Optimal. Leaf size=177 \[ \frac {e^{-\frac {a}{b n}} (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b^2 e^2 n^2}+\frac {2 g e^{-\frac {2 a}{b n}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {Ei}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^2 e^2 n^2}-\frac {(d+e x) (f+g x)}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )} \]
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Rubi [A] time = 0.25, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {2400, 2399, 2389, 2300, 2178, 2390, 2310} \[ \frac {e^{-\frac {a}{b n}} (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b^2 e^2 n^2}+\frac {2 g e^{-\frac {2 a}{b n}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {Ei}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^2 e^2 n^2}-\frac {(d+e x) (f+g x)}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )} \]
Antiderivative was successfully verified.
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Rule 2178
Rule 2300
Rule 2310
Rule 2389
Rule 2390
Rule 2399
Rule 2400
Rubi steps
\begin {align*} \int \frac {f+g x}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx &=-\frac {(d+e x) (f+g x)}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {2 \int \frac {f+g x}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b n}-\frac {(e f-d g) \int \frac {1}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b e n}\\ &=-\frac {(d+e x) (f+g x)}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {2 \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}{b n}-\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 )}{b e^2 n}\\ &=-\frac {(d+e x) (f+g x)}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {(2 g) \int \frac {d+e x}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b e n}+\frac {(2 (e f-d g)) \int \frac {1}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b e 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 )}{b e^2 n^2}\\ &=-\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^2 e^2 n^2}-\frac {(d+e x) (f+g x)}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {(2 g) \operatorname {Subst}\left (\int \frac {x}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{b e^2 n}+\frac {(2 (e f-d g)) \operatorname {Subst}\left (\int \frac {1}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{b 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^2 e^2 n^2}-\frac {(d+e x) (f+g x)}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {\left (2 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 )}{b e^2 n^2}+\frac {\left (2 (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 )}{b e^2 n^2}\\ &=\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^2 e^2 n^2}+\frac {2 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^2 e^2 n^2}-\frac {(d+e x) (f+g x)}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 208, normalized size = 1.18 \[ -\frac {e^{-\frac {2 a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-2/n} \left (-e^{\frac {a}{b n}} (e f-d g) \left (c (d+e x)^n\right )^{\frac {1}{n}} \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )-2 g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Ei}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )+b e n e^{\frac {2 a}{b n}} (f+g x) \left (c (d+e x)^n\right )^{2/n}\right )}{b^2 e^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 239, normalized size = 1.35 \[ \frac {{\left ({\left (a e f - a d g + {\left (b e f - b d g\right )} n \log \left (e x + d\right ) + {\left (b e f - b d g\right )} \log \relax (c)\right )} e^{\left (\frac {b \log \relax (c) + a}{b n}\right )} \operatorname {log\_integral}\left ({\left (e x + d\right )} e^{\left (\frac {b \log \relax (c) + a}{b n}\right )}\right ) - {\left (b e^{2} g n x^{2} + b d e f n + {\left (b e^{2} f + b d e g\right )} n x\right )} e^{\left (\frac {2 \, {\left (b \log \relax (c) + a\right )}}{b n}\right )} + 2 \, {\left (b g n \log \left (e x + d\right ) + b g \log \relax (c) + a g\right )} \operatorname {log\_integral}\left ({\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )} e^{\left (\frac {2 \, {\left (b \log \relax (c) + a\right )}}{b n}\right )}\right )\right )} e^{\left (-\frac {2 \, {\left (b \log \relax (c) + a\right )}}{b n}\right )}}{b^{3} e^{2} n^{3} \log \left (e x + d\right ) + b^{3} e^{2} n^{2} \log \relax (c) + a b^{2} e^{2} n^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.37, size = 984, normalized size = 5.56 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 4.83, size = 0, normalized size = 0.00 \[ \int \frac {g x +f}{\left (b \ln \left (c \left (e x +d \right )^{n}\right )+a \right )^{2}}\, dx \]
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 \[ -\frac {e g x^{2} + d f + {\left (e f + d g\right )} x}{b^{2} e n \log \left ({\left (e x + d\right )}^{n}\right ) + b^{2} e n \log \relax (c) + a b e n} + \int \frac {2 \, e g x + e f + d g}{b^{2} e n \log \left ({\left (e x + d\right )}^{n}\right ) + b^{2} e n \log \relax (c) + a b e n}\,{d x} \]
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 \[ \int \frac {f+g\,x}{{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2} \,d x \]
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 \[ \int \frac {f + g x}{\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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