3.2491 \(\int \left (a+b x^n\right )^{5/2} \, dx\)

Optimal. Leaf size=39 \[ \frac{x \left (a+b x^n\right )^{7/2} \, _2F_1\left (1,\frac{7}{2}+\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a} \]

[Out]

(x*(a + b*x^n)^(7/2)*Hypergeometric2F1[1, 7/2 + n^(-1), 1 + n^(-1), -((b*x^n)/a)
])/a

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Rubi [A]  time = 0.0339041, antiderivative size = 51, normalized size of antiderivative = 1.31, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{a^2 x \sqrt{a+b x^n} \, _2F_1\left (-\frac{5}{2},\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{\sqrt{\frac{b x^n}{a}+1}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^n)^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 3.79977, size = 44, normalized size = 1.13 \[ \frac{a^{2} x \sqrt{a + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{2}, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{\sqrt{1 + \frac{b x^{n}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**n)**(5/2),x)

[Out]

a**2*x*sqrt(a + b*x**n)*hyper((-5/2, 1/n), (1 + 1/n,), -b*x**n/a)/sqrt(1 + b*x**
n/a)

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Mathematica [B]  time = 0.221857, size = 135, normalized size = 3.46 \[ \frac{x \left (15 a^3 n^3 \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )+2 \left (a+b x^n\right ) \left (a^2 \left (23 n^2+18 n+4\right )+a b \left (11 n^2+26 n+8\right ) x^n+b^2 \left (3 n^2+8 n+4\right ) x^{2 n}\right )\right )}{(n+2) (3 n+2) (5 n+2) \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^n)^(5/2),x]

[Out]

(x*(2*(a + b*x^n)*(a^2*(4 + 18*n + 23*n^2) + a*b*(8 + 26*n + 11*n^2)*x^n + b^2*(
4 + 8*n + 3*n^2)*x^(2*n)) + 15*a^3*n^3*Sqrt[1 + (b*x^n)/a]*Hypergeometric2F1[1/2
, n^(-1), 1 + n^(-1), -((b*x^n)/a)]))/((2 + n)*(2 + 3*n)*(2 + 5*n)*Sqrt[a + b*x^
n])

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Maple [F]  time = 0.053, size = 0, normalized size = 0. \[ \int \left ( a+b{x}^{n} \right ) ^{{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^n)^(5/2),x)

[Out]

int((a+b*x^n)^(5/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + a\right )}^{\frac{5}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^(5/2),x, algorithm="maxima")

[Out]

integrate((b*x^n + a)^(5/2), 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*x^n + a)^(5/2),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((a+b*x**n)**(5/2),x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + a\right )}^{\frac{5}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^(5/2),x, algorithm="giac")

[Out]

integrate((b*x^n + a)^(5/2), x)