3.2505 \(\int \frac{x}{\left (a+b x^n\right )^{5/2}} \, dx\)

Optimal. Leaf size=48 \[ \frac{x^2 \, _2F_1\left (1,\frac{2}{n}-\frac{3}{2};\frac{n+2}{n};-\frac{b x^n}{a}\right )}{2 a \left (a+b x^n\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0563192, antiderivative size = 60, normalized size of antiderivative = 1.25, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{x^2 \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{5}{2},\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right )}{2 a^2 \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b*x^n)^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 6.66869, size = 46, normalized size = 0.96 \[ \frac{x^{2} \sqrt{a + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} \frac{5}{2}, \frac{2}{n} \\ \frac{n + 2}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{2 a^{3} \sqrt{1 + \frac{b x^{n}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.15881, size = 100, normalized size = 2.08 \[ \frac{x^2 \left (\left (3 n^2-16 n+16\right ) \left (a+b x^n\right ) \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right )+4 (3 n-4) \left (a+b x^n\right )+4 a n\right )}{6 a^2 n^2 \left (a+b x^n\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(x^2*(4*a*n + 4*(-4 + 3*n)*(a + b*x^n) + (16 - 16*n + 3*n^2)*(a + b*x^n)*Sqrt[1
+ (b*x^n)/a]*Hypergeometric2F1[1/2, 2/n, (2 + n)/n, -((b*x^n)/a)]))/(6*a^2*n^2*(
a + b*x^n)^(3/2))

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Maple [F]  time = 0.041, size = 0, normalized size = 0. \[ \int{x \left ( a+b{x}^{n} \right ) ^{-{\frac{5}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(a+b*x^n)^(5/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x/(b*x^n + a)^(5/2), x)