Optimal. Leaf size=400 \[ -\frac {b d n \sqrt {d+e x^2}}{x}-\frac {1}{4} b e n x \sqrt {d+e x^2}+\frac {3 b \sqrt {d} \sqrt {e} n \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{4 \sqrt {1+\frac {e x^2}{d}}}+\frac {3 b \sqrt {d} \sqrt {e} n \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{4 \sqrt {1+\frac {e x^2}{d}}}-\frac {3 b \sqrt {d} \sqrt {e} n \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{2 \sqrt {1+\frac {e x^2}{d}}}+\frac {3}{2} e x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x}+\frac {3 \sqrt {d} \sqrt {e} \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {1+\frac {e x^2}{d}}}-\frac {3 b \sqrt {d} \sqrt {e} n \sqrt {d+e x^2} \text {Li}_2\left (e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{4 \sqrt {1+\frac {e x^2}{d}}} \]
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Rubi [A]
time = 0.32, antiderivative size = 400, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 12, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {2386, 283,
201, 221, 2392, 12, 14, 5775, 3797, 2221, 2317, 2438} \begin {gather*} -\frac {3 b \sqrt {d} \sqrt {e} n \sqrt {d+e x^2} \text {PolyLog}\left (2,e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{4 \sqrt {\frac {e x^2}{d}+1}}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x}+\frac {3}{2} e x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac {3 \sqrt {d} \sqrt {e} \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {\frac {e x^2}{d}+1}}-\frac {1}{4} b e n x \sqrt {d+e x^2}-\frac {b d n \sqrt {d+e x^2}}{x}+\frac {3 b \sqrt {d} \sqrt {e} n \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{4 \sqrt {\frac {e x^2}{d}+1}}+\frac {3 b \sqrt {d} \sqrt {e} n \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{4 \sqrt {\frac {e x^2}{d}+1}}-\frac {3 b \sqrt {d} \sqrt {e} n \sqrt {d+e x^2} \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 )}{2 \sqrt {\frac {e x^2}{d}+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 201
Rule 221
Rule 283
Rule 2221
Rule 2317
Rule 2386
Rule 2392
Rule 2438
Rule 3797
Rule 5775
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx &=\frac {\left (d \sqrt {d+e x^2}\right ) \int \frac {\left (1+\frac {e x^2}{d}\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx}{\sqrt {1+\frac {e x^2}{d}}}\\ &=\frac {3}{2} e x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x}+\frac {3 \sqrt {d} \sqrt {e} \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {1+\frac {e x^2}{d}}}-\frac {\left (b d n \sqrt {d+e x^2}\right ) \int \frac {\left (-2 d+e x^2\right ) \sqrt {1+\frac {e x^2}{d}}+3 \sqrt {d} \sqrt {e} x \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d x^2} \, dx}{\sqrt {1+\frac {e x^2}{d}}}\\ &=\frac {3}{2} e x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x}+\frac {3 \sqrt {d} \sqrt {e} \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {1+\frac {e x^2}{d}}}-\frac {\left (b n \sqrt {d+e x^2}\right ) \int \frac {\left (-2 d+e x^2\right ) \sqrt {1+\frac {e x^2}{d}}+3 \sqrt {d} \sqrt {e} x \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{x^2} \, dx}{2 \sqrt {1+\frac {e x^2}{d}}}\\ &=\frac {3}{2} e x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x}+\frac {3 \sqrt {d} \sqrt {e} \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {1+\frac {e x^2}{d}}}-\frac {\left (b n \sqrt {d+e x^2}\right ) \int \left (e \sqrt {1+\frac {e x^2}{d}}-\frac {2 d \sqrt {1+\frac {e x^2}{d}}}{x^2}+\frac {3 \sqrt {d} \sqrt {e} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{x}\right ) \, dx}{2 \sqrt {1+\frac {e x^2}{d}}}\\ &=\frac {3}{2} e x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x}+\frac {3 \sqrt {d} \sqrt {e} \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {1+\frac {e x^2}{d}}}+\frac {\left (b d n \sqrt {d+e x^2}\right ) \int \frac {\sqrt {1+\frac {e x^2}{d}}}{x^2} \, dx}{\sqrt {1+\frac {e x^2}{d}}}-\frac {\left (3 b \sqrt {d} \sqrt {e} n \sqrt {d+e x^2}\right ) \int \frac {\sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{2 \sqrt {1+\frac {e x^2}{d}}}-\frac {\left (b e n \sqrt {d+e x^2}\right ) \int \sqrt {1+\frac {e x^2}{d}} \, dx}{2 \sqrt {1+\frac {e x^2}{d}}}\\ &=-\frac {b d n \sqrt {d+e x^2}}{x}-\frac {1}{4} b e n x \sqrt {d+e x^2}+\frac {3}{2} e x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x}+\frac {3 \sqrt {d} \sqrt {e} \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {1+\frac {e x^2}{d}}}-\frac {\left (3 b \sqrt {d} \sqrt {e} n \sqrt {d+e x^2}\right ) \text {Subst}\left (\int x \coth (x) \, dx,x,\sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right )}{2 \sqrt {1+\frac {e x^2}{d}}}-\frac {\left (b e n \sqrt {d+e x^2}\right ) \int \frac {1}{\sqrt {1+\frac {e x^2}{d}}} \, dx}{4 \sqrt {1+\frac {e x^2}{d}}}+\frac {\left (b e n \sqrt {d+e x^2}\right ) \int \frac {1}{\sqrt {1+\frac {e x^2}{d}}} \, dx}{\sqrt {1+\frac {e x^2}{d}}}\\ &=-\frac {b d n \sqrt {d+e x^2}}{x}-\frac {1}{4} b e n x \sqrt {d+e x^2}+\frac {3 b \sqrt {d} \sqrt {e} n \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{4 \sqrt {1+\frac {e x^2}{d}}}+\frac {3 b \sqrt {d} \sqrt {e} n \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{4 \sqrt {1+\frac {e x^2}{d}}}+\frac {3}{2} e x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x}+\frac {3 \sqrt {d} \sqrt {e} \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {1+\frac {e x^2}{d}}}+\frac {\left (3 b \sqrt {d} \sqrt {e} n \sqrt {d+e x^2}\right ) \text {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 )}{\sqrt {1+\frac {e x^2}{d}}}\\ &=-\frac {b d n \sqrt {d+e x^2}}{x}-\frac {1}{4} b e n x \sqrt {d+e x^2}+\frac {3 b \sqrt {d} \sqrt {e} n \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{4 \sqrt {1+\frac {e x^2}{d}}}+\frac {3 b \sqrt {d} \sqrt {e} n \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{4 \sqrt {1+\frac {e x^2}{d}}}-\frac {3 b \sqrt {d} \sqrt {e} n \sqrt {d+e x^2} \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 )}{2 \sqrt {1+\frac {e x^2}{d}}}+\frac {3}{2} e x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x}+\frac {3 \sqrt {d} \sqrt {e} \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {1+\frac {e x^2}{d}}}+\frac {\left (3 b \sqrt {d} \sqrt {e} n \sqrt {d+e x^2}\right ) \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right )}{2 \sqrt {1+\frac {e x^2}{d}}}\\ &=-\frac {b d n \sqrt {d+e x^2}}{x}-\frac {1}{4} b e n x \sqrt {d+e x^2}+\frac {3 b \sqrt {d} \sqrt {e} n \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{4 \sqrt {1+\frac {e x^2}{d}}}+\frac {3 b \sqrt {d} \sqrt {e} n \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{4 \sqrt {1+\frac {e x^2}{d}}}-\frac {3 b \sqrt {d} \sqrt {e} n \sqrt {d+e x^2} \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 )}{2 \sqrt {1+\frac {e x^2}{d}}}+\frac {3}{2} e x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x}+\frac {3 \sqrt {d} \sqrt {e} \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {1+\frac {e x^2}{d}}}+\frac {\left (3 b \sqrt {d} \sqrt {e} n \sqrt {d+e x^2}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{4 \sqrt {1+\frac {e x^2}{d}}}\\ &=-\frac {b d n \sqrt {d+e x^2}}{x}-\frac {1}{4} b e n x \sqrt {d+e x^2}+\frac {3 b \sqrt {d} \sqrt {e} n \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{4 \sqrt {1+\frac {e x^2}{d}}}+\frac {3 b \sqrt {d} \sqrt {e} n \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{4 \sqrt {1+\frac {e x^2}{d}}}-\frac {3 b \sqrt {d} \sqrt {e} n \sqrt {d+e x^2} \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 )}{2 \sqrt {1+\frac {e x^2}{d}}}+\frac {3}{2} e x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x}+\frac {3 \sqrt {d} \sqrt {e} \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {1+\frac {e x^2}{d}}}-\frac {3 b \sqrt {d} \sqrt {e} n \sqrt {d+e x^2} \text {Li}_2\left (e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{4 \sqrt {1+\frac {e x^2}{d}}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 0.64, size = 329, normalized size = 0.82 \begin {gather*} -\frac {b \sqrt {d} n \sqrt {d+e x^2} \left (\sqrt {d} \, _3F_2\left (-\frac {1}{2},-\frac {1}{2},-\frac {1}{2};\frac {1}{2},\frac {1}{2};-\frac {e x^2}{d}\right )+\left (\sqrt {d} \sqrt {1+\frac {e x^2}{d}}-\sqrt {e} x \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right ) \log (x)\right )}{x \sqrt {1+\frac {e x^2}{d}}}+\frac {b \sqrt {e} n \sqrt {d+e x^2} \left (-2 \sqrt {e} x \, _3F_2\left (\frac {1}{2},\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};-\frac {e x^2}{d}\right )+\left (\sqrt {e} x \sqrt {1+\frac {e x^2}{d}}+\sqrt {d} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right ) (-1+2 \log (x))\right )}{4 \sqrt {1+\frac {e x^2}{d}}}-\frac {\left (2 d-e x^2\right ) \sqrt {d+e x^2} \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{2 x}+\frac {3}{2} d \sqrt {e} \left (a-b n \log (x)+b \log \left (c x^n\right )\right ) \log \left (e x+\sqrt {e} \sqrt {d+e x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \ln \left (c \,x^{n}\right )\right )}{x^{2}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\left (a + b \log {\left (c x^{n} \right )}\right ) \left (d + e x^{2}\right )^{\frac {3}{2}}}{x^{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (e\,x^2+d\right )}^{3/2}\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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