Optimal. Leaf size=67 \[ \frac {e^{\frac {a}{2 b n}} \left (c x^n\right )^{\left .\frac {1}{2}\right /n} \text {Ei}\left (\frac {-a-b \log \left (c x^n\right )}{2 b n}\right )}{b d n \sqrt {d x}} \]
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Rubi [A] time = 0.06, antiderivative size = 64, normalized size of antiderivative = 0.96, 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} \[ \frac {e^{\frac {a}{2 b n}} \left (c x^n\right )^{\left .\frac {1}{2}\right /n} \text {Ei}\left (-\frac {a+b \log \left (c x^n\right )}{2 b n}\right )}{b d n \sqrt {d x}} \]
Antiderivative was successfully verified.
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Rule 2178
Rule 2310
Rubi steps
\begin {align*} \int \frac {1}{(d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )} \, dx &=\frac {\left (c x^n\right )^{\left .\frac {1}{2}\right /n} \operatorname {Subst}\left (\int \frac {e^{-\frac {x}{2 n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{d n \sqrt {d x}}\\ &=\frac {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 \sqrt {d x}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 62, normalized size = 0.93 \[ \frac {x e^{\frac {a}{2 b n}} \left (c x^n\right )^{\left .\frac {1}{2}\right /n} \text {Ei}\left (-\frac {a+b \log \left (c x^n\right )}{2 b n}\right )}{b n (d x)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d x}}{b d^{2} x^{2} \log \left (c x^{n}\right ) + a d^{2} x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 49, normalized size = 0.73 \[ \frac {c^{\frac {1}{2 \, n}} {\rm Ei}\left (-\frac {\log \relax (c)}{2 \, n} - \frac {a}{2 \, b n} - \frac {1}{2} \, \log \relax (x)\right ) e^{\left (\frac {a}{2 \, b n}\right )}}{b d^{\frac {3}{2}} n} \]
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 \[ \int \frac {1}{\left (d x \right )^{\frac {3}{2}} \left (b \ln \left (c \,x^{n}\right )+a \right )}\, dx \]
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 \[ -2 \, b n \int \frac {1}{{\left (b^{2} d^{\frac {3}{2}} \log \relax (c)^{2} + b^{2} d^{\frac {3}{2}} \log \left (x^{n}\right )^{2} + 2 \, a b d^{\frac {3}{2}} \log \relax (c) + a^{2} d^{\frac {3}{2}} + 2 \, {\left (b^{2} d^{\frac {3}{2}} \log \relax (c) + a b d^{\frac {3}{2}}\right )} \log \left (x^{n}\right )\right )} x^{\frac {3}{2}}}\,{d x} - \frac {2}{{\left (b d^{\frac {3}{2}} \log \relax (c) + b d^{\frac {3}{2}} \log \left (x^{n}\right ) + a d^{\frac {3}{2}}\right )} \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (d\,x\right )}^{3/2}\,\left (a+b\,\ln \left (c\,x^n\right )\right )} \,d x \]
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 \[ \int \frac {1}{\left (d x\right )^{\frac {3}{2}} \left (a + b \log {\left (c x^{n} \right )}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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