Optimal. Leaf size=746 \[ \frac {\log \left (c \left (d+e x^2\right )^p\right )}{4 g^{3/2} \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{4 g^{3/2} \left (\sqrt {-f}+\sqrt {g} x\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 \sqrt {f} g^{3/2}}+\frac {i p \text {Li}_2\left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}-\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}+1\right )}{4 \sqrt {f} g^{3/2}}+\frac {i p \text {Li}_2\left (1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{4 \sqrt {f} g^{3/2}}-\frac {e \sqrt {-f} p \log \left (\sqrt {-f}-\sqrt {g} x\right )}{2 g^{3/2} (e f-d g)}+\frac {e \sqrt {-f} p \log \left (\sqrt {-f}+\sqrt {g} x\right )}{2 g^{3/2} (e f-d g)}-\frac {p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (-\sqrt {-d} \sqrt {g}+i \sqrt {e} \sqrt {f}\right )}\right )}{2 \sqrt {f} g^{3/2}}-\frac {p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (\sqrt {-d} \sqrt {g}+i \sqrt {e} \sqrt {f}\right )}\right )}{2 \sqrt {f} g^{3/2}}-\frac {\sqrt {d} \sqrt {e} p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{g (e f-d g)}-\frac {i p \text {Li}_2\left (1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{2 \sqrt {f} g^{3/2}}+\frac {p \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right ) \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} g^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.51, antiderivative size = 746, normalized size of antiderivative = 1.00, number of steps used = 40, number of rules used = 14, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.560, Rules used = {2476, 2471, 2463, 801, 635, 205, 260, 2470, 12, 4928, 4856, 2402, 2315, 2447} \[ \frac {i p \text {PolyLog}\left (2,1+\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (-\sqrt {-d} \sqrt {g}+i \sqrt {e} \sqrt {f}\right )}\right )}{4 \sqrt {f} g^{3/2}}+\frac {i p \text {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (\sqrt {-d} \sqrt {g}+i \sqrt {e} \sqrt {f}\right )}\right )}{4 \sqrt {f} g^{3/2}}-\frac {i p \text {PolyLog}\left (2,1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{2 \sqrt {f} g^{3/2}}+\frac {\log \left (c \left (d+e x^2\right )^p\right )}{4 g^{3/2} \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{4 g^{3/2} \left (\sqrt {-f}+\sqrt {g} x\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 \sqrt {f} g^{3/2}}-\frac {e \sqrt {-f} p \log \left (\sqrt {-f}-\sqrt {g} x\right )}{2 g^{3/2} (e f-d g)}+\frac {e \sqrt {-f} p \log \left (\sqrt {-f}+\sqrt {g} x\right )}{2 g^{3/2} (e f-d g)}-\frac {p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (-\sqrt {-d} \sqrt {g}+i \sqrt {e} \sqrt {f}\right )}\right )}{2 \sqrt {f} g^{3/2}}-\frac {p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (\sqrt {-d} \sqrt {g}+i \sqrt {e} \sqrt {f}\right )}\right )}{2 \sqrt {f} g^{3/2}}-\frac {\sqrt {d} \sqrt {e} p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{g (e f-d g)}+\frac {p \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right ) \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} g^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 205
Rule 260
Rule 635
Rule 801
Rule 2315
Rule 2402
Rule 2447
Rule 2463
Rule 2470
Rule 2471
Rule 2476
Rule 4856
Rule 4928
Rubi steps
