3.709 \(\int x^m \sqrt{a+b x} \, dx\)

Optimal. Leaf size=48 \[ \frac{2 x^m (a+b x)^{3/2} \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (\frac{3}{2},-m;\frac{5}{2};\frac{b x}{a}+1\right )}{3 b} \]

[Out]

(2*x^m*(a + b*x)^(3/2)*Hypergeometric2F1[3/2, -m, 5/2, 1 + (b*x)/a])/(3*b*(-((b*
x)/a))^m)

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Rubi [A]  time = 0.0379839, antiderivative size = 48, 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 \[ \frac{2 x^m (a+b x)^{3/2} \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (\frac{3}{2},-m;\frac{5}{2};\frac{b x}{a}+1\right )}{3 b} \]

Antiderivative was successfully verified.

[In]  Int[x^m*Sqrt[a + b*x],x]

[Out]

(2*x^m*(a + b*x)^(3/2)*Hypergeometric2F1[3/2, -m, 5/2, 1 + (b*x)/a])/(3*b*(-((b*
x)/a))^m)

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Rubi in Sympy [A]  time = 6.14811, size = 37, normalized size = 0.77 \[ \frac{2 x^{m} \left (- \frac{b x}{a}\right )^{- m} \left (a + b x\right )^{\frac{3}{2}}{{}_{2}F_{1}\left (\begin{matrix} - m, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{1 + \frac{b x}{a}} \right )}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x**m*(-b*x/a)**(-m)*(a + b*x)**(3/2)*hyper((-m, 3/2), (5/2,), 1 + b*x/a)/(3*b)

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Mathematica [A]  time = 0.018694, size = 50, normalized size = 1.04 \[ \frac{x^{m+1} \sqrt{a+b x} \, _2F_1\left (-\frac{1}{2},m+1;m+2;-\frac{b x}{a}\right )}{(m+1) \sqrt{\frac{a+b x}{a}}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^m*Sqrt[a + b*x],x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(b*x+a)^(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(b*x + a)*x^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(b*x + a)*x^m, x)

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Sympy [A]  time = 4.90698, size = 37, normalized size = 0.77 \[ \frac{\sqrt{a} x x^{m} \Gamma \left (m + 1\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{b x e^{i \pi }}{a}} \right )}}{\Gamma \left (m + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(b*x + a)*x^m, x)