Optimal. Leaf size=65 \[ -2 a b n x+2 b^2 n^2 x-\frac {2 b^2 n (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e} \]
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Rubi [A]
time = 0.03, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps
used = 4, 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*} \frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-2 a b n x-\frac {2 b^2 n (d+e x) \log \left (c (d+e x)^n\right )}{e}+2 b^2 n^2 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 )^2 \, dx &=\frac {\text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e}\\ &=\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac {(2 b n) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e}\\ &=-2 a b n x+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac {\left (2 b^2 n\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e}\\ &=-2 a b n x+2 b^2 n^2 x-\frac {2 b^2 n (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 59, normalized size = 0.91 \begin {gather*} \frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-2 b n \left (a x-b n x+\frac {b (d+e x) \log \left (c (d+e x)^n\right )}{e}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.30, size = 130, normalized size = 2.00
method | result | size |
norman | \(\left (2 b^{2} n^{2}-2 b a n +a^{2}\right ) x +b^{2} x \ln \left (c \,{\mathrm e}^{n \ln \left (e x +d \right )}\right )^{2}+\left (-2 b^{2} n +2 b a \right ) x \ln \left (c \,{\mathrm e}^{n \ln \left (e x +d \right )}\right )+\frac {b^{2} d \ln \left (c \,{\mathrm e}^{n \ln \left (e x +d \right )}\right )^{2}}{e}+\frac {n \left (-2 b^{2} d n +2 a d b \right ) \ln \left (e x +d \right )}{e}\) | \(111\) |
default | \(a^{2} x +b^{2} x \ln \left (c \,{\mathrm e}^{n \ln \left (e x +d \right )}\right )^{2}+\frac {b^{2} d \ln \left (c \,{\mathrm e}^{n \ln \left (e x +d \right )}\right )^{2}}{e}+2 b^{2} n^{2} x -2 b^{2} n x \ln \left (c \,{\mathrm e}^{n \ln \left (e x +d \right )}\right )-\frac {2 n^{2} b^{2} d \ln \left (e x +d \right )}{e}+2 b a \ln \left (c \left (e x +d \right )^{n}\right ) x -2 a b n x +\frac {2 b a n d \ln \left (e x +d \right )}{e}\) | \(130\) |
risch | \(\text {Expression too large to display}\) | \(1125\) |
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. 136 vs.
\(2 (67) = 134\).
time = 0.27, size = 136, normalized size = 2.09 \begin {gather*} 2 \, {\left (d e^{\left (-2\right )} \log \left (x e + d\right ) - x e^{\left (-1\right )}\right )} a b n e + b^{2} x \log \left ({\left (x e + d\right )}^{n} c\right )^{2} + 2 \, a b x \log \left ({\left (x e + d\right )}^{n} c\right ) - {\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 )} b^{2} + a^{2} 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. 142 vs.
\(2 (67) = 134\).
time = 0.40, size = 142, normalized size = 2.18 \begin {gather*} {\left (b^{2} x e \log \left (c\right )^{2} - 2 \, {\left (b^{2} n - a b\right )} x e \log \left (c\right ) + {\left (2 \, b^{2} n^{2} - 2 \, a b n + a^{2}\right )} x e + {\left (b^{2} n^{2} x e + b^{2} d n^{2}\right )} \log \left (x e + d\right )^{2} - 2 \, {\left (b^{2} d n^{2} - a b d n + {\left (b^{2} n^{2} - a b n\right )} x e - {\left (b^{2} n x e + b^{2} d n\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. 146 vs.
\(2 (63) = 126\).
time = 0.28, size = 146, normalized size = 2.25 \begin {gather*} \begin {cases} a^{2} x + \frac {2 a b d \log {\left (c \left (d + e x\right )^{n} \right )}}{e} - 2 a b n x + 2 a b x \log {\left (c \left (d + e x\right )^{n} \right )} - \frac {2 b^{2} d n \log {\left (c \left (d + e x\right )^{n} \right )}}{e} + \frac {b^{2} d \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{e} + 2 b^{2} n^{2} x - 2 b^{2} n x \log {\left (c \left (d + e x\right )^{n} \right )} + b^{2} x \log {\left (c \left (d + e x\right )^{n} \right )}^{2} & \text {for}\: e \neq 0 \\x \left (a + b \log {\left (c d^{n} \right )}\right )^{2} & \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. 178 vs.
\(2 (67) = 134\).
time = 3.45, size = 178, normalized size = 2.74 \begin {gather*} {\left (x e + d\right )} b^{2} n^{2} e^{\left (-1\right )} \log \left (x e + d\right )^{2} - 2 \, {\left (x e + d\right )} b^{2} n^{2} e^{\left (-1\right )} \log \left (x e + d\right ) + 2 \, {\left (x e + d\right )} b^{2} n e^{\left (-1\right )} \log \left (x e + d\right ) \log \left (c\right ) + 2 \, {\left (x e + d\right )} b^{2} n^{2} e^{\left (-1\right )} + 2 \, {\left (x e + d\right )} a b n e^{\left (-1\right )} \log \left (x e + d\right ) - 2 \, {\left (x e + d\right )} b^{2} n e^{\left (-1\right )} \log \left (c\right ) + {\left (x e + d\right )} b^{2} e^{\left (-1\right )} \log \left (c\right )^{2} - 2 \, {\left (x e + d\right )} a b n e^{\left (-1\right )} + 2 \, {\left (x e + d\right )} a b e^{\left (-1\right )} \log \left (c\right ) + {\left (x e + d\right )} a^{2} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.20, size = 94, normalized size = 1.45 \begin {gather*} x\,\left (a^2-2\,a\,b\,n+2\,b^2\,n^2\right )+{\ln \left (c\,{\left (d+e\,x\right )}^n\right )}^2\,\left (b^2\,x+\frac {b^2\,d}{e}\right )-\frac {\ln \left (d+e\,x\right )\,\left (2\,b^2\,d\,n^2-2\,a\,b\,d\,n\right )}{e}+2\,b\,x\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\,\left (a-b\,n\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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