3.233 \(\int \frac{1}{x^3 \sqrt{a+b x^3+c x^6}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.431053, 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 \[ -\frac{\sqrt{\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1} F_1\left (-\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{1}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right )}{2 x^2 \sqrt{a+b x^3+c x^6}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 32.7759, size = 126, normalized size = 0.9 \[ - \frac{\sqrt{a + b x^{3} + c x^{6}} \operatorname{appellf_{1}}{\left (- \frac{2}{3},\frac{1}{2},\frac{1}{2},\frac{1}{3},- \frac{2 c x^{3}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{3}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{2 a x^{2} \sqrt{\frac{2 c x^{3}}{b - \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{\frac{2 c x^{3}}{b + \sqrt{- 4 a c + b^{2}}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(a + b*x**3 + c*x**6)*appellf1(-2/3, 1/2, 1/2, 1/3, -2*c*x**3/(b - sqrt(-4*
a*c + b**2)), -2*c*x**3/(b + sqrt(-4*a*c + b**2)))/(2*a*x**2*sqrt(2*c*x**3/(b -
sqrt(-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 = 1.15922, size = 705, normalized size = 5.04 \[ \frac{-\frac{2 a b x^3 \left (-\sqrt{b^2-4 a c}+b+2 c x^3\right ) \left (\sqrt{b^2-4 a c}+b+2 c x^3\right ) F_1\left (\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )}{c \left (16 a F_1\left (\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )-3 x^3 \left (\left (\sqrt{b^2-4 a c}+b\right ) F_1\left (\frac{4}{3};\frac{1}{2},\frac{3}{2};\frac{7}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )+\left (b-\sqrt{b^2-4 a c}\right ) F_1\left (\frac{4}{3};\frac{3}{2},\frac{1}{2};\frac{7}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )\right )\right )}+\frac{7 a x^6 \left (-\sqrt{b^2-4 a c}+b+2 c x^3\right ) \left (\sqrt{b^2-4 a c}+b+2 c x^3\right ) F_1\left (\frac{4}{3};\frac{1}{2},\frac{1}{2};\frac{7}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )}{4 \left (28 a F_1\left (\frac{4}{3};\frac{1}{2},\frac{1}{2};\frac{7}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )-3 x^3 \left (\left (\sqrt{b^2-4 a c}+b\right ) F_1\left (\frac{7}{3};\frac{1}{2},\frac{3}{2};\frac{10}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )+\left (b-\sqrt{b^2-4 a c}\right ) F_1\left (\frac{7}{3};\frac{3}{2},\frac{1}{2};\frac{10}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )\right )\right )}-\left (a+b x^3+c x^6\right )^2}{2 a x^2 \left (a+b x^3+c x^6\right )^{3/2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-(a + b*x^3 + c*x^6)^2 - (2*a*b*x^3*(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])])/(c*(16*a*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])] - 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 - Sqrt[b^2 - 4*a*c])*App
ellF1[4/3, 3/2, 1/2, 7/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sq
rt[b^2 - 4*a*c])]))) + (7*a*x^6*(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])])/(4*(28*a*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])] - 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])*AppellF
1[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])]))))/(2*a*x^2*(a + b*x^3 + c*x^6)^(3/2))

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Maple [F]  time = 0.046, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3}}{\frac{1}{\sqrt{c{x}^{6}+b{x}^{3}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(1/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. \[ \int \frac{1}{\sqrt{c x^{6} + b x^{3} + a} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^6 + b*x^3 + a)*x^3),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(c*x^6 + b*x^3 + a)*x^3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{c x^{6} + b x^{3} + a} x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^6 + b*x^3 + a)*x^3),x, algorithm="fricas")

[Out]

integral(1/(sqrt(c*x^6 + b*x^3 + a)*x^3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{3} \sqrt{a + b x^{3} + c x^{6}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(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. \[ \int \frac{1}{\sqrt{c x^{6} + b x^{3} + a} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^6 + b*x^3 + a)*x^3),x, algorithm="giac")

[Out]

integrate(1/(sqrt(c*x^6 + b*x^3 + a)*x^3), x)