3.2480 \(\int x \sqrt{a+b x^n} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0547318, antiderivative size = 57, normalized size of antiderivative = 1.19, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{x^2 \sqrt{a+b x^n} \, _2F_1\left (-\frac{1}{2},\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right )}{2 \sqrt{\frac{b x^n}{a}+1}} \]

Antiderivative was successfully verified.

[In]  Int[x*Sqrt[a + b*x^n],x]

[Out]

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

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Rubi in Sympy [A]  time = 6.22069, size = 44, normalized size = 0.92 \[ \frac{x^{2} \sqrt{a + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{2}{n} \\ \frac{n + 2}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{2 \sqrt{1 + \frac{b x^{n}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0886558, size = 75, normalized size = 1.56 \[ \frac{x^2 \left (a n \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right )+4 \left (a+b x^n\right )\right )}{2 (n+4) \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

[In]  Integrate[x*Sqrt[a + b*x^n],x]

[Out]

(x^2*(4*(a + b*x^n) + a*n*Sqrt[1 + (b*x^n)/a]*Hypergeometric2F1[1/2, 2/n, (2 + n
)/n, -((b*x^n)/a)]))/(2*(4 + n)*Sqrt[a + b*x^n])

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Maple [F]  time = 0.072, size = 0, normalized size = 0. \[ \int x\sqrt{a+b{x}^{n}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(a+b*x^n)^(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(b*x^n + a)*x, 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,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*(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. \[ \int \sqrt{b x^{n} + a} x\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(b*x^n + a)*x, x)