Optimal. Leaf size=260 \[ -\frac{35 i b e^{3/2} n \text{PolyLog}\left (2,-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{16 d^{9/2}}+\frac{35 i b e^{3/2} n \text{PolyLog}\left (2,\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{16 d^{9/2}}+\frac{e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (35 a+35 b \log \left (c x^n\right )-12 b n\right )}{8 d^{9/2}}+\frac{e \left (35 a+35 b \log \left (c x^n\right )-12 b n\right )}{8 d^4 x}+\frac{7 a+7 b \log \left (c x^n\right )-b n}{8 d^2 x^3 \left (d+e x^2\right )}-\frac{35 a+35 b \log \left (c x^n\right )-12 b n}{24 d^3 x^3}+\frac{a+b \log \left (c x^n\right )}{4 d x^3 \left (d+e x^2\right )^2}+\frac{35 b e n}{8 d^4 x}-\frac{35 b n}{72 d^3 x^3} \]
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Rubi [A] time = 0.420994, antiderivative size = 260, normalized size of antiderivative = 1., 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 2340
Rule 325
Rule 205
Rule 2351
Rule 2304
Rule 2324
Rule 12
Rule 4848
Rule 2391
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*}
Mathematica [B] time = 1.84227, size = 584, normalized size = 2.25 \[ \frac{1}{144} \left (\frac{315 b e^{3/2} n \text{PolyLog}\left (2,\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{(-d)^{9/2}}-\frac{315 b e^{3/2} n \text{PolyLog}\left (2,\frac{d \sqrt{e} x}{(-d)^{3/2}}\right )}{(-d)^{9/2}}-\frac{99 e^{3/2} \left (a+b \log \left (c x^n\right )\right )}{d^4 \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{99 e^{3/2} \left (a+b \log \left (c x^n\right )\right )}{d^4 \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{432 e \left (a+b \log \left (c x^n\right )\right )}{d^4 x}-\frac{48 \left (a+b \log \left (c x^n\right )\right )}{d^3 x^3}-\frac{9 e^{3/2} \left (a+b \log \left (c x^n\right )\right )}{(-d)^{7/2} \left (\sqrt{-d}-\sqrt{e} x\right )^2}+\frac{9 e^{3/2} \left (a+b \log \left (c x^n\right )\right )}{(-d)^{7/2} \left (\sqrt{-d}+\sqrt{e} x\right )^2}-\frac{315 e^{3/2} \log \left (\frac{\sqrt{e} x}{\sqrt{-d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{(-d)^{9/2}}+\frac{315 e^{3/2} \log \left (\frac{d \sqrt{e} x}{(-d)^{3/2}}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{(-d)^{9/2}}+\frac{432 b e n}{d^4 x}-\frac{16 b n}{d^3 x^3}+\frac{99 b e^{3/2} n \left (\log (x)-\log \left (\sqrt{-d}-\sqrt{e} x\right )\right )}{(-d)^{9/2}}-\frac{99 b e^{3/2} n \left (\log (x)-\log \left (\sqrt{-d}+\sqrt{e} x\right )\right )}{(-d)^{9/2}}-\frac{9 b e^{3/2} n \left (\frac{1}{\sqrt{-d} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{\log \left (\sqrt{-d}+\sqrt{e} x\right )}{d}-\frac{\log (x)}{d}\right )}{(-d)^{7/2}}+\frac{9 b e^{3/2} n \left (\frac{1}{\sqrt{-d} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{\log \left (d \sqrt{e} x+(-d)^{3/2}\right )}{d}-\frac{\log (x)}{d}\right )}{(-d)^{7/2}}\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.316, size = 1729, normalized size = 6.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \log \left (c x^{n}\right ) + a}{{\left (e x^{2} + d\right )}^{3} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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