Optimal. Leaf size=581 \[ -\frac {\sqrt {g} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{f^{3/2}}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{f x}-\frac {i \sqrt {g} p \text {Li}_2\left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}-\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}+1\right )}{2 f^{3/2}}-\frac {i \sqrt {g} p \text {Li}_2\left (1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 f^{3/2}}+\frac {\sqrt {g} p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (-\sqrt {-d} \sqrt {g}+i \sqrt {e} \sqrt {f}\right )}\right )}{f^{3/2}}+\frac {\sqrt {g} p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (\sqrt {-d} \sqrt {g}+i \sqrt {e} \sqrt {f}\right )}\right )}{f^{3/2}}+\frac {2 \sqrt {e} p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} f}+\frac {i \sqrt {g} p \text {Li}_2\left (1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{f^{3/2}}-\frac {2 \sqrt {g} p \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right ) \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{f^{3/2}} \]
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Rubi [A] time = 0.60, antiderivative size = 581, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2476, 2455, 205, 2470, 12, 4928, 4856, 2402, 2315, 2447} \[ -\frac {i \sqrt {g} p \text {PolyLog}\left (2,1+\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (-\sqrt {-d} \sqrt {g}+i \sqrt {e} \sqrt {f}\right )}\right )}{2 f^{3/2}}-\frac {i \sqrt {g} p \text {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (\sqrt {-d} \sqrt {g}+i \sqrt {e} \sqrt {f}\right )}\right )}{2 f^{3/2}}+\frac {i \sqrt {g} p \text {PolyLog}\left (2,1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{f^{3/2}}-\frac {\sqrt {g} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{f^{3/2}}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{f x}+\frac {\sqrt {g} p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (-\sqrt {-d} \sqrt {g}+i \sqrt {e} \sqrt {f}\right )}\right )}{f^{3/2}}+\frac {\sqrt {g} p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (\sqrt {-d} \sqrt {g}+i \sqrt {e} \sqrt {f}\right )}\right )}{f^{3/2}}+\frac {2 \sqrt {e} p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} f}-\frac {2 \sqrt {g} p \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right ) \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{f^{3/2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 205
Rule 2315
Rule 2402
Rule 2447
Rule 2455
Rule 2470
Rule 2476
Rule 4856
Rule 4928
Rubi steps
\begin {align*} \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{x^2 \left (f+g x^2\right )} \, dx &=\int \left (\frac {\log \left (c \left (d+e x^2\right )^p\right )}{f x^2}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{f \left (f+g x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{x^2} \, dx}{f}-\frac {g \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx}{f}\\ &=-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{f x}-\frac {\sqrt {g} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{f^{3/2}}+\frac {(2 e p) \int \frac {1}{d+e x^2} \, dx}{f}+\frac {(2 e g 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}{f}\\ &=\frac {2 \sqrt {e} p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} f}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{f x}-\frac {\sqrt {g} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{f^{3/2}}+\frac {\left (2 e \sqrt {g} p\right ) \int \frac {x \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{d+e x^2} \, dx}{f^{3/2}}\\ &=\frac {2 \sqrt {e} p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} f}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{f x}-\frac {\sqrt {g} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{f^{3/2}}+\frac {\left (2 e \sqrt {g} p\right ) \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}{f^{3/2}}\\ &=\frac {2 \sqrt {e} p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} f}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{f x}-\frac {\sqrt {g} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{f^{3/2}}-\frac {\left (\sqrt {e} \sqrt {g} p\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {-d}-\sqrt {e} x} \, dx}{f^{3/2}}+\frac {\left (\sqrt {e} \sqrt {g} p\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {-d}+\sqrt {e} x} \, dx}{f^{3/2}}\\ &=\frac {2 \sqrt {e} p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} f}-\frac {2 \sqrt {g} p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{f^{3/2}}+\frac {\sqrt {g} 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 )}{f^{3/2}}+\frac {\sqrt {g} 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 )}{f^{3/2}}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{f x}-\frac {\sqrt {g} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{f^{3/2}}+2 \frac {(g p) \int \frac {\log \left (\frac {2}{1-\frac {i \sqrt {g} x}{\sqrt {f}}}\right )}{1+\frac {g x^2}{f}} \, dx}{f^2}-\frac {(g 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^2}-\frac {(g 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^2}\\ &=\frac {2 \sqrt {e} p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} f}-\frac {2 \sqrt {g} p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{f^{3/2}}+\frac {\sqrt {g} 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 )}{f^{3/2}}+\frac {\sqrt {g} 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 )}{f^{3/2}}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{f x}-\frac {\sqrt {g} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{f^{3/2}}-\frac {i \sqrt {g} 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 f^{3/2}}-\frac {i \sqrt {g} 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 f^{3/2}}+2 \frac {\left (i \sqrt {g} p\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\frac {i \sqrt {g} x}{\sqrt {f}}}\right )}{f^{3/2}}\\ &=\frac {2 \sqrt {e} p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} f}-\frac {2 \sqrt {g} p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{f^{3/2}}+\frac {\sqrt {g} 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 )}{f^{3/2}}+\frac {\sqrt {g} 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 )}{f^{3/2}}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{f x}-\frac {\sqrt {g} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{f^{3/2}}+\frac {i \sqrt {g} p \text {Li}_2\left (1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{f^{3/2}}-\frac {i \sqrt {g} 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 f^{3/2}}-\frac {i \sqrt {g} 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 f^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 673, normalized size = 1.16 \[ \frac {-2 \sqrt {g} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )-\frac {2 \sqrt {f} \log \left (c \left (d+e x^2\right )^p\right )}{x}+i \sqrt {g} p \text {Li}_2\left (\frac {\sqrt {e} \left (\sqrt {f}-i \sqrt {g} x\right )}{\sqrt {e} \sqrt {f}-i \sqrt {-d} \sqrt {g}}\right )+i \sqrt {g} p \text {Li}_2\left (\frac {\sqrt {e} \left (\sqrt {f}-i \sqrt {g} x\right )}{\sqrt {e} \sqrt {f}+i \sqrt {-d} \sqrt {g}}\right )-i \sqrt {g} p \text {Li}_2\left (\frac {\sqrt {e} \left (i \sqrt {g} x+\sqrt {f}\right )}{\sqrt {e} \sqrt {f}-i \sqrt {-d} \sqrt {g}}\right )-i \sqrt {g} p \text {Li}_2\left (\frac {\sqrt {e} \left (i \sqrt {g} x+\sqrt {f}\right )}{\sqrt {e} \sqrt {f}+i \sqrt {-d} \sqrt {g}}\right )+i \sqrt {g} p \log \left (1-\frac {i \sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {\sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\sqrt {-d} \sqrt {g}+i \sqrt {e} \sqrt {f}}\right )+i \sqrt {g} p \log \left (1-\frac {i \sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {\sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\sqrt {-d} \sqrt {g}-i \sqrt {e} \sqrt {f}}\right )-i \sqrt {g} p \log \left (1+\frac {i \sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {\sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\sqrt {-d} \sqrt {g}-i \sqrt {e} \sqrt {f}}\right )-i \sqrt {g} p \log \left (1+\frac {i \sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {\sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\sqrt {-d} \sqrt {g}+i \sqrt {e} \sqrt {f}}\right )+\frac {4 \sqrt {e} \sqrt {f} p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d}}}{2 f^{3/2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{g x^{4} + f x^{2}}, x\right ) \]
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 \[ \int \frac {\log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{{\left (g x^{2} + f\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.40, size = 755, normalized size = 1.30 \[ \text {result too large to display} \]
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 \[ \int \frac {\log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{{\left (g x^{2} + f\right )} x^{2}}\,{d x} \]
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 \[ \int \frac {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}{x^2\,\left (g\,x^2+f\right )} \,d x \]
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 \[ \int \frac {\log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{x^{2} \left (f + g x^{2}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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