Optimal. Leaf size=110 \[ -((b f+a g) n x)+2 b g n^2 x-\frac {2 b g n (d+e x) \log \left (c (d+e x)^n\right )}{e}+x \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )+\frac {d \left (b f+a g+2 b g \log \left (c (d+e x)^n\right )\right )^2}{4 b e g} \]
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Rubi [A]
time = 0.12, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2478, 2458,
2388, 2338, 2332} \begin {gather*} \frac {d \left (a g+2 b g \log \left (c (d+e x)^n\right )+b f\right )^2}{4 b e g}+x \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (g \log \left (c (d+e x)^n\right )+f\right )-n x (a g+b f)-\frac {2 b g n (d+e x) \log \left (c (d+e x)^n\right )}{e}+2 b g n^2 x \end {gather*}
Antiderivative was successfully verified.
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Rule 2332
Rule 2338
Rule 2388
Rule 2458
Rule 2478
Rubi steps
\begin {align*} \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right ) \, dx &=x \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )-(b e n) \int \frac {x \left (f+g \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx-(e g n) \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx\\ &=x \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )-(b n) \text {Subst}\left (\int \frac {\left (-\frac {d}{e}+\frac {x}{e}\right ) \left (f+g \log \left (c x^n\right )\right )}{x} \, dx,x,d+e x\right )-(g n) \text {Subst}\left (\int \frac {\left (-\frac {d}{e}+\frac {x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx,x,d+e x\right )\\ &=x \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )-\frac {(b n) \text {Subst}\left (\int \left (f+g \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e}+\frac {(b d n) \text {Subst}\left (\int \frac {f+g \log \left (c x^n\right )}{x} \, dx,x,d+e x\right )}{e}-\frac {(g n) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e}+\frac {(d g n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x} \, dx,x,d+e x\right )}{e}\\ &=-b f n x-a g n x+\frac {d g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 b e}+x \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )+\frac {b d \left (f+g \log \left (c (d+e x)^n\right )\right )^2}{2 e g}-2 \frac {(b g n) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e}\\ &=-b f n x-a g n x+\frac {d g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 b e}+x \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )+\frac {b d \left (f+g \log \left (c (d+e x)^n\right )\right )^2}{2 e g}-2 \left (-b g n^2 x+\frac {b g n (d+e x) \log \left (c (d+e x)^n\right )}{e}\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 76, normalized size = 0.69 \begin {gather*} \frac {e (a (f-g n)+b n (-f+2 g n)) x+(a g+b (f-2 g n)) (d+e x) \log \left (c (d+e x)^n\right )+b g (d+e x) \log ^2\left (c (d+e x)^n\right )}{e} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.41, size = 156, normalized size = 1.42
method | result | size |
norman | \(\left (2 b g \,n^{2}-n a g -n b f +a f \right ) x +\left (-2 b g n +a g +b f \right ) x \ln \left (c \,{\mathrm e}^{n \ln \left (e x +d \right )}\right )+b g x \ln \left (c \,{\mathrm e}^{n \ln \left (e x +d \right )}\right )^{2}+\frac {n \left (-2 b d g n +a d g +b d f \right ) \ln \left (e x +d \right )}{e}+\frac {b d g \ln \left (c \,{\mathrm e}^{n \ln \left (e x +d \right )}\right )^{2}}{e}\) | \(116\) |
default | \(x a f +x a g \ln \left (c \left (e x +d \right )^{n}\right )-a g n x +\frac {a g n d \ln \left (e x +d \right )}{e}+x b \ln \left (c \left (e x +d \right )^{n}\right ) f -b f n x +\frac {b f n d \ln \left (e x +d \right )}{e}+b g x \ln \left (c \,{\mathrm e}^{n \ln \left (e x +d \right )}\right )^{2}+\frac {b d g \ln \left (c \,{\mathrm e}^{n \ln \left (e x +d \right )}\right )^{2}}{e}+2 b g \,n^{2} x -\frac {2 n^{2} b d g \ln \left (e x +d \right )}{e}-2 b g n x \ln \left (c \,{\mathrm e}^{n \ln \left (e x +d \right )}\right )\) | \(156\) |
