Optimal. Leaf size=254 \[ \frac {d f g p x^n}{e n}+\frac {d^3 g^2 p x^n}{4 e^3 n}-\frac {f g p x^{2 n}}{2 n}-\frac {d^2 g^2 p x^{2 n}}{8 e^2 n}+\frac {d g^2 p x^{3 n}}{12 e n}-\frac {g^2 p x^{4 n}}{16 n}-\frac {d^2 f g p \log \left (d+e x^n\right )}{e^2 n}-\frac {d^4 g^2 p \log \left (d+e x^n\right )}{4 e^4 n}+\frac {f g x^{2 n} \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {g^2 x^{4 n} \log \left (c \left (d+e x^n\right )^p\right )}{4 n}+\frac {f^2 \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f^2 p \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{n} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.17, antiderivative size = 254, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2525, 272, 45,
2463, 2441, 2352, 2442} \begin {gather*} \frac {f^2 p \text {PolyLog}\left (2,\frac {e x^n}{d}+1\right )}{n}+\frac {f^2 \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f g x^{2 n} \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {g^2 x^{4 n} \log \left (c \left (d+e x^n\right )^p\right )}{4 n}-\frac {d^4 g^2 p \log \left (d+e x^n\right )}{4 e^4 n}+\frac {d^3 g^2 p x^n}{4 e^3 n}-\frac {d^2 f g p \log \left (d+e x^n\right )}{e^2 n}-\frac {d^2 g^2 p x^{2 n}}{8 e^2 n}+\frac {d f g p x^n}{e n}+\frac {d g^2 p x^{3 n}}{12 e n}-\frac {f g p x^{2 n}}{2 n}-\frac {g^2 p x^{4 n}}{16 n} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 272
Rule 2352
Rule 2441
Rule 2442
Rule 2463
Rule 2525
Rubi steps
\begin {align*} \int \frac {\left (f+g x^{2 n}\right )^2 \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 )^2 \log \left (c (d+e x)^p\right )}{x} \, dx,x,x^n\right )}{n}\\ &=\frac {\text {Subst}\left (\int \left (\frac {f^2 \log \left (c (d+e x)^p\right )}{x}+2 f g x \log \left (c (d+e x)^p\right )+g^2 x^3 \log \left (c (d+e x)^p\right )\right ) \, dx,x,x^n\right )}{n}\\ &=\frac {f^2 \text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{x} \, dx,x,x^n\right )}{n}+\frac {(2 f g) \text {Subst}\left (\int x \log \left (c (d+e x)^p\right ) \, dx,x,x^n\right )}{n}+\frac {g^2 \text {Subst}\left (\int x^3 \log \left (c (d+e x)^p\right ) \, dx,x,x^n\right )}{n}\\ &=\frac {f g x^{2 n} \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {g^2 x^{4 n} \log \left (c \left (d+e x^n\right )^p\right )}{4 n}+\frac {f^2 \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}-\frac {\left (e f^2 p\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx,x,x^n\right )}{n}-\frac {(e f g p) \text {Subst}\left (\int \frac {x^2}{d+e x} \, dx,x,x^n\right )}{n}-\frac {\left (e g^2 p\right ) \text {Subst}\left (\int \frac {x^4}{d+e x} \, dx,x,x^n\right )}{4 n}\\ &=\frac {f g x^{2 n} \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {g^2 x^{4 n} \log \left (c \left (d+e x^n\right )^p\right )}{4 n}+\frac {f^2 \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f^2 p \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{n}-\frac {(e f 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 )}{n}-\frac {\left (e g^2 p\right ) \text {Subst}\left (\int \left (-\frac {d^3}{e^4}+\frac {d^2 x}{e^3}-\frac {d x^2}{e^2}+\frac {x^3}{e}+\frac {d^4}{e^4 (d+e x)}\right ) \, dx,x,x^n\right )}{4 n}\\ &=\frac {d f g p x^n}{e n}+\frac {d^3 g^2 p x^n}{4 e^3 n}-\frac {f g p x^{2 n}}{2 n}-\frac {d^2 g^2 p x^{2 n}}{8 e^2 n}+\frac {d g^2 p x^{3 n}}{12 e n}-\frac {g^2 p x^{4 n}}{16 n}-\frac {d^2 f g p \log \left (d+e x^n\right )}{e^2 n}-\frac {d^4 g^2 p \log \left (d+e x^n\right )}{4 e^4 n}+\frac {f g x^{2 n} \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {g^2 x^{4 n} \log \left (c \left (d+e x^n\right )^p\right )}{4 n}+\frac {f^2 \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f^2 p \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.15, size = 291, normalized size = 1.15 \begin {gather*} \frac {48 d e^3 