Optimal. Leaf size=330 \[ \frac {d^{3/2} \sqrt {\frac {e x^2}{d}+1} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {e} \sqrt {d+e x^2}}+\frac {1}{2} x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {b d^{3/2} n \sqrt {\frac {e x^2}{d}+1} \text {Li}_2\left (e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{4 \sqrt {e} \sqrt {d+e x^2}}+\frac {b d^{3/2} n \sqrt {\frac {e x^2}{d}+1} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{4 \sqrt {e} \sqrt {d+e x^2}}-\frac {b d^{3/2} n \sqrt {\frac {e x^2}{d}+1} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{2 \sqrt {e} \sqrt {d+e x^2}}-\frac {1}{4} b n x \sqrt {d+e x^2}-\frac {b d n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{4 \sqrt {e}} \]
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Rubi [A] time = 0.20, antiderivative size = 330, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2321, 195, 217, 206, 2327, 2325, 5659, 3716, 2190, 2279, 2391} \[ -\frac {b d^{3/2} n \sqrt {\frac {e x^2}{d}+1} \text {PolyLog}\left (2,e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{4 \sqrt {e} \sqrt {d+e x^2}}+\frac {d^{3/2} \sqrt {\frac {e x^2}{d}+1} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {e} \sqrt {d+e x^2}}+\frac {1}{2} x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac {b d^{3/2} n \sqrt {\frac {e x^2}{d}+1} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{4 \sqrt {e} \sqrt {d+e x^2}}-\frac {b d^{3/2} n \sqrt {\frac {e x^2}{d}+1} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{2 \sqrt {e} \sqrt {d+e x^2}}-\frac {1}{4} b n x \sqrt {d+e x^2}-\frac {b d n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{4 \sqrt {e}} \]
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rule 2190
Rule 2279
Rule 2321
Rule 2325
Rule 2327
Rule 2391
Rule 3716
Rule 5659
Rubi steps
\begin {align*} \int \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {1}{2} x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac {1}{2} d \int \frac {a+b \log \left (c x^n\right )}{\sqrt {d+e x^2}} \, dx-\frac {1}{2} (b n) \int \sqrt {d+e x^2} \, dx\\ &=-\frac {1}{4} b n x \sqrt {d+e x^2}+\frac {1}{2} x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} (b d n) \int \frac {1}{\sqrt {d+e x^2}} \, dx+\frac {\left (d \sqrt {1+\frac {e x^2}{d}}\right ) \int \frac {a+b \log \left (c x^n\right )}{\sqrt {1+\frac {e x^2}{d}}} \, dx}{2 \sqrt {d+e x^2}}\\ &=-\frac {1}{4} b n x \sqrt {d+e x^2}+\frac {1}{2} x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac {d^{3/2} \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 )}{2 \sqrt {e} \sqrt {d+e x^2}}-\frac {1}{4} (b d n) \operatorname {Subst}\left (\int \frac {1}{1-e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )-\frac {\left (b d^{3/2} n \sqrt {1+\frac {e x^2}{d}}\right ) \int \frac {\sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{2 \sqrt {e} \sqrt {d+e x^2}}\\ &=-\frac {1}{4} b n x \sqrt {d+e x^2}-\frac {b d n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{4 \sqrt {e}}+\frac {1}{2} x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac {d^{3/2} \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 )}{2 \sqrt {e} \sqrt {d+e x^2}}-\frac {\left (b d^{3/2} 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 )}{2 \sqrt {e} \sqrt {d+e x^2}}\\ &=-\frac {1}{4} b n x \sqrt {d+e x^2}+\frac {b d^{3/2} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{4 \sqrt {e} \sqrt {d+e x^2}}-\frac {b d n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{4 \sqrt {e}}+\frac {1}{2} x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac {d^{3/2} \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 )}{2 \sqrt {e} \sqrt {d+e x^2}}+\frac {\left (b d^{3/2} 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 )}{\sqrt {e} \sqrt {d+e x^2}}\\ &=-\frac {1}{4} b n x \sqrt {d+e x^2}+\frac {b d^{3/2} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{4 \sqrt {e} \sqrt {d+e x^2}}-\frac {b d n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{4 \sqrt {e}}-\frac {b d^{3/2} 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 )}{2 \sqrt {e} \sqrt {d+e x^2}}+\frac {1}{2} x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac {d^{3/2} \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 )}{2 \sqrt {e} \sqrt {d+e x^2}}+\frac {\left (b d^{3/2} 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 )}{2 \sqrt {e} \sqrt {d+e x^2}}\\ &=-\frac {1}{4} b n x \sqrt {d+e x^2}+\frac {b d^{3/2} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{4 \sqrt {e} \sqrt {d+e x^2}}-\frac {b d n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{4 \sqrt {e}}-\frac {b d^{3/2} 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 )}{2 \sqrt {e} \sqrt {d+e x^2}}+\frac {1}{2} x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac {d^{3/2} \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 )}{2 \sqrt {e} \sqrt {d+e x^2}}+\frac {\left (b d^{3/2} 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 )}{4 \sqrt {e} \sqrt {d+e x^2}}\\ &=-\frac {1}{4} b n x \sqrt {d+e x^2}+\frac {b d^{3/2} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{4 \sqrt {e} \sqrt {d+e x^2}}-\frac {b d n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{4 \sqrt {e}}-\frac {b d^{3/2} 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 )}{2 \sqrt {e} \sqrt {d+e x^2}}+\frac {1}{2} x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac {d^{3/2} \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 )}{2 \sqrt {e} \sqrt {d+e x^2}}-\frac {b d^{3/2} 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 )}{4 \sqrt {e} \sqrt {d+e x^2}}\\ \end {align*}
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Mathematica [C] time = 0.36, size = 237, normalized size = 0.72 \[ \frac {-2 b \sqrt {e} n x \sqrt {d+e x^2} \, _3F_2\left (\frac {1}{2},\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};-\frac {e x^2}{d}\right )+\sqrt {\frac {e x^2}{d}+1} \left (\sqrt {e} x (2 a-b n) \sqrt {d+e x^2}+2 d \log \left (\sqrt {e} \sqrt {d+e x^2}+e x\right ) (a-b n \log (x))+2 b \log \left (c x^n\right ) \left (\sqrt {e} x \sqrt {d+e x^2}+d \log \left (\sqrt {e} \sqrt {d+e x^2}+e x\right )\right )\right )+b \sqrt {d} n (2 \log (x)-1) \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{4 \sqrt {e} \sqrt {\frac {e x^2}{d}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {e x^{2} + d} b \log \left (c x^{n}\right ) + \sqrt {e x^{2} + d} a, x\right ) \]
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 \[ \int \sqrt {e x^{2} + d} {\left (b \log \left (c x^{n}\right ) + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.38, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \,x^{n}\right )+a \right ) \sqrt {e \,x^{2}+d}\, dx \]
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 \[ \frac {1}{2} \, {\left (\sqrt {e x^{2} + d} x + \frac {d \operatorname {arsinh}\left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {e}}\right )} a + b \int \sqrt {e x^{2} + d} {\left (\log \relax (c) + \log \left (x^{n}\right )\right )}\,{d x} \]
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 \[ \int \sqrt {e\,x^2+d}\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]
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 \[ \int \left (a + b \log {\left (c x^{n} \right )}\right ) \sqrt {d + e x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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