3.261 \(\int x^3 \left (a+b x^3+c x^6\right )^p \, dx\)

Optimal. Leaf size=138 \[ \frac{1}{4} x^4 \left (\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1\right )^{-p} \left (\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1\right )^{-p} \left (a+b x^3+c x^6\right )^p F_1\left (\frac{4}{3};-p,-p;\frac{7}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.282396, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{1}{4} x^4 \left (\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1\right )^{-p} \left (\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1\right )^{-p} \left (a+b x^3+c x^6\right )^p F_1\left (\frac{4}{3};-p,-p;\frac{7}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 33.5669, size = 116, normalized size = 0.84 \[ \frac{x^{4} \left (\frac{2 c x^{3}}{b - \sqrt{- 4 a c + b^{2}}} + 1\right )^{- p} \left (\frac{2 c x^{3}}{b + \sqrt{- 4 a c + b^{2}}} + 1\right )^{- p} \left (a + b x^{3} + c x^{6}\right )^{p} \operatorname{appellf_{1}}{\left (\frac{4}{3},- p,- p,\frac{7}{3},- \frac{2 c x^{3}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{3}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 3.45189, size = 456, normalized size = 3.3 \[ \frac{7 c 2^{-p-3} x^4 \left (\sqrt{b^2-4 a c}+b\right ) \left (x^3 \left (\sqrt{b^2-4 a c}-b\right )-2 a\right )^2 \left (\frac{b-\sqrt{b^2-4 a c}}{2 c}+x^3\right )^{-p} \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{c}\right )^{p+1} \left (a+b x^3+c x^6\right )^{p-1} F_1\left (\frac{4}{3};-p,-p;\frac{7}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )}{\left (\sqrt{b^2-4 a c}-b\right ) \left (\sqrt{b^2-4 a c}+b+2 c x^3\right ) \left (3 p x^3 \left (\left (\sqrt{b^2-4 a c}-b\right ) F_1\left (\frac{7}{3};1-p,-p;\frac{10}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )-\left (\sqrt{b^2-4 a c}+b\right ) F_1\left (\frac{7}{3};-p,1-p;\frac{10}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )\right )-14 a F_1\left (\frac{4}{3};-p,-p;\frac{7}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(7*2^(-3 - p)*c*(b + Sqrt[b^2 - 4*a*c])*x^4*((b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/c
)^(1 + p)*(-2*a + (-b + Sqrt[b^2 - 4*a*c])*x^3)^2*(a + b*x^3 + c*x^6)^(-1 + p)*A
ppellF1[4/3, -p, -p, 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])*((b - Sqrt[b^2 - 4*a*c])/(2*c) + x^
3)^p*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)*(-14*a*AppellF1[4/3, -p, -p, 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*p*x^3*((-b
+ Sqrt[b^2 - 4*a*c])*AppellF1[7/3, 1 - p, -p, 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, -p, 1 - p, 10/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^
2 - 4*a*c])])))

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Maple [F]  time = 0.038, size = 0, normalized size = 0. \[ \int{x}^{3} \left ( c{x}^{6}+b{x}^{3}+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(c*x^6+b*x^3+a)^p,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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