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

Optimal. Leaf size=139 \[ \frac{x \sqrt{a+b x^n+c x^{2 n}} F_1\left (\frac{1}{n};-\frac{1}{2},-\frac{1}{2};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{\sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1}} \]

[Out]

(x*Sqrt[a + b*x^n + c*x^(2*n)]*AppellF1[n^(-1), -1/2, -1/2, 1 + n^(-1), (-2*c*x^
n)/(b - Sqrt[b^2 - 4*a*c]), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(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 = 0.221845, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{x \sqrt{a+b x^n+c x^{2 n}} F_1\left (\frac{1}{n};-\frac{1}{2},-\frac{1}{2};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{\sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1}} \]

Antiderivative was successfully verified.

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

[Out]

(x*Sqrt[a + b*x^n + c*x^(2*n)]*AppellF1[n^(-1), -1/2, -1/2, 1 + n^(-1), (-2*c*x^
n)/(b - Sqrt[b^2 - 4*a*c]), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(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 = 37.1299, size = 124, normalized size = 0.89 \[ \frac{x \sqrt{a + b x^{n} + c x^{2 n}} \operatorname{appellf_{1}}{\left (\frac{1}{n},- \frac{1}{2},- \frac{1}{2},1 + \frac{1}{n},- \frac{2 c x^{n}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{n}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{\sqrt{\frac{2 c x^{n}}{b - \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{\frac{2 c x^{n}}{b + \sqrt{- 4 a c + b^{2}}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*sqrt(a + b*x**n + c*x**(2*n))*appellf1(1/n, -1/2, -1/2, 1 + 1/n, -2*c*x**n/(b
- sqrt(-4*a*c + b**2)), -2*c*x**n/(b + sqrt(-4*a*c + b**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 = 8.73882, size = 786, normalized size = 5.65 \[ \frac{x \left (\frac{2 a^2 b n (2 n+1) x^n \left (-\sqrt{b^2-4 a c}+b+2 c x^n\right ) \left (\sqrt{b^2-4 a c}+b+2 c x^n\right ) F_1\left (1+\frac{1}{n};\frac{1}{2},\frac{1}{2};2+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )}{(n+1)^2 \left (\sqrt{b^2-4 a c}-b\right ) \left (\sqrt{b^2-4 a c}+b\right ) \left (n x^n \left (\left (\sqrt{b^2-4 a c}+b\right ) F_1\left (2+\frac{1}{n};\frac{1}{2},\frac{3}{2};3+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )+\left (b-\sqrt{b^2-4 a c}\right ) F_1\left (2+\frac{1}{n};\frac{3}{2},\frac{1}{2};3+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )\right )-4 (2 a n+a) F_1\left (1+\frac{1}{n};\frac{1}{2},\frac{1}{2};2+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )\right )}+\frac{a^2 n \left (-\sqrt{b^2-4 a c}+b+2 c x^n\right ) \left (\sqrt{b^2-4 a c}+b+2 c x^n\right ) F_1\left (\frac{1}{n};\frac{1}{2},\frac{1}{2};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )}{c \left (n x^n \left (-\left (\sqrt{b^2-4 a c}+b\right )\right ) F_1\left (1+\frac{1}{n};\frac{1}{2},\frac{3}{2};2+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )+n x^n \left (\sqrt{b^2-4 a c}-b\right ) F_1\left (1+\frac{1}{n};\frac{3}{2},\frac{1}{2};2+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )+4 a (n+1) F_1\left (\frac{1}{n};\frac{1}{2},\frac{1}{2};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )\right )}+\frac{\left (a+x^n \left (b+c x^n\right )\right )^2}{n+1}\right )}{\left (a+x^n \left (b+c x^n\right )\right )^{3/2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.077, size = 0, normalized size = 0. \[ \int \sqrt{a+b{x}^{n}+c{x}^{2\,n}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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