3.403 \(\int (a+c x^2+b x^4)^p \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0580843, 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 = {1105, 429} \[ x \left (\frac{2 b x^2}{c-\sqrt{c^2-4 a b}}+1\right )^{-p} \left (\frac{2 b x^2}{\sqrt{c^2-4 a b}+c}+1\right )^{-p} \left (a+b x^4+c x^2\right )^p F_1\left (\frac{1}{2};-p,-p;\frac{3}{2};-\frac{2 b x^2}{c-\sqrt{c^2-4 a b}},-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2 + b*x^4)^p,x]

[Out]

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

Rule 1105

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

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+c x^2+b x^4\right )^p \, dx &=\left (\left (1+\frac{2 b x^2}{c-\sqrt{-4 a b+c^2}}\right )^{-p} \left (1+\frac{2 b x^2}{c+\sqrt{-4 a b+c^2}}\right )^{-p} \left (a+c x^2+b x^4\right )^p\right ) \int \left (1+\frac{2 b x^2}{c-\sqrt{-4 a b+c^2}}\right )^p \left (1+\frac{2 b x^2}{c+\sqrt{-4 a b+c^2}}\right )^p \, dx\\ &=x \left (1+\frac{2 b x^2}{c-\sqrt{-4 a b+c^2}}\right )^{-p} \left (1+\frac{2 b x^2}{c+\sqrt{-4 a b+c^2}}\right )^{-p} \left (a+c x^2+b x^4\right )^p F_1\left (\frac{1}{2};-p,-p;\frac{3}{2};-\frac{2 b x^2}{c-\sqrt{-4 a b+c^2}},-\frac{2 b x^2}{c+\sqrt{-4 a b+c^2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.180575, size = 161, normalized size = 1.21 \[ x \left (\frac{-\sqrt{c^2-4 a b}+2 b x^2+c}{c-\sqrt{c^2-4 a b}}\right )^{-p} \left (\frac{\sqrt{c^2-4 a b}+2 b x^2+c}{\sqrt{c^2-4 a b}+c}\right )^{-p} \left (a+b x^4+c x^2\right )^p F_1\left (\frac{1}{2};-p,-p;\frac{3}{2};-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}},\frac{2 b x^2}{\sqrt{c^2-4 a b}-c}\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + c*x^2 + b*x^4)^p,x]

[Out]

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

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Maple [F]  time = 0.013, size = 0, normalized size = 0. \begin{align*} \int \left ( b{x}^{4}+c{x}^{2}+a \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^4+c*x^2+a)^p,x)

[Out]

int((b*x^4+c*x^2+a)^p,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{4} + c x^{2} + a\right )}^{p}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+c*x^2+a)^p,x, algorithm="maxima")

[Out]

integrate((b*x^4 + c*x^2 + a)^p, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b x^{4} + c x^{2} + a\right )}^{p}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+c*x^2+a)^p,x, algorithm="fricas")

[Out]

integral((b*x^4 + c*x^2 + a)^p, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b x^{4} + c x^{2}\right )^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**4+c*x**2+a)**p,x)

[Out]

Integral((a + b*x**4 + c*x**2)**p, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{4} + c x^{2} + a\right )}^{p}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+c*x^2+a)^p,x, algorithm="giac")

[Out]

integrate((b*x^4 + c*x^2 + a)^p, x)