b^{3} x \log {\left (c \left (d + e x\right )^{n} \right )}^{3} & \text {for}\: e \neq 0 \\x \left (a + b \log {\left (c d^{n} \right )}\right )^{3} & \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. 409 vs.
\(2 (102) = 204\).
time = 3.90, size = 409, normalized size = 4.13 \begin {gather*} {\left (x e + d\right )} b^{3} n^{3} e^{\left (-1\right )} \log \left (x e + d\right )^{3} - 3 \, {\left (x e + d\right )} b^{3} n^{3} e^{\left (-1\right )} \log \left (x e + d\right )^{2} + 3 \, {\left (x e + d\right )} b^{3} n^{2} e^{\left (-1\right )} \log \left (x e + d\right )^{2} \log \left (c\right ) + 6 \, {\left (x e + d\right )} b^{3} n^{3} e^{\left (-1\right )} \log \left (x e + d\right ) + 3 \, {\left (x e + d\right )} a b^{2} n^{2} e^{\left (-1\right )} \log \left (x e + d\right )^{2} - 6 \, {\left (x e + d\right )} b^{3} n^{2} e^{\left (-1\right )} \log \left (x e + d\right ) \log \left (c\right ) + 3 \, {\left (x e + d\right )} b^{3} n e^{\left (-1\right )} \log \left (x e + d\right ) \log \left (c\right )^{2} - 6 \, {\left (x e + d\right )} b^{3} n^{3} e^{\left (-1\right )} - 6 \, {\left (x e + d\right )} a b^{2} n^{2} e^{\left (-1\right )} \log \left (x e + d\right ) + 6 \, {\left (x e + d\right )} b^{3} n^{2} e^{\left (-1\right )} \log \left (c\right ) + 6 \, {\left (x e + d\right )} a b^{2} n e^{\left (-1\right )} \log \left (x e + d\right ) \log \left (c\right ) - 3 \, {\left (x e + d\right )} b^{3} n e^{\left (-1\right )} \log \left (c\right )^{2} + {\left (x e + d\right )} b^{3} e^{\left (-1\right )} \log \left (c\right )^{3} + 6 \, {\left (x e + d\right )} a b^{2} n^{2} e^{\left (-1\right )} + 3 \, {\left (x e + d\right )} a^{2} b n e^{\left (-1\right )} \log \left (x e + d\right ) - 6 \, {\left (x e + d\right )} a b^{2} n e^{\left (-1\right )} \log \left (c\right ) + 3 \, {\left (x e + d\right )} a b^{2} e^{\left (-1\right )} \log \left (c\right )^{2} - 3 \, {\left (x e + d\right )} a^{2} b n e^{\left (-1\right )} + 3 \, {\left (x e + d\right )} a^{2} b e^{\left (-1\right )} \log \left (c\right ) + {\left (x e + d\right )} a^{3} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.26, size = 172, normalized size = 1.74 \begin {gather*} x\,\left (a^3-3\,a^2\,b\,n+6\,a\,b^2\,n^2-6\,b^3\,n^3\right )+{\ln \left (c\,{\left (d+e\,x\right )}^n\right )}^3\,\left (b^3\,x+\frac {b^3\,d}{e}\right )+{\ln \left (c\,{\left (d+e\,x\right )}^n\right )}^2\,\left (\frac {3\,\left (a\,b^2\,d-b^3\,d\,n\right )}{e}+3\,b^2\,x\,\left (a-b\,n\right )\right )+\frac {\ln \left (d+e\,x\right )\,\left (3\,d\,a^2\,b\,n-6\,d\,a\,b^2\,n^2+6\,d\,b^3\,n^3\right )}{e}+3\,b\,x\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\,\left (a^2-2\,a\,b\,n+2\,b^2\,n^2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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