3.2672 \(\int \frac{x^{3+n}}{\sqrt{a+b x^n}} \, dx\)

Optimal. Leaf size=56 \[ \frac{x^{n+4} \sqrt{a+b x^n} \, _2F_1\left (1,\frac{1}{2} \left (3+\frac{8}{n}\right );2 \left (1+\frac{2}{n}\right );-\frac{b x^n}{a}\right )}{a (n+4)} \]

[Out]

(x^(4 + n)*Sqrt[a + b*x^n]*Hypergeometric2F1[1, (3 + 8/n)/2, 2*(1 + 2/n), -((b*x
^n)/a)])/(a*(4 + n))

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Rubi [A]  time = 0.0733811, antiderivative size = 65, normalized size of antiderivative = 1.16, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{x^{n+4} \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{n+4}{n};2 \left (1+\frac{2}{n}\right );-\frac{b x^n}{a}\right )}{(n+4) \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

[In]  Int[x^(3 + n)/Sqrt[a + b*x^n],x]

[Out]

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

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Rubi in Sympy [A]  time = 7.83545, size = 49, normalized size = 0.88 \[ \frac{x^{n + 4} \sqrt{a + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{n + 4}{n} \\ 2 + \frac{4}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a \sqrt{1 + \frac{b x^{n}}{a}} \left (n + 4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(3+n)/(a+b*x**n)**(1/2),x)

[Out]

x**(n + 4)*sqrt(a + b*x**n)*hyper((1/2, (n + 4)/n), (2 + 4/n,), -b*x**n/a)/(a*sq
rt(1 + b*x**n/a)*(n + 4))

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Mathematica [A]  time = 0.0815198, size = 73, normalized size = 1.3 \[ \frac{2 x^4 \left (-a \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{4}{n};\frac{n+4}{n};-\frac{b x^n}{a}\right )+a+b x^n\right )}{b (n+8) \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(3 + n)/Sqrt[a + b*x^n],x]

[Out]

(2*x^4*(a + b*x^n - a*Sqrt[1 + (b*x^n)/a]*Hypergeometric2F1[1/2, 4/n, (4 + n)/n,
-((b*x^n)/a)]))/(b*(8 + n)*Sqrt[a + b*x^n])

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Maple [F]  time = 0.064, size = 0, normalized size = 0. \[ \int{{x}^{3+n}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(3+n)/(a+b*x^n)^(1/2),x)

[Out]

int(x^(3+n)/(a+b*x^n)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{n + 3}}{\sqrt{b x^{n} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(n + 3)/sqrt(b*x^n + a),x, algorithm="maxima")

[Out]

integrate(x^(n + 3)/sqrt(b*x^n + a), 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(x^(n + 3)/sqrt(b*x^n + a),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**(3+n)/(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. \[ \int \frac{x^{n + 3}}{\sqrt{b x^{n} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(n + 3)/sqrt(b*x^n + a),x, algorithm="giac")

[Out]

integrate(x^(n + 3)/sqrt(b*x^n + a), x)