Optimal. Leaf size=501 \[ -\frac{3 i x^2 \text{PolyLog}\left (2,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b} \log ^2(f)}+\frac{3 i x^2 \text{PolyLog}\left (2,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b} \log ^2(f)}+\frac{3 i x \text{PolyLog}\left (2,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^3(f)}-\frac{3 i x \text{PolyLog}\left (2,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^3(f)}+\frac{3 i x \text{PolyLog}\left (3,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^3(f)}-\frac{3 i \text{PolyLog}\left (3,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^4(f)}-\frac{3 i x \text{PolyLog}\left (3,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^3(f)}+\frac{3 i \text{PolyLog}\left (3,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^4(f)}-\frac{3 i \text{PolyLog}\left (4,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^4(f)}+\frac{3 i \text{PolyLog}\left (4,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^4(f)}-\frac{3 x^2 \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^2(f)}+\frac{x^3 \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log (f)}+\frac{x^3 f^x}{2 a \log (f) \left (a+b f^{2 x}\right )} \]
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Rubi [A] time = 0.513821, antiderivative size = 501, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 11, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.611, Rules used = {2249, 199, 205, 2245, 14, 12, 5143, 2531, 2282, 6589, 6609} \[ -\frac{3 i x^2 \text{PolyLog}\left (2,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b} \log ^2(f)}+\frac{3 i x^2 \text{PolyLog}\left (2,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b} \log ^2(f)}+\frac{3 i x \text{PolyLog}\left (2,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^3(f)}-\frac{3 i x \text{PolyLog}\left (2,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^3(f)}+\frac{3 i x \text{PolyLog}\left (3,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^3(f)}-\frac{3 i \text{PolyLog}\left (3,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^4(f)}-\frac{3 i x \text{PolyLog}\left (3,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^3(f)}+\frac{3 i \text{PolyLog}\left (3,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^4(f)}-\frac{3 i \text{PolyLog}\left (4,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^4(f)}+\frac{3 i \text{PolyLog}\left (4,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^4(f)}-\frac{3 x^2 \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^2(f)}+\frac{x^3 \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log (f)}+\frac{x^3 f^x}{2 a \log (f) \left (a+b f^{2 x}\right )} \]
Antiderivative was successfully verified.
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Rule 2249
Rule 199
Rule 205
Rule 2245
Rule 14
Rule 12
Rule 5143
Rule 2531
Rule 2282
Rule 6589
Rule 6609
Rubi steps
\begin{align*} \int \frac{f^x x^3}{\left (a+b f^{2 x}\right )^2} \, dx &=\frac{f^x x^3}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac{x^3 \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log (f)}-3 \int x^2 \left (\frac{f^x}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log (f)}\right ) \, dx\\ &=\frac{f^x x^3}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac{x^3 \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log (f)}-3 \int \left (\frac{f^x x^2}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac{x^2 \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log (f)}\right ) \, dx\\ &=\frac{f^x x^3}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac{x^3 \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log (f)}-\frac{3 \int \frac{f^x x^2}{a+b f^{2 x}} \, dx}{2 a \log (f)}-\frac{3 \int x^2 \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right ) \, dx}{2 a^{3/2} \sqrt{b} \log (f)}\\ &=-\frac{3 x^2 \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^2(f)}+\frac{f^x x^3}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac{x^3 \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log (f)}+\frac{3 \int \frac{x \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \log (f)} \, dx}{a \log (f)}-\frac{(3 i) \int x^2 \log \left (1-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right ) \, dx}{4 a^{3/2} \sqrt{b} \log (f)}+\frac{(3 i) \int x^2 \log \left (1+\frac{i \sqrt{b} f^x}{\sqrt{a}}\right ) \, dx}{4 a^{3/2} \sqrt{b} \log (f)}\\ &=-\frac{3 x^2 \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^2(f)}+\frac{f^x x^3}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac{x^3 \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log (f)}-\frac{3 i x^2 \text{Li}_2\left (-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b} \log ^2(f)}+\frac{3 i x^2 \text{Li}_2\left (\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b} \log ^2(f)}+\frac{(3 i) \int x \text{Li}_2\left (-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right ) \, dx}{2 a^{3/2} \sqrt{b} \log ^2(f)}-\frac{(3 i) \int x \text{Li}_2\left (\frac{i \sqrt{b} f^x}{\sqrt{a}}\right ) \, dx}{2 a^{3/2} \sqrt{b} \log ^2(f)}+\frac{3 \int x \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right ) \, dx}{a^{3/2} \sqrt{b} \log ^2(f)}\\ &=-\frac{3 x^2 \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^2(f)}+\frac{f^x x^3}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac{x^3 \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log (f)}-\frac{3 i x^2 \text{Li}_2\left (-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b} \log ^2(f)}+\frac{3 i x^2 \text{Li}_2\left (\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b} \log ^2(f)}+\frac{3 i x \text{Li}_3\left (-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^3(f)}-\frac{3 i x \text{Li}_3\left (\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^3(f)}-\frac{(3 i) \int \text{Li}_3\left (-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right ) \, dx}{2 a^{3/2} \sqrt{b} \log ^3(f)}+\frac{(3 i) \int \text{Li}_3\left (\frac{i \sqrt{b} f^x}{\sqrt{a}}\right ) \, dx}{2 a^{3/2} \sqrt{b} \log ^3(f)}+\frac{(3 i) \int x \log \left (1-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right ) \, dx}{2 a^{3/2} \sqrt{b} \log ^2(f)}-\frac{(3 i) \int x \log \left (1+\frac{i \sqrt{b} f^x}{\sqrt{a}}\right ) \, dx}{2 a^{3/2} \sqrt{b} \log ^2(f)}\\ &=-\frac{3 x^2 \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^2(f)}+\frac{f^x x^3}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac{x^3 \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log (f)}+\frac{3 i x \text{Li}_2\left (-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^3(f)}-\frac{3 i x^2 \text{Li}_2\left (-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b} \log ^2(f)}-\frac{3 i x \text{Li}_2\left (\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^3(f)}+\frac{3 i x^2 \text{Li}_2\left (\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b} \log ^2(f)}+\frac{3 i x \text{Li}_3\left (-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^3(f)}-\frac{3 i x \text{Li}_3\left (\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^3(f)}-\frac{(3 i) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (-\frac{i \sqrt{b} x}{\sqrt{a}}\right )}{x} \, dx,x,f^x\right )}{2 a^{3/2} \sqrt{b} \log ^4(f)}+\frac{(3 i) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (\frac{i \sqrt{b} x}{\sqrt{a}}\right )}{x} \, dx,x,f^x\right )}{2 a^{3/2} \sqrt{b} \log ^4(f)}-\frac{(3 i) \int \text{Li}_2\left (-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right ) \, dx}{2 a^{3/2} \sqrt{b} \log ^3(f)}+\frac{(3 i) \int \text{Li}_2\left (\frac{i \sqrt{b} f^x}{\sqrt{a}}\right ) \, dx}{2 a^{3/2} \sqrt{b} \log ^3(f)}\\ &=-\frac{3 x^2 \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^2(f)}+\frac{f^x x^3}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac{x^3 \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log (f)}+\frac{3 i x \text{Li}_2\left (-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^3(f)}-\frac{3 i x^2 \text{Li}_2\left (-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b} \log ^2(f)}-\frac{3 i x \text{Li}_2\left (\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^3(f)}+\frac{3 i x^2 \text{Li}_2\left (\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b} \log ^2(f)}+\frac{3 i x \text{Li}_3\left (-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^3(f)}-\frac{3 i x \text{Li}_3\left (\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^3(f)}-\frac{3 i \text{Li}_4\left (-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^4(f)}+\frac{3 i \text{Li}_4\left (\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^4(f)}-\frac{(3 i) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{i \sqrt{b} x}{\sqrt{a}}\right )}{x} \, dx,x,f^x\right )}{2 a^{3/2} \sqrt{b} \log ^4(f)}+\frac{(3 i) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{i \sqrt{b} x}{\sqrt{a}}\right )}{x} \, dx,x,f^x\right )}{2 a^{3/2} \sqrt{b} \log ^4(f)}\\ &=-\frac{3 x^2 \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^2(f)}+\frac{f^x x^3}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac{x^3 \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log (f)}+\frac{3 i x \text{Li}_2\left (-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^3(f)}-\frac{3 i x^2 \text{Li}_2\left (-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b} \log ^2(f)}-\frac{3 i x \text{Li}_2\left (\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^3(f)}+\frac{3 i x^2 \text{Li}_2\left (\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b} \log ^2(f)}-\frac{3 i \text{Li}_3\left (-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^4(f)}+\frac{3 i x \text{Li}_3\left (-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^3(f)}+\frac{3 i \text{Li}_3\left (\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^4(f)}-\frac{3 i x \text{Li}_3\left (\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^3(f)}-\frac{3 i \text{Li}_4\left (-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^4(f)}+\frac{3 i \text{Li}_4\left (\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^4(f)}\\ \end{align*}
