Optimal. Leaf size=172 \[ -\frac {i \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {i \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b} \log ^2(f)}-\frac {\tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {x \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}+\frac {x f^x}{2 a \log (f) \left (a+b f^{2 x}\right )} \]
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Rubi [A] time = 0.16, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \[ -\frac {i \text {PolyLog}\left (2,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {i \text {PolyLog}\left (2,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b} \log ^2(f)}-\frac {\tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {x \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}+\frac {x f^x}{2 a \log (f) \left (a+b f^{2 x}\right )} \]
Antiderivative was successfully verified.
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Rule 199
Rule 205
Rule 2245
Rule 2249
Rule 2282
Rule 2391
Rule 4848
Rubi steps
\begin {align*} \int \frac {f^x x}{\left (a+b f^{2 x}\right )^2} \, dx &=\frac {f^x x}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac {x \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}-\int \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}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac {x \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}-\frac {\int \frac {f^x}{a+b f^{2 x}} \, dx}{2 a \log (f)}-\frac {\int \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{2 a^{3/2} \sqrt {b} \log (f)}\\ &=\frac {f^x x}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac {x \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}-\frac {\operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,f^x\right )}{2 a \log ^2(f)}-\frac {\operatorname {Subst}\left (\int \frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{x} \, dx,x,f^x\right )}{2 a^{3/2} \sqrt {b} \log ^2(f)}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {f^x x}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac {x \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}-\frac {i \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {i \sqrt {b} x}{\sqrt {a}}\right )}{x} \, dx,x,f^x\right )}{4 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {i \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {i \sqrt {b} x}{\sqrt {a}}\right )}{x} \, dx,x,f^x\right )}{4 a^{3/2} \sqrt {b} \log ^2(f)}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {f^x x}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac {x \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}-\frac {i \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {i \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b} \log ^2(f)}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 271, normalized size = 1.58 \[ \frac {\frac {-\frac {i \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \log ^2(f)}-\frac {i x \log \left (1+\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \log (f)}+\frac {i x^2}{2 \sqrt {a}}}{2 \sqrt {b}}+\frac {\frac {i \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \log ^2(f)}+\frac {i x \log \left (1-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \log (f)}-\frac {i x^2}{2 \sqrt {a}}}{2 \sqrt {b}}}{2 a}+\frac {x f^x}{2 a \log (f) \left (a+b f^{2 x}\right )}-\frac {\left (\frac {b f^{2 x}}{a}+1\right ) \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b} \log ^2(f) \left (a+b f^{2 x}\right )} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 311, normalized size = 1.81 \[ \frac {2 \, b f^{x} x \log \relax (f) + {\left (b f^{2 \, x} \sqrt {-\frac {b}{a}} + a \sqrt {-\frac {b}{a}}\right )} {\rm Li}_2\left (f^{x} \sqrt {-\frac {b}{a}}\right ) - {\left (b f^{2 \, x} \sqrt {-\frac {b}{a}} + a \sqrt {-\frac {b}{a}}\right )} {\rm Li}_2\left (-f^{x} \sqrt {-\frac {b}{a}}\right ) - {\left (b f^{2 \, x} \sqrt {-\frac {b}{a}} + a \sqrt {-\frac {b}{a}}\right )} \log \left (2 \, b f^{x} + 2 \, a \sqrt {-\frac {b}{a}}\right ) + {\left (b f^{2 \, x} \sqrt {-\frac {b}{a}} + a \sqrt {-\frac {b}{a}}\right )} \log \left (2 \, b f^{x} - 2 \, a \sqrt {-\frac {b}{a}}\right ) - {\left (b f^{2 \, x} x \sqrt {-\frac {b}{a}} \log \relax (f) + a x \sqrt {-\frac {b}{a}} \log \relax (f)\right )} \log \left (f^{x} \sqrt {-\frac {b}{a}} + 1\right ) + {\left (b f^{2 \, x} x \sqrt {-\frac {b}{a}} \log \relax (f) + a x \sqrt {-\frac {b}{a}} \log \relax (f)\right )} \log \left (-f^{x} \sqrt {-\frac {b}{a}} + 1\right )}{4 \, {\left (a b^{2} f^{2 \, x} \log \relax (f)^{2} + a^{2} b \log \relax (f)^{2}\right )}} \]
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 \[ \int \frac {f^{x} x}{{\left (b f^{2 \, x} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 195, normalized size = 1.13 \[ \frac {x \,f^{x}}{2 \left (b \,f^{2 x}+a \right ) a \ln \relax (f )}+\frac {x \ln \left (\frac {-b \,f^{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )}{4 \sqrt {-a b}\, a \ln \relax (f )}-\frac {x \ln \left (\frac {b \,f^{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )}{4 \sqrt {-a b}\, a \ln \relax (f )}+\frac {\dilog \left (\frac {-b \,f^{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )}{4 \sqrt {-a b}\, a \ln \relax (f )^{2}}-\frac {\dilog \left (\frac {b \,f^{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )}{4 \sqrt {-a b}\, a \ln \relax (f )^{2}}-\frac {\arctan \left (\frac {b \,f^{x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, a \ln \relax (f )^{2}} \]
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 \[ \frac {f^{x} x}{2 \, {\left (a b f^{2 \, x} \log \relax (f) + a^{2} \log \relax (f)\right )}} + \int \frac {{\left (x \log \relax (f) - 1\right )} f^{x}}{2 \, {\left (a b f^{2 \, x} \log \relax (f) + a^{2} \log \relax (f)\right )}}\,{d x} \]
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 \[ \int \frac {f^x\,x}{{\left (a+b\,f^{2\,x}\right )}^2} \,d x \]
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 \[ \frac {f^{x} x}{2 a^{2} \log {\relax (f )} + 2 a b f^{2 x} \log {\relax (f )}} + \frac {\int \left (- \frac {f^{x}}{a + b f^{2 x}}\right )\, dx + \int \frac {f^{x} x \log {\relax (f )}}{a + b f^{2 x}}\, dx}{2 a \log {\relax (f )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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