\(\int \frac {(a+b \sqrt {x})^n}{\sqrt {x}} \, dx\) [2246]

Optimal result

Integrand size = 17, antiderivative size = 23 \[ \int \frac {\left (a+b \sqrt {x}\right )^n}{\sqrt {x}} \, dx=\frac {2 \left (a+b \sqrt {x}\right )^{1+n}}{b (1+n)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {267} \[ \int \frac {\left (a+b \sqrt {x}\right )^n}{\sqrt {x}} \, dx=\frac {2 \left (a+b \sqrt {x}\right )^{n+1}}{b (n+1)} \]

[In]

Int[(a + b*Sqrt[x])^n/Sqrt[x],x]

[Out]

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

Rule 267

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*} \text {integral}& = \frac {2 \left (a+b \sqrt {x}\right )^{1+n}}{b (1+n)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \sqrt {x}\right )^n}{\sqrt {x}} \, dx=\frac {2 \left (a+b \sqrt {x}\right )^{1+n}}{b (1+n)} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 3.42 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96

method	result	size
derivativedivides	\(\frac {2 \left (a +b \sqrt {x}\right )^{1+n}}{b \left (1+n \right )}\)	\(22\)
default	\(\frac {2 \left (a +b \sqrt {x}\right )^{1+n}}{b \left (1+n \right )}\)	\(22\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {\left (a+b \sqrt {x}\right )^n}{\sqrt {x}} \, dx=\frac {2 \, {\left (b \sqrt {x} + a\right )} {\left (b \sqrt {x} + a\right )}^{n}}{b n + b} \]

[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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 182 vs. \(2 (17) = 34\).

Time = 0.32 (sec) , antiderivative size = 182, normalized size of antiderivative = 7.91 \[ \int \frac {\left (a+b \sqrt {x}\right )^n}{\sqrt {x}} \, dx=\begin {cases} \tilde {\infty } \sqrt {x} & \text {for}\: a = 0 \wedge b = 0 \wedge n = -1 \\2 \cdot 0^{n} \sqrt {x} & \text {for}\: a = - b \sqrt {x} \\2 a^{n} \sqrt {x} & \text {for}\: b = 0 \\\frac {2 \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{b} & \text {for}\: n = -1 \\\frac {2 a^{2} \left (a + b \sqrt {x}\right )^{n}}{a b n + a b + b^{2} n \sqrt {x} + b^{2} \sqrt {x}} + \frac {4 a b \sqrt {x} \left (a + b \sqrt {x}\right )^{n}}{a b n + a b + b^{2} n \sqrt {x} + b^{2} \sqrt {x}} + \frac {2 b^{2} x \left (a + b \sqrt {x}\right )^{n}}{a b n + a b + b^{2} n \sqrt {x} + b^{2} \sqrt {x}} & \text {otherwise} \end {cases} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+b \sqrt {x}\right )^n}{\sqrt {x}} \, dx=\frac {2 \, {\left (b \sqrt {x} + a\right )}^{n + 1}}{b {\left (n + 1\right )}} \]

[In]

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

[Out]

2*(b*sqrt(x) + a)^(n + 1)/(b*(n + 1))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+b \sqrt {x}\right )^n}{\sqrt {x}} \, dx=\frac {2 \, {\left (b \sqrt {x} + a\right )}^{n + 1}}{b {\left (n + 1\right )}} \]

[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))

Mupad [B] (verification not implemented)

Time = 5.75 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+b \sqrt {x}\right )^n}{\sqrt {x}} \, dx=\frac {2\,{\left (a+b\,\sqrt {x}\right )}^{n+1}}{b\,\left (n+1\right )} \]

[In]

int((a + b*x^(1/2))^n/x^(1/2),x)

[Out]

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