Mathematica [A] time = 0.259529, size = 434, normalized size = 0.87 \[ \frac{-\frac{6 i \text{PolyLog}\left (3,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{b}}+\frac{6 i \text{PolyLog}\left (3,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{b}}-\frac{6 i \text{PolyLog}\left (4,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{b}}+\frac{6 i \text{PolyLog}\left (4,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{b}}-\frac{3 i x \log (f) (x \log (f)-2) \text{PolyLog}\left (2,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{b}}+\frac{3 i x \log (f) (x \log (f)-2) \text{PolyLog}\left (2,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{b}}+\frac{6 i x \log (f) \text{PolyLog}\left (3,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{b}}-\frac{6 i x \log (f) \text{PolyLog}\left (3,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{b}}+\frac{2 \sqrt{a} x^3 f^x \log ^3(f)}{a+b f^{2 x}}+\frac{i x^3 \log ^3(f) \log \left (1-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{b}}-\frac{3 i x^2 \log ^2(f) \log \left (1-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{b}}-\frac{i x^3 \log ^3(f) \log \left (1+\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{b}}+\frac{3 i x^2 \log ^2(f) \log \left (1+\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{b}}}{4 a^{3/2} \log ^4(f)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.161, size = 0, normalized size = 0. \begin{align*} \int{\frac{{f}^{x}{x}^{3}}{ \left ( a+b{f}^{2\,x} \right ) ^{2}}}\, dx \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 [C] time = 1.64195, size = 1245, normalized size = 2.49 \begin{align*} \frac{2 \, b f^{x} x^{3} \log \left (f\right )^{3} + 3 \,{\left ({\left (b x^{2} \log \left (f\right )^{2} - 2 \, b x \log \left (f\right )\right )} f^{2 \, x} \sqrt{-\frac{b}{a}} +{\left (a x^{2} \log \left (f\right )^{2} - 2 \, a x \log \left (f\right )\right )} \sqrt{-\frac{b}{a}}\right )}{\rm Li}_2\left (f^{x} \sqrt{-\frac{b}{a}}\right ) - 3 \,{\left ({\left (b x^{2} \log \left (f\right )^{2} - 2 \, b x \log \left (f\right )\right )} f^{2 \, x} \sqrt{-\frac{b}{a}} +{\left (a x^{2} \log \left (f\right )^{2} - 2 \, a x \log \left (f\right )\right )} \sqrt{-\frac{b}{a}}\right )}{\rm Li}_2\left (-f^{x} \sqrt{-\frac{b}{a}}\right ) -{\left ({\left (b x^{3} \log \left (f\right )^{3} - 3 \, b x^{2} \log \left (f\right )^{2}\right )} f^{2 \, x} \sqrt{-\frac{b}{a}} +{\left (a x^{3} \log \left (f\right )^{3} - 3 \, a x^{2} \log \left (f\right )^{2}\right )} \sqrt{-\frac{b}{a}}\right )} \log \left (f^{x} \sqrt{-\frac{b}{a}} + 1\right ) +{\left ({\left (b x^{3} \log \left (f\right )^{3} - 3 \, b x^{2} \log \left (f\right )^{2}\right )} f^{2 \, x} \sqrt{-\frac{b}{a}} +{\left (a x^{3} \log \left (f\right )^{3} - 3 \, a x^{2} \log \left (f\right )^{2}\right )} \sqrt{-\frac{b}{a}}\right )} \log \left (-f^{x} \sqrt{-\frac{b}{a}} + 1\right ) + 6 \,{\left (b f^{2 \, x} \sqrt{-\frac{b}{a}} + a \sqrt{-\frac{b}{a}}\right )}{\rm polylog}\left (4, f^{x} \sqrt{-\frac{b}{a}}\right ) - 6 \,{\left (b f^{2 \, x} \sqrt{-\frac{b}{a}} + a \sqrt{-\frac{b}{a}}\right )}{\rm polylog}\left (4, -f^{x} \sqrt{-\frac{b}{a}}\right ) - 6 \,{\left ({\left (b x \log \left (f\right ) - b\right )} f^{2 \, x} \sqrt{-\frac{b}{a}} +{\left (a x \log \left (f\right ) - a\right )} \sqrt{-\frac{b}{a}}\right )}{\rm polylog}\left (3, f^{x} \sqrt{-\frac{b}{a}}\right ) + 6 \,{\left ({\left (b x \log \left (f\right ) - b\right )} f^{2 \, x} \sqrt{-\frac{b}{a}} +{\left (a x \log \left (f\right ) - a\right )} \sqrt{-\frac{b}{a}}\right )}{\rm polylog}\left (3, -f^{x} \sqrt{-\frac{b}{a}}\right )}{4 \,{\left (a b^{2} f^{2 \, x} \log \left (f\right )^{4} + a^{2} b \log \left (f\right )^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{f^{x} x^{3}}{2 a^{2} \log{\left (f \right )} + 2 a b f^{2 x} \log{\left (f \right )}} + \frac{\int - \frac{3 f^{x} x^{2}}{a + b f^{2 x}}\, dx + \int \frac{f^{x} x^{3} \log{\left (f \right )}}{a + b f^{2 x}}\, dx}{2 a \log{\left (f \right )}} \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{f^{x} x^{3}}{{\left (b f^{2 \, x} + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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