3.197 \(\int x \sqrt{a+b x^3+c x^6} \, dx\)

Optimal. Leaf size=140 \[ \frac{x^2 \sqrt{a+b x^3+c x^6} F_1\left (\frac{2}{3};-\frac{1}{2},-\frac{1}{2};\frac{5}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right )}{2 \sqrt{\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1}} \]

[Out]

(x^2*Sqrt[a + b*x^3 + c*x^6]*AppellF1[2/3, -1/2, -1/2, 5/3, (-2*c*x^3)/(b - Sqrt
[b^2 - 4*a*c]), (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])])/(2*Sqrt[1 + (2*c*x^3)/(b -
Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])])

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Rubi [A]  time = 0.354646, antiderivative size = 140, 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 \[ \frac{x^2 \sqrt{a+b x^3+c x^6} F_1\left (\frac{2}{3};-\frac{1}{2},-\frac{1}{2};\frac{5}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right )}{2 \sqrt{\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1}} \]

Antiderivative was successfully verified.

[In]  Int[x*Sqrt[a + b*x^3 + c*x^6],x]

[Out]

(x^2*Sqrt[a + b*x^3 + c*x^6]*AppellF1[2/3, -1/2, -1/2, 5/3, (-2*c*x^3)/(b - Sqrt
[b^2 - 4*a*c]), (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])])/(2*Sqrt[1 + (2*c*x^3)/(b -
Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])])

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Rubi in Sympy [A]  time = 28.7392, size = 124, normalized size = 0.89 \[ \frac{x^{2} \sqrt{a + b x^{3} + c x^{6}} \operatorname{appellf_{1}}{\left (\frac{2}{3},- \frac{1}{2},- \frac{1}{2},\frac{5}{3},- \frac{2 c x^{3}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{3}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{2 \sqrt{\frac{2 c x^{3}}{b - \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{\frac{2 c x^{3}}{b + \sqrt{- 4 a c + b^{2}}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(c*x**6+b*x**3+a)**(1/2),x)

[Out]

x**2*sqrt(a + b*x**3 + c*x**6)*appellf1(2/3, -1/2, -1/2, 5/3, -2*c*x**3/(b - sqr
t(-4*a*c + b**2)), -2*c*x**3/(b + sqrt(-4*a*c + b**2)))/(2*sqrt(2*c*x**3/(b - sq
rt(-4*a*c + b**2)) + 1)*sqrt(2*c*x**3/(b + sqrt(-4*a*c + b**2)) + 1))

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Mathematica [B]  time = 3.82652, size = 701, normalized size = 5.01 \[ \frac{x^2 \left (\frac{75 a^2 \left (-\sqrt{b^2-4 a c}+b+2 c x^3\right ) \left (\sqrt{b^2-4 a c}+b+2 c x^3\right ) F_1\left (\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )}{40 a c F_1\left (\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )-6 c x^3 \left (\left (\sqrt{b^2-4 a c}+b\right ) F_1\left (\frac{5}{3};\frac{1}{2},\frac{3}{2};\frac{8}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )+\left (b-\sqrt{b^2-4 a c}\right ) F_1\left (\frac{5}{3};\frac{3}{2},\frac{1}{2};\frac{8}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )\right )}+\frac{12 a b x^3 \left (-\sqrt{b^2-4 a c}+b+2 c x^3\right ) \left (\sqrt{b^2-4 a c}+b+2 c x^3\right ) F_1\left (\frac{5}{3};\frac{1}{2},\frac{1}{2};\frac{8}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )}{c \left (32 a F_1\left (\frac{5}{3};\frac{1}{2},\frac{1}{2};\frac{8}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )-3 x^3 \left (\left (\sqrt{b^2-4 a c}+b\right ) F_1\left (\frac{8}{3};\frac{1}{2},\frac{3}{2};\frac{11}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )+\left (b-\sqrt{b^2-4 a c}\right ) F_1\left (\frac{8}{3};\frac{3}{2},\frac{1}{2};\frac{11}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )\right )\right )}+5 \left (a+b x^3+c x^6\right )^2\right )}{25 \left (a+b x^3+c x^6\right )^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x*Sqrt[a + b*x^3 + c*x^6],x]

[Out]

(x^2*(5*(a + b*x^3 + c*x^6)^2 + (75*a^2*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)*(b + S
qrt[b^2 - 4*a*c] + 2*c*x^3)*AppellF1[2/3, 1/2, 1/2, 5/3, (-2*c*x^3)/(b + Sqrt[b^
2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])])/(40*a*c*AppellF1[2/3, 1/2, 1/2
, 5/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])] -
6*c*x^3*((b + Sqrt[b^2 - 4*a*c])*AppellF1[5/3, 1/2, 3/2, 8/3, (-2*c*x^3)/(b + S
qrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])] + (b - Sqrt[b^2 - 4*a*c])
*AppellF1[5/3, 3/2, 1/2, 8/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b
+ Sqrt[b^2 - 4*a*c])])) + (12*a*b*x^3*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)*(b + Sqr
t[b^2 - 4*a*c] + 2*c*x^3)*AppellF1[5/3, 1/2, 1/2, 8/3, (-2*c*x^3)/(b + Sqrt[b^2
- 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])])/(c*(32*a*AppellF1[5/3, 1/2, 1/2,
8/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])] -
3*x^3*((b + Sqrt[b^2 - 4*a*c])*AppellF1[8/3, 1/2, 3/2, 11/3, (-2*c*x^3)/(b + Sqr
t[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])] + (b - Sqrt[b^2 - 4*a*c])*A
ppellF1[8/3, 3/2, 1/2, 11/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b +
Sqrt[b^2 - 4*a*c])])))))/(25*(a + b*x^3 + c*x^6)^(3/2))

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Maple [F]  time = 0.041, size = 0, normalized size = 0. \[ \int x\sqrt{c{x}^{6}+b{x}^{3}+a}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(c*x^6+b*x^3+a)^(1/2),x)

[Out]

int(x*(c*x^6+b*x^3+a)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{c x^{6} + b x^{3} + a} x\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^6 + b*x^3 + a)*x,x, algorithm="maxima")

[Out]

integrate(sqrt(c*x^6 + b*x^3 + a)*x, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{c x^{6} + b x^{3} + a} x, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^6 + b*x^3 + a)*x,x, algorithm="fricas")

[Out]

integral(sqrt(c*x^6 + b*x^3 + a)*x, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x \sqrt{a + b x^{3} + c x^{6}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(c*x**6+b*x**3+a)**(1/2),x)

[Out]

Integral(x*sqrt(a + b*x**3 + c*x**6), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{c x^{6} + b x^{3} + a} x\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^6 + b*x^3 + a)*x,x, algorithm="giac")

[Out]

integrate(sqrt(c*x^6 + b*x^3 + a)*x, x)