3.3.57 \(\int x^8 (a+b x^3+c x^6)^p \, dx\) [257]

Optimal. Leaf size=224 \[ -\frac {b (2+p) \left (a+b x^3+c x^6\right )^{1+p}}{6 c^2 (1+p) (3+2 p)}+\frac {x^3 \left (a+b x^3+c x^6\right )^{1+p}}{3 c (3+2 p)}+\frac {2^p \left (2 a c-b^2 (2+p)\right ) \left (-\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{\sqrt {b^2-4 a c}}\right )^{-1-p} \left (a+b x^3+c x^6\right )^{1+p} \, _2F_1\left (-p,1+p;2+p;\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{2 \sqrt {b^2-4 a c}}\right )}{3 c^2 \sqrt {b^2-4 a c} (1+p) (3+2 p)} \]

[Out]

-1/6*b*(2+p)*(c*x^6+b*x^3+a)^(1+p)/c^2/(2*p^2+5*p+3)+1/3*x^3*(c*x^6+b*x^3+a)^(1+p)/c/(3+2*p)+1/3*2^p*(2*a*c-b^
2*(2+p))*(c*x^6+b*x^3+a)^(1+p)*hypergeom([-p, 1+p],[2+p],1/2*(b+2*c*x^3+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2)
)*((-b-2*c*x^3+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(-1-p)/c^2/(1+p)/(3+2*p)/(-4*a*c+b^2)^(1/2)

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Rubi [A]
time = 0.17, antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1371, 756, 654, 638} \begin {gather*} \frac {2^p \left (2 a c-b^2 (p+2)\right ) \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^2 (p+1) (2 p+3) \sqrt {b^2-4 a c}}-\frac {b (p+2) \left (a+b x^3+c x^6\right )^{p+1}}{6 c^2 (p+1) (2 p+3)}+\frac {x^3 \left (a+b x^3+c x^6\right )^{p+1}}{3 c (2 p+3)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^8*(a + b*x^3 + c*x^6)^p,x]

[Out]

-1/6*(b*(2 + p)*(a + b*x^3 + c*x^6)^(1 + p))/(c^2*(1 + p)*(3 + 2*p)) + (x^3*(a + b*x^3 + c*x^6)^(1 + p))/(3*c*
(3 + 2*p)) + (2^p*(2*a*c - b^2*(2 + p))*(-((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)*Hypergeometric2F1[-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^2*Sqrt[b^2 - 4*a*c]*(1 + p)*(3 + 2*p))

Rule 638

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(-(a + b*x + c*
x^2)^(p + 1)/(q*(p + 1)*((q - b - 2*c*x)/(2*q))^(p + 1)))*Hypergeometric2F1[-p, p + 1, p + 2, (b + q + 2*c*x)/
(2*q)], x]] /; FreeQ[{a, b, c, p}, x] && NeQ[b^2 - 4*a*c, 0] &&  !IntegerQ[4*p]

Rule 654

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*((a + b*x + c*x^2)^(p +
1)/(2*c*(p + 1))), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 756

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*
((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Dist[1/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 2)*Simp[c*d^2
*(m + 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]*(a + b*x + c*x^2)^p, x], x] /;
FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0]
&& If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, c, d, e, m,
p, x]

Rule 1371

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[
b^2 - 4*a*c, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int x^8 \left (a+b x^3+c x^6\right )^p \, dx &=\frac {1}{3} \text {Subst}\left (\int x^2 \left (a+b x+c x^2\right )^p \, dx,x,x^3\right )\\ &=\frac {x^3 \left (a+b x^3+c x^6\right )^{1+p}}{3 c (3+2 p)}+\frac {\text {Subst}\left (\int (-a-b (2+p) x) \left (a+b x+c x^2\right )^p \, dx,x,x^3\right )}{3 c (3+2 p)}\\ &=-\frac {b (2+p) \left (a+b x^3+c x^6\right )^{1+p}}{6 c^2 (1+p) (3+2 p)}+\frac {x^3 \left (a+b x^3+c x^6\right )^{1+p}}{3 c (3+2 p)}-\frac {\left (2 a c-b^2 (2+p)\right ) \text {Subst}\left (\int \left (a+b x+c x^2\right )^p \, dx,x,x^3\right )}{6 c^2 (3+2 p)}\\ &=-\frac {b (2+p) \left (a+b x^3+c x^6\right )^{1+p}}{6 c^2 (1+p) (3+2 p)}+\frac {x^3 \left (a+b x^3+c x^6\right )^{1+p}}{3 c (3+2 p)}+\frac {2^p \left (2 a c-b^2 (2+p)\right ) \left (-\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{\sqrt {b^2-4 a c}}\right )^{-1-p} \left (a+b x^3+c x^6\right )^{1+p} \, _2F_1\left (-p,1+p;2+p;\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{2 \sqrt {b^2-4 a c}}\right )}{3 c^2 \sqrt {b^2-4 a c} (1+p) (3+2 p)}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
time = 0.33, size = 162, normalized size = 0.72 \begin {gather*} \frac {1}{9} x^9 \left (\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x^3+c x^6\right )^p F_1\left (3;-p,-p;4;-\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 ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[x^8*(a + b*x^3 + c*x^6)^p,x]

[Out]

(x^9*(a + b*x^3 + c*x^6)^p*AppellF1[3, -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])])/(9*((b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c]))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^
3)/(b + Sqrt[b^2 - 4*a*c]))^p)

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int x^{8} \left (c \,x^{6}+b \,x^{3}+a \right )^{p}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8*(c*x^6+b*x^3+a)^p,x)

[Out]

int(x^8*(c*x^6+b*x^3+a)^p,x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(c*x^6+b*x^3+a)^p,x, algorithm="maxima")

[Out]

integrate((c*x^6 + b*x^3 + a)^p*x^8, x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(c*x^6+b*x^3+a)^p,x, algorithm="fricas")

[Out]

integral((c*x^6 + b*x^3 + a)^p*x^8, x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8*(c*x**6+b*x**3+a)**p,x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(c*x^6+b*x^3+a)^p,x, algorithm="giac")

[Out]

integrate((c*x^6 + b*x^3 + a)^p*x^8, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^8\,{\left (c\,x^6+b\,x^3+a\right )}^p \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8*(a + b*x^3 + c*x^6)^p,x)

[Out]

int(x^8*(a + b*x^3 + c*x^6)^p, x)

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