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

Optimal. Leaf size=274 \[ \frac{1}{3} e x^3 \left (\frac{2 b x^2}{c-\sqrt{c^2-4 a b}}+1\right )^{-p} \left (a+b x^4+c x^2\right )^p \left (\frac{2 b x^2}{\sqrt{c^2-4 a b}+c}+1\right )^{-p} F_1\left (\frac{3}{2};-p,-p;\frac{5}{2};-\frac{2 b x^2}{c-\sqrt{c^2-4 a b}},-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}}\right )+c x \left (\frac{2 b x^2}{c-\sqrt{c^2-4 a b}}+1\right )^{-p} \left (a+b x^4+c x^2\right )^p \left (\frac{2 b x^2}{\sqrt{c^2-4 a b}+c}+1\right )^{-p} F_1\left (\frac{1}{2};-p,-p;\frac{3}{2};-\frac{2 b x^2}{c-\sqrt{c^2-4 a b}},-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}}\right ) \]

[Out]

(c*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 + Sqr
t[-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) + (
e*x^3*(a + c*x^2 + b*x^4)^p*AppellF1[3/2, -p, -p, 5/2, (-2*b*x^2)/(c - Sqrt[-4*a*b + c^2]), (-2*b*x^2)/(c + Sq
rt[-4*a*b + c^2])])/(3*(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 = 0.222344, antiderivative size = 274, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {1203, 1105, 429, 1141, 510} \[ \frac{1}{3} e x^3 \left (\frac{2 b x^2}{c-\sqrt{c^2-4 a b}}+1\right )^{-p} \left (a+b x^4+c x^2\right )^p \left (\frac{2 b x^2}{\sqrt{c^2-4 a b}+c}+1\right )^{-p} F_1\left (\frac{3}{2};-p,-p;\frac{5}{2};-\frac{2 b x^2}{c-\sqrt{c^2-4 a b}},-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}}\right )+c x \left (\frac{2 b x^2}{c-\sqrt{c^2-4 a b}}+1\right )^{-p} \left (a+b x^4+c x^2\right )^p \left (\frac{2 b x^2}{\sqrt{c^2-4 a b}+c}+1\right )^{-p} F_1\left (\frac{1}{2};-p,-p;\frac{3}{2};-\frac{2 b x^2}{c-\sqrt{c^2-4 a b}},-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(c + e*x^2)*(a + c*x^2 + b*x^4)^p,x]

[Out]

(c*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 + Sqr
t[-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) + (
e*x^3*(a + c*x^2 + b*x^4)^p*AppellF1[3/2, -p, -p, 5/2, (-2*b*x^2)/(c - Sqrt[-4*a*b + c^2]), (-2*b*x^2)/(c + Sq
rt[-4*a*b + c^2])])/(3*(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 1203

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[(d + e*x
^2)*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a
*e^2, 0]

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])

Rule 1141

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^2 +
c*x^4)^FracPart[p])/((1 + (2*c*x^2)/(b + Rt[b^2 - 4*a*c, 2]))^FracPart[p]*(1 + (2*c*x^2)/(b - Rt[b^2 - 4*a*c,
2]))^FracPart[p]), Int[(d*x)^m*(1 + (2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]))^p*(1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c
]))^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x]

Rule 510

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

Rubi steps

\begin{align*} \int \left (c+e x^2\right ) \left (a+c x^2+b x^4\right )^p \, dx &=\int \left (c \left (a+c x^2+b x^4\right )^p+e x^2 \left (a+c x^2+b x^4\right )^p\right ) \, dx\\ &=c \int \left (a+c x^2+b x^4\right )^p \, dx+e \int x^2 \left (a+c x^2+b x^4\right )^p \, dx\\ &=\left (c \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+\left (e \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 x^2 \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\\ &=c 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 )+\frac{1}{3} e x^3 \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{3}{2};-p,-p;\frac{5}{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 = 0.244317, size = 232, normalized size = 0.85 \[ \frac{1}{3} x \left (\frac{-\sqrt{c^2-4 a b}+2 b x^2+c}{c-\sqrt{c^2-4 a b}}\right )^{-p} \left (\frac{\sqrt{c^2-4 a b}+2 b x^2+c}{\sqrt{c^2-4 a b}+c}\right )^{-p} \left (a+b x^4+c x^2\right )^p \left (e x^2 F_1\left (\frac{3}{2};-p,-p;\frac{5}{2};-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}},\frac{2 b x^2}{\sqrt{c^2-4 a b}-c}\right )+3 c F_1\left (\frac{1}{2};-p,-p;\frac{3}{2};-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}},\frac{2 b x^2}{\sqrt{c^2-4 a b}-c}\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(c + e*x^2)*(a + c*x^2 + b*x^4)^p,x]

[Out]

(x*(a + c*x^2 + b*x^4)^p*(3*c*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])] + e*x^2*AppellF1[3/2, -p, -p, 5/2, (-2*b*x^2)/(c + Sqrt[-4*a*b + c^2]), (2*b*x^2)/(-c + S
qrt[-4*a*b + c^2])]))/(3*((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 = 0.025, size = 0, normalized size = 0. \begin{align*} \int \left ( e{x}^{2}+c \right ) \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((e*x^2+c)*(b*x^4+c*x^2+a)^p,x)

[Out]

int((e*x^2+c)*(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 (e x^{2} + c\right )}{\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((e*x^2+c)*(b*x^4+c*x^2+a)^p,x, algorithm="maxima")

[Out]

integrate((e*x^2 + c)*(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 (e x^{2} + c\right )}{\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((e*x^2+c)*(b*x^4+c*x^2+a)^p,x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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

[Out]

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