Optimal. Leaf size=316 \[ \frac {i \text {Li}_3\left (-\frac {i \sqrt {a} f^x}{\sqrt {b}}\right )}{8 a^{3/2} b^{3/2} \log ^3(f)}-\frac {i \text {Li}_3\left (\frac {i \sqrt {a} f^x}{\sqrt {b}}\right )}{8 a^{3/2} b^{3/2} \log ^3(f)}-\frac {i x \text {Li}_2\left (-\frac {i \sqrt {a} f^x}{\sqrt {b}}\right )}{8 a^{3/2} b^{3/2} \log ^2(f)}+\frac {i x \text {Li}_2\left (\frac {i \sqrt {a} f^x}{\sqrt {b}}\right )}{8 a^{3/2} b^{3/2} \log ^2(f)}+\frac {x^2 \tan ^{-1}\left (\frac {\sqrt {a} f^x}{\sqrt {b}}\right )}{8 a^{3/2} b^{3/2} \log (f)}-\frac {\tan ^{-1}\left (\frac {\sqrt {a} f^x}{\sqrt {b}}\right )}{4 a^{3/2} b^{3/2} \log ^3(f)}+\frac {x^2 f^x}{8 a b \log (f) \left (a f^{2 x}+b\right )}-\frac {x^2 f^x}{4 a \log (f) \left (a f^{2 x}+b\right )^2}+\frac {x f^x}{4 a b \log ^2(f) \left (a f^{2 x}+b\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.16, antiderivative size = 316, normalized size of antiderivative = 1.00, number of steps used = 43, number of rules used = 14, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.737, Rules used = {2283, 2254, 2249, 199, 205, 2245, 14, 2282, 4848, 2391, 12, 5143, 2531, 6589} \[ -\frac {i x \text {PolyLog}\left (2,-\frac {i \sqrt {a} f^x}{\sqrt {b}}\right )}{8 a^{3/2} b^{3/2} \log ^2(f)}+\frac {i x \text {PolyLog}\left (2,\frac {i \sqrt {a} f^x}{\sqrt {b}}\right )}{8 a^{3/2} b^{3/2} \log ^2(f)}+\frac {i \text {PolyLog}\left (3,-\frac {i \sqrt {a} f^x}{\sqrt {b}}\right )}{8 a^{3/2} b^{3/2} \log ^3(f)}-\frac {i \text {PolyLog}\left (3,\frac {i \sqrt {a} f^x}{\sqrt {b}}\right )}{8 a^{3/2} b^{3/2} \log ^3(f)}+\frac {x^2 \tan ^{-1}\left (\frac {\sqrt {a} f^x}{\sqrt {b}}\right )}{8 a^{3/2} b^{3/2} \log (f)}-\frac {\tan ^{-1}\left (\frac {\sqrt {a} f^x}{\sqrt {b}}\right )}{4 a^{3/2} b^{3/2} \log ^3(f)}+\frac {x^2 f^x}{8 a b \log (f) \left (a f^{2 x}+b\right )}-\frac {x^2 f^x}{4 a \log (f) \left (a f^{2 x}+b\right )^2}+\frac {x f^x}{4 a b \log ^2(f) \left (a f^{2 x}+b\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 14
Rule 199
Rule 205
Rule 2245
Rule 2249
Rule 2254
Rule 2282
Rule 2283
Rule 2391
Rule 2531
Rule 4848
Rule 5143
Rule 6589
Rubi steps
\begin {align*} \int \frac {x^2}{\left (b f^{-x}+a f^x\right )^3} \, dx &=\int \frac {f^{3 x} x^2}{\left (b+a f^{2 x}\right )^3} \, dx\\ &=\int \left (-\frac {b f^x x^2}{a \left (b+a f^{2 x}\right )^3}+\frac {f^x x^2}{a \left (b+a f^{2 x}\right )^2}\right ) \, dx\\ &=\frac {\int \frac {f^x x^2}{\left (b+a f^{2 x}\right )^2} \, dx}{a}-\frac {b \int \frac {f^x x^2}{\left (b+a f^{2 x}\right )^3} \, dx}{a}\\ &=-\frac {f^x x^2}{4 a \left (b+a f^{2 x}\right )^2 \log (f)}+\frac {f^x x^2}{8 a b \left (b+a f^{2 x}\right ) \log (f)}+\frac {x^2 \tan ^{-1}\left (\frac {\sqrt {a} f^x}{\sqrt {b}}\right )}{8 a^{3/2} b^{3/2} \log (f)}-\frac {2 \int x \left (\frac {f^x}{2 b \left (b+a f^{2 x}\right ) \log (f)}+\frac {\tan ^{-1}\left (\frac {\sqrt {a} f^x}{\sqrt {b}}\right )}{2 \sqrt {a} b^{3/2} \log (f)}\right ) \, dx}{a}+\frac {(2 b) \int x \left (\frac {f^x}{4 b \left (b+a f^{2 x}\right )^2 \log (f)}+\frac {3 f^x}{8 b^2 \left (b+a f^{2 x}\right ) \log (f)}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {a} f^x}{\sqrt {b}}\right )}{8 \sqrt {a} b^{5/2} \log (f)}\right ) \, dx}{a}\\ &=-\frac {f^x x^2}{4 a \left (b+a f^{2 x}\right )^2 \log (f)}+\frac {f^x x^2}{8 a b \left (b+a f^{2 x}\right ) \log (f)}+\frac {x^2 \tan ^{-1}\left (\frac {\sqrt {a} f^x}{\sqrt {b}}\right )}{8 a^{3/2} b^{3/2} \log (f)}-\frac {2 \int \left (\frac {f^x x}{2 b \left (b+a f^{2 x}\right ) \log (f)}+\frac {x \tan ^{-1}\left (\frac {\sqrt {a} f^x}{\sqrt {b}}\right )}{2 \sqrt {a} b^{3/2} \log (f)}\right ) \, dx}{a}+\frac {(2 b) \int \left (\frac {f^x x}{4 b \left (b+a f^{2 x}\right )^2 \log (f)}+\frac {3 f^x x}{8 b^2 \left (b+a f^{2 x}\right ) \log (f)}+\frac {3 x \tan ^{-1}\left (\frac {\sqrt {a} f^x}{\sqrt {b}}\right )}{8 \sqrt {a} b^{5/2} \log (f)}\right ) \, dx}{a}\\ &=-\frac {f^x x^2}{4 a \left (b+a f^{2 x}\right )^2 \log (f)}+\frac {f^x x^2}{8 a b \left (b+a f^{2 x}\right ) \log (f)}+\frac {x^2 \tan ^{-1}\left (\frac {\sqrt {a} f^x}{\sqrt {b}}\right )}{8 a^{3/2} b^{3/2} \log (f)}+\frac {\int \frac {f^x x}{\left (b+a f^{2 x}\right )^2} \, dx}{2 a \log (f)}+\frac {3 \int x \tan ^{-1}\left (\frac {\sqrt {a} f^x}{\sqrt {b}}\right ) \, dx}{4 a^{3/2} b^{3/2} \log (f)}-\frac {\int x \tan ^{-1}\left (\frac {\sqrt {a} f^x}{\sqrt {b}}\right ) \, dx}{a^{3/2} b^{3/2} \log (f)}+\frac {3 \int \frac {f^x x}{b+a f^{2 x}} \, dx}{4 a b \log (f)}-\frac {\int \frac {f^x x}{b+a f^{2 x}} \, dx}{a b \log (f)}\\ &=\frac {f^x x}{4 a b \left (b+a f^{2 x}\right ) \log ^2(f)}-\frac {f^x x^2}{4 a \left (b+a f^{2 x}\right )^2 \log (f)}+\frac {f^x x^2}{8 a b \left (b+a f^{2 x}\right ) \log (f)}+\frac {x^2 \tan ^{-1}\left (\frac {\sqrt {a} f^x}{\sqrt {b}}\right )}{8 a^{3/2} b^{3/2} \log (f)}-\frac {\int \left (\frac {f^x}{2 b \left (b+a f^{2 x}\right ) \log (f)}+\frac {\tan ^{-1}\left (\frac {\sqrt {a} f^x}{\sqrt {b}}\right )}{2 \sqrt {a} b^{3/2} \log (f)}\right ) \, dx}{2 a \log (f)}+\frac {(3 i) \int x \log \left (1-\frac {i \sqrt {a} f^x}{\sqrt {b}}\right ) \, dx}{8 a^{3/2} b^{3/2} \log (f)}-\frac {(3 i) \int x \log \left (1+\frac {i \sqrt {a} f^x}{\sqrt {b}}\right ) \, dx}{8 