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

Optimal. Leaf size=43 \[ 2 \sqrt{x} (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};-\frac{b x}{a}\right ) \]

[Out]

(2*Sqrt[x]*(a + b*x)^n*Hypergeometric2F1[1/2, -n, 3/2, -((b*x)/a)])/(1 + (b*x)/a
)^n

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Rubi [A]  time = 0.0266629, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ 2 \sqrt{x} (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};-\frac{b x}{a}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^n/Sqrt[x],x]

[Out]

(2*Sqrt[x]*(a + b*x)^n*Hypergeometric2F1[1/2, -n, 3/2, -((b*x)/a)])/(1 + (b*x)/a
)^n

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Rubi in Sympy [A]  time = 5.29669, size = 34, normalized size = 0.79 \[ 2 \sqrt{x} \left (1 + \frac{b x}{a}\right )^{- n} \left (a + b x\right )^{n}{{}_{2}F_{1}\left (\begin{matrix} - n, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{- \frac{b x}{a}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(x)*(1 + b*x/a)**(-n)*(a + b*x)**n*hyper((-n, 1/2), (3/2,), -b*x/a)

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Mathematica [A]  time = 0.0153464, size = 43, normalized size = 1. \[ 2 \sqrt{x} (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};-\frac{b x}{a}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[x]*(a + b*x)^n*Hypergeometric2F1[1/2, -n, 3/2, -((b*x)/a)])/(1 + (b*x)/a
)^n

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Maple [F]  time = 0.028, size = 0, normalized size = 0. \[ \int{ \left ( bx+a \right ) ^{n}{\frac{1}{\sqrt{x}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^n/x^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{n}}{\sqrt{x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x + a)^n/sqrt(x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x + a\right )}^{n}}{\sqrt{x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x + a)^n/sqrt(x), x)

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Sympy [A]  time = 5.195, size = 26, normalized size = 0.6 \[ 2 a^{n} \sqrt{x}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, - n \\ \frac{3}{2} \end{matrix}\middle |{\frac{b x e^{i \pi }}{a}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**n*sqrt(x)*hyper((1/2, -n), (3/2,), b*x*exp_polar(I*pi)/a)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{n}}{\sqrt{x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x + a)^n/sqrt(x), x)