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

Optimal. Leaf size=136 \[ \frac{a x \sqrt{a+b x^3+c x^6} F_1\left (\frac{1}{3};-\frac{3}{2},-\frac{3}{2};\frac{4}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right )}{\sqrt{\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1}} \]

[Out]

(a*x*Sqrt[a + b*x^3 + c*x^6]*AppellF1[1/3, -3/2, -3/2, 4/3, (-2*c*x^3)/(b - Sqrt
[b^2 - 4*a*c]), (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])])/(Sqrt[1 + (2*c*x^3)/(b - Sq
rt[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 = 0.200282, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{a x \sqrt{a+b x^3+c x^6} F_1\left (\frac{1}{3};-\frac{3}{2},-\frac{3}{2};\frac{4}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right )}{\sqrt{\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1}} \]

Antiderivative was successfully verified.

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

[Out]

(a*x*Sqrt[a + b*x^3 + c*x^6]*AppellF1[1/3, -3/2, -3/2, 4/3, (-2*c*x^3)/(b - Sqrt
[b^2 - 4*a*c]), (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])])/(Sqrt[1 + (2*c*x^3)/(b - Sq
rt[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 = 35.5643, size = 122, normalized size = 0.9 \[ \frac{a x \sqrt{a + b x^{3} + c x^{6}} \operatorname{appellf_{1}}{\left (\frac{1}{3},- \frac{3}{2},- \frac{3}{2},\frac{4}{3},- \frac{2 c x^{3}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{3}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{\sqrt{\frac{2 c x^{3}}{b - \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{\frac{2 c x^{3}}{b + \sqrt{- 4 a c + b^{2}}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*x*sqrt(a + b*x**3 + c*x**6)*appellf1(1/3, -3/2, -3/2, 4/3, -2*c*x**3/(b - sqrt
(-4*a*c + b**2)), -2*c*x**3/(b + sqrt(-4*a*c + b**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 = 3.41176, size = 1389, normalized size = 10.21 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

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

[Out]

(x*(8*c*(a + b*x^3 + c*x^6)^2*(27*b^2 + 184*b*c*x^3 + 28*c*(13*a + 4*c*x^6)) - (
864*a^2*b^2*(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])])/(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])])) + (24192
*a^3*c*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)*Appel
lF1[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])])/(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])*App
ellF1[4/3, 1/2, 3/2, 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])] + (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])])) + (8316*a^2*b
*c*x^3*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)*Appel
lF1[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])*App
ellF1[7/3, 1/2, 3/2, 10/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + S
qrt[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])])) + (945*a*b^
3*x^3*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)*Appell
F1[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])*App
ellF1[7/3, 1/2, 3/2, 10/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + S
qrt[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])]))))/(8960*c^2
*(a + b*x^3 + c*x^6)^(3/2))

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Maple [F]  time = 0.041, size = 0, normalized size = 0. \[ \int \left ( c{x}^{6}+b{x}^{3}+a \right ) ^{{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^6+b*x^3+a)^(3/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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