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

Optimal. Leaf size=141 \[ -\frac{a \sqrt{a+b x^3+c x^6} F_1\left (-\frac{2}{3};-\frac{3}{2},-\frac{3}{2};\frac{1}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right )}{2 x^2 \sqrt{\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1}} \]

[Out]

-(a*Sqrt[a + b*x^3 + c*x^6]*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*x^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 = 0.416652, antiderivative size = 141, 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 \[ -\frac{a \sqrt{a+b x^3+c x^6} F_1\left (-\frac{2}{3};-\frac{3}{2},-\frac{3}{2};\frac{1}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right )}{2 x^2 \sqrt{\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1}} \]

Antiderivative was successfully verified.

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

[Out]

-(a*Sqrt[a + b*x^3 + c*x^6]*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*x^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 = 36.4707, size = 129, normalized size = 0.91 \[ - \frac{a \sqrt{a + b x^{3} + c x^{6}} \operatorname{appellf_{1}}{\left (- \frac{2}{3},- \frac{3}{2},- \frac{3}{2},\frac{1}{3},- \frac{2 c x^{3}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{3}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{2 x^{2} \sqrt{\frac{2 c x^{3}}{b - \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{\frac{2 c x^{3}}{b + \sqrt{- 4 a c + b^{2}}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*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*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 = 1.86669, size = 1054, normalized size = 7.48 \[ \frac{\frac{378 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{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{189 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}{4 c \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 (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 )\right )}+\frac{648 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}{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 (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 )\right )}+\frac{2 \left (c x^6+b x^3+a\right )^2 \left (8 c x^6+17 b x^3-28 a\right )}{x^2}}{112 \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^3,x]

[Out]

((2*(a + b*x^3 + c*x^6)^2*(-28*a + 17*b*x^3 + 8*c*x^6))/x^2 + (648*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
])])/(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])]))) + (378*a^2*x^4*(b - S
qrt[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*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 + Sq
rt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])])) + (189*a*b^2*x^4*(b - Sq
rt[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*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
+ Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])]))))/(112*(a + b*x^3 +
c*x^6)^(3/2))

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Maple [F]  time = 0.045, 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((c*x^6+b*x^3+a)^(3/2)/x^3,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((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. \[{\rm integral}\left (\frac{{\left (c x^{6} + b x^{3} + a\right )}^{\frac{3}{2}}}{x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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