Optimal. Leaf size=129 \[ -\frac {b n x^2}{4 e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac {d x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^2 \left (d+e x^2\right )}-\frac {b d n \log \left (d+e x^2\right )}{4 e^3}-\frac {d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x^2}{d}\right )}{e^3}-\frac {b d n \text {Li}_2\left (-\frac {e x^2}{d}\right )}{2 e^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.15, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {272, 45, 2393,
2341, 2373, 266, 2375, 2438} \begin {gather*} -\frac {b d n \text {PolyLog}\left (2,-\frac {e x^2}{d}\right )}{2 e^3}-\frac {d \log \left (\frac {e x^2}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac {d x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^2 \left (d+e x^2\right )}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac {b d n \log \left (d+e x^2\right )}{4 e^3}-\frac {b n x^2}{4 e^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 266
Rule 272
Rule 2341
Rule 2373
Rule 2375
Rule 2393
Rule 2438
Rubi steps
\begin {align*} \int \frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2} \, dx &=\int \left (\frac {x \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {d^2 x \left (a+b \log \left (c x^n\right )\right )}{e^2 \left (d+e x^2\right )^2}-\frac {2 d x \left (a+b \log \left (c x^n\right )\right )}{e^2 \left (d+e x^2\right )}\right ) \, dx\\ &=\frac {\int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{e^2}-\frac {(2 d) \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx}{e^2}+\frac {d^2 \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2} \, dx}{e^2}\\ &=-\frac {b n x^2}{4 e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac {d x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^2 \left (d+e x^2\right )}-\frac {d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x^2}{d}\right )}{e^3}+\frac {(b d n) \int \frac {\log \left (1+\frac {e x^2}{d}\right )}{x} \, dx}{e^3}-\frac {(b d n) \int \frac {x}{d+e x^2} \, dx}{2 e^2}\\ &=-\frac {b n x^2}{4 e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac {d x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^2 \left (d+e x^2\right )}-\frac {b d n \log \left (d+e x^2\right )}{4 e^3}-\frac {d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x^2}{d}\right )}{e^3}-\frac {b d n \text {Li}_2\left (-\frac {e x^2}{d}\right )}{2 e^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 0.33, size = 287, normalized size = 2.22 \begin {gather*} \frac {2 e x^2 \left (a-b n \log (x)+b \log \left (c x^n\right )\right )-\frac {2 d^2 \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{d+e x^2}-4 d \left (a-b n \log (x)+b \log \left (c x^n\right )\right ) \log \left (d+e x^2\right )+b n \left (\frac {d \sqrt {e} x \log (x)}{-i \sqrt {d}+\sqrt {e} x}+\frac {d \sqrt {e} x \log (x)}{i \sqrt {d}+\sqrt {e} x}+e x^2 (-1+2 \log (x))-d \log \left (i \sqrt {d}-\sqrt {e} x\right )-d \log \left (i \sqrt {d}+\sqrt {e} x\right )-4 d \left (\log (x) \log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )+\text {Li}_2\left (-\frac {i \sqrt {e} x}{\sqrt {d}}\right )\right )-4 d \left (\log (x) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )+\text {Li}_2\left (\frac {i \sqrt {e} x}{\sqrt {d}}\right )\right )\right )}{4 e^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.13, size = 687, normalized size = 5.33
method | result | size |
risch | \(\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} d \ln \left (e \,x^{2}+d \right )}{2 e^{3}}-\frac {b n d \dilog \left (\frac {-e x +\sqrt {-e d}}{\sqrt {-e d}}\right )}{e^{3}}+\frac {b \ln \left (x^{n}\right ) x^{2}}{2 e^{2}}+\frac {a \,x^{2}}{2 e^{2}}+\frac {b \ln \left (c \right ) x^{2}}{2 e^{2}}-\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} d \ln \left (e \,x^{2}+d \right )}{2 e^{3}}+\frac {b n d \ln \left (x \right )}{2 e^{3}}-\frac {b n d \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-e d}}{\sqrt {-e d}}\right )}{e^{3}}+\frac {b n d \ln \left (x \right ) \ln \left (e \,x^{2}+d \right )}{e^{3}}-\frac {b n d \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-e d}}{\sqrt {-e d}}\right )}{e^{3}}-\frac {a \,d^{2}}{2 e^{3} \left (e \,x^{2}+d \right )}-\frac {a d \ln \left (e \,x^{2}+d \right )}{e^{3}}-\frac {b \ln \left (c \right ) d^{2}}{2 e^{3} \left (e \,x^{2}+d \right )}-\frac {b n d \dilog \left (\frac {e x +\sqrt {-e d}}{\sqrt {-e d}}\right )}{e^{3}}-\frac {b \ln \left (x^{n}\right ) d^{2}}{2 e^{3} \left (e \,x^{2}+d \right )}-\frac {b \ln \left (x^{n}\right ) d \ln \left (e \,x^{2}+d \right )}{e^{3}}-\frac {b \ln \left (c \right ) d \ln \left (e \,x^{2}+d \right )}{e^{3}}-\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} x^{2}}{4 e^{2}}-\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) x^{2}}{4 e^{2}}-\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} d^{2}}{4 e^{3} \left (e \,x^{2}+d \right )}-\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} d \ln \left (e \,x^{2}+d \right )}{2 e^{3}}-\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} d^{2}}{4 e^{3} \left (e \,x^{2}+d \right )}+\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{2}}{4 e^{2}}+\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{2}}{4 e^{2}}+\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} d^{2}}{4 e^{3} \left (e \,x^{2}+d \right )}+\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) d \ln \left (e \,x^{2}+d \right )}{2 e^{3}}-\frac {b n \,x^{2}}{4 e^{2}}+\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) d^{2}}{4 e^{3} \left (e \,x^{2}+d \right )}-\frac {b d n \ln \left (e \,x^{2}+d \right )}{4 e^{3}}\) | \(687\) |
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 [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]
________________________________________________________________________________________
Sympy [A]
time = 53.80, size = 316, normalized size = 2.45 \begin {gather*} \frac {a d^{2} \left (\begin {cases} \frac {x^{2}}{d^{2}} & \text {for}\: e = 0 \\- \frac {1}{d e + e^{2} x^{2}} & \text {otherwise} \end {cases}\right )}{2 e^{2}} - \frac {a d \left (\begin {cases} \frac {x^{2}}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x^{2} \right )}}{e} & \text {otherwise} \end {cases}\right )}{e^{2}} + \frac {a x^{2}}{2 e^{2}} - \frac {b d^{2} n \left (\begin {cases} \frac {x^{2}}{2 d^{2}} & \text {for}\: e = 0 \\- \frac {\log {\left (x \right )}}{d e} + \frac {\log {\left (\frac {d}{e} + x^{2} \right )}}{2 d e} & \text {otherwise} \end {cases}\right )}{2 e^{2}} + \frac {b d^{2} \left (\begin {cases} \frac {x^{2}}{d^{2}} & \text {for}\: e = 0 \\- \frac {1}{d e + e^{2} x^{2}} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{2 e^{2}} + \frac {b d n \left (\begin {cases} \frac {x^{2}}{2 d} & \text {for}\: e = 0 \\\frac {\begin {cases} - \frac {\operatorname {Li}_{2}\left (\frac {e x^{2} e^{i \pi }}{d}\right )}{2} & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\log {\left (d \right )} \log {\left (x \right )} - \frac {\operatorname {Li}_{2}\left (\frac {e x^{2} e^{i \pi }}{d}\right )}{2} & \text {for}\: \left |{x}\right | < 1 \\- \log {\left (d \right )} \log {\left (\frac {1}{x} \right )} - \frac {\operatorname {Li}_{2}\left (\frac {e x^{2} e^{i \pi }}{d}\right )}{2} & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\left (d \right )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\left (d \right )} - \frac {\operatorname {Li}_{2}\left (\frac {e x^{2} e^{i \pi }}{d}\right )}{2} & \text {otherwise} \end {cases}}{e} & \text {otherwise} \end {cases}\right )}{e^{2}} - \frac {b d \left (\begin {cases} \frac {x^{2}}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x^{2} \right )}}{e} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{e^{2}} - \frac {b n x^{2}}{4 e^{2}} + \frac {b x^{2} \log {\left (c x^{n} \right )}}{2 e^{2}} \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 = -0.01 \begin {gather*} \int \frac {x^5\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (e\,x^2+d\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________