Optimal. Leaf size=255 \[ -\frac{\sqrt{\pi } \sqrt{b} \sqrt{n} e^{-\frac{a}{b n}} (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{2 e^2}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} g \sqrt{n} e^{-\frac{2 a}{b n}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{4 e^2}+\frac{(d+e x) (e f-d g) \sqrt{a+b \log \left (c (d+e x)^n\right )}}{e^2}+\frac{g (d+e x)^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}{2 e^2} \]
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Rubi [A] time = 0.340551, antiderivative size = 255, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {2401, 2389, 2296, 2300, 2180, 2204, 2390, 2305, 2310} \[ -\frac{\sqrt{\pi } \sqrt{b} \sqrt{n} e^{-\frac{a}{b n}} (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{2 e^2}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} g \sqrt{n} e^{-\frac{2 a}{b n}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{4 e^2}+\frac{(d+e x) (e f-d g) \sqrt{a+b \log \left (c (d+e x)^n\right )}}{e^2}+\frac{g (d+e x)^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}{2 e^2} \]
Antiderivative was successfully verified.
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Rule 2401
Rule 2389
Rule 2296
Rule 2300
Rule 2180
Rule 2204
Rule 2390
Rule 2305
Rule 2310
Rubi steps
\begin{align*} \int (f+g x) \sqrt{a+b \log \left (c (d+e x)^n\right )} \, dx &=\int \left (\frac{(e f-d g) \sqrt{a+b \log \left (c (d+e x)^n\right )}}{e}+\frac{g (d+e x) \sqrt{a+b \log \left (c (d+e x)^n\right )}}{e}\right ) \, dx\\ &=\frac{g \int (d+e x) \sqrt{a+b \log \left (c (d+e x)^n\right )} \, dx}{e}+\frac{(e f-d g) \int \sqrt{a+b \log \left (c (d+e x)^n\right )} \, dx}{e}\\ &=\frac{g \operatorname{Subst}\left (\int x \sqrt{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{e^2}+\frac{(e f-d g) \operatorname{Subst}\left (\int \sqrt{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{e^2}\\ &=\frac{(e f-d g) (d+e x) \sqrt{a+b \log \left (c (d+e x)^n\right )}}{e^2}+\frac{g (d+e x)^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}{2 e^2}-\frac{(b g n) \operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{4 e^2}-\frac{(b (e f-d g) n) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{2 e^2}\\ &=\frac{(e f-d g) (d+e x) \sqrt{a+b \log \left (c (d+e x)^n\right )}}{e^2}+\frac{g (d+e x)^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}{2 e^2}-\frac{\left (b g (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{2 x}{n}}}{\sqrt{a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{4 e^2}-\frac{\left (b (e f-d g) (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^2}\\ &=\frac{(e f-d g) (d+e x) \sqrt{a+b \log \left (c (d+e x)^n\right )}}{e^2}+\frac{g (d+e x)^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}{2 e^2}-\frac{\left (g (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \operatorname{Subst}\left (\int e^{-\frac{2 a}{b n}+\frac{2 x^2}{b n}} \, dx,x,\sqrt{a+b \log \left (c (d+e x)^n\right )}\right )}{2 e^2}-\frac{\left ((e f-d g) (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^2}\\ &=-\frac{\sqrt{b} e^{-\frac{a}{b n}} (e f-d g) \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^2}-\frac{\sqrt{b} e^{-\frac{2 a}{b n}} g \sqrt{n} \sqrt{\frac{\pi }{2}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text{erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{4 e^2}+\frac{(e f-d g) (d+e x) \sqrt{a+b \log \left (c (d+e x)^n\right )}}{e^2}+\frac{g (d+e x)^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}{2 e^2}\\ \end{align*}
Mathematica [A] time = 0.286951, size = 235, normalized size = 0.92 \[ -\frac{e^{-\frac{2 a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-2/n} \left (4 \sqrt{\pi } \sqrt{b} \sqrt{n} e^{\frac{a}{b n}} (e f-d g) \left (c (d+e x)^n\right )^{\frac{1}{n}} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )+\sqrt{2 \pi } \sqrt{b} g \sqrt{n} (d+e x) \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )-4 e^{\frac{2 a}{b n}} \left (c (d+e x)^n\right )^{2/n} (-d g+2 e f+e g x) \sqrt{a+b \log \left (c (d+e x)^n\right )}\right )}{8 e^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.566, size = 0, normalized size = 0. \begin{align*} \int \left ( gx+f \right ) \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{\left (g x + f\right )} \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 )}} \left (f + g x\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{\left (g x + f\right )} \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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