3.587 \(\int \frac{1}{x^2 \sqrt{a+b x^n+c x^{2 n}}} \, dx\)

Optimal. Leaf size=149 \[ -\frac{\sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left (-\frac{1}{n};\frac{1}{2},\frac{1}{2};-\frac{1-n}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{x \sqrt{a+b x^n+c x^{2 n}}} \]

[Out]

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

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Rubi [A]  time = 0.463284, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{\sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left (-\frac{1}{n};\frac{1}{2},\frac{1}{2};-\frac{1-n}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{x \sqrt{a+b x^n+c x^{2 n}}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*Sqrt[a + b*x^n + c*x^(2*n)]),x]

[Out]

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

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Rubi in Sympy [A]  time = 33.2611, size = 126, normalized size = 0.85 \[ - \frac{\sqrt{a + b x^{n} + c x^{2 n}} \operatorname{appellf_{1}}{\left (- \frac{1}{n},\frac{1}{2},\frac{1}{2},\frac{n - 1}{n},- \frac{2 c x^{n}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{n}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{a x \sqrt{\frac{2 c x^{n}}{b - \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{\frac{2 c x^{n}}{b + \sqrt{- 4 a c + b^{2}}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(a + b*x**n + c*x**(2*n))*appellf1(-1/n, 1/2, 1/2, (n - 1)/n, -2*c*x**n/(b
- sqrt(-4*a*c + b**2)), -2*c*x**n/(b + sqrt(-4*a*c + b**2)))/(a*x*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 = 0.424412, size = 415, normalized size = 2.79 \[ -\frac{4 a^2 (n-1) \left (\sqrt{b^2-4 a c}-b-2 c x^n\right ) \left (\sqrt{b^2-4 a c}+b+2 c x^n\right ) F_1\left (-\frac{1}{n};\frac{1}{2},\frac{1}{2};\frac{n-1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )}{x \left (b-\sqrt{b^2-4 a c}\right ) \left (\sqrt{b^2-4 a c}+b\right ) \left (a+x^n \left (b+c x^n\right )\right )^{3/2} \left (n x^n \left (\left (\sqrt{b^2-4 a c}+b\right ) F_1\left (\frac{n-1}{n};\frac{1}{2},\frac{3}{2};2-\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )+\left (b-\sqrt{b^2-4 a c}\right ) F_1\left (\frac{n-1}{n};\frac{3}{2},\frac{1}{2};2-\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )\right )-4 a (n-1) F_1\left (-\frac{1}{n};\frac{1}{2},\frac{1}{2};\frac{n-1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/(x^2*Sqrt[a + b*x^n + c*x^(2*n)]),x]

[Out]

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

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Maple [F]  time = 0.031, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2}}{\frac{1}{\sqrt{a+b{x}^{n}+c{x}^{2\,n}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(a+b*x^n+c*x^(2*n))^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c x^{2 \, n} + b x^{n} + a} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^(2*n) + b*x^n + a)*x^2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(c*x^(2*n) + b*x^n + a)*x^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(1/(sqrt(c*x^(2*n) + b*x^n + a)*x^2),x, algorithm="fricas")

[Out]

Exception raised: TypeError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{2} \sqrt{a + b x^{n} + c x^{2 n}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**2/(a+b*x**n+c*x**(2*n))**(1/2),x)

[Out]

Integral(1/(x**2*sqrt(a + b*x**n + c*x**(2*n))), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c x^{2 \, n} + b x^{n} + a} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^(2*n) + b*x^n + a)*x^2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(c*x^(2*n) + b*x^n + a)*x^2), x)