Optimal. Leaf size=223 \[ -\frac {3 i \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {b} \log ^2(f)}+\frac {3 i \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {b} \log ^2(f)}-\frac {\tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{5/2} \sqrt {b} \log ^2(f)}+\frac {3 x \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log (f)}-\frac {f^x}{8 a^2 \log ^2(f) \left (a+b f^{2 x}\right )}+\frac {3 x f^x}{8 a^2 \log (f) \left (a+b f^{2 x}\right )}+\frac {x f^x}{4 a \log (f) \left (a+b f^{2 x}\right )^2} \]
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Rubi [A] time = 0.22, antiderivative size = 223, normalized size of antiderivative = 1.00, 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 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 )^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*}
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Mathematica [A] time = 0.31, size = 184, normalized size = 0.83 \[ \frac {\frac {6 i \left (-\text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )+\text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )+x \log (f) \left (\log \left (1-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )-\log \left (1+\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )\right )\right )}{\sqrt {a} \sqrt {b}}-\frac {16 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}}+\frac {8 a x f^x \log (f)}{\left (a+b f^{2 x}\right )^2}+\frac {4 f^x (3 x \log (f)-1)}{a+b f^{2 x}}}{32 a^2 \log ^2(f)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 494, normalized size = 2.22 \[ \frac {2 \, {\left (3 \, b^{2} x \log \relax (f) - b^{2}\right )} f^{3 \, x} + 2 \, {\left (5 \, a b x \log \relax (f) - a b\right )} f^{x} + 3 \, {\left (b^{2} f^{4 \, x} \sqrt {-\frac {b}{a}} + 2 \, a b f^{2 \, x} \sqrt {-\frac {b}{a}} + a^{2} \sqrt {-\frac {b}{a}}\right )} {\rm Li}_2\left (f^{x} \sqrt {-\frac {b}{a}}\right ) - 3 \, {\left (b^{2} f^{4 \, x} \sqrt {-\frac {b}{a}} + 2 \, a b f^{2 \, x} \sqrt {-\frac {b}{a}} + a^{2} \sqrt {-\frac {b}{a}}\right )} {\rm Li}_2\left (-f^{x} \sqrt {-\frac {b}{a}}\right ) - 4 \, {\left (b^{2} f^{4 \, x} \sqrt {-\frac {b}{a}} + 2 \, a b f^{2 \, x} \sqrt {-\frac {b}{a}} + a^{2} \sqrt {-\frac {b}{a}}\right )} \log \left (2 \, b f^{x} + 2 \, a \sqrt {-\frac {b}{a}}\right ) + 4 \, {\left (b^{2} f^{4 \, x} \sqrt {-\frac {b}{a}} + 2 \, a b f^{2 \, x} \sqrt {-\frac {b}{a}} + a^{2} \sqrt {-\frac {b}{a}}\right )} \log \left (2 \, b f^{x} - 2 \, a \sqrt {-\frac {b}{a}}\right ) - 3 \, {\left (b^{2} f^{4 \, x} x \sqrt {-\frac {b}{a}} \log \relax (f) + 2 \, a b f^{2 \, x} x \sqrt {-\frac {b}{a}} \log \relax (f) + a^{2} x \sqrt {-\frac {b}{a}} \log \relax (f)\right )} \log \left (f^{x} \sqrt {-\frac {b}{a}} + 1\right ) + 3 \, {\left (b^{2} f^{4 \, x} x \sqrt {-\frac {b}{a}} \log \relax (f) + 2 \, a b f^{2 \, x} x \sqrt {-\frac {b}{a}} \log \relax (f) + a^{2} x \sqrt {-\frac {b}{a}} \log \relax (f)\right )} \log \left (-f^{x} \sqrt {-\frac {b}{a}} + 1\right )}{16 \, {\left (a^{2} b^{3} f^{4 \, x} \log \relax (f)^{2} + 2 \, a^{3} b^{2} f^{2 \, x} \log \relax (f)^{2} + a^{4} b \log \relax (f)^{2}\right )}} \]
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 \[ \int \frac {f^{x} x}{{\left (b f^{2 \, x} + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 223, normalized size = 1.00 \[ \frac {3 x \ln \left (\frac {-b \,f^{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )}{16 \sqrt {-a b}\, a^{2} \ln \relax (f )}-\frac {3 x \ln \left (\frac {b \,f^{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )}{16 \sqrt {-a b}\, a^{2} \ln \relax (f )}+\frac {\left (3 b x \,f^{2 x} \ln \relax (f )+5 a x \ln \relax (f )-b \,f^{2 x}-a \right ) f^{x}}{8 \left (b \,f^{2 x}+a \right )^{2} a^{2} \ln \relax (f )^{2}}+\frac {3 \dilog \left (\frac {-b \,f^{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )}{16 \sqrt {-a b}\, a^{2} \ln \relax (f )^{2}}-\frac {3 \dilog \left (\frac {b \,f^{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )}{16 \sqrt {-a b}\, a^{2} \ln \relax (f )^{2}}-\frac {\arctan \left (\frac {b \,f^{x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, a^{2} \ln \relax (f )^{2}} \]
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 \[ \frac {{\left (3 \, b x \log \relax (f) - b\right )} f^{3 \, x} + {\left (5 \, a x \log \relax (f) - a\right )} f^{x}}{8 \, {\left (a^{2} b^{2} f^{4 \, x} \log \relax (f)^{2} + 2 \, a^{3} b f^{2 \, x} \log \relax (f)^{2} + a^{4} \log \relax (f)^{2}\right )}} + \int \frac {{\left (3 \, x \log \relax (f) - 4\right )} f^{x}}{8 \, {\left (a^{2} b f^{2 \, x} \log \relax (f) + a^{3} \log \relax (f)\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {f^x\,x}{{\left (a+b\,f^{2\,x}\right )}^3} \,d x \]
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 \[ \frac {f^{3 x} \left (3 b x \log {\relax (f )} - b\right ) + f^{x} \left (5 a x \log {\relax (f )} - a\right )}{8 a^{4} \log {\relax (f )}^{2} + 16 a^{3} b f^{2 x} \log {\relax (f )}^{2} + 8 a^{2} b^{2} f^{4 x} \log {\relax (f )}^{2}} + \frac {\int \left (- \frac {4 f^{x}}{a + b f^{2 x}}\right )\, dx + \int \frac {3 f^{x} x \log {\relax (f )}}{a + b f^{2 x}}\, dx}{8 a^{2} \log {\relax (f )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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