Optimal. Leaf size=295 \[ -b e n \sqrt {d+e x^2}-\frac {b d n \sqrt {d+e x^2}}{4 x^2}+\frac {3}{4} b \sqrt {d} e n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )+\frac {3}{4} b \sqrt {d} e n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )^2+\frac {3}{2} e \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {3}{2} \sqrt {d} e \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {3}{2} b \sqrt {d} e n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^2}}\right )-\frac {3}{4} b \sqrt {d} e n \text {Li}_2\left (1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^2}}\right ) \]
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Rubi [A]
time = 0.31, antiderivative size = 295, normalized size of antiderivative = 1.00, number of steps
used = 18, number of rules used = 12, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {272, 43, 52,
65, 214, 2392, 12, 14, 6131, 6055, 2449, 2352} \begin {gather*} -\frac {3}{4} b \sqrt {d} e n \text {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^2}}\right )-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}+\frac {3}{2} e \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {3}{2} \sqrt {d} e \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )-b e n \sqrt {d+e x^2}-\frac {b d n \sqrt {d+e x^2}}{4 x^2}+\frac {3}{4} b \sqrt {d} e n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )^2+\frac {3}{4} b \sqrt {d} e n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )-\frac {3}{2} b \sqrt {d} e n \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^2}}\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 43
Rule 52
Rule 65
Rule 214
Rule 272
Rule 2352
Rule 2392
Rule 2449
Rule 6055
Rule 6131
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^3} \, dx &=\frac {3}{2} e \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 )}{2 x^2}-\frac {3}{2} \sqrt {d} e \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {-\left (d-2 e x^2\right ) \sqrt {d+e x^2}-3 \sqrt {d} e x^2 \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{2 x^3} \, dx\\ &=\frac {3}{2} e \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 )}{2 x^2}-\frac {3}{2} \sqrt {d} e \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} (b n) \int \frac {-\left (d-2 e x^2\right ) \sqrt {d+e x^2}-3 \sqrt {d} e x^2 \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{x^3} \, dx\\ &=\frac {3}{2} e \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 )}{2 x^2}-\frac {3}{2} \sqrt {d} e \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} (b n) \int \left (-\frac {d \sqrt {d+e x^2}}{x^3}+\frac {2 e \sqrt {d+e x^2}}{x}-\frac {3 \sqrt {d} e \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{x}\right ) \, dx\\ &=\frac {3}{2} e \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 )}{2 x^2}-\frac {3}{2} \sqrt {d} e \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {1}{2} (b d n) \int \frac {\sqrt {d+e x^2}}{x^3} \, dx-(b e n) \int \frac {\sqrt {d+e x^2}}{x} \, dx+\frac {1}{2} \left (3 b \sqrt {d} e n\right ) \int \frac {\tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{x} \, dx\\ &=\frac {3}{2} e \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 )}{2 x^2}-\frac {3}{2} \sqrt {d} e \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {1}{4} (b d n) \text {Subst}\left (\int \frac {\sqrt {d+e x}}{x^2} \, dx,x,x^2\right )-\frac {1}{2} (b e n) \text {Subst}\left (\int \frac {\sqrt {d+e x}}{x} \, dx,x,x^2\right )+\frac {1}{4} \left (3 b \sqrt {d} e n\right ) \text {Subst}\left (\int \frac {\tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{x} \, dx,x,x^2\right )\\ &=-b e n \sqrt {d+e x^2}-\frac {b d n \sqrt {d+e x^2}}{4 x^2}+\frac {3}{2} e \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 )}{2 x^2}-\frac {3}{2} \sqrt {d} e \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {1}{2} \left (3 b \sqrt {d} e n\right ) \text {Subst}\left (\int \frac {x \tanh ^{-1}\left (\frac {x}{\sqrt {d}}\right )}{-d+x^2} \, dx,x,\sqrt {d+e x^2}\right )+\frac {1}{8} (b d e n) \text {Subst}\left (\int \frac {1}{x \sqrt {d+e