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

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

[Out]

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

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Rubi [A]  time = 0.468941, 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 \[ -\frac{\sqrt{a+b x^n+c x^{2 n}} F_1\left (-\frac{1}{n};-\frac{1}{2},-\frac{1}{2};-\frac{1-n}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{x \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 33.6334, size = 128, normalized size = 0.86 \[ - \frac{\sqrt{a + b x^{n} + c x^{2 n}} \operatorname{appellf_{1}}{\left (- \frac{1}{n},- \frac{1}{2},- \frac{1}{2},\frac{n - 1}{n},- \frac{2 c x^{n}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{n}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{x \sqrt{\frac{2 c x^{n}}{b - \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{\frac{2 c x^{n}}{b + \sqrt{- 4 a c + b^{2}}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**n+c*x**(2*n))**(1/2)/x**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)))/(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 = 8.58844, size = 821, normalized size = 5.51 \[ \frac{\frac{2 a^2 b n (2 n-1) \left (2 c x^n+b-\sqrt{b^2-4 a c}\right ) \left (2 c x^n+b+\sqrt{b^2-4 a c}\right ) F_1\left (\frac{n-1}{n};\frac{1}{2},\frac{1}{2};2-\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right ) x^n}{\left (\sqrt{b^2-4 a c}-b\right ) \left (b+\sqrt{b^2-4 a c}\right ) (n-1)^2 \left (\left (b+\sqrt{b^2-4 a c}\right ) n F_1\left (2-\frac{1}{n};\frac{1}{2},\frac{3}{2};3-\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right ) x^n-\left (\sqrt{b^2-4 a c}-b\right ) n F_1\left (2-\frac{1}{n};\frac{3}{2},\frac{1}{2};3-\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right ) x^n+4 a (1-2 n) F_1\left (\frac{n-1}{n};\frac{1}{2},\frac{1}{2};2-\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )\right )}+\frac{\left (\left (c x^n+b\right ) x^n+a\right )^2}{n-1}+\frac{a^2 n \left (-2 c x^n-b+\sqrt{b^2-4 a c}\right ) \left (2 c x^n+b+\sqrt{b^2-4 a c}\right ) F_1\left (-\frac{1}{n};\frac{1}{2},\frac{1}{2};\frac{n-1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )}{c \left (4 a (n-1) F_1\left (-\frac{1}{n};\frac{1}{2},\frac{1}{2};\frac{n-1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )-n x^n \left (\left (b+\sqrt{b^2-4 a c}\right ) F_1\left (\frac{n-1}{n};\frac{1}{2},\frac{3}{2};2-\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )+\left (b-\sqrt{b^2-4 a c}\right ) F_1\left (\frac{n-1}{n};\frac{3}{2},\frac{1}{2};2-\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )\right )\right )}}{x \left (\left (c x^n+b\right ) x^n+a\right )^{3/2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.084, 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]  int((a+b*x^n+c*x^(2*n))^(1/2)/x^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(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(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. \[ \int \frac{\sqrt{a + b x^{n} + c x^{2 n}}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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