Optimal. Leaf size=420 \[ \frac {i \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{5/2} \sqrt {b} \log ^3(f)}-\frac {i \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{5/2} \sqrt {b} \log ^3(f)}+\frac {3 i \text {Li}_3\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log ^3(f)}-\frac {3 i \text {Li}_3\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log ^3(f)}-\frac {3 i x \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log ^2(f)}+\frac {3 i x \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log ^2(f)}+\frac {3 x^2 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log (f)}+\frac {\tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{5/2} \sqrt {b} \log ^3(f)}-\frac {x \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{a^{5/2} \sqrt {b} \log ^2(f)}+\frac {3 x^2 f^x}{8 a^2 \log (f) \left (a+b f^{2 x}\right )}-\frac {x f^x}{4 a^2 \log ^2(f) \left (a+b f^{2 x}\right )}+\frac {x^2 f^x}{4 a \log (f) \left (a+b f^{2 x}\right )^2} \]
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Rubi [A] time = 0.58, antiderivative size = 420, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 12, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {2249, 199, 205, 2245, 14, 2282, 4848, 2391, 12, 5143, 2531, 6589} \[ -\frac {3 i x \text {PolyLog}\left (2,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log ^2(f)}+\frac {i \text {PolyLog}\left (2,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{5/2} \sqrt {b} \log ^3(f)}+\frac {3 i x \text {PolyLog}\left (2,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log ^2(f)}-\frac {i \text {PolyLog}\left (2,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{5/2} \sqrt {b} \log ^3(f)}+\frac {3 i \text {PolyLog}\left (3,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log ^3(f)}-\frac {3 i \text {PolyLog}\left (3,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log ^3(f)}+\frac {3 x^2 f^x}{8 a^2 \log (f) \left (a+b f^{2 x}\right )}+\frac {3 x^2 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log (f)}-\frac {x f^x}{4 a^2 \log ^2(f) \left (a+b f^{2 x}\right )}-\frac {x \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{a^{5/2} \sqrt {b} \log ^2(f)}+\frac {\tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{5/2} \sqrt {b} \log ^3(f)}+\frac {x^2 f^x}{4 a \log (f) \left (a+b f^{2 x}\right )^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 199
Rule 205
Rule 2245
Rule 2249
Rule 2282
Rule 2391
Rule 2531
Rule 4848
Rule 5143
Rule 6589
Rubi steps
\begin {align*} \int \frac {f^x x^2}{\left (a+b f^{2 x}\right )^3} \, dx &=\frac {f^x x^2}{4 a \left (a+b f^{2 x}\right )^2 \log (f)}+\frac {3 f^x x^2}{8 a^2 \left (a+b f^{2 x}\right ) \log (f)}+\frac {3 x^2 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log (f)}-2 \int x \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^2}{4 a \left (a+b f^{2 x}\right )^2 \log (f)}+\frac {3 f^x x^2}{8 a^2 \left (a+b f^{2 x}\right ) \log (f)}+\frac {3 x^2 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log (f)}-2 \int \left (\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)}\right ) \, dx\\ &=\frac {f^x x^2}{4 a \left (a+b f^{2 x}\right )^2 \log (f)}+\frac {3 f^x x^2}{8 a^2 \left (a+b f^{2 x}\right ) \log (f)}+\frac {3 x^2 \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 x}{a+b f^{2 x}} \, dx}{4 a^2 \log (f)}-\frac {\int \frac {f^x x}{\left (a+b f^{2 x}\right )^2} \, dx}{2 a \log (f)}-\frac {3 \int x \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{4 a^{5/2} \sqrt {b} \log (f)}\\ &=-\frac {f^x x}{4 a^2 \left (a+b f^{2 x}\right ) \log ^2(f)}-\frac {x \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{a^{5/2} \sqrt {b} \log ^2(f)}+\frac {f^x x^2}{4 a \left (a+b f^{2 x}\right )^2 \log (f)}+\frac {3 f^x x^2}{8 a^2 \left (a+b f^{2 x}\right ) \log (f)}+\frac {3 x^2 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log (f)}+\frac {3 \int \frac {\tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \log (f)} \, dx}{4 a^2 \log (f)}+\frac {\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}{2 a \log (f)}-\frac {(3 i) \int x \log \left (1-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{8 a^{5/2} \sqrt {b} \log (f)}+\frac {(3 i) \int x \log \left (1+\frac {i \sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{8 a^{5/2} \sqrt {b} \log (f)}\\ &=-\frac {f^x x}{4 a^2 \left (a+b f^{2 x}\right ) \log ^2(f)}-\frac {x \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{a^{5/2} \sqrt {b} \log ^2(f)}+\frac {f^x x^2}{4 a \left (a+b f^{2 x}\right )^2 \log (f)}+\frac {3 f^x x^2}{8 a^2 \left (a+b f^{2 x}\right ) \log (f)}+\frac {3 x^2 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log (f)}-\frac {3 i x \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log ^2(f)}+\frac {3 i x \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log ^2(f)}+\frac {\int \frac {f^x}{a+b f^{2 x}} \, dx}{4 a^2 \log ^2(f)}+\frac {(3 i) \int \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{8 a^{5/2} \sqrt {b} \log ^2(f)}-\frac {(3 i) \int \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{8 a^{5/2} \sqrt {b} \log ^2(f)}+\frac {\int \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{4 a^{5/2} \sqrt {b} \log ^2(f)}+\frac {3 \int \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{4 a^{5/2} \sqrt {b} \log ^2(f)}\\ &=-\frac {f^x x}{4 a^2 \left (a+b f^{2 x}\right ) \log ^2(f)}-\frac {x \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{a^{5/2} \sqrt {b} \log ^2(f)}+\frac {f^x x^2}{4 a \left (a+b f^{2 x}\right )^2 \log (f)}+\frac {3 f^x x^2}{8 a^2 \left (a+b f^{2 x}\right ) \log (f)}+\frac {3 x^2 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log (f)}-\frac {3 i x \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log ^2(f)}+\frac {3 i x \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log ^2(f)}+\frac {\operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,f^x\right )}{4 a^2 \log ^3(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 )}{8 a^{5/2} \sqrt {b} \log ^3(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 )}{8 a^{5/2} \sqrt {b} \log ^3(f)}+\frac {\operatorname {Subst}\left (\int \frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{x} \, dx,x,f^x\right )}{4 a^{5/2} \sqrt {b} \log ^3(f)}+\frac {3 \operatorname {Subst}\left (\int \frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{x} \, dx,x,f^x\right )}{4 a^{5/2} \sqrt {b} \log ^3(f)}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{5/2} \sqrt {b} \log ^3(f)}-\frac {f^x x}{4 a^2 \left (a+b f^{2 x}\right ) \log ^2(f)}-\frac {x \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{a^{5/2} \sqrt {b} \log ^2(f)}+\frac {f^x x^2}{4 a \left (a+b f^{2 x}\right )^2 \log (f)}+\frac {3 f^x x^2}{8 a^2 \left (a+b f^{2 x}\right ) \log (f)}+\frac {3 x^2 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log (f)}-\frac {3 i x \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log ^2(f)}+\frac {3 i x \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log ^2(f)}+\frac {3 i \text {Li}_3\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log ^3(f)}-\frac {3 i \text {Li}_3\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log ^3(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 )}{8 a^{5/2} \sqrt {b} \log ^3(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 )}{8 a^{5/2} \sqrt {b} \log ^3(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 )}{8 a^{5/2} \sqrt {b} \log ^3(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 )}{8 a^{5/2} \sqrt {b} \log ^3(f)}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{5/2} \sqrt {b} \log ^3(f)}-\frac {f^x x}{4 a^2 \left (a+b f^{2 x}\right ) \log ^2(f)}-\frac {x \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{a^{5/2} \sqrt {b} \log ^2(f)}+\frac {f^x x^2}{4 a \left (a+b f^{2 x}\right )^2 \log (f)}+\frac {3 f^x x^2}{8 a^2 \left (a+b f^{2 x}\right ) \log (f)}+\frac {3 x^2 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log (f)}+\frac {i \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{5/2} \sqrt {b} \log ^3(f)}-\frac {3 i x \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log ^2(f)}-\frac {i \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{5/2} \sqrt {b} \log ^3(f)}+\frac {3 i x \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log ^2(f)}+\frac {3 i \text {Li}_3\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log ^3(f)}-\frac {3 i \text {Li}_3\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log ^3(f)}\\ \end {align*}
