3.2500 \(\int \frac{x}{\left (a+b x^n\right )^{3/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0564546, 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 \[ \frac{x^2 \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{3}{2},\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right )}{2 a \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b*x^n)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 6.56893, size = 46, normalized size = 0.96 \[ \frac{x^{2} \sqrt{a + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{2}{n} \\ \frac{n + 2}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{2 a^{2} \sqrt{1 + \frac{b x^{n}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.101374, size = 69, normalized size = 1.44 \[ \frac{x^2 \left ((n-4) \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right )+4\right )}{2 a n \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.042, size = 0, normalized size = 0. \[ \int{x \left ( a+b{x}^{n} \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(a+b*x^n)^(3/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x/(b*x^n + a)^(3/2), x)