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

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 = 0.0588891, 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, Rules used = {1348, 429} \[ x \left (\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1\right )^{-p} \left (\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1\right )^{-p} \left (a+b x^3+c x^6\right )^p F_1\left (\frac{1}{3};-p,-p;\frac{4}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right ) \]

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)

Rule 1348

Int[((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n + c*x^(2*n))
^FracPart[p])/((1 + (2*c*x^n)/(b + Rt[b^2 - 4*a*c, 2]))^FracPart[p]*(1 + (2*c*x^n)/(b - Rt[b^2 - 4*a*c, 2]))^F
racPart[p]), Int[(1 + (2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]))^p*(1 + (2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]))^p, x], x] /
; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p]

Rule 429

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p,
-q, 1 + 1/n, -((b*x^n)/a), -((d*x^n)/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n
, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rubi steps

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

Mathematica [A]  time = 0.178785, size = 161, normalized size = 1.21 \[ x \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}\right )^{-p} \left (\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}\right )^{-p} \left (a+b x^3+c x^6\right )^p F_1\left (\frac{1}{3};-p,-p;\frac{4}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(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])])/(((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 = 0.016, size = 0, normalized size = 0. \begin{align*} \int \left ( c{x}^{6}+b{x}^{3}+a \right ) ^{p}\, dx \end{align*}

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. \begin{align*} \int{\left (c x^{6} + b x^{3} + a\right )}^{p}\,{d x} \end{align*}

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. \begin{align*}{\rm integral}\left ({\left (c x^{6} + b x^{3} + a\right )}^{p}, x\right ) \end{align*}

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. \begin{align*} \text{Timed out} \end{align*}

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

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)