Optimal. Leaf size=187 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 d^{3/2} e^{3/2}}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{8 d e \left (d+e x^2\right )}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 e \left (d+e x^2\right )^2}-\frac {i b n \text {Li}_2\left (-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{3/2} e^{3/2}}+\frac {i b n \text {Li}_2\left (\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{3/2} e^{3/2}}+\frac {b n x}{8 d e \left (d+e x^2\right )} \]
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Rubi [A] time = 0.37, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {288, 199, 205, 2351, 2323, 2324, 12, 4848, 2391} \[ -\frac {i b n \text {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{3/2} e^{3/2}}+\frac {i b n \text {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{3/2} e^{3/2}}+\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 d^{3/2} e^{3/2}}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{8 d e \left (d+e x^2\right )}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 e \left (d+e x^2\right )^2}+\frac {b n x}{8 d e \left (d+e x^2\right )} \]
Antiderivative was successfully verified.
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Rule 12
Rule 199
Rule 205
Rule 288
Rule 2323
Rule 2324
Rule 2351
Rule 2391
Rule 4848
Rubi steps
\begin {align*} \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx &=\int \left (-\frac {d \left (a+b \log \left (c x^n\right )\right )}{e \left (d+e x^2\right )^3}+\frac {a+b \log \left (c x^n\right )}{e \left (d+e x^2\right )^2}\right ) \, dx\\ &=\frac {\int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^2} \, dx}{e}-\frac {d \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^3} \, dx}{e}\\ &=-\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 e \left (d+e x^2\right )^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{2 d e \left (d+e x^2\right )}-\frac {3 \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^2} \, dx}{4 e}+\frac {\int \frac {a+b \log \left (c x^n\right )}{d+e x^2} \, dx}{2 d e}+\frac {(b n) \int \frac {1}{\left (d+e x^2\right )^2} \, dx}{4 e}-\frac {(b n) \int \frac {1}{d+e x^2} \, dx}{2 d e}\\ &=\frac {b n x}{8 d e \left (d+e x^2\right )}-\frac {b n \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} e^{3/2}}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 e \left (d+e x^2\right )^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{8 d e \left (d+e x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^{3/2} e^{3/2}}-\frac {3 \int \frac {a+b \log \left (c x^n\right )}{d+e x^2} \, dx}{8 d e}+\frac {(b n) \int \frac {1}{d+e x^2} \, dx}{8 d e}+\frac {(3 b n) \int \frac {1}{d+e x^2} \, dx}{8 d e}-\frac {(b n) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e} x} \, dx}{2 d e}\\ &=\frac {b n x}{8 d e \left (d+e x^2\right )}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 e \left (d+e x^2\right )^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{8 d e \left (d+e x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 d^{3/2} e^{3/2}}-\frac {(b n) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{2 d^{3/2} e^{3/2}}+\frac {(3 b n) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e} x} \, dx}{8 d e}\\ &=\frac {b n x}{8 d e \left (d+e x^2\right )}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 e \left (d+e x^2\right )^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{8 d e \left (d+e x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 d^{3/2} e^{3/2}}-\frac {(i b n) \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{4 d^{3/2} e^{3/2}}+\frac {(i b n) \int \frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{4 d^{3/2} e^{3/2}}+\frac {(3 b n) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{8 d^{3/2} e^{3/2}}\\ &=\frac {b n x}{8 d e \left (d+e x^2\right )}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 e \left (d+e x^2\right )^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{8 d e \left (d+e x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 d^{3/2} e^{3/2}}-\frac {i b n \text {Li}_2\left (-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{4 d^{3/2} e^{3/2}}+\frac {i b n \text {Li}_2\left (\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{4 d^{3/2} e^{3/2}}+\frac {(3 i b n) \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{16 d^{3/2} e^{3/2}}-\frac {(3 i b n) \int \frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{16 d^{3/2} e^{3/2}}\\ &=\frac {b n x}{8 d e \left (d+e x^2\right )}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 e \left (d+e x^2\right )^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{8 d e \left (d+e x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 d^{3/2} e^{3/2}}-\frac {i b n \text {Li}_2\left (-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{3/2} e^{3/2}}+\frac {i b n \text {Li}_2\left (\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{3/2} e^{3/2}}\\ \end {align*}
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Mathematica [B] time = 1.08, size = 497, normalized size = 2.66 \[ \frac {\frac {\log \left (\frac {\sqrt {e} x}{\sqrt {-d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{(-d)^{3/2}}+\frac {d \log \left (\frac {d \sqrt {e} x}{(-d)^{3/2}}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{(-d)^{5/2}}-\frac {a+b \log \left (c x^n\right )}{\sqrt {-d} d-d \sqrt {e} x}+\frac {a+b \log \left (c x^n\right )}{d \sqrt {e} x+\sqrt {-d} d}+\frac {d \left (a+b \log \left (c x^n\right )\right )}{(-d)^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )^2}+\frac {a+b \log \left (c x^n\right )}{\sqrt {-d} \left (\sqrt {-d}+\sqrt {e} x\right )^2}+\frac {b n \left (\log (x) \left (d-\sqrt {-d} \sqrt {e} x\right )+\left (\sqrt {-d} \sqrt {e} x-d\right ) \log \left (\sqrt {-d}+\sqrt {e} x\right )+d\right )}{d^2 \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {b n \left (\log (x) \left (\sqrt {-d} \sqrt {e} x+d\right )-\left (\sqrt {-d} \sqrt {e} x+d\right ) \log \left (d \sqrt {e} x+(-d)^{3/2}\right )+d\right )}{d^2 \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {b d n \text {Li}_2\left (\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{(-d)^{5/2}}+\frac {b n \text {Li}_2\left (\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )}{(-d)^{3/2}}+\frac {b d n \left (\log (x)-\log \left (\sqrt {-d}-\sqrt {e} x\right )\right )}{(-d)^{5/2}}+\frac {b n \left (\log (x)-\log \left (\sqrt {-d}+\sqrt {e} x\right )\right )}{(-d)^{3/2}}}{16 e^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x^{2} \log \left (c x^{n}\right ) + a x^{2}}{e^{3} x^{6} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{2} + d^{3}}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.36, size = 1247, normalized size = 6.67 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \]
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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