Integrand size = 16, antiderivative size = 223 \[ \int \frac {f^x x}{\left (a+b f^{2 x}\right )^3} \, dx=-\frac {f^x}{8 a^2 \left (a+b f^{2 x}\right ) \log ^2(f)}-\frac {\arctan \left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{5/2} \sqrt {b} \log ^2(f)}+\frac {f^x x}{4 a \left (a+b f^{2 x}\right )^2 \log (f)}+\frac {3 f^x x}{8 a^2 \left (a+b f^{2 x}\right ) \log (f)}+\frac {3 x \arctan \left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log (f)}-\frac {3 i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {b} \log ^2(f)}+\frac {3 i \operatorname {PolyLog}\left (2,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {b} \log ^2(f)} \]
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Time = 0.14 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {2281, 205, 211, 2277, 2320, 4940, 2438} \[ \int \frac {f^x x}{\left (a+b f^{2 x}\right )^3} \, dx=-\frac {\arctan \left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{5/2} \sqrt {b} \log ^2(f)}+\frac {3 x \arctan \left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log (f)}-\frac {3 i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {b} \log ^2(f)}+\frac {3 i \operatorname {PolyLog}\left (2,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {b} \log ^2(f)}-\frac {f^x}{8 a^2 \log ^2(f) \left (a+b f^{2 x}\right )}+\frac {3 x f^x}{8 a^2 \log (f) \left (a+b f^{2 x}\right )}+\frac {x f^x}{4 a \log (f) \left (a+b f^{2 x}\right )^2} \]
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Rule 205
Rule 211
Rule 2277
Rule 2281
Rule 2320
Rule 2438
Rule 4940
Rubi steps \begin{align*} \text {integral}& = \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 \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,f^x\right )}{8 a^2 \log ^2(f)}-\frac {\text {Subst}\left (\int \frac {1}{\left (a+b x^2\right )^2} \, dx,x,f^x\right )}{4 a \log ^2(f)}-\frac {3 \text {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 {\text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,f^x\right )}{8 a^2 \log ^2(f)}-\frac {(3 i) \text {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) \text {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*}
Time = 0.26 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.83 \[ \int \frac {f^x x}{\left (a+b f^{2 x}\right )^3} \, dx=\frac {-\frac {16 \arctan \left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}}+\frac {8 a f^x x \log (f)}{\left (a+b f^{2 x}\right )^2}+\frac {4 f^x (-1+3 x \log (f))}{a+b f^{2 x}}+\frac {6 i \left (x \log (f) \left (\log \left (1-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )-\log \left (1+\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )\right )-\operatorname {PolyLog}\left (2,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )+\operatorname {PolyLog}\left (2,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )\right )}{\sqrt {a} \sqrt {b}}}{32 a^2 \log ^2(f)} \]
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Time = 0.34 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.00
method | result | size |
risch | \(\frac {f^{x} \left (3 x b \,f^{2 x} \ln \left (f \right )+5 \ln \left (f \right ) a x -b \,f^{2 x}-a \right )}{8 \ln \left (f \right )^{2} a^{2} \left (a +b \,f^{2 x}\right )^{2}}-\frac {\arctan \left (\frac {b \,f^{x}}{\sqrt {a b}}\right )}{2 a^{2} \ln \left (f \right )^{2} \sqrt {a b}}+\frac {3 x \ln \left (\frac {-b \,f^{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )}{16 a^{2} \ln \left (f \right ) \sqrt {-a b}}-\frac {3 x \ln \left (\frac {b \,f^{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )}{16 a^{2} \ln \left (f \right ) \sqrt {-a b}}+\frac {3 \operatorname {dilog}\left (\frac {-b \,f^{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )}{16 a^{2} \ln \left (f \right )^{2} \sqrt {-a b}}-\frac {3 \operatorname {dilog}\left (\frac {b \,f^{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )}{16 a^{2} \ln \left (f \right )^{2} \sqrt {-a b}}\) | \(223\) |
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Leaf count of result is larger than twice the leaf count of optimal. 494 vs. \(2 (167) = 334\).
