Optimal. Leaf size=139 \[ \frac {e^{-\frac {a}{b n}} (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b e^2 n}+\frac {g e^{-\frac {2 a}{b n}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {Ei}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b e^2 n} \]
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Rubi [A] time = 0.16, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2399, 2389, 2300, 2178, 2390, 2310} \[ \frac {e^{-\frac {a}{b n}} (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b e^2 n}+\frac {g e^{-\frac {2 a}{b n}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {Ei}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b e^2 n} \]
Antiderivative was successfully verified.
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Rule 2178
Rule 2300
Rule 2310
Rule 2389
Rule 2390
Rule 2399
Rubi steps
\begin {align*} \int \frac {f+g x}{a+b \log \left (c (d+e x)^n\right )} \, dx &=\int \left (\frac {e f-d g}{e \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {g (d+e x)}{e \left (a+b \log \left (c (d+e x)^n\right )\right )}\right ) \, dx\\ &=\frac {g \int \frac {d+e x}{a+b \log \left (c (d+e x)^n\right )} \, dx}{e}+\frac {(e f-d g) \int \frac {1}{a+b \log \left (c (d+e x)^n\right )} \, dx}{e}\\ &=\frac {g \operatorname {Subst}\left (\int \frac {x}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{e^2}+\frac {(e f-d g) \operatorname {Subst}\left (\int \frac {1}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{e^2}\\ &=\frac {\left (g (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {2 x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e^2 n}+\frac {\left ((e f-d g) (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e^2 n}\\ &=\frac {e^{-\frac {a}{b n}} (e f-d g) (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b e^2 n}+\frac {e^{-\frac {2 a}{b n}} g (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {Ei}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b e^2 n}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 126, normalized size = 0.91 \[ \frac {e^{-\frac {2 a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-2/n} \left (e^{\frac {a}{b n}} (e f-d g) \left (c (d+e x)^n\right )^{\frac {1}{n}} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )+g (d+e x) \text {Ei}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )\right )}{b e^2 n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 105, normalized size = 0.76 \[ \frac {{\left ({\left (e f - d g\right )} e^{\left (\frac {b \log \relax (c) + a}{b n}\right )} \operatorname {log\_integral}\left ({\left (e x + d\right )} e^{\left (\frac {b \log \relax (c) + a}{b n}\right )}\right ) + g \operatorname {log\_integral}\left ({\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )} e^{\left (\frac {2 \, {\left (b \log \relax (c) + a\right )}}{b n}\right )}\right )\right )} e^{\left (-\frac {2 \, {\left (b \log \relax (c) + a\right )}}{b n}\right )}}{b e^{2} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 159, normalized size = 1.14 \[ -\frac {d g {\rm Ei}\left (\frac {\log \relax (c)}{n} + \frac {a}{b n} + \log \left (x e + d\right )\right ) e^{\left (-\frac {a}{b n} - 2\right )}}{b c^{\left (\frac {1}{n}\right )} n} + \frac {f {\rm Ei}\left (\frac {\log \relax (c)}{n} + \frac {a}{b n} + \log \left (x e + d\right )\right ) e^{\left (-\frac {a}{b n} - 1\right )}}{b c^{\left (\frac {1}{n}\right )} n} + \frac {g {\rm Ei}\left (\frac {2 \, \log \relax (c)}{n} + \frac {2 \, a}{b n} + 2 \, \log \left (x e + d\right )\right ) e^{\left (-\frac {2 \, a}{b n} - 2\right )}}{b c^{\frac {2}{n}} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.41, size = 0, normalized size = 0.00 \[ \int \frac {g x +f}{b \ln \left (c \left (e x +d \right )^{n}\right )+a}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {g x + f}{b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {f+g\,x}{a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f + g x}{a + b \log {\left (c \left (d + e x\right )^{n} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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