3.2261 \(\int \left (a+b \sqrt{x}\right )^p x^m \, dx\)

Optimal. Leaf size=52 \[ -\frac{2 x^{m+1} \left (a+b \sqrt{x}\right )^{p+1} \, _2F_1\left (1,2 m+p+3;p+2;\frac{a+b \sqrt{x}}{a}\right )}{a (p+1)} \]

[Out]

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

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Rubi [A]  time = 0.0714407, antiderivative size = 63, normalized size of antiderivative = 1.21, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{x^{m+1} \left (a+b \sqrt{x}\right )^p \left (\frac{b \sqrt{x}}{a}+1\right )^{-p} \, _2F_1\left (2 (m+1),-p;2 m+3;-\frac{b \sqrt{x}}{a}\right )}{m+1} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^p*x^m,x]

[Out]

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

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Rubi in Sympy [A]  time = 9.57257, size = 49, normalized size = 0.94 \[ \frac{x^{m + 1} \left (1 + \frac{b \sqrt{x}}{a}\right )^{- p} \left (a + b \sqrt{x}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, 2 m + 2 \\ 2 m + 3 \end{matrix}\middle |{- \frac{b \sqrt{x}}{a}} \right )}}{m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0742418, size = 65, normalized size = 1.25 \[ \frac{x^{m+1} \left (a+b \sqrt{x}\right )^p \left (\frac{b \sqrt{x}}{a}+1\right )^{-p} \, _2F_1\left (2 (m+1),-p;2 (m+1)+1;-\frac{b \sqrt{x}}{a}\right )}{m+1} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.024, size = 0, normalized size = 0. \[ \int{x}^{m} \left ( a+b\sqrt{x} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(a+b*x^(1/2))^p,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*sqrt(x) + a)^p*x^m, x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*sqrt(x) + a)^p*x^m, x)