3.714 \(\int \frac{x^{1+m}}{\sqrt{a+b x}} \, dx\)

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

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0421082, 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 \[ -\frac{2 a x^m \sqrt{a+b x} \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (\frac{1}{2},-m-1;\frac{3}{2};\frac{b x}{a}+1\right )}{b^2} \]

Antiderivative was successfully verified.

[In]  Int[x^(1 + m)/Sqrt[a + b*x],x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 7.29981, size = 42, normalized size = 0.86 \[ - \frac{2 a x^{m} \left (- \frac{b x}{a}\right )^{- m} \sqrt{a + b x}{{}_{2}F_{1}\left (\begin{matrix} - m - 1, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{1 + \frac{b x}{a}} \right )}}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0538295, size = 72, normalized size = 1.47 \[ \frac{x^{m+1} \sqrt{a+b x} \left (\, _2F_1\left (-\frac{1}{2},m+1;m+2;-\frac{b x}{a}\right )-\, _2F_1\left (\frac{1}{2},m+1;m+2;-\frac{b x}{a}\right )\right )}{b (m+1) \sqrt{\frac{b x}{a}+1}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(1 + m)/Sqrt[a + b*x],x]

[Out]

(x^(1 + m)*Sqrt[a + b*x]*(Hypergeometric2F1[-1/2, 1 + m, 2 + m, -((b*x)/a)] - Hy
pergeometric2F1[1/2, 1 + m, 2 + m, -((b*x)/a)]))/(b*(1 + m)*Sqrt[1 + (b*x)/a])

_______________________________________________________________________________________

Maple [F]  time = 0.036, size = 0, normalized size = 0. \[ \int{{x}^{1+m}{\frac{1}{\sqrt{bx+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1+m)/(b*x+a)^(1/2),x)

[Out]

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

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m + 1}}{\sqrt{b x + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(m + 1)/sqrt(b*x + a),x, algorithm="maxima")

[Out]

integrate(x^(m + 1)/sqrt(b*x + a), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{m + 1}}{\sqrt{b x + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(m + 1)/sqrt(b*x + a),x, algorithm="fricas")

[Out]

integral(x^(m + 1)/sqrt(b*x + a), x)

_______________________________________________________________________________________

Sympy [A]  time = 10.1239, size = 37, normalized size = 0.76 \[ \frac{x^{2} x^{m} \Gamma \left (m + 2\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, m + 2 \\ m + 3 \end{matrix}\middle |{\frac{b x e^{i \pi }}{a}} \right )}}{\sqrt{a} \Gamma \left (m + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**2*x**m*gamma(m + 2)*hyper((1/2, m + 2), (m + 3,), b*x*exp_polar(I*pi)/a)/(sqr
t(a)*gamma(m + 3))

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m + 1}}{\sqrt{b x + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(m + 1)/sqrt(b*x + a),x, algorithm="giac")

[Out]

integrate(x^(m + 1)/sqrt(b*x + a), x)