\begin {align*} \int \frac {x^2 \log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx &=\int \left (-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{g \left (f+g x^2\right )^2}+\frac {\log \left (c \left (d+e x^2\right )^p\right )}{g \left (f+g x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx}{g}-\frac {f \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx}{g}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {f} g^{3/2}}-\frac {f \int \left (-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{4 f \left (\sqrt {-f} \sqrt {g}-g x\right )^2}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{4 f \left (\sqrt {-f} \sqrt {g}+g x\right )^2}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{2 f \left (-f g-g^2 x^2\right )}\right ) \, dx}{g}-\frac {(2 e p) \int \frac {x \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g} \left (d+e x^2\right )} \, dx}{g}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {f} g^{3/2}}+\frac {1}{4} \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{\left (\sqrt {-f} \sqrt {g}-g x\right )^2} \, dx+\frac {1}{4} \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{\left (\sqrt {-f} \sqrt {g}+g x\right )^2} \, dx+\frac {1}{2} \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{-f g-g^2 x^2} \, dx-\frac {(2 e p) \int \frac {x \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{d+e x^2} \, dx}{\sqrt {f} g^{3/2}}\\ &=\frac {\log \left (c \left (d+e x^2\right )^p\right )}{4 g^{3/2} \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{4 g^{3/2} \left (\sqrt {-f}+\sqrt {g} x\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 \sqrt {f} g^{3/2}}+(e p) \int \frac {x \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} g^{3/2} \left (d+e x^2\right )} \, dx-\frac {(2 e p) \int \left (-\frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{\sqrt {f} g^{3/2}}-\frac {(e p) \int \frac {x}{\left (\sqrt {-f} \sqrt {g}-g x\right ) \left (d+e x^2\right )} \, dx}{2 g}+\frac {(e p) \int \frac {x}{\left (\sqrt {-f} \sqrt {g}+g x\right ) \left (d+e x^2\right )} \, dx}{2 g}\\ &=\frac {\log \left (c \left (d+e x^2\right )^p\right )}{4 g^{3/2} \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{4 g^{3/2} \left (\sqrt {-f}+\sqrt {g} x\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 \sqrt {f} g^{3/2}}+\frac {\left (\sqrt {e} p\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {-d}-\sqrt {e} x} \, dx}{\sqrt {f} g^{3/2}}-\frac {\left (\sqrt {e} p\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {-d}+\sqrt {e} x} \, dx}{\sqrt {f} g^{3/2}}+\frac {(e p) \int \frac {x \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{d+e x^2} \, dx}{\sqrt {f} g^{3/2}}+\frac {(e p) \int \left (\frac {\sqrt {-f}}{(e f-d g) \left (\sqrt {-f}+\sqrt {g} x\right )}-\frac {-d \sqrt {g}-e \sqrt {-f} x}{\sqrt {g} (-e f+d g) \left (d+e x^2\right )}\right ) \, dx}{2 g}-\frac {(e p) \int \left (\frac {\sqrt {-f}}{(e f-d g) \left (-\sqrt {-f}+\sqrt {g} x\right )}-\frac {d \sqrt {g}-e \sqrt {-f} x}{\sqrt {g} (-e f+d g) \left (d+e x^2\right )}\right ) \, dx}{2 g}\\ &=-\frac {e \sqrt {-f} p \log \left (\sqrt {-f}-\sqrt {g} x\right )}{2 g^{3/2} (e f-d g)}+\frac {2 p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{\sqrt {f} g^{3/2}}-\frac {p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}-\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{\sqrt {f} g^{3/2}}-\frac {p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{\sqrt {f} g^{3/2}}+\frac {e \sqrt {-f} p \log \left (\sqrt {-f}+\sqrt {g} x\right )}{2 g^{3/2} (e f-d g)}+\frac {\log \left (c \left (d+e x^2\right )^p\right )}{4 g^{3/2} \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{4 g^{3/2} \left (\sqrt {-f}+\sqrt {g} x\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 \sqrt {f} g^{3/2}}+\frac {(e p) \int \left (-\frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{\sqrt {f} g^{3/2}}-2 \frac {p \int \frac {\log \left (\frac {2}{1-\frac {i \sqrt {g} x}{\sqrt {f}}}\right )}{1+\frac {g x^2}{f}} \, dx}{f g}+\frac {p \int \frac {\log \left (\frac {2 \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\sqrt {f} \left (-i \sqrt {e}+\frac {\sqrt {-d} \sqrt {g}}{\sqrt {f}}\right ) \left (1-\frac {i \sqrt {g} x}{\sqrt {f}}\right )}\right )}{1+\frac {g x^2}{f}} \, dx}{f g}+\frac {p \int \frac {\log \left (\frac {2 \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\sqrt {f} \left (i \sqrt {e}+\frac {\sqrt {-d} \sqrt {g}}{\sqrt {f}}\right ) \left (1-\frac {i \sqrt {g} x}{\sqrt {f}}\right )}\right )}{1+\frac {g x^2}{f}} \, dx}{f g}+\frac {(e p) \int \frac {-d \sqrt {g}-e \sqrt {-f} x}{d+e x^2} \, dx}{2 g^{3/2} (e f-d g)}-\frac {(e p) \int \frac {d \sqrt {g}-e \sqrt {-f} x}{d+e x^2} \, dx}{2 g^{3/2} (e f-d g)}\\ &=-\frac {e \sqrt {-f} p \log \left (\sqrt {-f}-\sqrt {g} x\right )}{2 g^{3/2} (e f-d g)}+\frac {2 p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{\sqrt {f} g^{3/2}}-\frac {p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}-\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{\sqrt {f} g^{3/2}}-\frac {p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{\sqrt {f} g^{3/2}}+\frac {e \sqrt {-f} p \log \left (\sqrt {-f}+\sqrt {g} x\right )}{2 g^{3/2} (e f-d g)}+\frac {\log \left (c \left (d+e x^2\right )^p\right )}{4 g^{3/2} \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{4 g^{3/2} \left (\sqrt {-f}+\sqrt {g} x\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 \sqrt {f} g^{3/2}}+\frac {i p \text {Li}_2\left (1+\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}-\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 \sqrt {f} g^{3/2}}+\frac {i p \text {Li}_2\left (1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 \sqrt {f} g^{3/2}}-2 \frac {(i p) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\frac {i \sqrt {g} x}{\sqrt {f}}}\right )}{\sqrt {f} g^{3/2}}-\frac {\left (\sqrt {e} p\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {-d}-\sqrt {e} x} \, dx}{2 \sqrt {f} g^{3/2}}+\frac {\left (\sqrt {e} p\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 \sqrt {f} g^{3/2}}-2 \frac {(d e p) \int \frac {1}{d+e x^2} \, dx}{2 g (e f-d g)}\\ &=-\frac {\sqrt {d} \sqrt {e} p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{g (e f-d g)}-\frac {e \sqrt {-f} p \log \left (\sqrt {-f}-\sqrt {g} x\right )}{2 g^{3/2} (e f-d g)}+\frac {p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{\sqrt {f} g^{3/2}}-\frac {p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}-\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 \sqrt {f} g^{3/2}}-\frac {p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 \sqrt {f} g^{3/2}}+\frac {e \sqrt {-f} p \log \left (\sqrt {-f}+\sqrt {g} x\right )}{2 g^{3/2} (e f-d g)}+\frac {\log \left (c \left (d+e x^2\right )^p\right )}{4 g^{3/2} \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{4 g^{3/2} \left (\sqrt {-f}+\sqrt {g} x\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 \sqrt {f} g^{3/2}}-\frac {i p \text {Li}_2\left (1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{\sqrt {f} g^{3/2}}+\frac {i p \text {Li}_2\left (1+\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}-\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 \sqrt {f} g^{3/2}}+\frac {i p \text {Li}_2\left (1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 \sqrt {f} g^{3/2}}+2 \frac {p \int \frac {\log \left (\frac {2}{1-\frac {i \sqrt {g} x}{\sqrt {f}}}\right )}{1+\frac {g x^2}{f}} \, dx}{2 f g}-\frac {p \int \frac {\log \left (\frac {2 \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\sqrt {f} \left (-i \sqrt {e}+\frac {\sqrt {-d} \sqrt {g}}{\sqrt {f}}\right ) \left (1-\frac {i \sqrt {g} x}{\sqrt {f}}\right )}\right )}{1+\frac {g x^2}{f}} \, dx}{2 f g}-\frac {p \int \frac {\log \left (\frac {2 \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\sqrt {f} \left (i \sqrt {e}+\frac {\sqrt {-d} \sqrt {g}}{\sqrt {f}}\right ) \left (1-\frac {i \sqrt {g} x}{\sqrt {f}}\right )}\right )}{1+\frac {g x^2}{f}} \, dx}{2 f g}\\ &=-\frac {\sqrt {d} \sqrt {e} p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{g (e f-d g)}-\frac {e \sqrt {-f} p \log \left (\sqrt {-f}-\sqrt {g} x\right )}{2 g^{3/2} (e f-d g)}+\frac {p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{\sqrt {f} g^{3/2}}-\frac {p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}-\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 \sqrt {f} g^{3/2}}-\frac {p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 \sqrt {f} g^{3/2}}+\frac {e \sqrt {-f} p \log \left (\sqrt {-f}+\sqrt {g} x\right )}{2 g^{3/2} (e f-d g)}+\frac {\log \left (c \left (d+e x^2\right )^p\right )}{4 g^{3/2} \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{4 g^{3/2} \left (\sqrt {-f}+\sqrt {g} x\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 \sqrt {f} g^{3/2}}-\frac {i p \text {Li}_2\left (1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{\sqrt {f} g^{3/2}}+\frac {i p \text {Li}_2\left (1+\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}-\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{4 \sqrt {f} g^{3/2}}+\frac {i p \text {Li}_2\left (1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{4 \sqrt {f} g^{3/2}}+2 \frac {(i p) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\frac {i \sqrt {g} x}{\sqrt {f}}}\right )}{2 \sqrt {f} g^{3/2}}\\ &=-\frac {\sqrt {d} \sqrt {e} p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{g (e f-d g)}-\frac {e \sqrt {-f} p \log \left (\sqrt {-f}-\sqrt {g} x\right )}{2 g^{3/2} (e f-d g)}+\frac {p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{\sqrt {f} g^{3/2}}-\frac {p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}-\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 \sqrt {f} g^{3/2}}-\frac {p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 \sqrt {f} g^{3/2}}+\frac {e \sqrt {-f} p \log \left (\sqrt {-f}+\sqrt {g} x\right )}{2 g^{3/2} (e f-d g)}+\frac {\log \left (c \left (d+e x^2\right )^p\right )}{4 g^{3/2} \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{4 g^{3/2} \left (\sqrt {-f}+\sqrt {g} x\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 \sqrt {f} g^{3/2}}-\frac {i p \text {Li}_2\left (1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{2 \sqrt {f} g^{3/2}}+\frac {i p \text {Li}_2\left (1+\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}-\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{4 \sqrt {f} g^{3/2}}+\frac {i p \text {Li}_2\left (1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{4 \sqrt {f} g^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 3.52, size = 1231, normalized size = 1.65 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \left (\log \left (c \left (e x^2+d\right )^p\right )-p \log \left (e x^2+d\right )\right )}{2 \sqrt {f} g^{3/2}}+\frac {p x \log \left (e x^2+d\right )-x \log \left (c \left (e x^2+d\right )^p\right )}{2 