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

Optimal. Leaf size=139 \[ -\frac{a \sqrt{a+b x^3+c x^6} F_1\left (-\frac{1}{3};-\frac{3}{2},-\frac{3}{2};\frac{2}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right )}{x \sqrt{\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.420774, antiderivative size = 139, 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 \[ -\frac{a \sqrt{a+b x^3+c x^6} F_1\left (-\frac{1}{3};-\frac{3}{2},-\frac{3}{2};\frac{2}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right )}{x \sqrt{\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 36.849, size = 126, normalized size = 0.91 \[ - \frac{a \sqrt{a + b x^{3} + c x^{6}} \operatorname{appellf_{1}}{\left (- \frac{1}{3},- \frac{3}{2},- \frac{3}{2},\frac{2}{3},- \frac{2 c x^{3}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{3}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{x \sqrt{\frac{2 c x^{3}}{b - \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{\frac{2 c x^{3}}{b + \sqrt{- 4 a c + b^{2}}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [B]  time = 1.85329, size = 1058, normalized size = 7.61 \[ \frac{\frac{540 a^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{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 ) x^5}{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 (b+\sqrt{b^2-4 a c}\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 )}+\frac{27 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{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 ) x^5}{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 (b+\sqrt{b^2-4 a c}\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 )}+\frac{2025 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{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 ) x^2}{4 c \left (20 a 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 )-3 x^3 \left (\left (b+\sqrt{b^2-4 a c}\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 )\right )}+\frac{5 \left (c x^6+b x^3+a\right )^2 \left (10 c x^6+19 b x^3-80 a\right )}{4 x}}{100 \left (c x^6+b x^3+a\right )^{3/2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((5*(a + b*x^3 + c*x^6)^2*(-80*a + 19*b*x^3 + 10*c*x^6))/(4*x) + (2025*a^2*b*x^2
*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)*(b + Sqrt[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])])/(4*c*(20*a*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])] - 3*x^3*((b + Sqrt[b^2 - 4*a*c])*Appe
llF1[5/3, 1/2, 3/2, 8/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqr
t[b^2 - 4*a*c])] + (b - 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])]))) + (540*a^2*x^
5*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)*(b + Sqrt[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])])/(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 + 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[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])])) + (27*a*b^2*x^5*
(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)*(b + Sqrt[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])*AppellF
1[8/3, 1/2, 3/2, 11/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[
b^2 - 4*a*c])] + (b - Sqrt[b^2 - 4*a*c])*AppellF1[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])]))))/(100*(a + b*x
^3 + c*x^6)^(3/2))

_______________________________________________________________________________________

Maple [F]  time = 0.049, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2}} \left ( c{x}^{6}+b{x}^{3}+a \right ) ^{{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (c x^{6} + b x^{3} + a\right )}^{\frac{3}{2}}}{x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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