Optimal. Leaf size=124 \[ \frac {d g p x^n}{2 e n}-\frac {g p x^{2 n}}{4 n}-\frac {d^2 g p \log \left (d+e x^n\right )}{2 e^2 n}+\frac {g x^{2 n} \log \left (c \left (d+e x^n\right )^p\right )}{2 n}+\frac {f \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f p \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{n} \]
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Rubi [A]
time = 0.10, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {2525, 14, 2463,
2441, 2352, 2442, 45} \begin {gather*} \frac {f p \text {PolyLog}\left (2,\frac {e x^n}{d}+1\right )}{n}+\frac {f \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {g x^{2 n} \log \left (c \left (d+e x^n\right )^p\right )}{2 n}-\frac {d^2 g p \log \left (d+e x^n\right )}{2 e^2 n}+\frac {d g p x^n}{2 e n}-\frac {g p x^{2 n}}{4 n} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 45
Rule 2352
Rule 2441
Rule 2442
Rule 2463
Rule 2525
Rubi steps
\begin {align*} \int \frac {\left (f+g x^{2 n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (f+g x^2\right ) \log \left (c (d+e x)^p\right )}{x} \, dx,x,x^n\right )}{n}\\ &=\frac {\text {Subst}\left (\int \left (\frac {f \log \left (c (d+e x)^p\right )}{x}+g x \log \left (c (d+e x)^p\right )\right ) \, dx,x,x^n\right )}{n}\\ &=\frac {f \text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{x} \, dx,x,x^n\right )}{n}+\frac {g \text {Subst}\left (\int x \log \left (c (d+e x)^p\right ) \, dx,x,x^n\right )}{n}\\ &=\frac {g x^{2 n} \log \left (c \left (d+e x^n\right )^p\right )}{2 n}+\frac {f \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}-\frac {(e f p) \text {Subst}\left (\int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx,x,x^n\right )}{n}-\frac {(e g p) \text {Subst}\left (\int \frac {x^2}{d+e x} \, dx,x,x^n\right )}{2 n}\\ &=\frac {g x^{2 n} \log \left (c \left (d+e x^n\right )^p\right )}{2 n}+\frac {f \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f p \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{n}-\frac {(e g p) \text {Subst}\left (\int \left (-\frac {d}{e^2}+\frac {x}{e}+\frac {d^2}{e^2 (d+e x)}\right ) \, dx,x,x^n\right )}{2 n}\\ &=\frac {d g p x^n}{2 e n}-\frac {g p x^{2 n}}{4 n}-\frac {d^2 g p \log \left (d+e x^n\right )}{2 e^2 n}+\frac {g x^{2 n} \log \left (c \left (d+e x^n\right )^p\right )}{2 n}+\frac {f \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f p \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{n}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 173, normalized size = 1.40 \begin {gather*} \frac {2 d e g p x^n-e^2 g p x^{2 n}+2 d^2 g p \log \left (x^n\right )+4 e^2 f p \log \left (-\frac {e x^n}{d}\right ) \log \left (d+e x^n\right )-2 d^2 g p \log \left (d n \left (d+e x^n\right )\right )+2 e^2 g x^{2 n} \log \left (c \left (d+e x^n\right )^p\right )-2 n \log (x) \left (d^2 g p+2 e^2 f p \log \left (d+e x^n\right )-2 e^2 f \log \left (c \left (d+e x^n\right )^p\right )\right )+4 e^2 f p \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{4 e^2 n} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.10, size = 410, normalized size = 3.31
method | result | size |
risch | \(\frac {\left (2 f \ln \left (x \right ) n +g \,x^{2 n}\right ) \ln \left (\left (d +e \,x^{n}\right )^{p}\right )}{2 n}-\frac {i \pi \,\mathrm {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right ) \mathrm {csgn}\left (i c \right ) g \,x^{2 n}}{4 n}+\frac {i \pi \,\mathrm {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{2} f \ln \left (x \right )}{2}+\frac {i \pi \,\mathrm {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{2} g \,x^{2 n}}{4 n}-\frac {i \pi \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{3} g \,x^{2 n}}{4 n}+\frac {i \pi \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right ) g \,x^{2 n}}{4 n}-\frac {i \pi \,\mathrm {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right ) \mathrm {csgn}\left (i c \right ) f \ln \left (x \right )}{2}+\frac {i \pi \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right ) f \ln \left (x \right )}{2}-\frac {i \pi \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{3} f \ln \left (x \right )}{2}+\ln \left (c \right ) f \ln \left (x \right )+\frac {\ln \left (c \right ) g \,x^{2 n}}{2 n}-\frac {g p \,x^{2 n}}{4 n}+\frac {d g p \,x^{n}}{2 e n}-\frac {d^{2} g p \ln \left (d +e \,x^{n}\right )}{2 e^{2} n}-\frac {p f \dilog \left (\frac {d +e \,x^{n}}{d}\right )}{n}-p f \ln \left (x \right ) \ln \left (\frac {d +e \,x^{n}}{d}\right )\) | \(410\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 129, normalized size = 1.04 \begin {gather*} -\frac {{\left (4 \, f n p e^{2} \log \left (x\right ) \log \left (\frac {x^{n} e + d}{d}\right ) - 2 \, d g p x^{n} e - 4 \, f n e^{2} \log \left (c\right ) \log \left (x\right ) + 4 \, f p {\rm Li}_2\left (-\frac {x^{n} e + d}{d} + 1\right ) e^{2} + {\left (g p e^{2} - 2 \, g e^{2} \log \left (c\right )\right )} x^{2 \, n} - 2 \, {\left (2 \, f n p e^{2} \log \left (x\right ) - d^{2} g p + g p x^{2 \, n} e^{2}\right )} \log \left (x^{n} e + d\right )\right )} e^{\left (-2\right )}}{4 \, n} \end {gather*}
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 \begin {gather*} \int \frac {\left (f + g x^{2 n}\right ) \log {\left (c \left (d + e x^{n}\right )^{p} \right )}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )\,\left (f+g\,x^{2\,n}\right )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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