Optimal. Leaf size=73 \[ -\frac {b n \sqrt {d+e x^2}}{e}+\frac {b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{e}+\frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e} \]
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Rubi [A]
time = 0.05, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2376, 272, 52,
65, 214} \begin {gather*} \frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {b n \sqrt {d+e x^2}}{e}+\frac {b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{e} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 214
Rule 272
Rule 2376
Rubi steps
\begin {align*} \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d+e x^2}} \, dx &=\frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {(b n) \int \frac {\sqrt {d+e x^2}}{x} \, dx}{e}\\ &=\frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {(b n) \text {Subst}\left (\int \frac {\sqrt {d+e x}}{x} \, dx,x,x^2\right )}{2 e}\\ &=-\frac {b n \sqrt {d+e x^2}}{e}+\frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {(b d n) \text {Subst}\left (\int \frac {1}{x \sqrt {d+e x}} \, dx,x,x^2\right )}{2 e}\\ &=-\frac {b n \sqrt {d+e x^2}}{e}+\frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {(b d n) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x^2}\right )}{e^2}\\ &=-\frac {b n \sqrt {d+e x^2}}{e}+\frac {b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{e}+\frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 91, normalized size = 1.25 \begin {gather*} \frac {a \sqrt {d+e x^2}-b n \sqrt {d+e x^2}-b \sqrt {d} n \log (x)+b \sqrt {d+e x^2} \log \left (c x^n\right )+b \sqrt {d} n \log \left (d+\sqrt {d} \sqrt {d+e x^2}\right )}{e} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {x \left (a +b \ln \left (c \,x^{n}\right )\right )}{\sqrt {e \,x^{2}+d}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 68, normalized size = 0.93 \begin {gather*} {\left (\sqrt {d} \operatorname {arsinh}\left (\frac {\sqrt {d} e^{\left (-\frac {1}{2}\right )}}{{\left | x \right |}}\right ) - \sqrt {x^{2} e + d}\right )} b n e^{\left (-1\right )} + \sqrt {x^{2} e + d} b e^{\left (-1\right )} \log \left (c x^{n}\right ) + \sqrt {x^{2} e + d} a e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 127, normalized size = 1.74 \begin {gather*} \left [\frac {1}{2} \, {\left (b \sqrt {d} n \log \left (-\frac {x^{2} e + 2 \, \sqrt {x^{2} e + d} \sqrt {d} + 2 \, d}{x^{2}}\right ) + 2 \, \sqrt {x^{2} e + d} {\left (b n \log \left (x\right ) - b n + b \log \left (c\right ) + a\right )}\right )} e^{\left (-1\right )}, -{\left (b \sqrt {-d} n \arctan \left (\frac {\sqrt {-d}}{\sqrt {x^{2} e + d}}\right ) - \sqrt {x^{2} e + d} {\left (b n \log \left (x\right ) - b n + b \log \left (c\right ) + a\right )}\right )} e^{\left (-1\right )}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.67, size = 126, normalized size = 1.73 \begin {gather*} a \left (\begin {cases} \frac {x^{2}}{2 \sqrt {d}} & \text {for}\: e = 0 \\\frac {\sqrt {d + e x^{2}}}{e} & \text {otherwise} \end {cases}\right ) - b n \left (\begin {cases} \frac {x^{2}}{4 \sqrt {d}} & \text {for}\: e = 0 \\- \frac {\sqrt {d} \operatorname {asinh}{\left (\frac {\sqrt {d}}{\sqrt {e} x} \right )}}{e} + \frac {d}{e^{\frac {3}{2}} x \sqrt {\frac {d}{e x^{2}} + 1}} + \frac {x}{\sqrt {e} \sqrt {\frac {d}{e x^{2}} + 1}} & \text {otherwise} \end {cases}\right ) + b \left (\begin {cases} \frac {x^{2}}{2 \sqrt {d}} & \text {for}\: e = 0 \\\frac {\sqrt {d + e x^{2}}}{e} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} \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 = -0.01 \begin {gather*} \int \frac {x\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{\sqrt {e\,x^2+d}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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