3.717 \(\int \frac{x^{-2+m}}{\sqrt{a+b x}} \, dx\)

Optimal. Leaf size=49 \[ \frac{2 b x^m \sqrt{a+b x} \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (\frac{1}{2},2-m;\frac{3}{2};\frac{b x}{a}+1\right )}{a^2} \]

[Out]

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

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Rubi [A]  time = 0.0446194, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{2 b x^m \sqrt{a+b x} \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (\frac{1}{2},2-m;\frac{3}{2};\frac{b x}{a}+1\right )}{a^2} \]

Antiderivative was successfully verified.

[In]  Int[x^(-2 + m)/Sqrt[a + b*x],x]

[Out]

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

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Rubi in Sympy [A]  time = 7.60031, size = 39, normalized size = 0.8 \[ \frac{2 b x^{m} \left (- \frac{b x}{a}\right )^{- m} \sqrt{a + b x}{{}_{2}F_{1}\left (\begin{matrix} - m + 2, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{1 + \frac{b x}{a}} \right )}}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.13131, size = 114, normalized size = 2.33 \[ \frac{x^{m-1} \sqrt{a+b x} \left (a^2 m (m+1) \, _2F_1\left (-\frac{1}{2},m-1;m;-\frac{b x}{a}\right )-b (m-1) x \left (a (m+1) \, _2F_1\left (-\frac{1}{2},m;m+1;-\frac{b x}{a}\right )-b m x \, _2F_1\left (\frac{1}{2},m+1;m+2;-\frac{b x}{a}\right )\right )\right )}{a^3 m \left (m^2-1\right ) \sqrt{\frac{b x}{a}+1}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-2 + m)/Sqrt[a + b*x],x]

[Out]

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

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Maple [F]  time = 0.03, size = 0, normalized size = 0. \[ \int{{x}^{-2+m}{\frac{1}{\sqrt{bx+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-2+m)/(b*x+a)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m - 2}}{\sqrt{b x + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(m - 2)/sqrt(b*x + a),x, algorithm="maxima")

[Out]

integrate(x^(m - 2)/sqrt(b*x + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(m - 2)/sqrt(b*x + a),x, algorithm="fricas")

[Out]

integral(x^(m - 2)/sqrt(b*x + a), x)

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Sympy [A]  time = 148.243, size = 32, normalized size = 0.65 \[ \frac{x^{m} \Gamma \left (m - 1\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, m - 1 \\ m \end{matrix}\middle |{\frac{b x e^{i \pi }}{a}} \right )}}{\sqrt{a} x \Gamma \left (m\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**m*gamma(m - 1)*hyper((1/2, m - 1), (m,), b*x*exp_polar(I*pi)/a)/(sqrt(a)*x*ga
mma(m))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m - 2}}{\sqrt{b x + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(m - 2)/sqrt(b*x + a),x, algorithm="giac")

[Out]

integrate(x^(m - 2)/sqrt(b*x + a), x)