Optimal. Leaf size=102 \[ \frac{1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )+\frac{b d^3 n \sqrt{x}}{2 e^3}-\frac{b d^2 n x}{4 e^2}-\frac{b d^4 n \log \left (d+e \sqrt{x}\right )}{2 e^4}+\frac{b d n x^{3/2}}{6 e}-\frac{1}{8} b n x^2 \]
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Rubi [A] time = 0.0721291, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {2454, 2395, 43} \[ \frac{1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )+\frac{b d^3 n \sqrt{x}}{2 e^3}-\frac{b d^2 n x}{4 e^2}-\frac{b d^4 n \log \left (d+e \sqrt{x}\right )}{2 e^4}+\frac{b d n x^{3/2}}{6 e}-\frac{1}{8} b n x^2 \]
Antiderivative was successfully verified.
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Rule 2454
Rule 2395
Rule 43
Rubi steps
\begin{align*} \int x \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right ) \, dx &=2 \operatorname{Subst}\left (\int x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )-\frac{1}{2} (b e n) \operatorname{Subst}\left (\int \frac{x^4}{d+e x} \, dx,x,\sqrt{x}\right )\\ &=\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )-\frac{1}{2} (b e n) \operatorname{Subst}\left (\int \left (-\frac{d^3}{e^4}+\frac{d^2 x}{e^3}-\frac{d x^2}{e^2}+\frac{x^3}{e}+\frac{d^4}{e^4 (d+e x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{b d^3 n \sqrt{x}}{2 e^3}-\frac{b d^2 n x}{4 e^2}+\frac{b d n x^{3/2}}{6 e}-\frac{1}{8} b n x^2-\frac{b d^4 n \log \left (d+e \sqrt{x}\right )}{2 e^4}+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )\\ \end{align*}
Mathematica [A] time = 0.0329987, size = 107, normalized size = 1.05 \[ \frac{a x^2}{2}+\frac{1}{2} b x^2 \log \left (c \left (d+e \sqrt{x}\right )^n\right )+\frac{b d^3 n \sqrt{x}}{2 e^3}-\frac{b d^2 n x}{4 e^2}-\frac{b d^4 n \log \left (d+e \sqrt{x}\right )}{2 e^4}+\frac{b d n x^{3/2}}{6 e}-\frac{1}{8} b n x^2 \]
Antiderivative was successfully verified.
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Maple [F] time = 0.098, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b\ln \left ( c \left ( d+e\sqrt{x} \right ) ^{n} \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04189, size = 113, normalized size = 1.11 \begin{align*} -\frac{1}{24} \, b e n{\left (\frac{12 \, d^{4} \log \left (e \sqrt{x} + d\right )}{e^{5}} + \frac{3 \, e^{3} x^{2} - 4 \, d e^{2} x^{\frac{3}{2}} + 6 \, d^{2} e x - 12 \, d^{3} \sqrt{x}}{e^{4}}\right )} + \frac{1}{2} \, b x^{2} \log \left ({\left (e \sqrt{x} + d\right )}^{n} c\right ) + \frac{1}{2} \, a x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74638, size = 224, normalized size = 2.2 \begin{align*} \frac{12 \, b e^{4} x^{2} \log \left (c\right ) - 6 \, b d^{2} e^{2} n x - 3 \,{\left (b e^{4} n - 4 \, a e^{4}\right )} x^{2} + 12 \,{\left (b e^{4} n x^{2} - b d^{4} n\right )} \log \left (e \sqrt{x} + d\right ) + 4 \,{\left (b d e^{3} n x + 3 \, b d^{3} e n\right )} \sqrt{x}}{24 \, e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.7292, size = 100, normalized size = 0.98 \begin{align*} \frac{a x^{2}}{2} + b \left (- \frac{e n \left (\frac{2 d^{4} \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^{4}} - \frac{2 d^{3} \sqrt{x}}{e^{4}} + \frac{d^{2} x}{e^{3}} - \frac{2 d x^{\frac{3}{2}}}{3 e^{2}} + \frac{x^{2}}{2 e}\right )}{4} + \frac{x^{2} \log{\left (c \left (d + e \sqrt{x}\right )^{n} \right )}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.24899, size = 385, normalized size = 3.77 \begin{align*} \frac{1}{24} \,{\left ({\left (12 \,{\left (\sqrt{x} e + d\right )}^{4} e^{\left (-2\right )} \log \left (\sqrt{x} e + d\right ) - 48 \,{\left (\sqrt{x} e + d\right )}^{3} d e^{\left (-2\right )} \log \left (\sqrt{x} e + d\right ) + 72 \,{\left (\sqrt{x} e + d\right )}^{2} d^{2} e^{\left (-2\right )} \log \left (\sqrt{x} e + d\right ) - 48 \,{\left (\sqrt{x} e + d\right )} d^{3} e^{\left (-2\right )} \log \left (\sqrt{x} e + d\right ) - 3 \,{\left (\sqrt{x} e + d\right )}^{4} e^{\left (-2\right )} + 16 \,{\left (\sqrt{x} e + d\right )}^{3} d e^{\left (-2\right )} - 36 \,{\left (\sqrt{x} e + d\right )}^{2} d^{2} e^{\left (-2\right )} + 48 \,{\left (\sqrt{x} e + d\right )} d^{3} e^{\left (-2\right )}\right )} b n e^{\left (-1\right )} + 12 \,{\left ({\left (\sqrt{x} e + d\right )}^{4} - 4 \,{\left (\sqrt{x} e + d\right )}^{3} d + 6 \,{\left (\sqrt{x} e + d\right )}^{2} d^{2} - 4 \,{\left (\sqrt{x} e + d\right )} d^{3}\right )} b e^{\left (-3\right )} \log \left (c\right ) + 12 \,{\left ({\left (\sqrt{x} e + d\right )}^{4} - 4 \,{\left (\sqrt{x} e + d\right )}^{3} d + 6 \,{\left (\sqrt{x} e + d\right )}^{2} d^{2} - 4 \,{\left (\sqrt{x} e + d\right )} d^{3}\right )} a e^{\left (-3\right )}\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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