3.258 \(\int x^5 \left (a+b x^3+c x^6\right )^p \, dx\)

Optimal. Leaf size=161 \[ \frac{b 2^p \left (a+b x^3+c x^6\right )^{p+1} \left (-\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}}\right )^{-p-1} \, _2F_1\left (-p,p+1;p+2;\frac{2 c x^3+b+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )}{3 c (p+1) \sqrt{b^2-4 a c}}+\frac{\left (a+b x^3+c x^6\right )^{p+1}}{6 c (p+1)} \]

[Out]

(a + b*x^3 + c*x^6)^(1 + p)/(6*c*(1 + p)) + (2^p*b*(-((b - Sqrt[b^2 - 4*a*c] + 2
*c*x^3)/Sqrt[b^2 - 4*a*c]))^(-1 - p)*(a + b*x^3 + c*x^6)^(1 + p)*Hypergeometric2
F1[-p, 1 + p, 2 + p, (b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(2*Sqrt[b^2 - 4*a*c])])/(
3*c*Sqrt[b^2 - 4*a*c]*(1 + p))

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Rubi [A]  time = 0.253238, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{b 2^p \left (a+b x^3+c x^6\right )^{p+1} \left (-\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}}\right )^{-p-1} \, _2F_1\left (-p,p+1;p+2;\frac{2 c x^3+b+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )}{3 c (p+1) \sqrt{b^2-4 a c}}+\frac{\left (a+b x^3+c x^6\right )^{p+1}}{6 c (p+1)} \]

Antiderivative was successfully verified.

[In]  Int[x^5*(a + b*x^3 + c*x^6)^p,x]

[Out]

(a + b*x^3 + c*x^6)^(1 + p)/(6*c*(1 + p)) + (2^p*b*(-((b - Sqrt[b^2 - 4*a*c] + 2
*c*x^3)/Sqrt[b^2 - 4*a*c]))^(-1 - p)*(a + b*x^3 + c*x^6)^(1 + p)*Hypergeometric2
F1[-p, 1 + p, 2 + p, (b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(2*Sqrt[b^2 - 4*a*c])])/(
3*c*Sqrt[b^2 - 4*a*c]*(1 + p))

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Rubi in Sympy [A]  time = 22.1113, size = 136, normalized size = 0.84 \[ \frac{b \left (\frac{- \frac{b}{2} - c x^{3} + \frac{\sqrt{- 4 a c + b^{2}}}{2}}{\sqrt{- 4 a c + b^{2}}}\right )^{- p - 1} \left (a + b x^{3} + c x^{6}\right )^{p + 1}{{}_{2}F_{1}\left (\begin{matrix} - p, p + 1 \\ p + 2 \end{matrix}\middle |{\frac{\frac{b}{2} + c x^{3} + \frac{\sqrt{- 4 a c + b^{2}}}{2}}{\sqrt{- 4 a c + b^{2}}}} \right )}}{6 c \left (p + 1\right ) \sqrt{- 4 a c + b^{2}}} + \frac{\left (a + b x^{3} + c x^{6}\right )^{p + 1}}{6 c \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(c*x**6+b*x**3+a)**p,x)

[Out]

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

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Mathematica [C]  time = 0.903061, size = 439, normalized size = 2.73 \[ \frac{c 2^{-p-2} x^6 \left (\sqrt{b^2-4 a c}+b\right ) \left (x^3 \left (b-\sqrt{b^2-4 a c}\right )+2 a\right )^2 \left (\frac{b-\sqrt{b^2-4 a c}}{2 c}+x^3\right )^{-p} \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{c}\right )^{p+1} \left (a+x^3 \left (b+c x^3\right )\right )^{p-1} F_1\left (2;-p,-p;3;-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )}{\left (\sqrt{b^2-4 a c}-b\right ) \left (\sqrt{b^2-4 a c}+b+2 c x^3\right ) \left (p x^3 \left (\left (\sqrt{b^2-4 a c}-b\right ) F_1\left (3;1-p,-p;4;-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )-\left (\sqrt{b^2-4 a c}+b\right ) F_1\left (3;-p,1-p;4;-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )\right )-6 a F_1\left (2;-p,-p;3;-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x^5*(a + b*x^3 + c*x^6)^p,x]

[Out]

(2^(-2 - p)*c*(b + Sqrt[b^2 - 4*a*c])*x^6*((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 + x^3*(b + c*x^3))^(-1 + p)*App
ellF1[2, -p, -p, 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)*(-6*a*AppellF1[2, -p, -p, 3, (-2*c*x^3)/(b + Sq
rt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])] + p*x^3*((-b + Sqrt[b^2 -
4*a*c])*AppellF1[3, 1 - p, -p, 4, (-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[3, -p, 1 - p, 4, (-
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 = 0.047, size = 0, normalized size = 0. \[ \int{x}^{5} \left ( c{x}^{6}+b{x}^{3}+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(c*x^6+b*x^3+a)^p,x)

[Out]

int(x^5*(c*x^6+b*x^3+a)^p,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{6} + b x^{3} + a\right )}^{p} x^{5}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)^p*x^5,x, algorithm="maxima")

[Out]

integrate((c*x^6 + b*x^3 + a)^p*x^5, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (c x^{6} + b x^{3} + a\right )}^{p} x^{5}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)^p*x^5,x, algorithm="fricas")

[Out]

integral((c*x^6 + b*x^3 + a)^p*x^5, 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**5*(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. \[ \int{\left (c x^{6} + b x^{3} + a\right )}^{p} x^{5}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)^p*x^5,x, algorithm="giac")

[Out]

integrate((c*x^6 + b*x^3 + a)^p*x^5, x)