Optimal. Leaf size=260 \[ \frac {e^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (35 a+35 b \log \left (c x^n\right )-12 b n\right )}{8 d^{9/2}}+\frac {e \left (35 a+35 b \log \left (c x^n\right )-12 b n\right )}{8 d^4 x}-\frac {35 a+35 b \log \left (c x^n\right )-12 b n}{24 d^3 x^3}+\frac {7 a+7 b \log \left (c x^n\right )-b n}{8 d^2 x^3 \left (d+e x^2\right )}+\frac {a+b \log \left (c x^n\right )}{4 d x^3 \left (d+e x^2\right )^2}-\frac {35 i b e^{3/2} n \text {Li}_2\left (-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{9/2}}+\frac {35 i b e^{3/2} n \text {Li}_2\left (\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{9/2}}+\frac {35 b e n}{8 d^4 x}-\frac {35 b n}{72 d^3 x^3} \]
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Rubi [A] time = 0.42, antiderivative size = 260, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {2340, 325, 205, 2351, 2304, 2324, 12, 4848, 2391} \[ -\frac {35 i b e^{3/2} n \text {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{9/2}}+\frac {35 i b e^{3/2} n \text {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{9/2}}+\frac {e^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (35 a+35 b \log \left (c x^n\right )-12 b n\right )}{8 d^{9/2}}+\frac {e \left (35 a+35 b \log \left (c x^n\right )-12 b n\right )}{8 d^4 x}+\frac {7 a+7 b \log \left (c x^n\right )-b n}{8 d^2 x^3 \left (d+e x^2\right )}-\frac {35 a+35 b \log \left (c x^n\right )-12 b n}{24 d^3 x^3}+\frac {a+b \log \left (c x^n\right )}{4 d x^3 \left (d+e x^2\right )^2}+\frac {35 b e n}{8 d^4 x}-\frac {35 b n}{72 d^3 x^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 205
Rule 325
Rule 2304
Rule 2324
Rule 2340
Rule 2351
Rule 2391
Rule 4848
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^3} \, dx &=\frac {a+b \log \left (c x^n\right )}{4 d x^3 \left (d+e x^2\right )^2}-\frac {\int \frac {-7 a+b n-7 b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^2} \, dx}{4 d}\\ &=\frac {a+b \log \left (c x^n\right )}{4 d x^3 \left (d+e x^2\right )^2}+\frac {7 a-b n+7 b \log \left (c x^n\right )}{8 d^2 x^3 \left (d+e x^2\right )}+\frac {\int \frac {-7 b n-5 (-7 a+b n)+35 b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )} \, dx}{8 d^2}\\ &=\frac {a+b \log \left (c x^n\right )}{4 d x^3 \left (d+e x^2\right )^2}+\frac {7 a-b n+7 b \log \left (c x^n\right )}{8 d^2 x^3 \left (d+e x^2\right )}+\frac {\int \left (\frac {-7 b n-5 (-7 a+b n)+35 b \log \left (c x^n\right )}{d x^4}-\frac {e \left (-7 b n-5 (-7 a+b n)+35 b \log \left (c x^n\right )\right )}{d^2 x^2}+\frac {e^2 \left (-7 b n-5 (-7 a+b n)+35 b \log \left (c x^n\right )\right )}{d^2 \left (d+e x^2\right )}\right ) \, dx}{8 d^2}\\ &=\frac {a+b \log \left (c x^n\right )}{4 d x^3 \left (d+e x^2\right )^2}+\frac {7 a-b n+7 b \log \left (c x^n\right )}{8 d^2 x^3 \left (d+e x^2\right )}+\frac {\int \frac {-7 b n-5 (-7 a+b n)+35 b \log \left (c x^n\right )}{x^4} \, dx}{8 d^3}-\frac {e \int \frac {-7 b n-5 (-7 a+b n)+35 b \log \left (c x^n\right )}{x^2} \, dx}{8 d^4}+\frac {e^2 \int \frac {-7 b n-5 (-7 a+b n)+35 b \log \left (c x^n\right )}{d+e x^2} \, dx}{8 d^4}\\ &=-\frac {35 b n}{72 d^3 x^3}+\frac {35 b e n}{8 d^4 x}+\frac {a+b \log \left (c x^n\right )}{4 d x^3 \left (d+e x^2\right )^2}+\frac {7 a-b n+7 b \log \left (c x^n\right )}{8 d^2 x^3 \left (d+e x^2\right )}-\frac {35 a-12 b n+35 b \log \left (c x^n\right )}{24 d^3 x^3}+\frac {e \left (35 a-12 b n+35 b \log \left (c x^n\right )\right )}{8 d^4 x}+\frac {e^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (35 a-12 b n+35 b \log \left (c x^n\right )\right )}{8 d^{9/2}}-\frac {\left (35 b e^2 n\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e} x} \, dx}{8 d^4}\\ &=-\frac {35 b n}{72 d^3 x^3}+\frac {35 b e n}{8 d^4 x}+\frac {a+b \log \left (c x^n\right )}{4 d x^3 \left (d+e x^2\right )^2}+\frac {7 a-b n+7 b \log \left (c x^n\right )}{8 d^2 x^3 \left (d+e x^2\right )}-\frac {35 a-12 b n+35 b \log \left (c x^n\right )}{24 d^3 x^3}+\frac {e \left (35 a-12 b n+35 b \log \left (c x^n\right )\right )}{8 d^4 x}+\frac {e^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (35 a-12 b n+35 b \log \left (c x^n\right )\right )}{8 d^{9/2}}-\frac {\left (35 b e^{3/2} n\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{8 d^{9/2}}\\ &=-\frac {35 b n}{72 d^3 x^3}+\frac {35 b e n}{8 d^4 x}+\frac {a+b \log \left (c x^n\right )}{4 d x^3 \left (d+e x^2\right )^2}+\frac {7 a-b n+7 b \log \left (c x^n\right )}{8 d^2 x^3 \left (d+e x^2\right )}-\frac {35 a-12 b n+35 b \log \left (c x^n\right )}{24 d^3 x^3}+\frac {e \left (35 a-12 b n+35 b \log \left (c x^n\right )\right )}{8 d^4 x}+\frac {e^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (35 a-12 b n+35 b \log \left (c x^n\right )\right )}{8 d^{9/2}}-\frac {\left (35 i b e^{3/2} n\right ) \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{16 d^{9/2}}+\frac {\left (35 i b e^{3/2} n\right ) \int \frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{16 d^{9/2}}\\ &=-\frac {35 b n}{72 d^3 x^3}+\frac {35 b e n}{8 d^4 x}+\frac {a+b \log \left (c x^n\right )}{4 d x^3 \left (d+e x^2\right )^2}+\frac {7 a-b n+7 b \log \left (c x^n\right )}{8 d^2 x^3 \left (d+e x^2\right )}-\frac {35 a-12 b n+35 b \log \left (c x^n\right )}{24 d^3 x^3}+\frac {e \left (35 a-12 b n+35 b \log \left (c x^n\right )\right )}{8 d^4 x}+\frac {e^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (35 a-12 b n+35 b \log \left (c x^n\right )\right )}{8 d^{9/2}}-\frac {35 i b e^{3/2} n \text {Li}_2\left (-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{9/2}}+\frac {35 i b e^{3/2} n \text {Li}_2\left (\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{9/2}}\\ \end {align*}
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Mathematica [B] time = 1.67, size = 584, normalized size = 2.25 \[ \frac {1}{144} \left (-\frac {99 e^{3/2} \left (a+b \log \left (c x^n\right )\right )}{d^4 \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {99 e^{3/2} \left (a+b \log \left (c x^n\right )\right )}{d^4 \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {432 e \left (a+b \log \left (c x^n\right )\right )}{d^4 x}-\frac {48 \left (a+b \log \left (c x^n\right )\right )}{d^3 x^3}-\frac {9 e^{3/2} \left (a+b \log \left (c x^n\right )\right )}{(-d)^{7/2} \left (\sqrt {-d}-\sqrt {e} x\right )^2}+\frac {9 e^{3/2} \left (a+b \log \left (c x^n\right )\right )}{(-d)^{7/2} \left (\sqrt {-d}+\sqrt {e} x\right )^2}-\frac {315 e^{3/2} \log \left (\frac {\sqrt {e} x}{\sqrt {-d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{(-d)^{9/2}}+\frac {315 e^{3/2} \log \left (\frac {d \sqrt {e} x}{(-d)^{3/2}}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{(-d)^{9/2}}+\frac {432 b e n}{d^4 x}-\frac {16 b n}{d^3 x^3}+\frac {315 b e^{3/2} n \text {Li}_2\left (\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{(-d)^{9/2}}-\frac {315 b e^{3/2} n \text {Li}_2\left (\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )}{(-d)^{9/2}}+\frac {99 b e^{3/2} n \left (\log (x)-\log \left (\sqrt {-d}-\sqrt {e} x\right )\right )}{(-d)^{9/2}}-\frac {99 b e^{3/2} n \left (\log (x)-\log \left (\sqrt {-d}+\sqrt {e} x\right )\right )}{(-d)^{9/2}}-\frac {9 b e^{3/2} n \left (\frac {1}{\sqrt {-d} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {\log \left (\sqrt {-d}+\sqrt {e} x\right )}{d}-\frac {\log (x)}{d}\right )}{(-d)^{7/2}}+\frac {9 b e^{3/2} n \left (\frac {1}{\sqrt {-d} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\log \left (d \sqrt {e} x+(-d)^{3/2}\right )}{d}-\frac {\log (x)}{d}\right )}{(-d)^{7/2}}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \log \left (c x^{n}\right ) + a}{e^{3} x^{10} + 3 \, d e^{2} x^{8} + 3 \, d^{2} e x^{6} + d^{3} x^{4}}, x\right ) \]
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 \[ \int \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{2} + d\right )}^{3} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.40, size = 1729, normalized size = 6.65 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int \frac {a+b\,\ln \left (c\,x^n\right )}{x^4\,{\left (e\,x^2+d\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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