3.748 \(\int \frac{(a+b x)^n}{x^{3/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0277451, 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 \[ -\frac{2 (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (-\frac{1}{2},-n;\frac{1}{2};-\frac{b x}{a}\right )}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^n/x^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.3348, size = 37, normalized size = 0.86 \[ - \frac{2 \left (1 + \frac{b x}{a}\right )^{- n} \left (a + b x\right )^{n}{{}_{2}F_{1}\left (\begin{matrix} - n, - \frac{1}{2} \\ \frac{1}{2} \end{matrix}\middle |{- \frac{b x}{a}} \right )}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**n/x**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.0181434, size = 43, normalized size = 1. \[ -\frac{2 (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (-\frac{1}{2},-n;\frac{1}{2};-\frac{b x}{a}\right )}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^n/x^(3/2),x]

[Out]

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

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Maple [F]  time = 0.027, size = 0, normalized size = 0. \[ \int{ \left ( bx+a \right ) ^{n}{x}^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^n/x^(3/2),x)

[Out]

int((b*x+a)^n/x^(3/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{n}}{x^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^n/x^(3/2),x, algorithm="maxima")

[Out]

integrate((b*x + a)^n/x^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x + a\right )}^{n}}{x^{\frac{3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^n/x^(3/2),x, algorithm="fricas")

[Out]

integral((b*x + a)^n/x^(3/2), x)

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Sympy [A]  time = 12.7532, size = 29, normalized size = 0.67 \[ - \frac{2 a^{n}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, - n \\ \frac{1}{2} \end{matrix}\middle |{\frac{b x e^{i \pi }}{a}} \right )}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**n/x**(3/2),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{n}}{x^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^n/x^(3/2),x, algorithm="giac")

[Out]

integrate((b*x + a)^n/x^(3/2), x)