Optimal. Leaf size=223 \[ -\frac{3 i \text{PolyLog}\left (2,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{b} \log ^2(f)}+\frac{3 i \text{PolyLog}\left (2,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{b} \log ^2(f)}-\frac{f^x}{8 a^2 \log ^2(f) \left (a+b f^{2 x}\right )}+\frac{3 x f^x}{8 a^2 \log (f) \left (a+b f^{2 x}\right )}-\frac{\tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{5/2} \sqrt{b} \log ^2(f)}+\frac{3 x \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b} \log (f)}+\frac{x f^x}{4 a \log (f) \left (a+b f^{2 x}\right )^2} \]
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Rubi [A] time = 0.217052, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {2249, 199, 205, 2245, 2282, 4848, 2391} \[ -\frac{3 i \text{PolyLog}\left (2,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{b} \log ^2(f)}+\frac{3 i \text{PolyLog}\left (2,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{b} \log ^2(f)}-\frac{f^x}{8 a^2 \log ^2(f) \left (a+b f^{2 x}\right )}+\frac{3 x f^x}{8 a^2 \log (f) \left (a+b f^{2 x}\right )}-\frac{\tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{5/2} \sqrt{b} \log ^2(f)}+\frac{3 x \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b} \log (f)}+\frac{x f^x}{4 a \log (f) \left (a+b f^{2 x}\right )^2} \]
Antiderivative was successfully verified.
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Rule 2249
Rule 199
Rule 205
Rule 2245
Rule 2282
Rule 4848
Rule 2391
Rubi steps
\begin{align*} \int \frac{f^x x}{\left (a+b f^{2 x}\right )^3} \, dx &=\frac{f^x x}{4 a \left (a+b f^{2 x}\right )^2 \log (f)}+\frac{3 f^x x}{8 a^2 \left (a+b f^{2 x}\right ) \log (f)}+\frac{3 x \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b} \log (f)}-\int \left (\frac{f^x}{4 a \left (a+b f^{2 x}\right )^2 \log (f)}+\frac{3 f^x}{8 a^2 \left (a+b f^{2 x}\right ) \log (f)}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b} \log (f)}\right ) \, dx\\ &=\frac{f^x x}{4 a \left (a+b f^{2 x}\right )^2 \log (f)}+\frac{3 f^x x}{8 a^2 \left (a+b f^{2 x}\right ) \log (f)}+\frac{3 x \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b} \log (f)}-\frac{3 \int \frac{f^x}{a+b f^{2 x}} \, dx}{8 a^2 \log (f)}-\frac{\int \frac{f^x}{\left (a+b f^{2 x}\right )^2} \, dx}{4 a \log (f)}-\frac{3 \int \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right ) \, dx}{8 a^{5/2} \sqrt{b} \log (f)}\\ &=\frac{f^x x}{4 a \left (a+b f^{2 x}\right )^2 \log (f)}+\frac{3 f^x x}{8 a^2 \left (a+b f^{2 x}\right ) \log (f)}+\frac{3 x \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b} \log (f)}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,f^x\right )}{8 a^2 \log ^2(f)}-\frac{\operatorname{Subst}\left (\int \frac{1}{\left (a+b x^2\right )^2} \, dx,x,f^x\right )}{4 a \log ^2(f)}-\frac{3 \operatorname{Subst}\left (\int \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{x} \, dx,x,f^x\right )}{8 a^{5/2} \sqrt{b} \log ^2(f)}\\ &=-\frac{f^x}{8 a^2 \left (a+b f^{2 x}\right ) \log ^2(f)}-\frac{3 \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b} \log ^2(f)}+\frac{f^x x}{4 a \left (a+b f^{2 x}\right )^2 \log (f)}+\frac{3 f^x x}{8 a^2 \left (a+b f^{2 x}\right ) \log (f)}+\frac{3 x \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b} \log (f)}-\frac{\operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,f^x\right )}{8 a^2 \log ^2(f)}-\frac{(3 i) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{i \sqrt{b} x}{\sqrt{a}}\right )}{x} \, dx,x,f^x\right )}{16 a^{5/2} \sqrt{b} \log ^2(f)}+\frac{(3 i) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{i \sqrt{b} x}{\sqrt{a}}\right )}{x} \, dx,x,f^x\right )}{16 a^{5/2} \sqrt{b} \log ^2(f)}\\ &=-\frac{f^x}{8 a^2 \left (a+b f^{2 x}\right ) \log ^2(f)}-\frac{\tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{5/2} \sqrt{b} \log ^2(f)}+\frac{f^x x}{4 a \left (a+b f^{2 x}\right )^2 \log (f)}+\frac{3 f^x x}{8 a^2 \left (a+b f^{2 x}\right ) \log (f)}+\frac{3 x \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b} \log (f)}-\frac{3 i \text{Li}_2\left (-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{b} \log ^2(f)}+\frac{3 i \text{Li}_2\left (\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{b} \log ^2(f)}\\ \end{align*}
