Optimal. Leaf size=111 \[ \frac{(d+e x) \sqrt{a+b \log \left (c (d+e x)^n\right )}}{e}-\frac{\sqrt{\pi } \sqrt{b} \sqrt{n} e^{-\frac{a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{2 e} \]
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Rubi [A] time = 0.0866578, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {2389, 2296, 2300, 2180, 2204} \[ \frac{(d+e x) \sqrt{a+b \log \left (c (d+e x)^n\right )}}{e}-\frac{\sqrt{\pi } \sqrt{b} \sqrt{n} e^{-\frac{a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{2 e} \]
Antiderivative was successfully verified.
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Rule 2389
Rule 2296
Rule 2300
Rule 2180
Rule 2204
Rubi steps
\begin{align*} \int \sqrt{a+b \log \left (c (d+e x)^n\right )} \, dx &=\frac{\operatorname{Subst}\left (\int \sqrt{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{e}\\ &=\frac{(d+e x) \sqrt{a+b \log \left (c (d+e x)^n\right )}}{e}-\frac{(b n) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{2 e}\\ &=\frac{(d+e x) \sqrt{a+b \log \left (c (d+e x)^n\right )}}{e}-\frac{\left (b (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{x}{n}}}{\sqrt{a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{2 e}\\ &=\frac{(d+e x) \sqrt{a+b \log \left (c (d+e x)^n\right )}}{e}-\frac{\left ((d+e x) \left (c (d+e 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 (d+e x)^n\right )}\right )}{e}\\ &=-\frac{\sqrt{b} e^{-\frac{a}{b n}} \sqrt{n} \sqrt{\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text{erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{2 e}+\frac{(d+e x) \sqrt{a+b \log \left (c (d+e x)^n\right )}}{e}\\ \end{align*}
Mathematica [A] time = 0.0485265, size = 106, normalized size = 0.95 \[ \frac{(d+e x) \left (2 \sqrt{a+b \log \left (c (d+e x)^n\right )}-\sqrt{\pi } \sqrt{b} \sqrt{n} e^{-\frac{a}{b n}} \left (c (d+e x)^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )\right )}{2 e} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.06, size = 0, normalized size = 0. \begin{align*} \int \sqrt{a+b\ln \left ( c \left ( ex+d \right ) ^{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 ({\left (e x + d\right )}^{n} c\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 \left (d + e x\right )^{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 ({\left (e x + d\right )}^{n} c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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