Optimal. Leaf size=65 \[ \frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-2 a b n x-\frac {2 b^2 n (d+e x) \log \left (c (d+e x)^n\right )}{e}+2 b^2 n^2 x \]
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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 = {2389, 2296, 2295} \[ \frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-2 a b n x-\frac {2 b^2 n (d+e x) \log \left (c (d+e x)^n\right )}{e}+2 b^2 n^2 x \]
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 )^2 \, dx &=\frac {\operatorname {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) \operatorname {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 ) \operatorname {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 \[ \frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-2 b n \left (a x+\frac {b (d+e x) \log \left (c (d+e x)^n\right )}{e}-b n x\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 1.28, size = 140, normalized size = 2.15 \[ \frac {b^{2} e x \log \relax (c)^{2} + {\left (b^{2} e n^{2} x + b^{2} d n^{2}\right )} \log \left (e x + d\right )^{2} - 2 \, {\left (b^{2} e n - a b e\right )} x \log \relax (c) + {\left (2 \, b^{2} e n^{2} - 2 \, a b e n + a^{2} e\right )} x - 2 \, {\left (b^{2} d n^{2} - a b d n + {\left (b^{2} e n^{2} - a b e n\right )} x - {\left (b^{2} e n x + b^{2} d n\right )} \log \relax (c)\right )} \log \left (e x + d\right )}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 178, normalized size = 2.74 \[ {\left (x e + d\right )} b^{2} n^{2} e^{\left (-1\right )} \log \left (x e + d\right )^{2} - 2 \, {\left (x e + d\right )} b^{2} n^{2} e^{\left (-1\right )} \log \left (x e + d\right ) + 2 \, {\left (x e + d\right )} b^{2} n e^{\left (-1\right )} \log \left (x e + d\right ) \log \relax (c) + 2 \, {\left (x e + d\right )} b^{2} n^{2} e^{\left (-1\right )} + 2 \, {\left (x e + d\right )} a b n e^{\left (-1\right )} \log \left (x e + d\right ) - 2 \, {\left (x e + d\right )} b^{2} n e^{\left (-1\right )} \log \relax (c) + {\left (x e + d\right )} b^{2} e^{\left (-1\right )} \log \relax (c)^{2} - 2 \, {\left (x e + d\right )} a b n e^{\left (-1\right )} + 2 \, {\left (x e + d\right )} a b e^{\left (-1\right )} \log \relax (c) + {\left (x e + d\right )} a^{2} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 130, normalized size = 2.00 \[ -\frac {2 b^{2} d \,n^{2} \ln \left (e x +d \right )}{e}+2 b^{2} n^{2} x -2 b^{2} n x \ln \left (c \,{\mathrm e}^{n \ln \left (e x +d \right )}\right )+b^{2} x \ln \left (c \,{\mathrm e}^{n \ln \left (e x +d \right )}\right )^{2}+\frac {2 a b d n \ln \left (e x +d \right )}{e}-2 a b n x +2 a b x \ln \left (c \left (e x +d \right )^{n}\right )+\frac {b^{2} d \ln \left (c \,{\mathrm e}^{n \ln \left (e x +d \right )}\right )^{2}}{e}+a^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.60, size = 131, normalized size = 2.02 \[ -2 \, a b e n {\left (\frac {x}{e} - \frac {d \log \left (e x + d\right )}{e^{2}}\right )} + b^{2} x \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + 2 \, a b x \log \left ({\left (e x + d\right )}^{n} c\right ) - {\left (2 \, e n {\left (\frac {x}{e} - \frac {d \log \left (e x + d\right )}{e^{2}}\right )} \log \left ({\left (e x + d\right )}^{n} c\right ) + \frac {{\left (d \log \left (e x + d\right )^{2} - 2 \, e x + 2 \, d \log \left (e x + d\right )\right )} n^{2}}{e}\right )} b^{2} + a^{2} x \]
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 \[ x\,\left (a^2-2\,a\,b\,n+2\,b^2\,n^2\right )+{\ln \left (c\,{\left (d+e\,x\right )}^n\right )}^2\,\left (b^2\,x+\frac {b^2\,d}{e}\right )-\frac {\ln \left (d+e\,x\right )\,\left (2\,b^2\,d\,n^2-2\,a\,b\,d\,n\right )}{e}+2\,b\,x\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\,\left (a-b\,n\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.47, size = 211, normalized size = 3.25 \[ \begin {cases} a^{2} x + \frac {2 a b d n \log {\left (d + e x \right )}}{e} + 2 a b n x \log {\left (d + e x \right )} - 2 a b n x + 2 a b x \log {\relax (c )} + \frac {b^{2} d n^{2} \log {\left (d + e x \right )}^{2}}{e} - \frac {2 b^{2} d n^{2} \log {\left (d + e x \right )}}{e} + \frac {2 b^{2} d n \log {\relax (c )} \log {\left (d + e x \right )}}{e} + b^{2} n^{2} x \log {\left (d + e x \right )}^{2} - 2 b^{2} n^{2} x \log {\left (d + e x \right )} + 2 b^{2} n^{2} x + 2 b^{2} n x \log {\relax (c )} \log {\left (d + e x \right )} - 2 b^{2} n x \log {\relax (c )} + b^{2} x \log {\relax (c )}^{2} & \text {for}\: e \neq 0 \\x \left (a + b \log {\left (c d^{n} \right )}\right )^{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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