Optimal. Leaf size=139 \[ \frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )+\frac{b e^3 n x^{3/2}}{9 d^3}-\frac{b e^2 n x^2}{12 d^2}+\frac{b e^5 n \sqrt{x}}{3 d^5}-\frac{b e^4 n x}{6 d^4}-\frac{b e^6 n \log \left (d+\frac{e}{\sqrt{x}}\right )}{3 d^6}-\frac{b e^6 n \log (x)}{6 d^6}+\frac{b e n x^{5/2}}{15 d} \]
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Rubi [A] time = 0.0962967, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {2454, 2395, 44} \[ \frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )+\frac{b e^3 n x^{3/2}}{9 d^3}-\frac{b e^2 n x^2}{12 d^2}+\frac{b e^5 n \sqrt{x}}{3 d^5}-\frac{b e^4 n x}{6 d^4}-\frac{b e^6 n \log \left (d+\frac{e}{\sqrt{x}}\right )}{3 d^6}-\frac{b e^6 n \log (x)}{6 d^6}+\frac{b e n x^{5/2}}{15 d} \]
Antiderivative was successfully verified.
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Rule 2454
Rule 2395
Rule 44
Rubi steps
\begin{align*} \int x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right ) \, dx &=-\left (2 \operatorname{Subst}\left (\int \frac{a+b \log \left (c (d+e x)^n\right )}{x^7} \, dx,x,\frac{1}{\sqrt{x}}\right )\right )\\ &=\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )-\frac{1}{3} (b e n) \operatorname{Subst}\left (\int \frac{1}{x^6 (d+e x)} \, dx,x,\frac{1}{\sqrt{x}}\right )\\ &=\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )-\frac{1}{3} (b e n) \operatorname{Subst}\left (\int \left (\frac{1}{d x^6}-\frac{e}{d^2 x^5}+\frac{e^2}{d^3 x^4}-\frac{e^3}{d^4 x^3}+\frac{e^4}{d^5 x^2}-\frac{e^5}{d^6 x}+\frac{e^6}{d^6 (d+e x)}\right ) \, dx,x,\frac{1}{\sqrt{x}}\right )\\ &=\frac{b e^5 n \sqrt{x}}{3 d^5}-\frac{b e^4 n x}{6 d^4}+\frac{b e^3 n x^{3/2}}{9 d^3}-\frac{b e^2 n x^2}{12 d^2}+\frac{b e n x^{5/2}}{15 d}-\frac{b e^6 n \log \left (d+\frac{e}{\sqrt{x}}\right )}{3 d^6}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )-\frac{b e^6 n \log (x)}{6 d^6}\\ \end{align*}
Mathematica [A] time = 0.0862148, size = 130, normalized size = 0.94 \[ \frac{a x^3}{3}+\frac{1}{3} b x^3 \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )-\frac{1}{3} b e n \left (-\frac{e^2 x^{3/2}}{3 d^3}-\frac{e^4 \sqrt{x}}{d^5}+\frac{e^3 x}{2 d^4}+\frac{e^5 \log \left (d+\frac{e}{\sqrt{x}}\right )}{d^6}+\frac{e^5 \log (x)}{2 d^6}+\frac{e x^2}{4 d^2}-\frac{x^{5/2}}{5 d}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.331, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( a+b\ln \left ( c \left ( d+{e{\frac{1}{\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 = 1.029, size = 130, normalized size = 0.94 \begin{align*} \frac{1}{3} \, b x^{3} \log \left (c{\left (d + \frac{e}{\sqrt{x}}\right )}^{n}\right ) + \frac{1}{3} \, a x^{3} - \frac{1}{180} \, b e n{\left (\frac{60 \, e^{5} \log \left (d \sqrt{x} + e\right )}{d^{6}} - \frac{12 \, d^{4} x^{\frac{5}{2}} - 15 \, d^{3} e x^{2} + 20 \, d^{2} e^{2} x^{\frac{3}{2}} - 30 \, d e^{3} x + 60 \, e^{4} \sqrt{x}}{d^{5}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.93807, size = 369, normalized size = 2.65 \begin{align*} \frac{60 \, b d^{6} x^{3} \log \left (c\right ) - 15 \, b d^{4} e^{2} n x^{2} + 60 \, a d^{6} x^{3} - 30 \, b d^{2} e^{4} n x - 60 \, b d^{6} n \log \left (\sqrt{x}\right ) + 60 \,{\left (b d^{6} - b e^{6}\right )} n \log \left (d \sqrt{x} + e\right ) + 60 \,{\left (b d^{6} n x^{3} - b d^{6} n\right )} \log \left (\frac{d x + e \sqrt{x}}{x}\right ) + 4 \,{\left (3 \, b d^{5} e n x^{2} + 5 \, b d^{3} e^{3} n x + 15 \, b d e^{5} n\right )} \sqrt{x}}{180 \, d^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29016, size = 136, normalized size = 0.98 \begin{align*} \frac{1}{3} \, b x^{3} \log \left (c\right ) + \frac{1}{3} \, a x^{3} + \frac{1}{180} \,{\left (60 \, x^{3} \log \left (d + \frac{e}{\sqrt{x}}\right ) +{\left (\frac{12 \, d^{4} x^{\frac{5}{2}} - 15 \, d^{3} x^{2} e + 20 \, d^{2} x^{\frac{3}{2}} e^{2} - 30 \, d x e^{3} + 60 \, \sqrt{x} e^{4}}{d^{5}} - \frac{60 \, e^{5} \log \left ({\left | d \sqrt{x} + e \right |}\right )}{d^{6}}\right )} e\right )} b n \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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