Optimal. Leaf size=491 \[ \frac {a+b \log \left (c (d+e x)^n\right )}{4 g^{3/2} \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {a+b \log \left (c (d+e x)^n\right )}{4 g^{3/2} \left (\sqrt {-f}+\sqrt {g} x\right )}+\frac {\log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{d \sqrt {g}+e \sqrt {-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 \sqrt {-f} g^{3/2}}-\frac {\log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 \sqrt {-f} g^{3/2}}-\frac {b n \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{4 \sqrt {-f} g^{3/2}}+\frac {b n \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right )}{4 \sqrt {-f} g^{3/2}}+\frac {b e n \log (d+e x)}{4 g^{3/2} \left (e \sqrt {-f}-d \sqrt {g}\right )}-\frac {b e n \log (d+e x)}{4 g^{3/2} \left (d \sqrt {g}+e \sqrt {-f}\right )}+\frac {b e n \log \left (\sqrt {-f}-\sqrt {g} x\right )}{4 g^{3/2} \left (d \sqrt {g}+e \sqrt {-f}\right )}-\frac {b e n \log \left (\sqrt {-f}+\sqrt {g} x\right )}{4 g^{3/2} \left (e \sqrt {-f}-d \sqrt {g}\right )} \]
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Rubi [A] time = 0.76, antiderivative size = 491, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 10, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {288, 205, 2416, 2409, 2395, 36, 31, 2394, 2393, 2391} \[ -\frac {b n \text {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{4 \sqrt {-f} g^{3/2}}+\frac {b n \text {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{d \sqrt {g}+e \sqrt {-f}}\right )}{4 \sqrt {-f} g^{3/2}}+\frac {a+b \log \left (c (d+e x)^n\right )}{4 g^{3/2} \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {a+b \log \left (c (d+e x)^n\right )}{4 g^{3/2} \left (\sqrt {-f}+\sqrt {g} x\right )}+\frac {\log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{d \sqrt {g}+e \sqrt {-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 \sqrt {-f} g^{3/2}}-\frac {\log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 \sqrt {-f} g^{3/2}}+\frac {b e n \log (d+e x)}{4 g^{3/2} \left (e \sqrt {-f}-d \sqrt {g}\right )}-\frac {b e n \log (d+e x)}{4 g^{3/2} \left (d \sqrt {g}+e \sqrt {-f}\right )}+\frac {b e n \log \left (\sqrt {-f}-\sqrt {g} x\right )}{4 g^{3/2} \left (d \sqrt {g}+e \sqrt {-f}\right )}-\frac {b e n \log \left (\sqrt {-f}+\sqrt {g} x\right )}{4 g^{3/2} \left (e \sqrt {-f}-d \sqrt {g}\right )} \]
Antiderivative was successfully verified.
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Rule 31
Rule 36
Rule 205
Rule 288
Rule 2391
Rule 2393
Rule 2394
Rule 2395
Rule 2409
Rule 2416
Rubi steps
\begin {align*} \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx &=\int \left (-\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \left (f+g x^2\right )^2}+\frac {a+b \log \left (c (d+e x)^n\right )}{g \left (f+g x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {a+b \log \left (c (d+e x)^n\right )}{f+g x^2} \, dx}{g}-\frac {f \int \frac {a+b \log \left (c (d+e x)^n\right )}{\left (f+g x^2\right )^2} \, dx}{g}\\ &=\frac {\int \left (\frac {\sqrt {-f} \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {\sqrt {-f} \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f \left (\sqrt {-f}+\sqrt {g} x\right )}\right ) \, dx}{g}-\frac {f \int \left (-\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 f \left (\sqrt {-f} \sqrt {g}-g x\right )^2}-\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 f \left (\sqrt {-f} \sqrt {g}+g x\right )^2}-\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f \left (-f g-g^2 x^2\right )}\right ) \, dx}{g}\\ &=\frac {1}{4} \int \frac {a+b \log \left (c (d+e x)^n\right )}{\left (\sqrt {-f} \sqrt {g}-g x\right )^2} \, dx+\frac {1}{4} \int \frac {a+b \log \left (c (d+e x)^n\right )}{\left (\sqrt {-f} \sqrt {g}+g x\right )^2} \, dx+\frac {1}{2} \int \frac {a+b \log \left (c (d+e x)^n\right )}{-f g-g^2 x^2} \, dx-\frac {\int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {-f}-\sqrt {g} x} \, dx}{2 \sqrt {-f} g}-\frac {\int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {-f}+\sqrt {g} x} \, dx}{2 \sqrt {-f} g}\\ &=\frac {a+b \log \left (c (d+e x)^n\right )}{4 g^{3/2} \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {a+b \log \left (c (d+e x)^n\right )}{4 g^{3/2} \left (\sqrt {-f}+\sqrt {g} x\right )}+\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 \sqrt {-f} g^{3/2}}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \sqrt {-f} g^{3/2}}+\frac {1}{2} \int \left (-\frac {\sqrt {-f} \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f g \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {\sqrt {-f} \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f g \left (\sqrt {-f}+\sqrt {g} x\right )}\right ) \, dx-\frac {(b e n) \int \frac {\log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{d+e x} \, dx}{2 \sqrt {-f} g^{3/2}}+\frac {(b e n) \int \frac {\log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{d+e x} \, dx}{2 \sqrt {-f} g^{3/2}}-\frac {(b e n) \int \frac {1}{(d+e x) \left (\sqrt {-f} \sqrt {g}-g x\right )} \, dx}{4 g}+\frac {(b e n) \int \frac {1}{(d+e x) \left (\sqrt {-f} \sqrt {g}+g x\right )} \, dx}{4 g}\\ &=\frac {a+b \log \left (c (d+e x)^n\right )}{4 g^{3/2} \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {a+b \log \left (c (d+e x)^n\right )}{4 g^{3/2} \left (\sqrt {-f}+\sqrt {g} x\right )}+\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 \sqrt {-f} g^{3/2}}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \sqrt {-f} g^{3/2}}+\frac {\int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {-f}-\sqrt {g} x} \, dx}{4 \sqrt {-f} g}+\frac {\int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {-f}+\sqrt {g} x} \, dx}{4 \sqrt {-f} g}+\frac {(b n) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {g} x}{e \sqrt {-f}-d \sqrt {g}}\right )}{x} \, dx,x,d+e x\right )}{2 \sqrt {-f} g^{3/2}}-\frac {(b n) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {g} x}{e \sqrt {-f}+d \sqrt {g}}\right )}{x} \, dx,x,d+e x\right )}{2 \sqrt {-f} g^{3/2}}+\frac {\left (b e^2 n\right ) \int \frac {1}{d+e x} \, dx}{4 \left (e \sqrt {-f}-d \sqrt {g}\right ) g^{3/2}}-\frac {\left (b e^2 n\right ) \int \frac {1}{d+e x} \, dx}{4 \left (e \sqrt {-f}+d \sqrt {g}\right ) g^{3/2}}-\frac {(b e n) \int \frac {1}{\sqrt {-f} \sqrt {g}+g x} \, dx}{4 \left (e \sqrt {-f}-d \sqrt {g}\right ) \sqrt {g}}-\frac {(b e n) \int \frac {1}{\sqrt {-f} \sqrt {g}-g x} \, dx}{4 \left (e \sqrt {-f}+d \sqrt {g}\right ) \sqrt {g}}\\ &=\frac {b e n \log (d+e x)}{4 \left (e \sqrt {-f}-d \sqrt {g}\right ) g^{3/2}}-\frac {b e n \log (d+e x)}{4 \left (e \sqrt {-f}+d \sqrt {g}\right ) g^{3/2}}+\frac {a+b \log \left (c (d+e x)^n\right )}{4 g^{3/2} \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {a+b \log \left (c (d+e x)^n\right )}{4 g^{3/2} \left (\sqrt {-f}+\sqrt {g} x\right )}+\frac {b e n \log \left (\sqrt {-f}-\sqrt {g} x\right )}{4 \left (e \sqrt {-f}+d \sqrt {g}\right ) g^{3/2}}+\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{4 \sqrt {-f} g^{3/2}}-\frac {b e n \log \left (\sqrt {-f}+\sqrt {g} x\right )}{4 \left (e \sqrt {-f}-d \sqrt {g}\right ) g^{3/2}}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{4 \sqrt {-f} g^{3/2}}-\frac {b n \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \sqrt {-f} g^{3/2}}+\frac {b n \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 \sqrt {-f} g^{3/2}}+\frac {(b e n) \int \frac {\log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{d+e x} \, dx}{4 \sqrt {-f} g^{3/2}}-\frac {(b e n) \int \frac {\log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{d+e x} \, dx}{4 \sqrt {-f} g^{3/2}}\\ &=\frac {b e n \log (d+e x)}{4 \left (e \sqrt {-f}-d \sqrt {g}\right ) g^{3/2}}-\frac {b e n \log (d+e