3.2259 \(\int \frac{x^m}{a+b \sqrt{x}} \, dx\)

Optimal. Leaf size=37 \[ \frac{x^{m+1} \, _2F_1\left (1,2 (m+1);2 m+3;-\frac{b \sqrt{x}}{a}\right )}{a (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.0499234, antiderivative size = 37, 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 \[ \frac{x^{m+1} \, _2F_1\left (1,2 (m+1);2 m+3;-\frac{b \sqrt{x}}{a}\right )}{a (m+1)} \]

Antiderivative was successfully verified.

[In]  Int[x^m/(a + b*Sqrt[x]),x]

[Out]

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

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Rubi in Sympy [A]  time = 6.6144, size = 27, normalized size = 0.73 \[ \frac{x^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, 2 m + 2 \\ 2 m + 3 \end{matrix}\middle |{- \frac{b \sqrt{x}}{a}} \right )}}{a \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0564639, size = 42, normalized size = 1.14 \[ -\frac{2 x^{m+\frac{1}{2}} \left (\, _2F_1\left (1,2 m+1;2 m+2;-\frac{b \sqrt{x}}{a}\right )-1\right )}{2 b m+b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.021, size = 0, normalized size = 0. \[ \int{{x}^{m} \left ( a+b\sqrt{x} \right ) ^{-1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m/(a+b*x^(1/2)),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^m/(b*sqrt(x) + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x^m/(b*sqrt(x) + a), x)

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Sympy [A]  time = 3.49082, size = 82, normalized size = 2.22 \[ \frac{4 m x x^{m} \Phi \left (\frac{b \sqrt{x} e^{i \pi }}{a}, 1, 2 m + 2\right ) \Gamma \left (2 m + 2\right )}{a \Gamma \left (2 m + 3\right )} + \frac{4 x x^{m} \Phi \left (\frac{b \sqrt{x} e^{i \pi }}{a}, 1, 2 m + 2\right ) \Gamma \left (2 m + 2\right )}{a \Gamma \left (2 m + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*m*x*x**m*lerchphi(b*sqrt(x)*exp_polar(I*pi)/a, 1, 2*m + 2)*gamma(2*m + 2)/(a*g
amma(2*m + 3)) + 4*x*x**m*lerchphi(b*sqrt(x)*exp_polar(I*pi)/a, 1, 2*m + 2)*gamm
a(2*m + 2)/(a*gamma(2*m + 3))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^m/(b*sqrt(x) + a), x)