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

Optimal. Leaf size=148 \[ \frac{x^3 \sqrt{a+b x^n+c x^{2 n}} F_1\left (\frac{3}{n};-\frac{1}{2},-\frac{1}{2};\frac{n+3}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{3 \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1}} \]

[Out]

(x^3*Sqrt[a + b*x^n + c*x^(2*n)]*AppellF1[3/n, -1/2, -1/2, (3 + n)/n, (-2*c*x^n)
/(b - Sqrt[b^2 - 4*a*c]), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(3*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 = 0.469974, antiderivative size = 148, 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 \[ \frac{x^3 \sqrt{a+b x^n+c x^{2 n}} F_1\left (\frac{3}{n};-\frac{1}{2},-\frac{1}{2};\frac{n+3}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{3 \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1}} \]

Antiderivative was successfully verified.

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

[Out]

(x^3*Sqrt[a + b*x^n + c*x^(2*n)]*AppellF1[3/n, -1/2, -1/2, (3 + n)/n, (-2*c*x^n)
/(b - Sqrt[b^2 - 4*a*c]), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(3*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 = 33.5349, size = 128, normalized size = 0.86 \[ \frac{x^{3} \sqrt{a + b x^{n} + c x^{2 n}} \operatorname{appellf_{1}}{\left (\frac{3}{n},- \frac{1}{2},- \frac{1}{2},\frac{n + 3}{n},- \frac{2 c x^{n}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{n}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{3 \sqrt{\frac{2 c x^{n}}{b - \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{\frac{2 c x^{n}}{b + \sqrt{- 4 a c + b^{2}}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**3*sqrt(a + b*x**n + c*x**(2*n))*appellf1(3/n, -1/2, -1/2, (n + 3)/n, -2*c*x**
n/(b - sqrt(-4*a*c + b**2)), -2*c*x**n/(b + sqrt(-4*a*c + b**2)))/(3*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 = 8.24288, size = 825, normalized size = 5.57 \[ \frac{x^3 \left (\frac{6 a^2 b n (2 n+3) \left (2 c x^n+b-\sqrt{b^2-4 a c}\right ) \left (2 c x^n+b+\sqrt{b^2-4 a c}\right ) F_1\left (\frac{n+3}{n};\frac{1}{2},\frac{1}{2};2+\frac{3}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right ) x^n}{\left (\sqrt{b^2-4 a c}-b\right ) \left (b+\sqrt{b^2-4 a c}\right ) (n+3)^2 \left (\left (b+\sqrt{b^2-4 a c}\right ) n F_1\left (2+\frac{3}{n};\frac{1}{2},\frac{3}{2};3+\frac{3}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right ) x^n-\left (\sqrt{b^2-4 a c}-b\right ) n F_1\left (2+\frac{3}{n};\frac{3}{2},\frac{1}{2};3+\frac{3}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right ) x^n-4 a (2 n+3) F_1\left (\frac{n+3}{n};\frac{1}{2},\frac{1}{2};2+\frac{3}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )\right )}+\frac{3 \left (\left (c x^n+b\right ) x^n+a\right )^2}{n+3}+\frac{a^2 n \left (2 c x^n+b-\sqrt{b^2-4 a c}\right ) \left (2 c x^n+b+\sqrt{b^2-4 a c}\right ) F_1\left (\frac{3}{n};\frac{1}{2},\frac{1}{2};\frac{n+3}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )}{c \left (4 a (n+3) F_1\left (\frac{3}{n};\frac{1}{2},\frac{1}{2};\frac{n+3}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )-n x^n \left (\left (b+\sqrt{b^2-4 a c}\right ) F_1\left (\frac{n+3}{n};\frac{1}{2},\frac{3}{2};2+\frac{3}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )+\left (b-\sqrt{b^2-4 a c}\right ) F_1\left (\frac{n+3}{n};\frac{3}{2},\frac{1}{2};2+\frac{3}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )\right )\right )}\right )}{3 \left (\left (c x^n+b\right ) x^n+a\right )^{3/2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(x^3*((3*(a + x^n*(b + c*x^n))^2)/(3 + n) + (6*a^2*b*n*(3 + 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[(3 + n)/n, 1/2
, 1/2, 2 + 3/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])*(3 + n)^2*((b + Sqrt[
b^2 - 4*a*c])*n*x^n*AppellF1[2 + 3/n, 1/2, 3/2, 3 + 3/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 + 3/n, 3/2, 1/2, 3 + 3/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*(3 + 2*n)*AppellF1[(3 + n)/n, 1/2, 1/2, 2 +
3/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)*App
ellF1[3/n, 1/2, 1/2, (3 + 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*(3 + n)*AppellF1[3/n, 1/2, 1/2, (3 + 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[(3 + n)/n, 1/2, 3/2, 2 + 3/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])*Ap
pellF1[(3 + n)/n, 3/2, 1/2, 2 + 3/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^
n)/(-b + Sqrt[b^2 - 4*a*c])])))))/(3*(a + x^n*(b + c*x^n))^(3/2))

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Maple [F]  time = 0.079, size = 0, normalized size = 0. \[ \int{x}^{2}\sqrt{a+b{x}^{n}+c{x}^{2\,n}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(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. \[ \int \sqrt{c x^{2 \, n} + b x^{n} + a} x^{2}\,{d x} \]

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)