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

Optimal. Leaf size=143 \[ \frac{x^4 \sqrt{\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{4}{3};\frac{3}{2},\frac{3}{2};\frac{7}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right )}{4 a \sqrt{a+b x^3+c x^6}} \]

[Out]

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

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Rubi [A]  time = 0.498967, 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 \[ \frac{x^4 \sqrt{\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{4}{3};\frac{3}{2},\frac{3}{2};\frac{7}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right )}{4 a \sqrt{a+b x^3+c x^6}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 44.7924, size = 124, normalized size = 0.87 \[ \frac{x^{4} \sqrt{a + b x^{3} + c x^{6}} \operatorname{appellf_{1}}{\left (\frac{4}{3},\frac{3}{2},\frac{3}{2},\frac{7}{3},- \frac{2 c x^{3}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{3}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{4 a^{2} \sqrt{\frac{2 c x^{3}}{b - \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{\frac{2 c x^{3}}{b + \sqrt{- 4 a c + b^{2}}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**4*sqrt(a + b*x**3 + c*x**6)*appellf1(4/3, 3/2, 3/2, 7/3, -2*c*x**3/(b - sqrt(
-4*a*c + b**2)), -2*c*x**3/(b + sqrt(-4*a*c + b**2)))/(4*a**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 = 1.71025, size = 711, normalized size = 4.97 \[ \frac{2 x \left (\frac{7 a x^3 \left (-\sqrt{b^2-4 a c}+b+2 c x^3\right ) \left (\sqrt{b^2-4 a c}+b+2 c x^3\right ) F_1\left (\frac{4}{3};\frac{1}{2},\frac{1}{2};\frac{7}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )}{56 a F_1\left (\frac{4}{3};\frac{1}{2},\frac{1}{2};\frac{7}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )-6 x^3 \left (\left (\sqrt{b^2-4 a c}+b\right ) F_1\left (\frac{7}{3};\frac{1}{2},\frac{3}{2};\frac{10}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )+\left (b-\sqrt{b^2-4 a c}\right ) F_1\left (\frac{7}{3};\frac{3}{2},\frac{1}{2};\frac{10}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )\right )}+\frac{4 a b \left (-\sqrt{b^2-4 a c}+b+2 c x^3\right ) \left (\sqrt{b^2-4 a c}+b+2 c x^3\right ) F_1\left (\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )}{c \left (16 a F_1\left (\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )-3 x^3 \left (\left (\sqrt{b^2-4 a c}+b\right ) F_1\left (\frac{4}{3};\frac{1}{2},\frac{3}{2};\frac{7}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )+\left (b-\sqrt{b^2-4 a c}\right ) F_1\left (\frac{4}{3};\frac{3}{2},\frac{1}{2};\frac{7}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )\right )\right )}-\left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )\right )}{3 \left (b^2-4 a c\right ) \left (a+b x^3+c x^6\right )^{3/2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2*x*(-((b + 2*c*x^3)*(a + b*x^3 + c*x^6)) + (4*a*b*(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])]))) + (7*a*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[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])])/(56*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
])] - 6*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])]))))/(3*(b^2 - 4*a*c)*(a + b*x^3 + c*x^6)^(3/2))

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Maple [F]  time = 0.023, size = 0, normalized size = 0. \[ \int{{x}^{3} \left ( c{x}^{6}+b{x}^{3}+a \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(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. \[ \int \frac{x^{3}}{{\left (c x^{6} + b x^{3} + a\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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