3.2494 \(\int \frac{\left (a+b x^n\right )^{5/2}}{x^3} \, dx\)

Optimal. Leaf size=51 \[ -\frac{\left (a+b x^n\right )^{7/2} \, _2F_1\left (1,\frac{7}{2}-\frac{2}{n};-\frac{2-n}{n};-\frac{b x^n}{a}\right )}{2 a x^2} \]

[Out]

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

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Rubi [A]  time = 0.0661571, antiderivative size = 63, normalized size of antiderivative = 1.24, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{a^2 \sqrt{a+b x^n} \, _2F_1\left (-\frac{5}{2},-\frac{2}{n};-\frac{2-n}{n};-\frac{b x^n}{a}\right )}{2 x^2 \sqrt{\frac{b x^n}{a}+1}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^n)^(5/2)/x^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 7.13268, size = 51, normalized size = 1. \[ - \frac{a^{2} \sqrt{a + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{2}, - \frac{2}{n} \\ \frac{n - 2}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{2 x^{2} \sqrt{1 + \frac{b x^{n}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**n)**(5/2)/x**3,x)

[Out]

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

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Mathematica [B]  time = 0.253055, size = 144, normalized size = 2.82 \[ \frac{4 \left (a+b x^n\right ) \left (a^2 \left (23 n^2-36 n+16\right )+a b \left (11 n^2-52 n+32\right ) x^n+b^2 \left (3 n^2-16 n+16\right ) x^{2 n}\right )-15 a^3 n^3 \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},-\frac{2}{n};\frac{n-2}{n};-\frac{b x^n}{a}\right )}{2 (n-4) (3 n-4) (5 n-4) x^2 \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^n)^(5/2)/x^3,x]

[Out]

(4*(a + b*x^n)*(a^2*(16 - 36*n + 23*n^2) + a*b*(32 - 52*n + 11*n^2)*x^n + b^2*(1
6 - 16*n + 3*n^2)*x^(2*n)) - 15*a^3*n^3*Sqrt[1 + (b*x^n)/a]*Hypergeometric2F1[1/
2, -2/n, (-2 + n)/n, -((b*x^n)/a)])/(2*(-4 + n)*(-4 + 3*n)*(-4 + 5*n)*x^2*Sqrt[a
+ b*x^n])

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Maple [F]  time = 0.069, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3}} \left ( a+b{x}^{n} \right ) ^{{\frac{5}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^n)^(5/2)/x^3,x)

[Out]

int((a+b*x^n)^(5/2)/x^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{n} + a\right )}^{\frac{5}{2}}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^(5/2)/x^3,x, algorithm="maxima")

[Out]

integrate((b*x^n + a)^(5/2)/x^3, 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^3,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**3,x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{n} + a\right )}^{\frac{5}{2}}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^(5/2)/x^3,x, algorithm="giac")

[Out]

integrate((b*x^n + a)^(5/2)/x^3, x)