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

Optimal. Leaf size=133 \[ x \left (\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1\right )^{-p} \left (\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1\right )^{-p} \left (a+b x^3+c x^6\right )^p F_1\left (\frac{1}{3};-p,-p;\frac{4}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right ) \]

[Out]

(x*(a + b*x^3 + c*x^6)^p*AppellF1[1/3, -p, -p, 4/3, (-2*c*x^3)/(b - Sqrt[b^2 - 4
*a*c]), (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])])/((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 = 0.150893, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ x \left (\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1\right )^{-p} \left (\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1\right )^{-p} \left (a+b x^3+c x^6\right )^p F_1\left (\frac{1}{3};-p,-p;\frac{4}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3 + c*x^6)^p,x]

[Out]

(x*(a + b*x^3 + c*x^6)^p*AppellF1[1/3, -p, -p, 4/3, (-2*c*x^3)/(b - Sqrt[b^2 - 4
*a*c]), (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])])/((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 = 37.5018, size = 112, normalized size = 0.84 \[ x \left (\frac{2 c x^{3}}{b - \sqrt{- 4 a c + b^{2}}} + 1\right )^{- p} \left (\frac{2 c x^{3}}{b + \sqrt{- 4 a c + b^{2}}} + 1\right )^{- p} \left (a + b x^{3} + c x^{6}\right )^{p} \operatorname{appellf_{1}}{\left (\frac{1}{3},- p,- p,\frac{4}{3},- \frac{2 c x^{3}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{3}}{b + \sqrt{- 4 a c + b^{2}}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 3.67082, size = 487, normalized size = 3.66 \[ \frac{2^{1-2 p} x \left (\sqrt{b^2-4 a c}+b\right ) \left (x^3 \left (\sqrt{b^2-4 a c}-b\right )-2 a\right )^2 \left (\frac{b-\sqrt{b^2-4 a c}}{2 c}+x^3\right )^{-p} \left (\frac{\sqrt{b^2-4 a c}+b}{2 c}+x^3\right )^{-p} \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{c}\right )^{p+1} \left (\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{c}\right )^{p-1} \left (a+b x^3+c x^6\right )^{p-1} F_1\left (\frac{1}{3};-p,-p;\frac{4}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )}{\left (\sqrt{b^2-4 a c}-b\right ) \left (3 p x^3 \left (\left (\sqrt{b^2-4 a c}-b\right ) F_1\left (\frac{4}{3};1-p,-p;\frac{7}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )-\left (\sqrt{b^2-4 a c}+b\right ) F_1\left (\frac{4}{3};-p,1-p;\frac{7}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )\right )-8 a F_1\left (\frac{1}{3};-p,-p;\frac{4}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2^(1 - 2*p)*(b + Sqrt[b^2 - 4*a*c])*x*((b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/c)^(1
+ p)*((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)*AppellF1[1/3, -p, -p, 4/3, (-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])*((b - Sqrt[b^2 - 4*a*c])/(2*c) + x^3)^p*((b + Sqrt[b^2 - 4*a*c])/(2*c) +
x^3)^p*(-8*a*AppellF1[1/3, -p, -p, 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*p*x^3*((-b + Sqrt[b^2 - 4*a*c])*AppellF1[4/3
, 1 - 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])] - (b + Sqrt[b^2 - 4*a*c])*AppellF1[4/3, -p, 1 - 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])])))

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Maple [F]  time = 0.021, size = 0, normalized size = 0. \[ \int \left ( c{x}^{6}+b{x}^{3}+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^6+b*x^3+a)^p,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^6 + b*x^3 + a)^p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^6 + b*x^3 + a)^p, x)