3.749 \(\int \frac{(a+b x)^n}{x^{5/2}} \, dx\)

Optimal. Leaf size=45 \[ -\frac{2 (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (-\frac{3}{2},-n;-\frac{1}{2};-\frac{b x}{a}\right )}{3 x^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0277944, antiderivative size = 45, 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 \[ -\frac{2 (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (-\frac{3}{2},-n;-\frac{1}{2};-\frac{b x}{a}\right )}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^n/x^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.37066, size = 41, normalized size = 0.91 \[ - \frac{2 \left (1 + \frac{b x}{a}\right )^{- n} \left (a + b x\right )^{n}{{}_{2}F_{1}\left (\begin{matrix} - n, - \frac{3}{2} \\ - \frac{1}{2} \end{matrix}\middle |{- \frac{b x}{a}} \right )}}{3 x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**n/x**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.0244816, size = 45, normalized size = 1. \[ -\frac{2 (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (-\frac{3}{2},-n;-\frac{1}{2};-\frac{b x}{a}\right )}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^n/x^(5/2),x]

[Out]

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

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Maple [F]  time = 0.025, size = 0, normalized size = 0. \[ \int{ \left ( bx+a \right ) ^{n}{x}^{-{\frac{5}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^n/x^(5/2),x)

[Out]

int((b*x+a)^n/x^(5/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{n}}{x^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^n/x^(5/2),x, algorithm="maxima")

[Out]

integrate((b*x + a)^n/x^(5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x + a\right )}^{n}}{x^{\frac{5}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^n/x^(5/2),x, algorithm="fricas")

[Out]

integral((b*x + a)^n/x^(5/2), x)

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Sympy [A]  time = 90.6005, size = 32, normalized size = 0.71 \[ - \frac{2 a^{n}{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, - n \\ - \frac{1}{2} \end{matrix}\middle |{\frac{b x e^{i \pi }}{a}} \right )}}{3 x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**n/x**(5/2),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{n}}{x^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^n/x^(5/2),x, algorithm="giac")

[Out]

integrate((b*x + a)^n/x^(5/2), x)