\(\int (c+e x^2) (a+c x^2+b x^4)^p \, dx\) [402]

Optimal result

Integrand size = 22, antiderivative size = 274 \[ \int \left (c+e x^2\right ) \left (a+c x^2+b x^4\right )^p \, dx=c x \left (1+\frac {2 b x^2}{c-\sqrt {-4 a b+c^2}}\right )^{-p} \left (1+\frac {2 b x^2}{c+\sqrt {-4 a b+c^2}}\right )^{-p} \left (a+c x^2+b x^4\right )^p \operatorname {AppellF1}\left (\frac {1}{2},-p,-p,\frac {3}{2},-\frac {2 b x^2}{c-\sqrt {-4 a b+c^2}},-\frac {2 b x^2}{c+\sqrt {-4 a b+c^2}}\right )+\frac {1}{3} e x^3 \left (1+\frac {2 b x^2}{c-\sqrt {-4 a b+c^2}}\right )^{-p} \left (1+\frac {2 b x^2}{c+\sqrt {-4 a b+c^2}}\right )^{-p} \left (a+c x^2+b x^4\right )^p \operatorname {AppellF1}\left (\frac {3}{2},-p,-p,\frac {5}{2},-\frac {2 b x^2}{c-\sqrt {-4 a b+c^2}},-\frac {2 b x^2}{c+\sqrt {-4 a b+c^2}}\right ) \]

[Out]

c*x*(b*x^4+c*x^2+a)^p*AppellF1(1/2,-p,-p,3/2,-2*b*x^2/(c-(-4*a*b+c^2)^(1/2)),-2*b*x^2/(c+(-4*a*b+c^2)^(1/2)))/
((1+2*b*x^2/(c-(-4*a*b+c^2)^(1/2)))^p)/((1+2*b*x^2/(c+(-4*a*b+c^2)^(1/2)))^p)+1/3*e*x^3*(b*x^4+c*x^2+a)^p*Appe
llF1(3/2,-p,-p,5/2,-2*b*x^2/(c-(-4*a*b+c^2)^(1/2)),-2*b*x^2/(c+(-4*a*b+c^2)^(1/2)))/((1+2*b*x^2/(c-(-4*a*b+c^2
)^(1/2)))^p)/((1+2*b*x^2/(c+(-4*a*b+c^2)^(1/2)))^p)

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {1217, 1119, 440, 1155, 524} \[ \int \left (c+e x^2\right ) \left (a+c x^2+b x^4\right )^p \, dx=\frac {1}{3} e x^3 \left (\frac {2 b x^2}{c-\sqrt {c^2-4 a b}}+1\right )^{-p} \left (a+b x^4+c x^2\right )^p \left (\frac {2 b x^2}{\sqrt {c^2-4 a b}+c}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {3}{2},-p,-p,\frac {5}{2},-\frac {2 b x^2}{c-\sqrt {c^2-4 a b}},-\frac {2 b x^2}{c+\sqrt {c^2-4 a b}}\right )+c x \left (\frac {2 b x^2}{c-\sqrt {c^2-4 a b}}+1\right )^{-p} \left (a+b x^4+c x^2\right )^p \left (\frac {2 b x^2}{\sqrt {c^2-4 a b}+c}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},-p,-p,\frac {3}{2},-\frac {2 b x^2}{c-\sqrt {c^2-4 a b}},-\frac {2 b x^2}{c+\sqrt {c^2-4 a b}}\right ) \]

[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 440

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 524

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)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], 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])

Rule 1119

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 1155

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 1217

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]

Rubi steps \begin{align*} \text {integral}& = \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] (warning: unable to verify)

Time = 0.48 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.85 \[ \int \left (c+e x^2\right ) \left (a+c x^2+b x^4\right )^p \, dx=\frac {1}{3} x \left (\frac {c-\sqrt {-4 a b+c^2}+2 b x^2}{c-\sqrt {-4 a b+c^2}}\right )^{-p} \left (\frac {c+\sqrt {-4 a b+c^2}+2 b x^2}{c+\sqrt {-4 a b+c^2}}\right )^{-p} \left (a+c x^2+b x^4\right )^p \left (3 c \operatorname {AppellF1}\left (\frac {1}{2},-p,-p,\frac {3}{2},-\frac {2 b x^2}{c+\sqrt {-4 a b+c^2}},\frac {2 b x^2}{-c+\sqrt {-4 a b+c^2}}\right )+e x^2 \operatorname {AppellF1}\left (\frac {3}{2},-p,-p,\frac {5}{2},-\frac {2 b x^2}{c+\sqrt {-4 a b+c^2}},\frac {2 b x^2}{-c+\sqrt {-4 a b+c^2}}\right )\right ) \]

[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)

Maple [F]

\[\int \left (e \,x^{2}+c \right ) \left (b \,x^{4}+c \,x^{2}+a \right )^{p}d x\]

[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)

Fricas [F]

\[ \int \left (c+e x^2\right ) \left (a+c x^2+b x^4\right )^p \, dx=\int { {\left (e x^{2} + c\right )} {\left (b x^{4} + c x^{2} + a\right )}^{p} \,d x } \]

[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)

Sympy [F]

\[ \int \left (c+e x^2\right ) \left (a+c x^2+b x^4\right )^p \, dx=\int \left (c + e x^{2}\right ) \left (a + b x^{4} + c x^{2}\right )^{p}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \left (c+e x^2\right ) \left (a+c x^2+b x^4\right )^p \, dx=\int { {\left (e x^{2} + c\right )} {\left (b x^{4} + c x^{2} + a\right )}^{p} \,d x } \]

[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)

Giac [F]

\[ \int \left (c+e x^2\right ) \left (a+c x^2+b x^4\right )^p \, dx=\int { {\left (e x^{2} + c\right )} {\left (b x^{4} + c x^{2} + a\right )}^{p} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \left (c+e x^2\right ) \left (a+c x^2+b x^4\right )^p \, dx=\int \left (e\,x^2+c\right )\,{\left (b\,x^4+c\,x^2+a\right )}^p \,d x \]

[In]

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

[Out]

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