3.2244 \(\int \frac{\left (a+b \sqrt{x}\right )^n}{\sqrt{x}} \, dx\)

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 = 0.0208757, 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 \[ \frac{2 \left (a+b \sqrt{x}\right )^{n+1}}{b (n+1)} \]

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))

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Rubi in Sympy [A]  time = 2.64388, size = 17, normalized size = 0.74 \[ \frac{2 \left (a + b \sqrt{x}\right )^{n + 1}}{b \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**(1/2))**n/x**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0139538, 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 = 0.005, size = 22, normalized size = 1. \[ 2\,{\frac{ \left ( a+b\sqrt{x} \right ) ^{1+n}}{b \left ( 1+n \right ) }} \]

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]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^n/sqrt(x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.276852, size = 34, normalized size = 1.48 \[ \frac{2 \,{\left (b \sqrt{x} + a\right )}{\left (b \sqrt{x} + a\right )}^{n}}{b n + b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^n/sqrt(x),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 4.74355, size = 51, normalized size = 2.22 \[ \frac{2 a a^{n} \left (1 + \frac{b \sqrt{x}}{a}\right )^{n}}{b n + b} + \frac{2 a^{n} b \sqrt{x} \left (1 + \frac{b \sqrt{x}}{a}\right )^{n}}{b n + b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.217978, size = 28, normalized size = 1.22 \[ \frac{2 \,{\left (b \sqrt{x} + a\right )}^{n + 1}}{b{\left (n + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^n/sqrt(x),x, algorithm="giac")

[Out]

2*(b*sqrt(x) + a)^(n + 1)/(b*(n + 1))