Optimal. Leaf size=23 \[ \frac{2 \left (a+b \sqrt{x}\right )^{n+1}}{b (n+1)} \]
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Rubi [A] time = 0.0051468, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {261} \[ \frac{2 \left (a+b \sqrt{x}\right )^{n+1}}{b (n+1)} \]
Antiderivative was successfully verified.
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Rule 261
Rubi steps
\begin{align*} \int \frac{\left (a+b \sqrt{x}\right )^n}{\sqrt{x}} \, dx &=\frac{2 \left (a+b \sqrt{x}\right )^{1+n}}{b (1+n)}\\ \end{align*}
Mathematica [A] time = 0.0052083, size = 23, normalized size = 1. \[ \frac{2 \left (a+b \sqrt{x}\right )^{n+1}}{b (n+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 22, normalized size = 1. \begin{align*} 2\,{\frac{ \left ( a+b\sqrt{x} \right ) ^{1+n}}{b \left ( 1+n \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.39203, size = 63, normalized size = 2.74 \begin{align*} \frac{2 \,{\left (b \sqrt{x} + a\right )}{\left (b \sqrt{x} + a\right )}^{n}}{b n + b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.885402, size = 182, normalized size = 7.91 \begin{align*} \begin{cases} \tilde{\infty } \sqrt{x} & \text{for}\: a = 0 \wedge b = 0 \wedge n = -1 \\2 \cdot 0^{n} \sqrt{x} & \text{for}\: a = - b \sqrt{x} \\2 a^{n} \sqrt{x} & \text{for}\: b = 0 \\\frac{2 \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{b} & \text{for}\: n = -1 \\\frac{2 a^{2} \left (a + b \sqrt{x}\right )^{n}}{a b n + a b + b^{2} n \sqrt{x} + b^{2} \sqrt{x}} + \frac{4 a b \sqrt{x} \left (a + b \sqrt{x}\right )^{n}}{a b n + a b + b^{2} n \sqrt{x} + b^{2} \sqrt{x}} + \frac{2 b^{2} x \left (a + b \sqrt{x}\right )^{n}}{a b n + a b + b^{2} n \sqrt{x} + b^{2} \sqrt{x}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08802, size = 28, normalized size = 1.22 \begin{align*} \frac{2 \,{\left (b \sqrt{x} + a\right )}^{n + 1}}{b{\left (n + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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