f g p x^n+12 d^3 e g^2 p x^n-24 e^4 f g p x^{2 n}-6 d^2 e^2 g^2 p x^{2 n}+4 d e^3 g^2 p x^{3 n}-3 e^4 g^2 p x^{4 n}-48 d^2 e^2 f g p \log \left (d-d x^n\right )-12 d^4 g^2 p \log \left (d-d x^n\right )+48 e^4 f^2 p \log \left (-\frac {e x^n}{d}\right ) \log \left (d+e x^n\right )+48 e^4 f g x^{2 n} \log \left (c \left (d+e x^n\right )^p\right )+12 e^4 g^2 x^{4 n} \log \left (c \left (d+e x^n\right )^p\right )-12 n \log (x) \left (d^2 g \left (4 e^2 f+d^2 g\right ) p+4 e^4 f^2 p \log \left (d+e x^n\right )-4 e^4 f^2 \log \left (c \left (d+e x^n\right )^p\right )\right )+48 e^4 f^2 p \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{48 e^4 n} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.77, size = 734, normalized size = 2.89
method | result | size |
risch | \(-\frac {g^{2} p \,x^{4 n}}{16 n}-\frac {d^{2} f g p \ln \left (d +e \,x^{n}\right )}{e^{2} n}-\frac {d^{4} g^{2} p \ln \left (d +e \,x^{n}\right )}{4 e^{4} n}+\frac {d^{3} g^{2} p \,x^{n}}{4 e^{3} n}-\frac {d^{2} g^{2} p \,x^{2 n}}{8 e^{2} n}+\frac {d \,g^{2} p \,x^{3 n}}{12 e n}+\frac {d f g p \,x^{n}}{e n}-\frac {f g p \,x^{2 n}}{2 n}-\frac {p \,f^{2} \dilog \left (\frac {d +e \,x^{n}}{d}\right )}{n}-p \,f^{2} \ln \left (x \right ) \ln \left (\frac {d +e \,x^{n}}{d}\right )+\frac {\ln \left (c \right ) f^{2} \ln \left (x^{n}\right )}{n}+\frac {i \pi \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right ) f^{2} \ln \left (x^{n}\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 )^{2} f^{2} \ln \left (x^{n}\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 ) f^{2} \ln \left (x^{n}\right )}{2 n}-\frac {i \pi \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{3} f^{2} \ln \left (x^{n}\right )}{2 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^{2} x^{4 n}}{8 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} g^{2} x^{4 n}}{8 n}-\frac {i \pi \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{3} f g \,x^{2 n}}{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 )^{2} f g \,x^{2 n}}{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^{2} x^{4 n}}{8 n}+\frac {i \pi \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right ) f g \,x^{2 n}}{2 n}-\frac {i \pi \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{3} g^{2} x^{4 n}}{8 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 g \,x^{2 n}}{2 n}+\frac {\ln \left (c \right ) f g \,x^{2 n}}{n}+\frac {\left (g^{2} x^{4 n}+4 f^{2} \ln \left (x \right ) n +4 f g \,x^{2 n}\right ) \ln \left (\left (d +e \,x^{n}\right )^{p}\right )}{4 n}+\frac {\ln \left (c \right ) g^{2} x^{4 n}}{4 n}\) | \(734\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.38, size = 229, normalized size = 0.90 \begin {gather*} -\frac {{\left (48 \, f^{2} n p e^{4} \log \left (x\right ) \log \left (\frac {x^{n} e + d}{d}\right ) - 4 \, d g^{2} p x^{3 \, n} e^{3} - 48 \, f^{2} n e^{4} \log \left (c\right ) \log \left (x\right ) + 48 \, f^{2} p {\rm Li}_2\left (-\frac {x^{n} e + d}{d} + 1\right ) e^{4} + 3 \, {\left (g^{2} p e^{4} - 4 \, g^{2} e^{4} \log \left (c\right )\right )} x^{4 \, n} + 6 \, {\left (d^{2} g^{2} p e^{2} + 4 \, f g p e^{4} - 8 \, f g e^{4} \log \left (c\right )\right )} x^{2 \, n} - 12 \, {\left (d^{3} g^{2} p e + 4 \, d f g p e^{3}\right )} x^{n} + 12 \, {\left (d^{4} g^{2} p + 4 \, d^{2} f g p e^{2} - 4 \, f^{2} n p e^{4} \log \left (x\right ) - g^{2} p x^{4 \, n} e^{4} - 4 \, f g p x^{2 \, n} e^{4}\right )} \log \left (x^{n} e + d\right )\right )} e^{\left (-4\right )}}{48 \, n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (f + g x^{2 n}\right )^{2} \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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )\,{\left (f+g\,x^{2\,n}\right )}^2}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________