Optimal. Leaf size=356 \[ -\frac {7 b^2 f m n^2}{32 e x^2}-\frac {b^2 f^2 m n^2 \log (x)}{16 e^2}-\frac {3 b f m n \left (a+b \log \left (c x^n\right )\right )}{8 e x^2}+\frac {b f^2 m n \log \left (1+\frac {e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 e^2}-\frac {f m \left (a+b \log \left (c x^n\right )\right )^2}{4 e x^2}+\frac {f^2 m \log \left (1+\frac {e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 e^2}+\frac {b^2 f^2 m n^2 \log \left (e+f x^2\right )}{32 e^2}-\frac {b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{32 x^4}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{8 x^4}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^4}-\frac {b^2 f^2 m n^2 \text {Li}_2\left (-\frac {e}{f x^2}\right )}{16 e^2}-\frac {b f^2 m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e}{f x^2}\right )}{4 e^2}-\frac {b^2 f^2 m n^2 \text {Li}_3\left (-\frac {e}{f x^2}\right )}{8 e^2} \]
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Rubi [A]
time = 0.38, antiderivative size = 356, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 10, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {2342, 2341,
2425, 272, 46, 2380, 2379, 2438, 2421, 6724} \begin {gather*} -\frac {b f^2 m n \text {PolyLog}\left (2,-\frac {e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 e^2}-\frac {b^2 f^2 m n^2 \text {PolyLog}\left (2,-\frac {e}{f x^2}\right )}{16 e^2}-\frac {b^2 f^2 m n^2 \text {PolyLog}\left (3,-\frac {e}{f x^2}\right )}{8 e^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^4}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{8 x^4}+\frac {f^2 m \log \left (\frac {e}{f x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 e^2}+\frac {b f^2 m n \log \left (\frac {e}{f x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{8 e^2}-\frac {f m \left (a+b \log \left (c x^n\right )\right )^2}{4 e x^2}-\frac {3 b f m n \left (a+b \log \left (c x^n\right )\right )}{8 e x^2}-\frac {b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{32 x^4}+\frac {b^2 f^2 m n^2 \log \left (e+f x^2\right )}{32 e^2}-\frac {b^2 f^2 m n^2 \log (x)}{16 e^2}-\frac {7 b^2 f m n^2}{32 e x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 272
Rule 2341
Rule 2342
Rule 2379
Rule 2380
Rule 2421
Rule 2425
Rule 2438
Rule 6724
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x^5} \, dx &=-\frac {b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{32 x^4}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{8 x^4}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^4}-(2 f m) \int \left (-\frac {b^2 n^2}{32 x^3 \left (e+f x^2\right )}-\frac {b n \left (a+b \log \left (c x^n\right )\right )}{8 x^3 \left (e+f x^2\right )}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{4 x^3 \left (e+f x^2\right )}\right ) \, dx\\ &=-\frac {b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{32 x^4}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{8 x^4}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^4}+\frac {1}{2} (f m) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3 \left (e+f x^2\right )} \, dx+\frac {1}{4} (b f m n) \int \frac {a+b \log \left (c x^n\right )}{x^3 \left (e+f x^2\right )} \, dx+\frac {1}{16} \left (b^2 f m n^2\right ) \int \frac {1}{x^3 \left (e+f x^2\right )} \, dx\\ &=-\frac {b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{32 x^4}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{8 x^4}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^4}+\frac {1}{2} (f m) \int \left (\frac {\left (a+b \log \left (c x^n\right )\right )^2}{e x^3}-\frac {f \left (a+b \log \left (c x^n\right )\right )^2}{e^2 x}+\frac {f^2 x \left (a+b \log \left (c x^n\right )\right )^2}{e^2 \left (e+f x^2\right )}\right ) \, dx+\frac {1}{4} (b f m n) \int \left (\frac {a+b \log \left (c x^n\right )}{e x^3}-\frac {f \left (a+b \log \left (c x^n\right )\right )}{e^2 x}+\frac {f^2 x \left (a+b \log \left (c x^n\right )\right )}{e^2 \left (e+f x^2\right )}\right ) \, dx+\frac {1}{32} \left (b^2 f m n^2\right ) \text {Subst}\left (\int \frac {1}{x^2 (e+f x)} \, dx,x,x^2\right )\\ &=-\frac {b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{32 x^4}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{8 x^4}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^4}+\frac {(f m) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx}{2 e}-\frac {\left (f^2 m\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx}{2 e^2}+\frac {\left (f^3 m\right ) \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{e+f x^2} \, dx}{2 e^2}+\frac {(b f m n) \int \frac {a+b \log \left (c x^n\right )}{x^3} \, dx}{4 e}-\frac {\left (b f^2 m n\right ) \int \frac {a+b \log \left (c x^n\right )}{x} \, dx}{4 e^2}+\frac {\left (b f^3 m n\right ) \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{e+f x^2} \, dx}{4 e^2}+\frac {1}{32} \left (b^2 f m n^2\right ) \text {Subst}\left (\int \left (\frac {1}{e x^2}-\frac {f}{e^2 x}+\frac {f^2}{e^2 (e+f x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {3 b^2 f m n^2}{32 e x^2}-\frac {b^2 f^2 m n^2 \log (x)}{16 e^2}-\frac {b f m n \left (a+b \log \left (c x^n\right )\right )}{8 e x^2}-\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}-\frac {f m \left (a+b \log \left (c x^n\right )\right )^2}{4 e x^2}+\frac {b^2 f^2 m n^2 \log \left (e+f x^2\right )}{32 e^2}-\frac {b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{32 x^4}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{8 x^4}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^4}+\frac {b f^2 m n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x^2}{e}\right )}{8 e^2}+\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x^2}{e}\right )}{4 e^2}-\frac {\left (f^2 m\right ) \text {Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )}{2 b e^2 n}+\frac {(b f m n) \int \frac {a+b \log \left (c x^n\right )}{x^3} \, dx}{2 e}-\frac {\left (b f^2 m n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x^2}{e}\right )}{x} \, dx}{2 e^2}-\frac {\left (b^2 f^2 m n^2\right ) \int \frac {\log \left (1+\frac {f x^2}{e}\right )}{x} \, dx}{8 e^2}\\ &=-\frac {7 b^2 f m n^2}{32 e x^2}-\frac {b^2 f^2 m n^2 \log (x)}{16 e^2}-\frac {3 b f m n \left (a+b \log \left (c x^n\right )\right )}{8 e x^2}-\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}-\frac {f m \left (a+b \log \left (c x^n\right )\right )^2}{4 e x^2}-\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )^3}{6 b e^2 n}+\frac {b^2 f^2 m n^2 \log \left (e+f x^2\right )}{32 e^2}-\frac {b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{32 x^4}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{8 x^4}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^4}+\frac {b f^2 m n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x^2}{e}\right )}{8 e^2}+\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x^2}{e}\right )}{4 e^2}+\frac {b^2 f^2 m n^2 \text {Li}_2\left (-\frac {f x^2}{e}\right )}{16 e^2}+\frac {b f^2 m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x^2}{e}\right )}{4 e^2}-\frac {\left (b^2 f^2 m n^2\right ) \int \frac {\text {Li}_2\left (-\frac {f x^2}{e}\right )}{x} \, dx}{4 e^2}\\ &=-\frac {7 b^2 f m n^2}{32 e x^2}-\frac {b^2 f^2 m n^2 \log (x)}{16 e^2}-\frac {3 b f m n \left (a+b \log \left (c x^n\right )\right )}{8 e x^2}-\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}-\frac {f m \left (a+b \log \left (c x^n\right )\right )^2}{4 e x^2}-\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )^3}{6 b e^2 n}+\frac {b^2 f^2 m n^2 \log \left (e+f x^2\right )}{32 e^2}-\frac {b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{32 x^4}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{8 x^4}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^4}+\frac {b f^2 m n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x^2}{e}\right )}{8 e^2}+\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x^2}{e}\right )}{4 e^2}+\frac {b^2 f^2 m n^2 \text {Li}_2\left (-\frac {f x^2}{e}\right )}{16 e^2}+\frac {b f^2 m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x^2}{e}\right )}{4 e^2}-\frac {b^2 f^2 m n^2 \text {Li}_3\left (-\frac {f x^2}{e}\right )}{8 e^2}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.28, size = 1111, normalized size = 3.12 \begin {gather*} -\frac {24 a^2 e f m x^2+36 a b e f m n x^2+21 b^2 e f m n^2 x^2+48 a^2 f^2 m x^4 \log (x)+24 a b f^2 m n x^4 \log (x)+6 b^2 f^2 m n^2 x^4 \log (x)-48 a b f^2 m n x^4 \log ^2(x)-12 b^2 f^2 m n^2 x^4 \log ^2(x)+16 b^2 f^2 m n^2 x^4 \log ^3(x)+48 a b e f m x^2 \log \left (c x^n\right )+36 b^2 e f m n x^2 \log \left (c x^n\right )+96 a b f^2 m x^4 \log (x) \log \left (c x^n\right )+24 b^2 f^2 m n x^4 \log (x) \log \left (c x^n\right )-48 b^2 f^2 m n x^4 \log ^2(x) \log \left (c x^n\right )+24 b^2 e f m x^2 \log ^2\left (c x^n\right )+48 b^2 f^2 m x^4 \log (x) \log ^2\left (c x^n\right )-48 a b f^2 m n x^4 \log (x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-12 b^2 f^2 m n^2 x^4 \log (x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+24 b^2 f^2 m n^2 x^4 \log ^2(x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-48 b^2 f^2 m n x^4 \log (x) \log \left (c x^n\right ) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-48 a b f^2 m n x^4 \log (x) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )-12 b^2 f^2 m n^2 x^4 \log (x) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )+24 b^2 f^2 m n^2 x^4 \log ^2(x) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )-48 b^2 f^2 m n x^4 \log (x) \log \left (c x^n\right ) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )-24 a^2 f^2 m x^4 \log \left (e+f x^2\right )-12 a b f^2 m n x^4 \log \left (e+f x^2\right )-3 b^2 f^2 m n^2 x^4 \log \left (e+f x^2\right )+48 a b f^2 m n x^4 \log (x) \log \left (e+f x^2\right )+12 b^2 f^2 m n^2 x^4 \log (x) \log \left (e+f x^2\right )-24 b^2 f^2 m n^2 x^4 \log ^2(x) \log \left (e+f x^2\right )-48 a b f^2 m x^4 \log \left (c x^n\right ) \log \left (e+f x^2\right )-12 b^2 f^2 m n x^4 \log \left (c x^n\right ) \log \left (e+f x^2\right )+48 b^2 f^2 m n x^4 \log (x) \log \left (c x^n\right ) \log \left (e+f x^2\right )-24 b^2 f^2 m x^4 \log ^2\left (c x^n\right ) \log \left (e+f x^2\right )+24 a^2 e^2 \log \left (d \left (e+f x^2\right )^m\right )+12 a b e^2 n \log \left (d \left (e+f x^2\right )^m\right )+3 b^2 e^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )+48 a b e^2 \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )+12 b^2 e^2 n \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )+24 b^2 e^2 \log ^2\left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )-12 b f^2 m n x^4 \left (4 a+b n+4 b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-12 b f^2 m n x^4 \left (4 a+b n+4 b \log \left (c x^n\right )\right ) \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )+48 b^2 f^2 m n^2 x^4 \text {Li}_3\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+48 b^2 f^2 m n^2 x^4 \text {Li}_3\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{96 e^2 x^4} \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 = 0.59, size = 13825, normalized size = 38.83
method | result | size |
risch | \(\text {Expression too large to display}\) | \(13825\) |
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 [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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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 = -0.00 \begin {gather*} \int \frac {\ln \left (d\,{\left (f\,x^2+e\right )}^m\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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