a^{3/2} b^{3/2} \log (f)}-\frac {i \int x \log \left (1-\frac {i \sqrt {a} f^x}{\sqrt {b}}\right ) \, dx}{2 a^{3/2} b^{3/2} \log (f)}+\frac {i \int x \log \left (1+\frac {i \sqrt {a} f^x}{\sqrt {b}}\right ) \, dx}{2 a^{3/2} b^{3/2} \log (f)}-\frac {3 \int \frac {\tan ^{-1}\left (\frac {\sqrt {a} f^x}{\sqrt {b}}\right )}{\sqrt {a} \sqrt {b} \log (f)} \, dx}{4 a b \log (f)}+\frac {\int \frac {\tan ^{-1}\left (\frac {\sqrt {a} f^x}{\sqrt {b}}\right )}{\sqrt {a} \sqrt {b} \log (f)} \, dx}{a b \log (f)}\\ &=\frac {f^x x}{4 a b \left (b+a f^{2 x}\right ) \log ^2(f)}-\frac {f^x x^2}{4 a \left (b+a f^{2 x}\right )^2 \log (f)}+\frac {f^x x^2}{8 a b \left (b+a f^{2 x}\right ) \log (f)}+\frac {x^2 \tan ^{-1}\left (\frac {\sqrt {a} f^x}{\sqrt {b}}\right )}{8 a^{3/2} b^{3/2} \log (f)}-\frac {i x \text {Li}_2\left (-\frac {i \sqrt {a} f^x}{\sqrt {b}}\right )}{8 a^{3/2} b^{3/2} \log ^2(f)}+\frac {i x \text {Li}_2\left (\frac {i \sqrt {a} f^x}{\sqrt {b}}\right )}{8 a^{3/2} b^{3/2} \log ^2(f)}-\frac {(3 i) \int \text {Li}_2\left (-\frac {i \sqrt {a} f^x}{\sqrt {b}}\right ) \, dx}{8 a^{3/2} b^{3/2} \log ^2(f)}+\frac {(3 i) \int \text {Li}_2\left (\frac {i \sqrt {a} f^x}{\sqrt {b}}\right ) \, dx}{8 a^{3/2} b^{3/2} \log ^2(f)}+\frac {i \int \text {Li}_2\left (-\frac {i \sqrt {a} f^x}{\sqrt {b}}\right ) \, dx}{2 a^{3/2} b^{3/2} \log ^2(f)}-\frac {i \int \text {Li}_2\left (\frac {i \sqrt {a} f^x}{\sqrt {b}}\right ) \, dx}{2 a^{3/2} b^{3/2} \log ^2(f)}-\frac {\int \tan ^{-1}\left (\frac {\sqrt {a} f^x}{\sqrt {b}}\right ) \, dx}{4 a^{3/2} b^{3/2} \log ^2(f)}-\frac {3 \int \tan ^{-1}\left (\frac {\sqrt {a} f^x}{\sqrt {b}}\right ) \, dx}{4 a^{3/2} b^{3/2} \log ^2(f)}+\frac {\int \tan ^{-1}\left (\frac {\sqrt {a} f^x}{\sqrt {b}}\right ) \, dx}{a^{3/2} b^{3/2} \log ^2(f)}-\frac {\int \frac {f^x}{b+a f^{2 x}} \, dx}{4 a b \log ^2(f)}\\ &=\frac {f^x x}{4 a b \left (b+a f^{2 x}\right ) \log ^2(f)}-\frac {f^x x^2}{4 a \left (b+a f^{2 x}\right )^2 \log (f)}+\frac {f^x x^2}{8 a b \left (b+a f^{2 x}\right ) \log (f)}+\frac {x^2 \tan ^{-1}\left (\frac {\sqrt {a} f^x}{\sqrt {b}}\right )}{8 a^{3/2} b^{3/2} \log (f)}-\frac {i x \text {Li}_2\left (-\frac {i \sqrt {a} f^x}{\sqrt {b}}\right )}{8 a^{3/2} b^{3/2} \log ^2(f)}+\frac {i x \text {Li}_2\left (\frac {i \sqrt {a} f^x}{\sqrt {b}}\right )}{8 a^{3/2} b^{3/2} \log ^2(f)}-\frac {(3 i) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {i \sqrt {a} x}{\sqrt {b}}\right )}{x} \, dx,x,f^x\right )}{8 a^{3/2} b^{3/2} \log ^3(f)}+\frac {(3 i) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {i \sqrt {a} x}{\sqrt {b}}\right )}{x} \, dx,x,f^x\right )}{8 a^{3/2} b^{3/2} \log ^3(f)}+\frac {i \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {i \sqrt {a} x}{\sqrt {b}}\right )}{x} \, dx,x,f^x\right )}{2 a^{3/2} b^{3/2} \log ^3(f)}-\frac {i \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {i \sqrt {a} x}{\sqrt {b}}\right )}{x} \, dx,x,f^x\right )}{2 a^{3/2} b^{3/2} \log ^3(f)}-\frac {\operatorname {Subst}\left (\int \frac {\tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{x} \, dx,x,f^x\right )}{4 a^{3/2} b^{3/2} \log ^3(f)}-\frac {3 \operatorname {Subst}\left (\int \frac {\tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{x} \, dx,x,f^x\right )}{4 a^{3/2} b^{3/2} \log ^3(f)}+\frac {\operatorname {Subst}\left (\int \frac {\tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{x} \, dx,x,f^x\right )}{a^{3/2} b^{3/2} \log ^3(f)}-\frac {\operatorname {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,f^x\right )}{4 a b \log ^3(f)}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {a} f^x}{\sqrt {b}}\right )}{4 a^{3/2} b^{3/2} \log ^3(f)}+\frac {f^x x}{4 a b \left (b+a f^{2 x}\right ) \log ^2(f)}-\frac {f^x x^2}{4 a \left (b+a f^{2 x}\right )^2 \log (f)}+\frac {f^x x^2}{8 a b \left (b+a f^{2 x}\right ) \log (f)}+\frac {x^2 \tan ^{-1}\left (\frac {\sqrt {a} f^x}{\sqrt {b}}\right )}{8 a^{3/2} b^{3/2} \log (f)}-\frac {i x \text {Li}_2\left (-\frac {i \sqrt {a} f^x}{\sqrt {b}}\right )}{8 a^{3/2} b^{3/2} \log ^2(f)}+\frac {i x \text {Li}_2\left (\frac {i \sqrt {a} f^x}{\sqrt {b}}\right )}{8 a^{3/2} b^{3/2} \log ^2(f)}+\frac {i \text {Li}_3\left (-\frac {i \sqrt {a} f^x}{\sqrt {b}}\right )}{8 a^{3/2} b^{3/2} \log ^3(f)}-\frac {i \text {Li}_3\left (\frac {i \sqrt {a} f^x}{\sqrt {b}}\right )}{8 a^{3/2} b^{3/2} \log ^3(f)}-\frac {i \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {i \sqrt {a} x}{\sqrt {b}}\right )}{x} \, dx,x,f^x\right )}{8 a^{3/2} b^{3/2} \log ^3(f)}+\frac {i \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {i \sqrt {a} x}{\sqrt {b}}\right )}{x} \, dx,x,f^x\right )}{8 a^{3/2} b^{3/2} \log ^3(f)}-\frac {(3 i) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {i \sqrt {a} x}{\sqrt {b}}\right )}{x} \, dx,x,f^x\right )}{8 a^{3/2} b^{3/2} \log ^3(f)}+\frac {(3 i) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {i \sqrt {a} x}{\sqrt {b}}\right )}{x} \, dx,x,f^x\right )}{8 a^{3/2} b^{3/2} \log ^3(f)}+\frac {i \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {i \sqrt {a} x}{\sqrt {b}}\right )}{x} \, dx,x,f^x\right )}{2 a^{3/2} b^{3/2} \log ^3(f)}-\frac {i \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {i \sqrt {a} x}{\sqrt {b}}\right )}{x} \, dx,x,f^x\right )}{2 a^{3/2} b^{3/2} \log ^3(f)}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {a} f^x}{\sqrt {b}}\right )}{4 a^{3/2} b^{3/2} \log ^3(f)}+\frac {f^x x}{4 a b \left (b+a f^{2 x}\right ) \log ^2(f)}-\frac {f^x x^2}{4 a \left (b+a f^{2 x}\right )^2 \log (f)}+\frac {f^x x^2}{8 a b \left (b+a f^{2 x}\right ) \log (f)}+\frac {x^2 \tan ^{-1}\left (\frac {\sqrt {a} f^x}{\sqrt {b}}\right )}{8 a^{3/2} b^{3/2} \log (f)}-\frac {i x \text {Li}_2\left (-\frac {i \sqrt {a} f^x}{\sqrt {b}}\right )}{8 a^{3/2} b^{3/2} \log ^2(f)}+\frac {i x \text {Li}_2\left (\frac {i \sqrt {a} f^x}{\sqrt {b}}\right )}{8 a^{3/2} b^{3/2} \log ^2(f)}+\frac {i \text {Li}_3\left (-\frac {i \sqrt {a} f^x}{\sqrt {b}}\right )}{8 a^{3/2} b^{3/2} \log ^3(f)}-\frac {i \text {Li}_3\left (\frac {i \sqrt {a} f^x}{\sqrt {b}}\right )}{8 a^{3/2} b^{3/2} \log ^3(f)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.52, size = 254, normalized size = 0.80 \[ \frac {\frac {3 i \left (2 \text {Li}_3\left (-\frac {i \sqrt {a} f^x}{\sqrt {b}}\right )-2 \text {Li}_3\left (\frac {i \sqrt {a} f^x}{\sqrt {b}}\right )-2 x \log (f) \text {Li}_2\left (-\frac {i \sqrt {a} f^x}{\sqrt {b}}\right )+2 x \log (f) \text {Li}_2\left (\frac {i \sqrt {a} f^x}{\sqrt {b}}\right )+x^2 \log ^2(f) \log \left (1-\frac {i \sqrt {a} f^x}{\sqrt {b}}\right )-x^2 \log ^2(f) \log \left (1+\frac {i \sqrt {a} f^x}{\sqrt {b}}\right )\right )}{b^{3/2}}-\frac {12 \tan ^{-1}\left (\frac {\sqrt {a} f^x}{\sqrt {b}}\right )}{b^{3/2}}-\frac {12 \sqrt {a} x^2 f^x \log ^2(f)}{\left (a f^{2 x}+b\right )^2}+\frac {6 \sqrt {a} x f^x \log (f) (x \log (f)+2)}{b \left (a f^{2 x}+b\right )}}{48 a^{3/2} \log ^3(f)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [C] time = 0.44, size = 674, normalized size = 2.13 \[ \frac {2 \, {\left (a^{2} x^{2} \log \relax (f)^{2} + 2 \, a^{2} x \log \relax (f)\right )} f^{3 \, x} - 2 \, {\left (a b x^{2} \log \relax (f)^{2} - 2 \, a b x \log \relax (f)\right )} f^{x} + 2 \, {\left (a^{2} f^{4 \, x} x \sqrt {-\frac {a}{b}} \log \relax (f) + 2 \, a b f^{2 \, x} x \sqrt {-\frac {a}{b}} \log \relax (f) + b^{2} x \sqrt {-\frac {a}{b}} \log \relax (f)\right )} {\rm Li}_2\left (f^{x} \sqrt {-\frac {a}{b}}\right ) - 2 \, {\left (a^{2} f^{4 \, x} x \sqrt {-\frac {a}{b}} \log \relax (f) + 2 \, a b f^{2 \, x} x \sqrt {-\frac {a}{b}} \log \relax (f) + b^{2} x \sqrt {-\frac {a}{b}} \log \relax (f)\right )} {\rm Li}_2\left (-f^{x} \sqrt {-\frac {a}{b}}\right ) - 2 \, {\left (a^{2} f^{4 \, x} \sqrt {-\frac {a}{b}} + 2 \, a b f^{2 \, x} \sqrt {-\frac {a}{b}} + b^{2} \sqrt {-\frac {a}{b}}\right )} \log \left (2 \, a f^{x} + 2 \, b \sqrt {-\frac {a}{b}}\right ) + 2 \, {\left (a^{2} f^{4 \, x} \sqrt {-\frac {a}{b}} + 2 \, a b f^{2 \, x} \sqrt {-\frac {a}{b}} + b^{2} \sqrt {-\frac {a}{b}}\right )} \log \left (2 \, a f^{x} - 2 \, b \sqrt {-\frac {a}{b}}\right ) - {\left (a^{2} f^{4 \, x} x^{2} \sqrt {-\frac {a}{b}} \log \relax (f)^{2} + 2 \, a b f^{2 \, x} x^{2} \sqrt {-\frac {a}{b}} \log \relax (f)^{2} + b^{2} x^{2} \sqrt {-\frac {a}{b}} \log \relax (f)^{2}\right )} \log \left (f^{x} \sqrt {-\frac {a}{b}} + 1\right ) + {\left (a^{2} f^{4 \, x} x^{2} \sqrt {-\frac {a}{b}} \log \relax (f)^{2} + 2 \, a b f^{2 \, x} x^{2} \sqrt {-\frac {a}{b}} \log \relax (f)^{2} + b^{2} x^{2} \sqrt {-\frac {a}{b}} \log \relax (f)^{2}\right )} \log \left (-f^{x} \sqrt {-\frac {a}{b}} + 1\right ) - 2 \, {\left (a^{2} f^{4 \, x} \sqrt {-\frac {a}{b}} + 2 \, a b f^{2 \, x} \sqrt {-\frac {a}{b}} + b^{2} \sqrt {-\frac {a}{b}}\right )} {\rm polylog}\left (3, f^{x} \sqrt {-\frac {a}{b}}\right ) + 2 \, {\left (a^{2} f^{4 \, x} \sqrt {-\frac {a}{b}} + 2 \, a b f^{2 \, x} \sqrt {-\frac {a}{b}} + b^{2} \sqrt {-\frac {a}{b}}\right )} {\rm polylog}\left (3, -f^{x} \sqrt {-\frac {a}{b}}\right )}{16 \, {\left (a^{4} b f^{4 \, x} \log \relax (f)^{3} + 2 \, a^{3} b^{2} f^{2 \, x} \log \relax (f)^{3} + a^{2} b^{3} \log \relax (f)^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{{\left (a f^{x} + \frac {b}{f^{x}}\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.17, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\left (a \,f^{x}+b \,f^{-x}\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (a x^{2} \log \relax (f) + 2 \, a x\right )} f^{3 \, x} - {\left (b x^{2} \log \relax (f) - 2 \, b x\right )} f^{x}}{8 \, {\left (a^{3} b f^{4 \, x} \log \relax (f)^{2} + 2 \, a^{2} b^{2} f^{2 \, x} \log \relax (f)^{2} + a b^{3} \log \relax (f)^{2}\right )}} + \int \frac {{\left (x^{2} \log \relax (f)^{2} - 2\right )} f^{x}}{8 \, {\left (a^{2} b f^{2 \, x} \log \relax (f)^{2} + a b^{2} \log \relax (f)^{2}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^2}{{\left (\frac {b}{f^x}+a\,f^x\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {f^{3 x} \left (a x^{2} \log {\relax (f )} + 2 a x\right ) + f^{x} \left (- b x^{2} \log {\relax (f )} + 2 b x\right )}{8 a^{3} b f^{4 x} \log {\relax (f )}^{2} + 16 a^{2} b^{2} f^{2 x} \log {\relax (f )}^{2} + 8 a b^{3} \log {\relax (f )}^{2}} + \frac {\int \left (- \frac {2 f^{x}}{a f^{2 x} + b}\right )\, dx + \int \frac {f^{x} x^{2} \log {\relax (f )}^{2}}{a f^{2 x} + b}\, dx}{8 a b \log {\relax (f )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________