3.2667 \(\int x^m \sqrt{a+b x^n} \, dx\)

Optimal. Leaf size=55 \[ \frac{x^{m+1} \left (a+b x^n\right )^{3/2} \, _2F_1\left (1,\frac{m+1}{n}+\frac{3}{2};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.0672495, antiderivative size = 64, normalized size of antiderivative = 1.16, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{x^{m+1} \sqrt{a+b x^n} \, _2F_1\left (-\frac{1}{2},\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{(m+1) \sqrt{\frac{b x^n}{a}+1}} \]

Antiderivative was successfully verified.

[In]  Int[x^m*Sqrt[a + b*x^n],x]

[Out]

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

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Rubi in Sympy [A]  time = 7.48744, size = 51, normalized size = 0.93 \[ \frac{x^{m + 1} \sqrt{a + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{m + 1}{n} \\ \frac{m + n + 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{\sqrt{1 + \frac{b x^{n}}{a}} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.114909, size = 88, normalized size = 1.6 \[ \frac{x^{m+1} \left (a n \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )+2 (m+1) \left (a+b x^n\right )\right )}{(m+1) (2 m+n+2) \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^m*Sqrt[a + b*x^n],x]

[Out]

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

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Maple [F]  time = 0.091, size = 0, normalized size = 0. \[ \int{x}^{m}\sqrt{a+b{x}^{n}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(a+b*x^n)^(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError

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Sympy [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(x**m*(a+b*x**n)**(1/2),x)

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(b*x^n + a)*x^m, x)