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Mathematica [A] time = 0.50, size = 353, normalized size = 0.84 \[ \frac {\frac {3 i \left (2 \text {Li}_3\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )-2 \text {Li}_3\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )-2 x \log (f) \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )+2 x \log (f) \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )+x^2 \log ^2(f) \log \left (1-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )-x^2 \log ^2(f) \log \left (1+\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )\right )}{\sqrt {a} \sqrt {b}}-\frac {8 i \left (-\text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )+\text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )+x \log (f) \left (\log \left (1-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )-\log \left (1+\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )\right )\right )}{\sqrt {a} \sqrt {b}}+\frac {4 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}}+\frac {4 a x^2 f^x \log ^2(f)}{\left (a+b f^{2 x}\right )^2}+\frac {2 x f^x \log (f) (3 x \log (f)-2)}{a+b f^{2 x}}}{16 a^2 \log ^3(f)} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.45, size = 786, normalized size = 1.87 \[ \frac {2 \, {\left (3 \, b^{2} x^{2} \log \relax (f)^{2} - 2 \, b^{2} x \log \relax (f)\right )} f^{3 \, x} + 2 \, {\left (5 \, a b x^{2} \log \relax (f)^{2} - 2 \, a b x \log \relax (f)\right )} f^{x} + 2 \, {\left ({\left (3 \, b^{2} x \log \relax (f) - 4 \, b^{2}\right )} f^{4 \, x} \sqrt {-\frac {b}{a}} + 2 \, {\left (3 \, a b x \log \relax (f) - 4 \, a b\right )} f^{2 \, x} \sqrt {-\frac {b}{a}} + {\left (3 \, a^{2} x \log \relax (f) - 4 \, a^{2}\right )} \sqrt {-\frac {b}{a}}\right )} {\rm Li}_2\left (f^{x} \sqrt {-\frac {b}{a}}\right ) - 2 \, {\left ({\left (3 \, b^{2} x \log \relax (f) - 4 \, b^{2}\right )} f^{4 \, x} \sqrt {-\frac {b}{a}} + 2 \, {\left (3 \, a b x \log \relax (f) - 4 \, a b\right )} f^{2 \, x} \sqrt {-\frac {b}{a}} + {\left (3 \, a^{2} x \log \relax (f) - 4 \, a^{2}\right )} \sqrt {-\frac {b}{a}}\right )} {\rm Li}_2\left (-f^{x} \sqrt {-\frac {b}{a}}\right ) + 2 \, {\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 ) - 2 \, {\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 ) - {\left ({\left (3 \, b^{2} x^{2} \log \relax (f)^{2} - 8 \, b^{2} x \log \relax (f)\right )} f^{4 \, x} \sqrt {-\frac {b}{a}} + 2 \, {\left (3 \, a b x^{2} \log \relax (f)^{2} - 8 \, a b x \log \relax (f)\right )} f^{2 \, x} \sqrt {-\frac {b}{a}} + {\left (3 \, a^{2} x^{2} \log \relax (f)^{2} - 8 \, a^{2} x \log \relax (f)\right )} \sqrt {-\frac {b}{a}}\right )} \log \left (f^{x} \sqrt {-\frac {b}{a}} + 1\right ) + {\left ({\left (3 \, b^{2} x^{2} \log \relax (f)^{2} - 8 \, b^{2} x \log \relax (f)\right )} f^{4 \, x} \sqrt {-\frac {b}{a}} + 2 \, {\left (3 \, a b x^{2} \log \relax (f)^{2} - 8 \, a b x \log \relax (f)\right )} f^{2 \, x} \sqrt {-\frac {b}{a}} + {\left (3 \, a^{2} x^{2} \log \relax (f)^{2} - 8 \, a^{2} x \log \relax (f)\right )} \sqrt {-\frac {b}{a}}\right )} \log \left (-f^{x} \sqrt {-\frac {b}{a}} + 1\right ) - 6 \, {\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 polylog}\left (3, f^{x} \sqrt {-\frac {b}{a}}\right ) + 6 \, {\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 polylog}\left (3, -f^{x} \sqrt {-\frac {b}{a}}\right )}{16 \, {\left (a^{2} b^{3} f^{4 \, x} \log \relax (f)^{3} + 2 \, a^{3} b^{2} f^{2 \, x} \log \relax (f)^{3} + a^{4} b \log \relax (f)^{3}\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 \[ \int \frac {f^{x} x^{2}}{{\left (b f^{2 \, x} + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.21, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} f^{x}}{\left (b \,f^{2 x}+a \right )^{3}}\, dx \]
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 \[ \frac {{\left (3 \, b x^{2} \log \relax (f) - 2 \, b x\right )} f^{3 \, x} + {\left (5 \, a x^{2} \log \relax (f) - 2 \, a x\right )} f^{x}}{8 \, {\left (a^{2} b^{2} f^{4 \, x} \log \relax (f)^{2} + 2 \, a^{3} b f^{2 \, x} \log \relax (f)^{2} + a^{4} \log \relax (f)^{2}\right )}} + \int \frac {{\left (3 \, x^{2} \log \relax (f)^{2} - 8 \, x \log \relax (f) + 2\right )} f^{x}}{8 \, {\left (a^{2} b f^{2 \, x} \log \relax (f)^{2} + a^{3} \log \relax (f)^{2}\right )}}\,{d x} \]
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 \[ \int \frac {f^x\,x^2}{{\left (a+b\,f^{2\,x}\right )}^3} \,d x \]
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 \[ \frac {f^{3 x} \left (3 b x^{2} \log {\relax (f )} - 2 b x\right ) + f^{x} \left (5 a x^{2} \log {\relax (f )} - 2 a x\right )}{8 a^{4} \log {\relax (f )}^{2} + 16 a^{3} b f^{2 x} \log {\relax (f )}^{2} + 8 a^{2} b^{2} f^{4 x} \log {\relax (f )}^{2}} + \frac {\int \frac {2 f^{x}}{a + b f^{2 x}}\, dx + \int \left (- \frac {8 f^{x} x \log {\relax (f )}}{a + b f^{2 x}}\right )\, dx + \int \frac {3 f^{x} x^{2} \log {\relax (f )}^{2}}{a + b f^{2 x}}\, dx}{8 a^{2} \log {\relax (f )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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