3.2506 \(\int \frac{1}{\left (a+b x^n\right )^{5/2}} \, dx\)

Optimal. Leaf size=39 \[ \frac{x \, _2F_1\left (1,\frac{1}{n}-\frac{3}{2};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a \left (a+b x^n\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0349105, 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 \[ \frac{x \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{5}{2},\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2 \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^n)^(-5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 3.88484, size = 42, normalized size = 1.08 \[ \frac{x \sqrt{a + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} \frac{5}{2}, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a^{3} \sqrt{1 + \frac{b x^{n}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b*x**n)**(5/2),x)

[Out]

x*sqrt(a + b*x**n)*hyper((5/2, 1/n), (1 + 1/n,), -b*x**n/a)/(a**3*sqrt(1 + b*x**
n/a))

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Mathematica [B]  time = 0.148702, size = 94, normalized size = 2.41 \[ \frac{x \left (\left (3 n^2-8 n+4\right ) \left (a+b x^n\right ) \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )+2 (3 n-2) \left (a+b x^n\right )+2 a n\right )}{3 a^2 n^2 \left (a+b x^n\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^n)^(-5/2),x]

[Out]

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

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Maple [F]  time = 0.04, 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(1/(a+b*x^n)^(5/2),x)

[Out]

int(1/(a+b*x^n)^(5/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{n} + a\right )}^{\frac{5}{2}}}\,{d x} \]

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(1/(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. \[ \int \frac{1}{{\left (b x^{n} + a\right )}^{\frac{5}{2}}}\,{d x} \]

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)