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

Optimal. Leaf size=140 \[ \frac{x^4 \sqrt{a+b x^3+c x^6} F_1\left (\frac{4}{3};-\frac{1}{2},-\frac{1}{2};\frac{7}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right )}{4 \sqrt{\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1}} \]

[Out]

(x^4*Sqrt[a + b*x^3 + c*x^6]*AppellF1[4/3, -1/2, -1/2, 7/3, (-2*c*x^3)/(b - Sqrt
[b^2 - 4*a*c]), (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])])/(4*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 = 0.529538, antiderivative size = 140, 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 \[ \frac{x^4 \sqrt{a+b x^3+c x^6} F_1\left (\frac{4}{3};-\frac{1}{2},-\frac{1}{2};\frac{7}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right )}{4 \sqrt{\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1}} \]

Antiderivative was successfully verified.

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

[Out]

(x^4*Sqrt[a + b*x^3 + c*x^6]*AppellF1[4/3, -1/2, -1/2, 7/3, (-2*c*x^3)/(b - Sqrt
[b^2 - 4*a*c]), (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])])/(4*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 = 37.1961, size = 124, normalized size = 0.89 \[ \frac{x^{4} \sqrt{a + b x^{3} + c x^{6}} \operatorname{appellf_{1}}{\left (\frac{4}{3},- \frac{1}{2},- \frac{1}{2},\frac{7}{3},- \frac{2 c x^{3}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{3}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{4 \sqrt{\frac{2 c x^{3}}{b - \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{\frac{2 c x^{3}}{b + \sqrt{- 4 a c + b^{2}}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**4*sqrt(a + b*x**3 + c*x**6)*appellf1(4/3, -1/2, -1/2, 7/3, -2*c*x**3/(b - sqr
t(-4*a*c + b**2)), -2*c*x**3/(b + sqrt(-4*a*c + b**2)))/(4*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 = 5.14938, size = 1043, normalized size = 7.45 \[ \frac{\frac{336 a^2 c \left (2 c x^3+b-\sqrt{b^2-4 a c}\right ) \left (2 c x^3+b+\sqrt{b^2-4 a c}\right ) F_1\left (\frac{4}{3};\frac{1}{2},\frac{1}{2};\frac{7}{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 ) x^4}{28 a F_1\left (\frac{4}{3};\frac{1}{2},\frac{1}{2};\frac{7}{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 (b+\sqrt{b^2-4 a c}\right ) F_1\left (\frac{7}{3};\frac{1}{2},\frac{3}{2};\frac{10}{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{7}{3};\frac{3}{2},\frac{1}{2};\frac{10}{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{105 a b^2 \left (2 c x^3+b-\sqrt{b^2-4 a c}\right ) \left (2 c x^3+b+\sqrt{b^2-4 a c}\right ) F_1\left (\frac{4}{3};\frac{1}{2},\frac{1}{2};\frac{7}{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 ) x^4}{3 x^3 \left (\left (b+\sqrt{b^2-4 a c}\right ) F_1\left (\frac{7}{3};\frac{1}{2},\frac{3}{2};\frac{10}{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{7}{3};\frac{3}{2},\frac{1}{2};\frac{10}{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 )-28 a F_1\left (\frac{4}{3};\frac{1}{2},\frac{1}{2};\frac{7}{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 )}+\frac{96 a^2 b \left (2 c x^3+b-\sqrt{b^2-4 a c}\right ) \left (2 c x^3+b+\sqrt{b^2-4 a c}\right ) F_1\left (\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{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 ) x}{3 x^3 \left (\left (b+\sqrt{b^2-4 a c}\right ) F_1\left (\frac{4}{3};\frac{1}{2},\frac{3}{2};\frac{7}{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{4}{3};\frac{3}{2},\frac{1}{2};\frac{7}{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 )-16 a F_1\left (\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{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 )}+8 c \left (8 c x^4+3 b x\right ) \left (c x^6+b x^3+a\right )^2}{448 c^2 \left (c x^6+b x^3+a\right )^{3/2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(8*c*(3*b*x + 8*c*x^4)*(a + b*x^3 + c*x^6)^2 + (96*a^2*b*x*(b - Sqrt[b^2 - 4*a*c
] + 2*c*x^3)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)*AppellF1[1/3, 1/2, 1/2, 4/3, (-2*
c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])])/(-16*a*Appe
llF1[1/3, 1/2, 1/2, 4/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqr
t[b^2 - 4*a*c])] + 3*x^3*((b + Sqrt[b^2 - 4*a*c])*AppellF1[4/3, 1/2, 3/2, 7/3, (
-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])] + (b - Sq
rt[b^2 - 4*a*c])*AppellF1[4/3, 3/2, 1/2, 7/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])
, (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])])) + (336*a^2*c*x^4*(b - Sqrt[b^2 - 4*a*c]
+ 2*c*x^3)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)*AppellF1[4/3, 1/2, 1/2, 7/3, (-2*c*
x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])])/(28*a*AppellF
1[4/3, 1/2, 1/2, 7/3, (-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[7/3, 1/2, 3/2, 10/3, (-2
*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])] + (b - Sqrt
[b^2 - 4*a*c])*AppellF1[7/3, 3/2, 1/2, 10/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]),
(2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])])) + (105*a*b^2*x^4*(b - Sqrt[b^2 - 4*a*c] +
2*c*x^3)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)*AppellF1[4/3, 1/2, 1/2, 7/3, (-2*c*x
^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])])/(-28*a*AppellF
1[4/3, 1/2, 1/2, 7/3, (-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[7/3, 1/2, 3/2, 10/3, (-2
*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])] + (b - Sqrt
[b^2 - 4*a*c])*AppellF1[7/3, 3/2, 1/2, 10/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]),
(2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])])))/(448*c^2*(a + b*x^3 + c*x^6)^(3/2))

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Maple [F]  time = 0.046, size = 0, normalized size = 0. \[ \int{x}^{3}\sqrt{c{x}^{6}+b{x}^{3}+a}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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