Optimal. Leaf size=191 \[ -\frac {3 \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^{5/2}}+\frac {d x \left (a+b \log \left (c x^n\right )\right )}{2 e^2 \left (d+e x^2\right )}+\frac {a x}{e^2}+\frac {b x \log \left (c x^n\right )}{e^2}+\frac {3 i b \sqrt {d} n \text {Li}_2\left (-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{4 e^{5/2}}-\frac {3 i b \sqrt {d} n \text {Li}_2\left (\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{4 e^{5/2}}-\frac {b \sqrt {d} n \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 e^{5/2}}-\frac {b n x}{e^2} \]
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Rubi [A] time = 0.30, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 10, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {288, 321, 205, 2351, 2295, 2323, 2324, 12, 4848, 2391} \[ \frac {3 i b \sqrt {d} n \text {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{4 e^{5/2}}-\frac {3 i b \sqrt {d} n \text {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{4 e^{5/2}}+\frac {d x \left (a+b \log \left (c x^n\right )\right )}{2 e^2 \left (d+e x^2\right )}-\frac {3 \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^{5/2}}+\frac {a x}{e^2}+\frac {b x \log \left (c x^n\right )}{e^2}-\frac {b \sqrt {d} n \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 e^{5/2}}-\frac {b n x}{e^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 205
Rule 288
Rule 321
Rule 2295
Rule 2323
Rule 2324
Rule 2351
Rule 2391
Rule 4848
Rubi steps
\begin {align*} \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2} \, dx &=\int \left (\frac {a+b \log \left (c x^n\right )}{e^2}+\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{e^2 \left (d+e x^2\right )^2}-\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{e^2 \left (d+e x^2\right )}\right ) \, dx\\ &=\frac {\int \left (a+b \log \left (c x^n\right )\right ) \, dx}{e^2}-\frac {(2 d) \int \frac {a+b \log \left (c x^n\right )}{d+e x^2} \, dx}{e^2}+\frac {d^2 \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^2} \, dx}{e^2}\\ &=\frac {a x}{e^2}+\frac {d x \left (a+b \log \left (c x^n\right )\right )}{2 e^2 \left (d+e x^2\right )}-\frac {2 \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{5/2}}+\frac {b \int \log \left (c x^n\right ) \, dx}{e^2}+\frac {d \int \frac {a+b \log \left (c x^n\right )}{d+e x^2} \, dx}{2 e^2}-\frac {(b d n) \int \frac {1}{d+e x^2} \, dx}{2 e^2}+\frac {(2 b d n) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e} x} \, dx}{e^2}\\ &=\frac {a x}{e^2}-\frac {b n x}{e^2}-\frac {b \sqrt {d} n \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 e^{5/2}}+\frac {b x \log \left (c x^n\right )}{e^2}+\frac {d x \left (a+b \log \left (c x^n\right )\right )}{2 e^2 \left (d+e x^2\right )}-\frac {3 \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^{5/2}}+\frac {\left (2 b \sqrt {d} n\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{e^{5/2}}-\frac {(b d n) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e} x} \, dx}{2 e^2}\\ &=\frac {a x}{e^2}-\frac {b n x}{e^2}-\frac {b \sqrt {d} n \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 e^{5/2}}+\frac {b x \log \left (c x^n\right )}{e^2}+\frac {d x \left (a+b \log \left (c x^n\right )\right )}{2 e^2 \left (d+e x^2\right )}-\frac {3 \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^{5/2}}+\frac {\left (i b \sqrt {d} n\right ) \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{e^{5/2}}-\frac {\left (i b \sqrt {d} n\right ) \int \frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{e^{5/2}}-\frac {\left (b \sqrt {d} n\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{2 e^{5/2}}\\ &=\frac {a x}{e^2}-\frac {b n x}{e^2}-\frac {b \sqrt {d} n \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 e^{5/2}}+\frac {b x \log \left (c x^n\right )}{e^2}+\frac {d x \left (a+b \log \left (c x^n\right )\right )}{2 e^2 \left (d+e x^2\right )}-\frac {3 \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^{5/2}}+\frac {i b \sqrt {d} n \text {Li}_2\left (-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{e^{5/2}}-\frac {i b \sqrt {d} n \text {Li}_2\left (\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{e^{5/2}}-\frac {\left (i b \sqrt {d} n\right ) \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{4 e^{5/2}}+\frac {\left (i b \sqrt {d} n\right ) \int \frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{4 e^{5/2}}\\ &=\frac {a x}{e^2}-\frac {b n x}{e^2}-\frac {b \sqrt {d} n \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 e^{5/2}}+\frac {b x \log \left (c x^n\right )}{e^2}+\frac {d x \left (a+b \log \left (c x^n\right )\right )}{2 e^2 \left (d+e x^2\right )}-\frac {3 \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^{5/2}}+\frac {3 i b \sqrt {d} n \text {Li}_2\left (-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{4 e^{5/2}}-\frac {3 i b \sqrt {d} n \text {Li}_2\left (\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{4 e^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.56, size = 296, normalized size = 1.55 \[ \frac {-\frac {d \left (a+b \log \left (c x^n\right )\right )}{\sqrt {-d}-\sqrt {e} x}+\frac {d \left (a+b \log \left (c x^n\right )\right )}{\sqrt {-d}+\sqrt {e} x}-3 \sqrt {-d} \log \left (\frac {\sqrt {e} x}{\sqrt {-d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )+3 \sqrt {-d} \log \left (\frac {d \sqrt {e} x}{(-d)^{3/2}}+1\right ) \left (a+b \log \left (c x^n\right )\right )+4 a \sqrt {e} x+4 b \sqrt {e} x \log \left (c x^n\right )+3 b \sqrt {-d} n \text {Li}_2\left (\frac {\sqrt {e} x}{\sqrt {-d}}\right )-3 b \sqrt {-d} n \text {Li}_2\left (\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )+\frac {b d n \left (\log (x)-\log \left (\sqrt {-d}-\sqrt {e} x\right )\right )}{\sqrt {-d}}+b \sqrt {-d} n \left (\log (x)-\log \left (\sqrt {-d}+\sqrt {e} x\right )\right )-4 b \sqrt {e} n x}{4 e^{5/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x^{4} \log \left (c x^{n}\right ) + a x^{4}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{4}}{{\left (e x^{2} + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.30, size = 913, normalized size = 4.78 \[ \text {result too large to display} \]
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 \[ \frac {1}{2} \, a {\left (\frac {d x}{e^{3} x^{2} + d e^{2}} - \frac {3 \, d \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e} e^{2}} + \frac {2 \, x}{e^{2}}\right )} + b \int \frac {x^{4} \log \relax (c) + x^{4} \log \left (x^{n}\right )}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}\,{d x} \]
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 \[ \int \frac {x^4\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (e\,x^2+d\right )}^2} \,d x \]
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 \[ \int \frac {x^{4} \left (a + b \log {\left (c x^{n} \right )}\right )}{\left (d + e x^{2}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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