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 = 0.03, antiderivative size = 53, normalized size of antiderivative = 1.00, 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 43
Rule 190
Rule 263
Rule 2448
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*}
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Mathematica [A] time = 0.03, 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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fricas [A] time = 0.46, size = 90, normalized size = 1.70 \[ \frac {b d^{2} x \log \relax (c) - b d^{2} n \log \left (\sqrt {x}\right ) + b d e n \sqrt {x} + a d^{2} x + {\left (b d^{2} - b e^{2}\right )} n \log \left (d \sqrt {x} + e\right ) + {\left (b d^{2} n x - b d^{2} n\right )} \log \left (\frac {d x + e \sqrt {x}}{x}\right )}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 56, normalized size = 1.06 \[ -{\left ({\left ({\left (\frac {e \log \left ({\left | d \sqrt {x} + e \right |}\right )}{d^{2}} - \frac {\sqrt {x}}{d}\right )} e - x \log \left (d + \frac {e}{\sqrt {x}}\right )\right )} n - x \log \relax (c)\right )} b + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 94, normalized size = 1.77 \[ -\frac {b \,e^{2} n \ln \left (d \sqrt {x}+e \right )}{2 d^{2}}+\frac {b \,e^{2} n \ln \left (d \sqrt {x}-e \right )}{2 d^{2}}-\frac {b \,e^{2} n \ln \left (d^{2} x -e^{2}\right )}{2 d^{2}}+b x \ln \left (c \left (\frac {d \sqrt {x}+e}{\sqrt {x}}\right )^{n}\right )+\frac {b e n \sqrt {x}}{d}+a x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.66, size = 48, normalized size = 0.91 \[ -{\left (e n {\left (\frac {e \log \left (d \sqrt {x} + e\right )}{d^{2}} - \frac {\sqrt {x}}{d}\right )} - x \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right )\right )} b + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.38, size = 44, normalized size = 0.83 \[ a\,x+b\,x\,\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )-\frac {b\,e\,n\,\left (e\,\ln \left (e+d\,\sqrt {x}\right )-d\,\sqrt {x}\right )}{d^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.42, size = 76, normalized size = 1.43 \[ a x + b \left (\frac {e n \left (\frac {2 \sqrt {x}}{d} - \frac {2 e^{2} \left (\begin {cases} \frac {1}{d \sqrt {x}} & \text {for}\: e = 0 \\\frac {\log {\left (d + \frac {e}{\sqrt {x}} \right )}}{e} & \text {otherwise} \end {cases}\right )}{d^{2}} + \frac {2 e \log {\left (\frac {1}{\sqrt {x}} \right )}}{d^{2}}\right )}{2} + x \log {\left (c \left (d + \frac {e}{\sqrt {x}}\right )^{n} \right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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