x)}{4 \left (e \sqrt {-f}+d \sqrt {g}\right ) g^{3/2}}+\frac {a+b \log \left (c (d+e x)^n\right )}{4 g^{3/2} \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {a+b \log \left (c (d+e x)^n\right )}{4 g^{3/2} \left (\sqrt {-f}+\sqrt {g} x\right )}+\frac {b e n \log \left (\sqrt {-f}-\sqrt {g} x\right )}{4 \left (e \sqrt {-f}+d \sqrt {g}\right ) g^{3/2}}+\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{4 \sqrt {-f} g^{3/2}}-\frac {b e n \log \left (\sqrt {-f}+\sqrt {g} x\right )}{4 \left (e \sqrt {-f}-d \sqrt {g}\right ) g^{3/2}}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{4 \sqrt {-f} g^{3/2}}-\frac {b n \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \sqrt {-f} g^{3/2}}+\frac {b n \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 \sqrt {-f} g^{3/2}}-\frac {(b n) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {g} x}{e \sqrt {-f}-d \sqrt {g}}\right )}{x} \, dx,x,d+e x\right )}{4 \sqrt {-f} g^{3/2}}+\frac {(b n) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {g} x}{e \sqrt {-f}+d \sqrt {g}}\right )}{x} \, dx,x,d+e x\right )}{4 \sqrt {-f} g^{3/2}}\\ &=\frac {b e n \log (d+e x)}{4 \left (e \sqrt {-f}-d \sqrt {g}\right ) g^{3/2}}-\frac {b e n \log (d+e x)}{4 \left (e \sqrt {-f}+d \sqrt {g}\right ) g^{3/2}}+\frac {a+b \log \left (c (d+e x)^n\right )}{4 g^{3/2} \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {a+b \log \left (c (d+e x)^n\right )}{4 g^{3/2} \left (\sqrt {-f}+\sqrt {g} x\right )}+\frac {b e n \log \left (\sqrt {-f}-\sqrt {g} x\right )}{4 \left (e \sqrt {-f}+d \sqrt {g}\right ) g^{3/2}}+\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{4 \sqrt {-f} g^{3/2}}-\frac {b e n \log \left (\sqrt {-f}+\sqrt {g} x\right )}{4 \left (e \sqrt {-f}-d \sqrt {g}\right ) g^{3/2}}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{4 \sqrt {-f} g^{3/2}}-\frac {b n \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{4 \sqrt {-f} g^{3/2}}+\frac {b n \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{4 \sqrt {-f} g^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.67, size = 383, normalized size = 0.78 \[ \frac {\frac {\log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{d \sqrt {g}+e \sqrt {-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {-f}}+\frac {f \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(-f)^{3/2}}+\frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {-f}-\sqrt {g} x}-\frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {-f}+\sqrt {g} x}+\frac {b f n \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{(-f)^{3/2}}+\frac {b n \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right )}{\sqrt {-f}}-\frac {b e n \left (\log (d+e x)-\log \left (\sqrt {-f}-\sqrt {g} x\right )\right )}{d \sqrt {g}+e \sqrt {-f}}+\frac {b e n \left (\log (d+e x)-\log \left (\sqrt {-f}+\sqrt {g} x\right )\right )}{e \sqrt {-f}-d \sqrt {g}}}{4 g^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x^{2} \log \left ({\left (e x + d\right )}^{n} c\right ) + a x^{2}}{g^{2} x^{4} + 2 \, f g x^{2} + f^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} x^{2}}{{\left (g x^{2} + f\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.45, size = 1781, normalized size = 3.63 \[ \text {result too large to display} \]
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 \[ -\frac {1}{2} \, a {\left (\frac {x}{g^{2} x^{2} + f g} - \frac {\arctan \left (\frac {g x}{\sqrt {f g}}\right )}{\sqrt {f g} g}\right )} + b \int \frac {x^{2} \log \left ({\left (e x + d\right )}^{n}\right ) + x^{2} \log \relax (c)}{g^{2} x^{4} + 2 \, f g x^{2} + f^{2}}\,{d x} \]
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 \[ \int \frac {x^2\,\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}{{\left (g\,x^2+f\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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