3.2246 \(\int \frac{(a+b \sqrt{x})^n}{\sqrt{x}} \, dx\)

Optimal. Leaf size=23 \[ \frac{2 \left (a+b \sqrt{x}\right )^{n+1}}{b (n+1)} \]

[Out]

(2*(a + b*Sqrt[x])^(1 + n))/(b*(1 + n))

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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.

[In]

Int[(a + b*Sqrt[x])^n/Sqrt[x],x]

[Out]

(2*(a + b*Sqrt[x])^(1 + n))/(b*(1 + n))

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

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.

[In]

Integrate[(a + b*Sqrt[x])^n/Sqrt[x],x]

[Out]

(2*(a + b*Sqrt[x])^(1 + n))/(b*(1 + n))

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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.

[In]

int((a+b*x^(1/2))^n/x^(1/2),x)

[Out]

2*(a+b*x^(1/2))^(1+n)/b/(1+n)

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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.

[In]

integrate((a+b*x^(1/2))^n/x^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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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.

[In]

integrate((a+b*x^(1/2))^n/x^(1/2),x, algorithm="fricas")

[Out]

2*(b*sqrt(x) + a)*(b*sqrt(x) + a)^n/(b*n + b)

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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.

[In]

integrate((a+b*x**(1/2))**n/x**(1/2),x)

[Out]

Piecewise((zoo*sqrt(x), Eq(a, 0) & Eq(b, 0) & Eq(n, -1)), (2*0**n*sqrt(x), Eq(a, -b*sqrt(x))), (2*a**n*sqrt(x)
, Eq(b, 0)), (2*log(a/b + sqrt(x))/b, Eq(n, -1)), (2*a**2*(a + b*sqrt(x))**n/(a*b*n + a*b + b**2*n*sqrt(x) + b
**2*sqrt(x)) + 4*a*b*sqrt(x)*(a + b*sqrt(x))**n/(a*b*n + a*b + b**2*n*sqrt(x) + b**2*sqrt(x)) + 2*b**2*x*(a +
b*sqrt(x))**n/(a*b*n + a*b + b**2*n*sqrt(x) + b**2*sqrt(x)), True))

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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.

[In]

integrate((a+b*x^(1/2))^n/x^(1/2),x, algorithm="giac")

[Out]

2*(b*sqrt(x) + a)^(n + 1)/(b*(n + 1))