Optimal. Leaf size=328 \[ \frac {\sqrt {d} \sqrt {\frac {e x^2}{d}+1} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{3/2} \sqrt {d+e x^2}}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{e \sqrt {d+e x^2}}-\frac {b \sqrt {d} n \sqrt {\frac {e x^2}{d}+1} \text {Li}_2\left (e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{2 e^{3/2} \sqrt {d+e x^2}}+\frac {b \sqrt {d} n \sqrt {\frac {e x^2}{d}+1} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{2 e^{3/2} \sqrt {d+e x^2}}+\frac {b \sqrt {d} n \sqrt {\frac {e x^2}{d}+1} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{e^{3/2} \sqrt {d+e x^2}}-\frac {b \sqrt {d} n \sqrt {\frac {e x^2}{d}+1} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{e^{3/2} \sqrt {d+e x^2}} \]
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Rubi [A] time = 0.47, antiderivative size = 328, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {2341, 288, 215, 2350, 14, 21, 5659, 3716, 2190, 2279, 2391} \[ -\frac {b \sqrt {d} n \sqrt {\frac {e x^2}{d}+1} \text {PolyLog}\left (2,e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{2 e^{3/2} \sqrt {d+e x^2}}+\frac {\sqrt {d} \sqrt {\frac {e x^2}{d}+1} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{3/2} \sqrt {d+e x^2}}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{e \sqrt {d+e x^2}}+\frac {b \sqrt {d} n \sqrt {\frac {e x^2}{d}+1} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{2 e^{3/2} \sqrt {d+e x^2}}+\frac {b \sqrt {d} n \sqrt {\frac {e x^2}{d}+1} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{e^{3/2} \sqrt {d+e x^2}}-\frac {b \sqrt {d} n \sqrt {\frac {e x^2}{d}+1} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{e^{3/2} \sqrt {d+e x^2}} \]
Antiderivative was successfully verified.
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Rule 14
Rule 21
Rule 215
Rule 288
Rule 2190
Rule 2279
Rule 2341
Rule 2350
Rule 2391
Rule 3716
Rule 5659
Rubi steps
\begin {align*} \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{3/2}} \, dx &=\frac {\sqrt {1+\frac {e x^2}{d}} \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\left (1+\frac {e x^2}{d}\right )^{3/2}} \, dx}{d \sqrt {d+e x^2}}\\ &=-\frac {x \left (a+b \log \left (c x^n\right )\right )}{e \sqrt {d+e x^2}}+\frac {\sqrt {d} \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{3/2} \sqrt {d+e x^2}}-\frac {\left (b n \sqrt {1+\frac {e x^2}{d}}\right ) \int \frac {-\frac {d x}{e \sqrt {1+\frac {e x^2}{d}}}+\frac {d^{3/2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{e^{3/2}}}{x} \, dx}{d \sqrt {d+e x^2}}\\ &=-\frac {x \left (a+b \log \left (c x^n\right )\right )}{e \sqrt {d+e x^2}}+\frac {\sqrt {d} \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{3/2} \sqrt {d+e x^2}}-\frac {\left (b n \sqrt {1+\frac {e x^2}{d}}\right ) \int \left (\frac {d^2 \sqrt {1+\frac {e x^2}{d}}}{e \left (-d-e x^2\right )}+\frac {d^{3/2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{e^{3/2} x}\right ) \, dx}{d \sqrt {d+e x^2}}\\ &=-\frac {x \left (a+b \log \left (c x^n\right )\right )}{e \sqrt {d+e x^2}}+\frac {\sqrt {d} \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{3/2} \sqrt {d+e x^2}}-\frac {\left (b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}}\right ) \int \frac {\sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{e^{3/2} \sqrt {d+e x^2}}-\frac {\left (b d n \sqrt {1+\frac {e x^2}{d}}\right ) \int \frac {\sqrt {1+\frac {e x^2}{d}}}{-d-e x^2} \, dx}{e \sqrt {d+e x^2}}\\ &=-\frac {x \left (a+b \log \left (c x^n\right )\right )}{e \sqrt {d+e x^2}}+\frac {\sqrt {d} \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{3/2} \sqrt {d+e x^2}}-\frac {\left (b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}}\right ) \operatorname {Subst}\left (\int x \coth (x) \, dx,x,\sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right )}{e^{3/2} \sqrt {d+e x^2}}+\frac {\left (b n \sqrt {1+\frac {e x^2}{d}}\right ) \int \frac {1}{\sqrt {1+\frac {e x^2}{d}}} \, dx}{e \sqrt {d+e x^2}}\\ &=\frac {b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{e^{3/2} \sqrt {d+e x^2}}+\frac {b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{2 e^{3/2} \sqrt {d+e x^2}}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{e \sqrt {d+e x^2}}+\frac {\sqrt {d} \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{3/2} \sqrt {d+e x^2}}+\frac {\left (2 b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}}\right ) \operatorname {Subst}\left (\int \frac {e^{2 x} x}{1-e^{2 x}} \, dx,x,\sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right )}{e^{3/2} \sqrt {d+e x^2}}\\ &=\frac {b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{e^{3/2} \sqrt {d+e x^2}}+\frac {b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{2 e^{3/2} \sqrt {d+e x^2}}-\frac {b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{e^{3/2} \sqrt {d+e x^2}}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{e \sqrt {d+e x^2}}+\frac {\sqrt {d} \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{3/2} \sqrt {d+e x^2}}+\frac {\left (b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}}\right ) \operatorname {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right )}{e^{3/2} \sqrt {d+e x^2}}\\ &=\frac {b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{e^{3/2} \sqrt {d+e x^2}}+\frac {b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{2 e^{3/2} \sqrt {d+e x^2}}-\frac {b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{e^{3/2} \sqrt {d+e x^2}}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{e \sqrt {d+e x^2}}+\frac {\sqrt {d} \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{3/2} \sqrt {d+e x^2}}+\frac {\left (b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}}\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{2 e^{3/2} \sqrt {d+e x^2}}\\ &=\frac {b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{e^{3/2} \sqrt {d+e x^2}}+\frac {b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{2 e^{3/2} \sqrt {d+e x^2}}-\frac {b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{e^{3/2} \sqrt {d+e x^2}}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{e \sqrt {d+e x^2}}+\frac {\sqrt {d} \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{3/2} \sqrt {d+e x^2}}-\frac {b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}} \text {Li}_2\left (e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{2 e^{3/2} \sqrt {d+e x^2}}\\ \end {align*}
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Mathematica [C] time = 0.48, size = 217, normalized size = 0.66 \[ -\frac {b n \sqrt {\frac {e x^2}{d}+1} \left (e^{3/2} x^3 \left (d+e x^2\right ) \, _3F_2\left (\frac {3}{2},\frac {3}{2},\frac {3}{2};\frac {5}{2},\frac {5}{2};-\frac {e x^2}{d}\right )-9 d^{3/2} \log (x) \left (d+e x^2\right ) \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )+9 d^2 \sqrt {e} x \log (x) \sqrt {\frac {e x^2}{d}+1}\right )}{9 d e^{3/2} \left (d+e x^2\right )^{3/2}}+\frac {\log \left (\sqrt {e} \sqrt {d+e x^2}+e x\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{e^{3/2}}-\frac {x \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{e \sqrt {d+e x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x^{2} + d} b x^{2} \log \left (c x^{n}\right ) + \sqrt {e x^{2} + d} a x^{2}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.32, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right ) x^{2}}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}\, dx \]
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 \[ -a {\left (\frac {x}{\sqrt {e x^{2} + d} e} - \frac {\operatorname {arsinh}\left (\frac {e x}{\sqrt {d e}}\right )}{e^{\frac {3}{2}}}\right )} + b \int \frac {x^{2} \log \relax (c) + x^{2} \log \left (x^{n}\right )}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x^2\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (e\,x^2+d\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{2} \left (a + b \log {\left (c x^{n} \right )}\right )}{\left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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