x}} \, dx,x,x^2\right )-\frac {1}{2} (b d e n) \text {Subst}\left (\int \frac {1}{x \sqrt {d+e x}} \, dx,x,x^2\right )\\ &=-b e n \sqrt {d+e x^2}-\frac {b d n \sqrt {d+e x^2}}{4 x^2}+\frac {3}{4} b \sqrt {d} e n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )^2+\frac {3}{2} e \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 )}{2 x^2}-\frac {3}{2} \sqrt {d} e \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {1}{4} (b d n) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x^2}\right )-(b d n) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x^2}\right )-\frac {1}{2} (3 b e n) \text {Subst}\left (\int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {d}}\right )}{1-\frac {x}{\sqrt {d}}} \, dx,x,\sqrt {d+e x^2}\right )\\ &=-b e n \sqrt {d+e x^2}-\frac {b d n \sqrt {d+e x^2}}{4 x^2}+\frac {3}{4} b \sqrt {d} e n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )+\frac {3}{4} b \sqrt {d} e n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )^2+\frac {3}{2} e \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 )}{2 x^2}-\frac {3}{2} \sqrt {d} e \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {3}{2} b \sqrt {d} e n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^2}}\right )+\frac {1}{2} (3 b e n) \text {Subst}\left (\int \frac {\log \left (\frac {2}{1-\frac {x}{\sqrt {d}}}\right )}{1-\frac {x^2}{d}} \, dx,x,\sqrt {d+e x^2}\right )\\ &=-b e n \sqrt {d+e x^2}-\frac {b d n \sqrt {d+e x^2}}{4 x^2}+\frac {3}{4} b \sqrt {d} e n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )+\frac {3}{4} b \sqrt {d} e n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )^2+\frac {3}{2} e \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 )}{2 x^2}-\frac {3}{2} \sqrt {d} e \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {3}{2} b \sqrt {d} e n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^2}}\right )-\frac {1}{2} \left (3 b \sqrt {d} e n\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\frac {\sqrt {d+e x^2}}{\sqrt {d}}}\right )\\ &=-b e n \sqrt {d+e x^2}-\frac {b d n \sqrt {d+e x^2}}{4 x^2}+\frac {3}{4} b \sqrt {d} e n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )+\frac {3}{4} b \sqrt {d} e n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )^2+\frac {3}{2} e \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 )}{2 x^2}-\frac {3}{2} \sqrt {d} e \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {3}{2} b \sqrt {d} e n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^2}}\right )-\frac {3}{4} b \sqrt {d} e n \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {d+e x^2}}{\sqrt {d}}}\right )\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 0.61, size = 349, normalized size = 1.18 \begin {gather*} \frac {b e n \sqrt {d+e x^2} \left (-\, _3F_2\left (-\frac {1}{2},-\frac {1}{2},-\frac {1}{2};\frac {1}{2},\frac {1}{2};-\frac {d}{e x^2}\right )+\sqrt {1+\frac {d}{e x^2}} \log (x)-\frac {\sqrt {d} \sinh ^{-1}\left (\frac {\sqrt {d}}{\sqrt {e} x}\right ) \log (x)}{\sqrt {e} x}\right )}{\sqrt {1+\frac {d}{e x^2}}}-\frac {b \sqrt {d} n \sqrt {d+e x^2} \left (2 \sqrt {d} \, _3F_2\left (\frac {1}{2},\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};-\frac {d}{e x^2}\right )+\left (\sqrt {d} \sqrt {1+\frac {d}{e x^2}}+\sqrt {e} x \sinh ^{-1}\left (\frac {\sqrt {d}}{\sqrt {e} x}\right )\right ) (1+2 \log (x))\right )}{4 \sqrt {1+\frac {d}{e x^2}} x^2}-\frac {\left (d-2 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^2}+\frac {3}{2} \sqrt {d} e \log (x) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )-\frac {3}{2} \sqrt {d} e \left (a-b n \log (x)+b \log \left (c x^n\right )\right ) \log \left (d+\sqrt {d} \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 \[\int \frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \ln \left (c \,x^{n}\right )\right )}{x^{3}}\, dx\]
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^{3}}\, 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^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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