3.247 \(\int \frac{1}{x^3 \left (a+b x^3+c x^6\right )^{3/2}} \, dx\)

Optimal. Leaf size=143 \[ -\frac{\sqrt{\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1} F_1\left (-\frac{2}{3};\frac{3}{2},\frac{3}{2};\frac{1}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right )}{2 a x^2 \sqrt{a+b x^3+c x^6}} \]

[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, 3/2, 3/2, 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*a*x^2*Sqrt[a + b*x^3 + c*x^6])

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Rubi [A]  time = 0.423321, antiderivative size = 143, 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 \[ -\frac{\sqrt{\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1} F_1\left (-\frac{2}{3};\frac{3}{2},\frac{3}{2};\frac{1}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right )}{2 a x^2 \sqrt{a+b x^3+c x^6}} \]

Antiderivative was successfully verified.

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

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Rubi in Sympy [A]  time = 44.8635, size = 128, normalized size = 0.9 \[ - \frac{\sqrt{a + b x^{3} + c x^{6}} \operatorname{appellf_{1}}{\left (- \frac{2}{3},\frac{3}{2},\frac{3}{2},\frac{1}{3},- \frac{2 c x^{3}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{3}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{2 a^{2} x^{2} \sqrt{\frac{2 c x^{3}}{b - \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{\frac{2 c x^{3}}{b + \sqrt{- 4 a c + b^{2}}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(a + b*x**3 + c*x**6)*appellf1(-2/3, 3/2, 3/2, 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**2*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 = 3.52696, size = 1593, normalized size = 11.14 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

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

[Out]

((4*x*(b^3 - 3*a*b*c + b^2*c*x^3 - 2*a*c^2*x^3)*(a + b*x^3 + c*x^6))/(a^2*(-b^2
+ 4*a*c)) - (3*(a + b*x^3 + c*x^6)^2)/(a^2*x^2) - (56*b^3*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])])/((b^2 - 4*
a*c)*(-b + Sqrt[b^2 - 4*a*c])*(b + Sqrt[b^2 - 4*a*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]
)*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])]))) + (288*a*b*c*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])])/((b^2 - 4*a*c)*(-b + Sqrt[b^2
- 4*a*c])*(b + Sqrt[b^2 - 4*a*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 + Sqr
t[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])*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
])]))) + (49*b^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])])/((b^2 - 4*a*c)*(-b + Sqrt[b^2 - 4*a*c])*(b + Sq
rt[b^2 - 4*a*c])*(-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])*Ap
pellF1[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])]))) - (140*a*
c^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)*Appe
llF1[4/3, 1/2, 1/2, 7/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqr
t[b^2 - 4*a*c])])/((b^2 - 4*a*c)*(-b + Sqrt[b^2 - 4*a*c])*(b + Sqrt[b^2 - 4*a*c]
)*(-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])*AppellF1[7/3, 3/2, 1/2, 10/3, (-2*c*x^3)/(b + Sqr
t[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])]))))/(6*(a + b*x^3 + c*x^6)^
(3/2))

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Maple [F]  time = 0.055, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3}} \left ( c{x}^{6}+b{x}^{3}+a \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{6} + b x^{3} + a\right )}^{\frac{3}{2}} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^6 + b*x^3 + a)^(3/2)*x^3),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (c x^{9} + b x^{6} + a x^{3}\right )} \sqrt{c x^{6} + b x^{3} + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^6 + b*x^3 + a)^(3/2)*x^3),x, algorithm="fricas")

[Out]

integral(1/((c*x^9 + b*x^6 + a*x^3)*sqrt(c*x^6 + b*x^3 + a)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{3} \left (a + b x^{3} + c x^{6}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{6} + b x^{3} + a\right )}^{\frac{3}{2}} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^6 + b*x^3 + a)^(3/2)*x^3),x, algorithm="giac")

[Out]

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