Optimal. Leaf size=60 \[ a x+b x \log \left (c \left (d+e \sqrt {x}\right )^n\right )-\frac {b d^2 n \log \left (d+e \sqrt {x}\right )}{e^2}+\frac {b d n \sqrt {x}}{e}-\frac {b n x}{2} \]
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Rubi [A] time = 0.04, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2448, 266, 43} \[ a x+b x \log \left (c \left (d+e \sqrt {x}\right )^n\right )-\frac {b d^2 n \log \left (d+e \sqrt {x}\right )}{e^2}+\frac {b d n \sqrt {x}}{e}-\frac {b n x}{2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 2448
Rubi steps
\begin {align*} \int \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \, dx &=a x+b \int \log \left (c \left (d+e \sqrt {x}\right )^n\right ) \, dx\\ &=a x+b x \log \left (c \left (d+e \sqrt {x}\right )^n\right )-\frac {1}{2} (b e n) \int \frac {\sqrt {x}}{d+e \sqrt {x}} \, dx\\ &=a x+b x \log \left (c \left (d+e \sqrt {x}\right )^n\right )-(b e n) \operatorname {Subst}\left (\int \frac {x^2}{d+e x} \, dx,x,\sqrt {x}\right )\\ &=a x+b x \log \left (c \left (d+e \sqrt {x}\right )^n\right )-(b e n) \operatorname {Subst}\left (\int \left (-\frac {d}{e^2}+\frac {x}{e}+\frac {d^2}{e^2 (d+e x)}\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {b d n \sqrt {x}}{e}+a x-\frac {b n x}{2}-\frac {b d^2 n \log \left (d+e \sqrt {x}\right )}{e^2}+b x \log \left (c \left (d+e \sqrt {x}\right )^n\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 60, normalized size = 1.00 \[ a x+b x \log \left (c \left (d+e \sqrt {x}\right )^n\right )-\frac {b d^2 n \log \left (d+e \sqrt {x}\right )}{e^2}+\frac {b d n \sqrt {x}}{e}-\frac {b n x}{2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 65, normalized size = 1.08 \[ \frac {2 \, b e^{2} x \log \relax (c) + 2 \, b d e n \sqrt {x} - {\left (b e^{2} n - 2 \, a e^{2}\right )} x + 2 \, {\left (b e^{2} n x - b d^{2} n\right )} \log \left (e \sqrt {x} + d\right )}{2 \, e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 107, normalized size = 1.78 \[ \frac {1}{2} \, {\left ({\left (2 \, {\left (\sqrt {x} e + d\right )}^{2} \log \left (\sqrt {x} e + d\right ) - 4 \, {\left (\sqrt {x} e + d\right )} d \log \left (\sqrt {x} e + d\right ) - {\left (\sqrt {x} e + d\right )}^{2} + 4 \, {\left (\sqrt {x} e + d\right )} d\right )} n e^{\left (-1\right )} + 2 \, {\left ({\left (\sqrt {x} e + d\right )}^{2} - 2 \, {\left (\sqrt {x} e + d\right )} d\right )} e^{\left (-1\right )} \log \relax (c)\right )} b e^{\left (-1\right )} + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 53, normalized size = 0.88 \[ -\frac {b \,d^{2} n \ln \left (e \sqrt {x}+d \right )}{e^{2}}-\frac {b n x}{2}+b x \ln \left (c \left (e \sqrt {x}+d \right )^{n}\right )+\frac {b d n \sqrt {x}}{e}+a x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 57, normalized size = 0.95 \[ -\frac {1}{2} \, {\left (e n {\left (\frac {2 \, d^{2} \log \left (e \sqrt {x} + d\right )}{e^{3}} + \frac {e x - 2 \, d \sqrt {x}}{e^{2}}\right )} - 2 \, x \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right )\right )} b + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.43, size = 52, normalized size = 0.87 \[ a\,x+b\,x\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )-\frac {b\,n\,\left (e^2\,x+2\,d^2\,\ln \left (d+e\,\sqrt {x}\right )-2\,d\,e\,\sqrt {x}\right )}{2\,e^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.86, size = 66, normalized size = 1.10 \[ a x + b \left (- \frac {e n \left (\frac {2 d^{2} \left (\begin {cases} \frac {\sqrt {x}}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e \sqrt {x} \right )}}{e} & \text {otherwise} \end {cases}\right )}{e^{2}} - \frac {2 d \sqrt {x}}{e^{2}} + \frac {x}{e}\right )}{2} + x \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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