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

Optimal. Leaf size=151 \[ \frac{x^2 \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{2}{n};\frac{3}{2},\frac{3}{2};\frac{n+2}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{2 a \sqrt{a+b x^n+c x^{2 n}}} \]

[Out]

(x^2*Sqrt[1 + (2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^n)/(b + Sqrt[b^
2 - 4*a*c])]*AppellF1[2/n, 3/2, 3/2, (2 + n)/n, (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c
]), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(2*a*Sqrt[a + b*x^n + c*x^(2*n)])

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Rubi [A]  time = 0.355707, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{x^2 \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{2}{n};\frac{3}{2},\frac{3}{2};\frac{n+2}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{2 a \sqrt{a+b x^n+c x^{2 n}}} \]

Antiderivative was successfully verified.

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

[Out]

(x^2*Sqrt[1 + (2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^n)/(b + Sqrt[b^
2 - 4*a*c])]*AppellF1[2/n, 3/2, 3/2, (2 + n)/n, (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c
]), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(2*a*Sqrt[a + b*x^n + c*x^(2*n)])

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Rubi in Sympy [A]  time = 28.7335, size = 128, normalized size = 0.85 \[ \frac{x^{2} \sqrt{a + b x^{n} + c x^{2 n}} \operatorname{appellf_{1}}{\left (\frac{2}{n},\frac{3}{2},\frac{3}{2},\frac{n + 2}{n},- \frac{2 c x^{n}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{n}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{2 a^{2} \sqrt{\frac{2 c x^{n}}{b - \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{\frac{2 c x^{n}}{b + \sqrt{- 4 a c + b^{2}}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**2*sqrt(a + b*x**n + c*x**(2*n))*appellf1(2/n, 3/2, 3/2, (n + 2)/n, -2*c*x**n/
(b - sqrt(-4*a*c + b**2)), -2*c*x**n/(b + sqrt(-4*a*c + b**2)))/(2*a**2*sqrt(2*c
*x**n/(b - sqrt(-4*a*c + b**2)) + 1)*sqrt(2*c*x**n/(b + sqrt(-4*a*c + b**2)) + 1
))

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Mathematica [B]  time = 5.43996, size = 1947, normalized size = 12.89 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2*x^2*(-(((b^2 - 2*a*c + b*c*x^n)*(a + x^n*(b + c*x^n)))/n) + (16*a^2*b*c*(1 +
n)*x^n*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)*Appel
lF1[(2 + n)/n, 1/2, 1/2, 2 + 2/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/
(-b + Sqrt[b^2 - 4*a*c])])/((-b + Sqrt[b^2 - 4*a*c])*(b + Sqrt[b^2 - 4*a*c])*n*(
2 + n)*((b + Sqrt[b^2 - 4*a*c])*n*x^n*AppellF1[2 + 2/n, 1/2, 3/2, 3 + 2/n, (-2*c
*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] - (-b + Sqrt[
b^2 - 4*a*c])*n*x^n*AppellF1[2 + 2/n, 3/2, 1/2, 3 + 2/n, (-2*c*x^n)/(b + Sqrt[b^
2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] - 8*a*(1 + n)*AppellF1[(2 + n)/
n, 1/2, 1/2, 2 + 2/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b
^2 - 4*a*c])])) + (a^2*(2 + n)*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)*(b + Sqrt[b^2 -
4*a*c] + 2*c*x^n)*AppellF1[2/n, 1/2, 1/2, (2 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 -
4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])/(4*a*(2 + n)*AppellF1[2/n, 1/2, 1
/2, (2 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*
a*c])] - n*x^n*((b + Sqrt[b^2 - 4*a*c])*AppellF1[(2 + n)/n, 1/2, 3/2, 2 + 2/n, (
-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] + (b - Sq
rt[b^2 - 4*a*c])*AppellF1[(2 + n)/n, 3/2, 1/2, 2 + 2/n, (-2*c*x^n)/(b + Sqrt[b^2
- 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])) - (a*b^2*(2 + n)*(b - Sqrt[b^2
- 4*a*c] + 2*c*x^n)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)*AppellF1[2/n, 1/2, 1/2, (
2 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])
])/(4*c*(4*a*(2 + n)*AppellF1[2/n, 1/2, 1/2, (2 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2
- 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] - n*x^n*((b + Sqrt[b^2 - 4*a*c])
*AppellF1[(2 + n)/n, 1/2, 3/2, 2 + 2/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c
*x^n)/(-b + Sqrt[b^2 - 4*a*c])] + (b - Sqrt[b^2 - 4*a*c])*AppellF1[(2 + n)/n, 3/
2, 1/2, 2 + 2/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 -
4*a*c])]))) - (2*a^2*(2 + n)*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)*(b + Sqrt[b^2 - 4
*a*c] + 2*c*x^n)*AppellF1[2/n, 1/2, 1/2, (2 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4
*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])/(n*(4*a*(2 + n)*AppellF1[2/n, 1/2,
1/2, (2 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4
*a*c])] - n*x^n*((b + Sqrt[b^2 - 4*a*c])*AppellF1[(2 + n)/n, 1/2, 3/2, 2 + 2/n,
(-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] + (b - S
qrt[b^2 - 4*a*c])*AppellF1[(2 + n)/n, 3/2, 1/2, 2 + 2/n, (-2*c*x^n)/(b + Sqrt[b^
2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])]))) + (a*b^2*(2 + n)*(b - Sqrt[b
^2 - 4*a*c] + 2*c*x^n)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)*AppellF1[2/n, 1/2, 1/2,
(2 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c
])])/(c*n*(4*a*(2 + n)*AppellF1[2/n, 1/2, 1/2, (2 + n)/n, (-2*c*x^n)/(b + Sqrt[b
^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] - n*x^n*((b + Sqrt[b^2 - 4*a*c
])*AppellF1[(2 + n)/n, 1/2, 3/2, 2 + 2/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2
*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] + (b - Sqrt[b^2 - 4*a*c])*AppellF1[(2 + n)/n,
3/2, 1/2, 2 + 2/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2
- 4*a*c])])))))/(a*(-b^2 + 4*a*c)*(a + x^n*(b + c*x^n))^(3/2))

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Maple [F]  time = 0.019, size = 0, normalized size = 0. \[ \int{x \left ( a+b{x}^{n}+c{x}^{2\,n} \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x/(c*x^(2*n) + 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/(c*x^(2*n) + b*x^n + a)^(3/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x/(a + b*x**n + c*x**(2*n))**(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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