g^2 x^2+2 f g}+\frac {1}{4} p \left (-\frac {i \left (\frac {\log \left (x-\frac {i \sqrt {d}}{\sqrt {e}}\right )}{i \sqrt {g} x+\sqrt {f}}+\frac {\sqrt {e} \left (\log \left (i \sqrt {f}-\sqrt {g} x\right )-\log \left (i \sqrt {d}-\sqrt {e} x\right )\right )}{\sqrt {e} \sqrt {f}-\sqrt {d} \sqrt {g}}\right )}{g^{3/2}}-\frac {i \left (\frac {\log \left (x+\frac {i \sqrt {d}}{\sqrt {e}}\right )}{i \sqrt {g} x+\sqrt {f}}+\frac {\sqrt {e} \left (\log \left (i \sqrt {f}-\sqrt {g} x\right )-\log \left (\sqrt {e} x+i \sqrt {d}\right )\right )}{\sqrt {e} \sqrt {f}+\sqrt {d} \sqrt {g}}\right )}{g^{3/2}}+\frac {\sqrt {e} \left (\sqrt {f}-i \sqrt {g} x\right ) \left (\log \left (i \sqrt {d}-\sqrt {e} x\right )-\log \left (\sqrt {g} x+i \sqrt {f}\right )\right )-\left (\sqrt {e} \sqrt {f}+\sqrt {d} \sqrt {g}\right ) \log \left (x-\frac {i \sqrt {d}}{\sqrt {e}}\right )}{\left (\sqrt {e} \sqrt {f}+\sqrt {d} \sqrt {g}\right ) g^{3/2} \left (\sqrt {g} x+i \sqrt {f}\right )}+\frac {-\frac {\log \left (x+\frac {i \sqrt {d}}{\sqrt {e}}\right )}{\sqrt {g} x+i \sqrt {f}}-\frac {i \sqrt {e} \left (\log \left (\sqrt {e} x+i \sqrt {d}\right )-\log \left (\sqrt {g} x+i \sqrt {f}\right )\right )}{\sqrt {e} \sqrt {f}-\sqrt {d} \sqrt {g}}}{g^{3/2}}+4 \left (\frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{2 \sqrt {f} g^{3/2}}-\frac {x}{2 g \left (g x^2+f\right )}\right ) \left (-\log \left (x-\frac {i \sqrt {d}}{\sqrt {e}}\right )-\log \left (x+\frac {i \sqrt {d}}{\sqrt {e}}\right )+\log \left (e x^2+d\right )\right )+\frac {i \left (\log \left (x+\frac {i \sqrt {d}}{\sqrt {e}}\right ) \log \left (\frac {\sqrt {e} \left (\sqrt {f}-i \sqrt {g} x\right )}{\sqrt {e} \sqrt {f}-\sqrt {d} \sqrt {g}}\right )+\text {Li}_2\left (-\frac {\sqrt {g} \left (\sqrt {d}-i \sqrt {e} x\right )}{\sqrt {e} \sqrt {f}-\sqrt {d} \sqrt {g}}\right )\right )}{\sqrt {f} g^{3/2}}-\frac {i \left (\log \left (x+\frac {i \sqrt {d}}{\sqrt {e}}\right ) \log \left (\frac {\sqrt {e} \left (i \sqrt {g} x+\sqrt {f}\right )}{\sqrt {e} \sqrt {f}+\sqrt {d} \sqrt {g}}\right )+\text {Li}_2\left (\frac {\sqrt {g} \left (\sqrt {d}-i \sqrt {e} x\right )}{\sqrt {e} \sqrt {f}+\sqrt {d} \sqrt {g}}\right )\right )}{\sqrt {f} g^{3/2}}-\frac {i \left (\log \left (x-\frac {i \sqrt {d}}{\sqrt {e}}\right ) \log \left (\frac {\sqrt {e} \left (i \sqrt {g} x+\sqrt {f}\right )}{\sqrt {e} \sqrt {f}-\sqrt {d} \sqrt {g}}\right )+\text {Li}_2\left (-\frac {\sqrt {g} \left (i \sqrt {e} x+\sqrt {d}\right )}{\sqrt {e} \sqrt {f}-\sqrt {d} \sqrt {g}}\right )\right )}{\sqrt {f} g^{3/2}}+\frac {i \left (\log \left (x-\frac {i \sqrt {d}}{\sqrt {e}}\right ) \log \left (\frac {\sqrt {e} \left (\sqrt {f}-i \sqrt {g} x\right )}{\sqrt {e} \sqrt {f}+\sqrt {d} \sqrt {g}}\right )+\text {Li}_2\left (\frac {\sqrt {g} \left (i \sqrt {e} x+\sqrt {d}\right )}{\sqrt {e} \sqrt {f}+\sqrt {d} \sqrt {g}}\right )\right )}{\sqrt {f} g^{3/2}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{2} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{g^{2} x^{4} + 2 \, f g x^{2} + f^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{{\left (g x^{2} + f\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 1.14, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{\left (g \,x^{2}+f \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{{\left (g x^{2} + f\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^2\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}{{\left (g\,x^2+f\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________