Optimal. Leaf size=53 \[ a x+b x \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )-\frac{b e^2 n \log \left (d \sqrt{x}+e\right )}{d^2}+\frac{b e n \sqrt{x}}{d} \]
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Rubi [A] time = 0.0343658, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2448, 263, 190, 43} \[ a x+b x \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )-\frac{b e^2 n \log \left (d \sqrt{x}+e\right )}{d^2}+\frac{b e n \sqrt{x}}{d} \]
Antiderivative was successfully verified.
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Rule 2448
Rule 263
Rule 190
Rule 43
Rubi steps
\begin{align*} \int \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right ) \, dx &=a x+b \int \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right ) \, dx\\ &=a x+b x \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )+\frac{1}{2} (b e n) \int \frac{1}{\left (d+\frac{e}{\sqrt{x}}\right ) \sqrt{x}} \, dx\\ &=a x+b x \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )+\frac{1}{2} (b e n) \int \frac{1}{e+d \sqrt{x}} \, dx\\ &=a x+b x \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )+(b e n) \operatorname{Subst}\left (\int \frac{x}{e+d x} \, dx,x,\sqrt{x}\right )\\ &=a x+b x \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )+(b e n) \operatorname{Subst}\left (\int \left (\frac{1}{d}-\frac{e}{d (e+d x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{b e n \sqrt{x}}{d}+a x+b x \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )-\frac{b e^2 n \log \left (e+d \sqrt{x}\right )}{d^2}\\ \end{align*}
Mathematica [A] time = 0.0314587, size = 62, normalized size = 1.17 \[ a x+b x \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )-b e n \left (\frac{e \log \left (d+\frac{e}{\sqrt{x}}\right )}{d^2}+\frac{e \log (x)}{2 d^2}-\frac{\sqrt{x}}{d}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.093, size = 94, normalized size = 1.8 \begin{align*} ax+xb\ln \left ( c \left ({ \left ( e+d\sqrt{x} \right ){\frac{1}{\sqrt{x}}}} \right ) ^{n} \right ) +{\frac{enb}{d}\sqrt{x}}+{\frac{b{e}^{2}n}{2\,{d}^{2}}\ln \left ( d\sqrt{x}-e \right ) }-{\frac{b{e}^{2}n}{2\,{d}^{2}}\ln \left ( e+d\sqrt{x} \right ) }-{\frac{b{e}^{2}n\ln \left ( x{d}^{2}-{e}^{2} \right ) }{2\,{d}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0404, size = 65, normalized size = 1.23 \begin{align*} -{\left (e n{\left (\frac{e \log \left (d \sqrt{x} + e\right )}{d^{2}} - \frac{\sqrt{x}}{d}\right )} - x \log \left (c{\left (d + \frac{e}{\sqrt{x}}\right )}^{n}\right )\right )} b + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.79058, size = 217, normalized size = 4.09 \begin{align*} \frac{b d^{2} x \log \left (c\right ) - b d^{2} n \log \left (\sqrt{x}\right ) + b d e n \sqrt{x} + a d^{2} x +{\left (b d^{2} - b e^{2}\right )} n \log \left (d \sqrt{x} + e\right ) +{\left (b d^{2} n x - b d^{2} n\right )} \log \left (\frac{d x + e \sqrt{x}}{x}\right )}{d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 16.9777, size = 76, normalized size = 1.43 \begin{align*} a x + b \left (\frac{e n \left (\frac{2 \sqrt{x}}{d} - \frac{2 e^{2} \left (\begin{cases} \frac{1}{d \sqrt{x}} & \text{for}\: e = 0 \\\frac{\log{\left (d + \frac{e}{\sqrt{x}} \right )}}{e} & \text{otherwise} \end{cases}\right )}{d^{2}} + \frac{2 e \log{\left (\frac{1}{\sqrt{x}} \right )}}{d^{2}}\right )}{2} + x \log{\left (c \left (d + \frac{e}{\sqrt{x}}\right )^{n} \right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.34533, size = 76, normalized size = 1.43 \begin{align*} -{\left ({\left ({\left (\frac{e \log \left ({\left | d \sqrt{x} + e \right |}\right )}{d^{2}} - \frac{\sqrt{x}}{d}\right )} e - x \log \left (d + \frac{e}{\sqrt{x}}\right )\right )} n - x \log \left (c\right )\right )} b + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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