Optimal. Leaf size=64 \[ \frac {\sqrt {d x} e^{-\frac {a}{2 b n}} \left (c x^n\right )^{\left .-\frac {1}{2}\right /n} \text {Ei}\left (\frac {a+b \log \left (c x^n\right )}{2 b n}\right )}{b d n} \]
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Rubi [A] time = 0.06, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2310, 2178} \[ \frac {\sqrt {d x} e^{-\frac {a}{2 b n}} \left (c x^n\right )^{\left .-\frac {1}{2}\right /n} \text {Ei}\left (\frac {a+b \log \left (c x^n\right )}{2 b n}\right )}{b d n} \]
Antiderivative was successfully verified.
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Rule 2178
Rule 2310
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {d x} \left (a+b \log \left (c x^n\right )\right )} \, dx &=\frac {\left (\sqrt {d x} \left (c x^n\right )^{\left .-\frac {1}{2}\right /n}\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {x}{2 n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{d n}\\ &=\frac {e^{-\frac {a}{2 b n}} \sqrt {d x} \left (c x^n\right )^{\left .-\frac {1}{2}\right /n} \text {Ei}\left (\frac {a+b \log \left (c x^n\right )}{2 b n}\right )}{b d n}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 62, normalized size = 0.97 \[ \frac {x e^{-\frac {a}{2 b n}} \left (c x^n\right )^{\left .-\frac {1}{2}\right /n} \text {Ei}\left (\frac {a+b \log \left (c x^n\right )}{2 b n}\right )}{b n \sqrt {d x}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d x}}{b d x \log \left (c x^{n}\right ) + a d x}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {d x} {\left (b \log \left (c x^{n}\right ) + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {d x}\, \left (b \ln \left (c \,x^{n}\right )+a \right )}\, dx \]
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 \[ 2 \, b n \int \frac {1}{{\left (b^{2} \sqrt {d} \log \relax (c)^{2} + b^{2} \sqrt {d} \log \left (x^{n}\right )^{2} + 2 \, a b \sqrt {d} \log \relax (c) + a^{2} \sqrt {d} + 2 \, {\left (b^{2} \sqrt {d} \log \relax (c) + a b \sqrt {d}\right )} \log \left (x^{n}\right )\right )} \sqrt {x}}\,{d x} + \frac {2 \, \sqrt {x}}{b \sqrt {d} \log \relax (c) + b \sqrt {d} \log \left (x^{n}\right ) + a \sqrt {d}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {d\,x}\,\left (a+b\,\ln \left (c\,x^n\right )\right )} \,d x \]
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 \[ \int \frac {1}{\sqrt {d x} \left (a + b \log {\left (c x^{n} \right )}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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