Optimal. Leaf size=139 \[ \frac{(e+f x) \sqrt{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{f}-\frac{\sqrt{\pi } \sqrt{b} \sqrt{m} \sqrt{n} (e+f x) e^{-\frac{a}{b m n}} \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac{1}{m n}} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{\sqrt{b} \sqrt{m} \sqrt{n}}\right )}{2 f} \]
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Rubi [A] time = 0.219506, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {2389, 2296, 2300, 2180, 2204, 2445} \[ \frac{(e+f x) \sqrt{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{f}-\frac{\sqrt{\pi } \sqrt{b} \sqrt{m} \sqrt{n} (e+f x) e^{-\frac{a}{b m n}} \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac{1}{m n}} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{\sqrt{b} \sqrt{m} \sqrt{n}}\right )}{2 f} \]
Antiderivative was successfully verified.
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Rule 2389
Rule 2296
Rule 2300
Rule 2180
Rule 2204
Rule 2445
Rubi steps
\begin{align*} \int \sqrt{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )} \, dx &=\operatorname{Subst}\left (\int \sqrt{a+b \log \left (c d^n (e+f x)^{m n}\right )} \, dx,c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right )\\ &=\operatorname{Subst}\left (\frac{\operatorname{Subst}\left (\int \sqrt{a+b \log \left (c d^n x^{m n}\right )} \, dx,x,e+f x\right )}{f},c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right )\\ &=\frac{(e+f x) \sqrt{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{f}-\operatorname{Subst}\left (\frac{(b m n) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \log \left (c d^n x^{m n}\right )}} \, dx,x,e+f x\right )}{2 f},c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right )\\ &=\frac{(e+f x) \sqrt{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{f}-\operatorname{Subst}\left (\frac{\left (b (e+f x) \left (c d^n (e+f x)^{m n}\right )^{-\frac{1}{m n}}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{x}{m n}}}{\sqrt{a+b x}} \, dx,x,\log \left (c d^n (e+f x)^{m n}\right )\right )}{2 f},c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right )\\ &=\frac{(e+f x) \sqrt{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{f}-\operatorname{Subst}\left (\frac{\left ((e+f x) \left (c d^n (e+f x)^{m n}\right )^{-\frac{1}{m n}}\right ) \operatorname{Subst}\left (\int e^{-\frac{a}{b m n}+\frac{x^2}{b m n}} \, dx,x,\sqrt{a+b \log \left (c d^n (e+f x)^{m n}\right )}\right )}{f},c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right )\\ &=-\frac{\sqrt{b} e^{-\frac{a}{b m n}} \sqrt{m} \sqrt{n} \sqrt{\pi } (e+f x) \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac{1}{m n}} \text{erfi}\left (\frac{\sqrt{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{\sqrt{b} \sqrt{m} \sqrt{n}}\right )}{2 f}+\frac{(e+f x) \sqrt{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{f}\\ \end{align*}
Mathematica [A] time = 0.062111, size = 134, normalized size = 0.96 \[ \frac{(e+f x) \left (2 \sqrt{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}-\sqrt{\pi } \sqrt{b} \sqrt{m} \sqrt{n} e^{-\frac{a}{b m n}} \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac{1}{m n}} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{\sqrt{b} \sqrt{m} \sqrt{n}}\right )\right )}{2 f} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.081, size = 0, normalized size = 0. \begin{align*} \int \sqrt{a+b\ln \left ( c \left ( d \left ( fx+e \right ) ^{m} \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 ({\left (f x + e\right )}^{m} 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 \left (e + f x\right )^{m}\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 ({\left (f x + e\right )}^{m} d\right )^{n} c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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