Time = 0.27 (sec) , antiderivative size = 494, normalized size of antiderivative = 2.22 \[ \int \frac {f^x x}{\left (a+b f^{2 x}\right )^3} \, dx=\frac {2 \, {\left (3 \, b^{2} x \log \left (f\right ) - b^{2}\right )} f^{3 \, x} + 2 \, {\left (5 \, a b x \log \left (f\right ) - a b\right )} f^{x} + 3 \, {\left (b^{2} f^{4 \, x} \sqrt {-\frac {b}{a}} + 2 \, a b f^{2 \, x} \sqrt {-\frac {b}{a}} + a^{2} \sqrt {-\frac {b}{a}}\right )} {\rm Li}_2\left (f^{x} \sqrt {-\frac {b}{a}}\right ) - 3 \, {\left (b^{2} f^{4 \, x} \sqrt {-\frac {b}{a}} + 2 \, a b f^{2 \, x} \sqrt {-\frac {b}{a}} + a^{2} \sqrt {-\frac {b}{a}}\right )} {\rm Li}_2\left (-f^{x} \sqrt {-\frac {b}{a}}\right ) - 4 \, {\left (b^{2} f^{4 \, x} \sqrt {-\frac {b}{a}} + 2 \, a b f^{2 \, x} \sqrt {-\frac {b}{a}} + a^{2} \sqrt {-\frac {b}{a}}\right )} \log \left (2 \, b f^{x} + 2 \, a \sqrt {-\frac {b}{a}}\right ) + 4 \, {\left (b^{2} f^{4 \, x} \sqrt {-\frac {b}{a}} + 2 \, a b f^{2 \, x} \sqrt {-\frac {b}{a}} + a^{2} \sqrt {-\frac {b}{a}}\right )} \log \left (2 \, b f^{x} - 2 \, a \sqrt {-\frac {b}{a}}\right ) - 3 \, {\left (b^{2} f^{4 \, x} x \sqrt {-\frac {b}{a}} \log \left (f\right ) + 2 \, a b f^{2 \, x} x \sqrt {-\frac {b}{a}} \log \left (f\right ) + a^{2} x \sqrt {-\frac {b}{a}} \log \left (f\right )\right )} \log \left (f^{x} \sqrt {-\frac {b}{a}} + 1\right ) + 3 \, {\left (b^{2} f^{4 \, x} x \sqrt {-\frac {b}{a}} \log \left (f\right ) + 2 \, a b f^{2 \, x} x \sqrt {-\frac {b}{a}} \log \left (f\right ) + a^{2} x \sqrt {-\frac {b}{a}} \log \left (f\right )\right )} \log \left (-f^{x} \sqrt {-\frac {b}{a}} + 1\right )}{16 \, {\left (a^{2} b^{3} f^{4 \, x} \log \left (f\right )^{2} + 2 \, a^{3} b^{2} f^{2 \, x} \log \left (f\right )^{2} + a^{4} b \log \left (f\right )^{2}\right )}} \]
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\[ \int \frac {f^x x}{\left (a+b f^{2 x}\right )^3} \, dx=\frac {f^{3 x} \left (3 b x \log {\left (f \right )} - b\right ) + f^{x} \left (5 a x \log {\left (f \right )} - a\right )}{8 a^{4} \log {\left (f \right )}^{2} + 16 a^{3} b f^{2 x} \log {\left (f \right )}^{2} + 8 a^{2} b^{2} f^{4 x} \log {\left (f \right )}^{2}} + \frac {\int \left (- \frac {4 f^{x}}{a + b f^{2 x}}\right )\, dx + \int \frac {3 f^{x} x \log {\left (f \right )}}{a + b f^{2 x}}\, dx}{8 a^{2} \log {\left (f \right )}} \]
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\[ \int \frac {f^x x}{\left (a+b f^{2 x}\right )^3} \, dx=\int { \frac {f^{x} x}{{\left (b f^{2 \, x} + a\right )}^{3}} \,d x } \]
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\[ \int \frac {f^x x}{\left (a+b f^{2 x}\right )^3} \, dx=\int { \frac {f^{x} x}{{\left (b f^{2 \, x} + a\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {f^x x}{\left (a+b f^{2 x}\right )^3} \, dx=\int \frac {f^x\,x}{{\left (a+b\,f^{2\,x}\right )}^3} \,d x \]
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