risch | \(\text {Expression too large to display}\) | \(1245\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 169, normalized size = 1.54 \begin {gather*} {\left (d e^{\left (-2\right )} \log \left (x e + d\right ) - x e^{\left (-1\right )}\right )} b f n e + {\left (d e^{\left (-2\right )} \log \left (x e + d\right ) - x e^{\left (-1\right )}\right )} a g n e + b g x \log \left ({\left (x e + d\right )}^{n} c\right )^{2} + b f x \log \left ({\left (x e + d\right )}^{n} c\right ) + a g 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 g + a f x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 156, normalized size = 1.42 \begin {gather*} {\left (b g x e \log \left (c\right )^{2} - {\left (2 \, b g n - b f - a g\right )} x e \log \left (c\right ) + {\left (2 \, b g n^{2} + a f - {\left (b f + a g\right )} n\right )} x e + {\left (b g n^{2} x e + b d g n^{2}\right )} \log \left (x e + d\right )^{2} - {\left (2 \, b d g n^{2} + {\left (2 \, b g n^{2} - {\left (b f + a g\right )} n\right )} x e - {\left (b d f + a d g\right )} n - 2 \, {\left (b g n x e + b d g 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 [A]
time = 0.31, size = 189, normalized size = 1.72 \begin {gather*} \begin {cases} \frac {a d g \log {\left (c \left (d + e x\right )^{n} \right )}}{e} + a f x - a g n x + a g x \log {\left (c \left (d + e x\right )^{n} \right )} + \frac {b d f \log {\left (c \left (d + e x\right )^{n} \right )}}{e} - \frac {2 b d g n \log {\left (c \left (d + e x\right )^{n} \right )}}{e} + \frac {b d g \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{e} - b f n x + b f x \log {\left (c \left (d + e x\right )^{n} \right )} + 2 b g n^{2} x - 2 b g n x \log {\left (c \left (d + e x\right )^{n} \right )} + b g 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 ) \left (f + g \log {\left (c d^{n} \right )}\right ) & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.58, size = 214, normalized size = 1.95 \begin {gather*} {\left (x e + d\right )} b g n^{2} e^{\left (-1\right )} \log \left (x e + d\right )^{2} - 2 \, {\left (x e + d\right )} b g n^{2} e^{\left (-1\right )} \log \left (x e + d\right ) + 2 \, {\left (x e + d\right )} b g n e^{\left (-1\right )} \log \left (x e + d\right ) \log \left (c\right ) + 2 \, {\left (x e + d\right )} b g n^{2} e^{\left (-1\right )} + {\left (x e + d\right )} b f n e^{\left (-1\right )} \log \left (x e + d\right ) + {\left (x e + d\right )} a g n e^{\left (-1\right )} \log \left (x e + d\right ) - 2 \, {\left (x e + d\right )} b g n e^{\left (-1\right )} \log \left (c\right ) + {\left (x e + d\right )} b g e^{\left (-1\right )} \log \left (c\right )^{2} - {\left (x e + d\right )} b f n e^{\left (-1\right )} - {\left (x e + d\right )} a g n e^{\left (-1\right )} + {\left (x e + d\right )} b f e^{\left (-1\right )} \log \left (c\right ) + {\left (x e + d\right )} a g e^{\left (-1\right )} \log \left (c\right ) + {\left (x e + d\right )} a f e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.27, size = 102, normalized size = 0.93 \begin {gather*} {\ln \left (c\,{\left (d+e\,x\right )}^n\right )}^2\,\left (b\,g\,x+\frac {b\,d\,g}{e}\right )+x\,\left (a\,f-a\,g\,n-b\,f\,n+2\,b\,g\,n^2\right )+x\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\,\left (a\,g+b\,f-2\,b\,g\,n\right )+\frac {\ln \left (d+e\,x\right )\,\left (a\,d\,g\,n-2\,b\,d\,g\,n^2+b\,d\,f\,n\right )}{e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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