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

Optimal. Leaf size=148 \[ \frac{x^2 \sqrt{a+b x^n+c x^{2 n}} F_1\left (\frac{2}{n};-\frac{1}{2},-\frac{1}{2};\frac{n+2}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{2 \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1}} \]

[Out]

(x^2*Sqrt[a + b*x^n + c*x^(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])])/(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])])

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Rubi [A]  time = 0.351197, antiderivative size = 148, 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 \[ \frac{x^2 \sqrt{a+b x^n+c x^{2 n}} F_1\left (\frac{2}{n};-\frac{1}{2},-\frac{1}{2};\frac{n+2}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{2 \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1}} \]

Antiderivative was successfully verified.

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

[Out]

(x^2*Sqrt[a + b*x^n + c*x^(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])])/(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])])

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Rubi in Sympy [A]  time = 28.4207, size = 128, normalized size = 0.86 \[ \frac{x^{2} \sqrt{a + b x^{n} + c x^{2 n}} \operatorname{appellf_{1}}{\left (\frac{2}{n},- \frac{1}{2},- \frac{1}{2},\frac{n + 2}{n},- \frac{2 c x^{n}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{n}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{2 \sqrt{\frac{2 c x^{n}}{b - \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{\frac{2 c x^{n}}{b + \sqrt{- 4 a c + b^{2}}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**2*sqrt(a + b*x**n + c*x**(2*n))*appellf1(2/n, -1/2, -1/2, (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*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 = 8.44303, size = 816, normalized size = 5.51 \[ \frac{x^2 \left (\frac{4 a^2 b n (n+1) \left (2 c x^n+b-\sqrt{b^2-4 a c}\right ) \left (2 c x^n+b+\sqrt{b^2-4 a c}\right ) F_1\left (\frac{n+2}{n};\frac{1}{2},\frac{1}{2};2+\frac{2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right ) x^n}{\left (\sqrt{b^2-4 a c}-b\right ) \left (b+\sqrt{b^2-4 a c}\right ) (n+2)^2 \left (\left (b+\sqrt{b^2-4 a c}\right ) n F_1\left (2+\frac{2}{n};\frac{1}{2},\frac{3}{2};3+\frac{2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right ) x^n-\left (\sqrt{b^2-4 a c}-b\right ) n F_1\left (2+\frac{2}{n};\frac{3}{2},\frac{1}{2};3+\frac{2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right ) x^n-8 a (n+1) F_1\left (\frac{n+2}{n};\frac{1}{2},\frac{1}{2};2+\frac{2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )\right )}+\frac{\left (\left (c x^n+b\right ) x^n+a\right )^2}{n+2}+\frac{a^2 n \left (2 c x^n+b-\sqrt{b^2-4 a c}\right ) \left (2 c x^n+b+\sqrt{b^2-4 a c}\right ) F_1\left (\frac{2}{n};\frac{1}{2},\frac{1}{2};\frac{n+2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )}{8 a c (n+2) F_1\left (\frac{2}{n};\frac{1}{2},\frac{1}{2};\frac{n+2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )-2 c n x^n \left (\left (b+\sqrt{b^2-4 a c}\right ) F_1\left (\frac{n+2}{n};\frac{1}{2},\frac{3}{2};2+\frac{2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )+\left (b-\sqrt{b^2-4 a c}\right ) F_1\left (\frac{n+2}{n};\frac{3}{2},\frac{1}{2};2+\frac{2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )\right )}\right )}{\left (\left (c x^n+b\right ) x^n+a\right )^{3/2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(x^2*((a + x^n*(b + c*x^n))^2/(2 + n) + (4*a^2*b*n*(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)*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])
])/((-b + Sqrt[b^2 - 4*a*c])*(b + Sqrt[b^2 - 4*a*c])*(2 + n)^2*((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*Appe
llF1[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*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])])/(8*a*c*(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])] - 2*c*n*x^n*((b + Sqr
t[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])*AppellF
1[(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 + x^n*(b + c*x^n))^(3/2)

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Maple [F]  time = 0.078, size = 0, normalized size = 0. \[ \int x\sqrt{a+b{x}^{n}+c{x}^{2\,n}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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