3.2671 \(\int \frac{x^{3+2 n}}{\sqrt{a+b x^n}} \, dx\)

Optimal. Leaf size=59 \[ \frac{x^{2 (n+2)} \sqrt{a+b x^n} \, _2F_1\left (1,\frac{1}{2} \left (5+\frac{8}{n}\right );3+\frac{4}{n};-\frac{b x^n}{a}\right )}{2 a (n+2)} \]

[Out]

(x^(2*(2 + n))*Sqrt[a + b*x^n]*Hypergeometric2F1[1, (5 + 8/n)/2, 3 + 4/n, -((b*x
^n)/a)])/(2*a*(2 + n))

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Rubi [A]  time = 0.076905, antiderivative size = 70, normalized size of antiderivative = 1.19, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{x^{2 (n+2)} \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},2 \left (1+\frac{2}{n}\right );3+\frac{4}{n};-\frac{b x^n}{a}\right )}{2 (n+2) \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

[In]  Int[x^(3 + 2*n)/Sqrt[a + b*x^n],x]

[Out]

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

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Rubi in Sympy [A]  time = 8.03668, size = 53, normalized size = 0.9 \[ \frac{x^{2 n + 4} \sqrt{a + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, 2 + \frac{4}{n} \\ 3 + \frac{4}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{2 a \sqrt{1 + \frac{b x^{n}}{a}} \left (n + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**(2*n + 4)*sqrt(a + b*x**n)*hyper((1/2, 2 + 4/n), (3 + 4/n,), -b*x**n/a)/(2*a*
sqrt(1 + b*x**n/a)*(n + 2))

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Mathematica [A]  time = 0.20525, size = 104, normalized size = 1.76 \[ \frac{2 x^4 \left (2 a^2 (n+4) \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{4}{n};\frac{n+4}{n};-\frac{b x^n}{a}\right )-\left (a+b x^n\right ) \left (2 a (n+4)-b (n+8) x^n\right )\right )}{b^2 (n+8) (3 n+8) \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(3 + 2*n)/Sqrt[a + b*x^n],x]

[Out]

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

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Maple [F]  time = 0.081, size = 0, normalized size = 0. \[ \int{{x}^{3+2\,n}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2 \, n + 3}}{\sqrt{b x^{n} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(2*n + 3)/sqrt(b*x^n + a),x, algorithm="maxima")

[Out]

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2 \, n + 3}}{\sqrt{b x^{n} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(2*n + 3)/sqrt(b*x^n + a),x, algorithm="giac")

[Out]

integrate(x^(2*n + 3)/sqrt(b*x^n + a), x)