Mathematica [A] time = 0.305697, size = 184, normalized size = 0.83 \[ \frac{\frac{6 i \left (-\text{PolyLog}\left (2,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )+\text{PolyLog}\left (2,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )+x \log (f) \left (\log \left (1-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )-\log \left (1+\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )\right )\right )}{\sqrt{a} \sqrt{b}}+\frac{8 a x f^x \log (f)}{\left (a+b f^{2 x}\right )^2}+\frac{4 f^x (3 x \log (f)-1)}{a+b f^{2 x}}-\frac{16 \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}}}{32 a^2 \log ^2(f)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.069, size = 223, normalized size = 1. \begin{align*}{\frac{{f}^{x} \left ( 3\,\ln \left ( f \right ) bx \left ({f}^{x} \right ) ^{2}+5\,\ln \left ( f \right ) ax-b \left ({f}^{x} \right ) ^{2}-a \right ) }{8\, \left ( \ln \left ( f \right ) \right ) ^{2}{a}^{2} \left ( a+b \left ({f}^{x} \right ) ^{2} \right ) ^{2}}}-{\frac{1}{2\, \left ( \ln \left ( f \right ) \right ) ^{2}{a}^{2}}\arctan \left ({b{f}^{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,x}{16\,\ln \left ( f \right ){a}^{2}}\ln \left ({ \left ( -b{f}^{x}+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}} \right ){\frac{1}{\sqrt{-ab}}}}-{\frac{3\,x}{16\,\ln \left ( f \right ){a}^{2}}\ln \left ({ \left ( b{f}^{x}+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}} \right ){\frac{1}{\sqrt{-ab}}}}+{\frac{3}{16\, \left ( \ln \left ( f \right ) \right ) ^{2}{a}^{2}}{\it dilog} \left ({ \left ( -b{f}^{x}+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}} \right ){\frac{1}{\sqrt{-ab}}}}-{\frac{3}{16\, \left ( \ln \left ( f \right ) \right ) ^{2}{a}^{2}}{\it dilog} \left ({ \left ( b{f}^{x}+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}} \right ){\frac{1}{\sqrt{-ab}}}} \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 [B] time = 1.60431, size = 1068, normalized size = 4.79 \begin{align*} \frac{2 \,{\left (3 \, b^{2} x \log \left (f\right ) - b^{2}\right )} f^{3 \, x} + 2 \,{\left (5 \, a b x \log \left (f\right ) - a b\right )} f^{x} + 3 \,{\left (b^{2} f^{4 \, x} \sqrt{-\frac{b}{a}} + 2 \, a b f^{2 \, x} \sqrt{-\frac{b}{a}} + a^{2} \sqrt{-\frac{b}{a}}\right )}{\rm Li}_2\left (f^{x} \sqrt{-\frac{b}{a}}\right ) - 3 \,{\left (b^{2} f^{4 \, x} \sqrt{-\frac{b}{a}} + 2 \, a b f^{2 \, x} \sqrt{-\frac{b}{a}} + a^{2} \sqrt{-\frac{b}{a}}\right )}{\rm Li}_2\left (-f^{x} \sqrt{-\frac{b}{a}}\right ) - 4 \,{\left (b^{2} f^{4 \, x} \sqrt{-\frac{b}{a}} + 2 \, a b f^{2 \, x} \sqrt{-\frac{b}{a}} + a^{2} \sqrt{-\frac{b}{a}}\right )} \log \left (2 \, b f^{x} + 2 \, a \sqrt{-\frac{b}{a}}\right ) + 4 \,{\left (b^{2} f^{4 \, x} \sqrt{-\frac{b}{a}} + 2 \, a b f^{2 \, x} \sqrt{-\frac{b}{a}} + a^{2} \sqrt{-\frac{b}{a}}\right )} \log \left (2 \, b f^{x} - 2 \, a \sqrt{-\frac{b}{a}}\right ) - 3 \,{\left (b^{2} f^{4 \, x} x \sqrt{-\frac{b}{a}} \log \left (f\right ) + 2 \, a b f^{2 \, x} x \sqrt{-\frac{b}{a}} \log \left (f\right ) + a^{2} x \sqrt{-\frac{b}{a}} \log \left (f\right )\right )} \log \left (f^{x} \sqrt{-\frac{b}{a}} + 1\right ) + 3 \,{\left (b^{2} f^{4 \, x} x \sqrt{-\frac{b}{a}} \log \left (f\right ) + 2 \, a b f^{2 \, x} x \sqrt{-\frac{b}{a}} \log \left (f\right ) + a^{2} x \sqrt{-\frac{b}{a}} \log \left (f\right )\right )} \log \left (-f^{x} \sqrt{-\frac{b}{a}} + 1\right )}{16 \,{\left (a^{2} b^{3} f^{4 \, x} \log \left (f\right )^{2} + 2 \, a^{3} b^{2} f^{2 \, x} \log \left (f\right )^{2} + a^{4} b \log \left (f\right )^{2}\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^{3 x} \left (3 b x \log{\left (f \right )} - b\right ) + f^{x} \left (5 a x \log{\left (f \right )} - a\right )}{8 a^{4} \log{\left (f \right )}^{2} + 16 a^{3} b f^{2 x} \log{\left (f \right )}^{2} + 8 a^{2} b^{2} f^{4 x} \log{\left (f \right )}^{2}} + \frac{\int - \frac{4 f^{x}}{a + b f^{2 x}}\, dx + \int \frac{3 f^{x} x \log{\left (f \right )}}{a + b f^{2 x}}\, dx}{8 a^{2} \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}{{\left (b f^{2 \, x} + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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