Optimal. Leaf size=85 \[ x \sqrt{a+b \log \left (c x^n\right )}-\frac{1}{2} \sqrt{\pi } \sqrt{b} \sqrt{n} x e^{-\frac{a}{b n}} \left (c x^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c x^n\right )}}{\sqrt{b} \sqrt{n}}\right ) \]
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Rubi [A] time = 0.0722577, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2296, 2300, 2180, 2204} \[ x \sqrt{a+b \log \left (c x^n\right )}-\frac{1}{2} \sqrt{\pi } \sqrt{b} \sqrt{n} x e^{-\frac{a}{b n}} \left (c x^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c x^n\right )}}{\sqrt{b} \sqrt{n}}\right ) \]
Antiderivative was successfully verified.
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Rule 2296
Rule 2300
Rule 2180
Rule 2204
Rubi steps
\begin{align*} \int \sqrt{a+b \log \left (c x^n\right )} \, dx &=x \sqrt{a+b \log \left (c x^n\right )}-\frac{1}{2} (b n) \int \frac{1}{\sqrt{a+b \log \left (c x^n\right )}} \, dx\\ &=x \sqrt{a+b \log \left (c x^n\right )}-\frac{1}{2} \left (b x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{x}{n}}}{\sqrt{a+b x}} \, dx,x,\log \left (c x^n\right )\right )\\ &=x \sqrt{a+b \log \left (c x^n\right )}-\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int e^{-\frac{a}{b n}+\frac{x^2}{b n}} \, dx,x,\sqrt{a+b \log \left (c x^n\right )}\right )\\ &=-\frac{1}{2} \sqrt{b} e^{-\frac{a}{b n}} \sqrt{n} \sqrt{\pi } x \left (c x^n\right )^{-1/n} \text{erfi}\left (\frac{\sqrt{a+b \log \left (c x^n\right )}}{\sqrt{b} \sqrt{n}}\right )+x \sqrt{a+b \log \left (c x^n\right )}\\ \end{align*}
Mathematica [A] time = 0.0376772, size = 85, normalized size = 1. \[ x \sqrt{a+b \log \left (c x^n\right )}-\frac{1}{2} \sqrt{\pi } \sqrt{b} \sqrt{n} x e^{-\frac{a}{b n}} \left (c x^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c x^n\right )}}{\sqrt{b} \sqrt{n}}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.485, size = 0, normalized size = 0. \begin{align*} \int \sqrt{a+b\ln \left ( c{x}^{n} \right ) }\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \log \left (c x^{n}\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a + b \log{\left (c x^{n} \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \log \left (c x^{n}\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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