\(\int (a+c x^2)^p (A+B x+C x^2) (d+f x^2)^q \, dx\) [400]

Optimal result

Integrand size = 29, antiderivative size = 252 \[ \int \left (a+c x^2\right )^p \left (A+B x+C x^2\right ) \left (d+f x^2\right )^q \, dx=A x \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \left (d+f x^2\right )^q \left (1+\frac {f x^2}{d}\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {c x^2}{a},-\frac {f x^2}{d}\right )+\frac {1}{3} C x^3 \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \left (d+f x^2\right )^q \left (1+\frac {f x^2}{d}\right )^{-q} \operatorname {AppellF1}\left (\frac {3}{2},-p,-q,\frac {5}{2},-\frac {c x^2}{a},-\frac {f x^2}{d}\right )+\frac {B \left (a+c x^2\right )^{1+p} \left (d+f x^2\right )^q \left (\frac {c \left (d+f x^2\right )}{c d-a f}\right )^{-q} \operatorname {Hypergeometric2F1}\left (1+p,-q,2+p,-\frac {f \left (a+c x^2\right )}{c d-a f}\right )}{2 c (1+p)} \]

[Out]

A*x*(c*x^2+a)^p*(f*x^2+d)^q*AppellF1(1/2,-p,-q,3/2,-c*x^2/a,-f*x^2/d)/((1+c*x^2/a)^p)/((1+f*x^2/d)^q)+1/3*C*x^
3*(c*x^2+a)^p*(f*x^2+d)^q*AppellF1(3/2,-p,-q,5/2,-c*x^2/a,-f*x^2/d)/((1+c*x^2/a)^p)/((1+f*x^2/d)^q)+1/2*B*(c*x
^2+a)^(p+1)*(f*x^2+d)^q*hypergeom([-q, p+1],[2+p],-f*(c*x^2+a)/(-a*f+c*d))/c/(p+1)/((c*(f*x^2+d)/(-a*f+c*d))^q
)

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {6874, 441, 440, 455, 72, 71, 525, 524} \[ \int \left (a+c x^2\right )^p \left (A+B x+C x^2\right ) \left (d+f x^2\right )^q \, dx=A x \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} \left (d+f x^2\right )^q \left (\frac {f x^2}{d}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {c x^2}{a},-\frac {f x^2}{d}\right )+\frac {1}{3} C x^3 \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} \left (d+f x^2\right )^q \left (\frac {f x^2}{d}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {3}{2},-p,-q,\frac {5}{2},-\frac {c x^2}{a},-\frac {f x^2}{d}\right )+\frac {B \left (a+c x^2\right )^{p+1} \left (d+f x^2\right )^q \left (\frac {c \left (d+f x^2\right )}{c d-a f}\right )^{-q} \operatorname {Hypergeometric2F1}\left (p+1,-q,p+2,-\frac {f \left (c x^2+a\right )}{c d-a f}\right )}{2 c (p+1)} \]

[In]

Int[(a + c*x^2)^p*(A + B*x + C*x^2)*(d + f*x^2)^q,x]

[Out]

(A*x*(a + c*x^2)^p*(d + f*x^2)^q*AppellF1[1/2, -p, -q, 3/2, -((c*x^2)/a), -((f*x^2)/d)])/((1 + (c*x^2)/a)^p*(1
+ (f*x^2)/d)^q) + (C*x^3*(a + c*x^2)^p*(d + f*x^2)^q*AppellF1[3/2, -p, -q, 5/2, -((c*x^2)/a), -((f*x^2)/d)])/
(3*(1 + (c*x^2)/a)^p*(1 + (f*x^2)/d)^q) + (B*(a + c*x^2)^(1 + p)*(d + f*x^2)^q*Hypergeometric2F1[1 + p, -q, 2
+ p, -((f*(a + c*x^2))/(c*d - a*f))])/(2*c*(1 + p)*((c*(d + f*x^2))/(c*d - a*f))^q)

Rule 71

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c
- a*d))^n))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-d/(b*c - a*d), 0]))

Rule 72

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Dist[(c + d*x)^FracPart[n]/((b/(b*c - a*d)
)^IntPart[n]*(b*((c + d*x)/(b*c - a*d)))^FracPart[n]), Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c -
a*d)), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] &&
(RationalQ[m] ||  !SimplerQ[n + 1, m + 1])

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 441

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

Rule 455

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 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 525

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPar
t[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]), Int[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x
] /; FreeQ[{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])

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \int \left (A \left (a+c x^2\right )^p \left (d+f x^2\right )^q+B x \left (a+c x^2\right )^p \left (d+f x^2\right )^q+C x^2 \left (a+c x^2\right )^p \left (d+f x^2\right )^q\right ) \, dx \\ & = A \int \left (a+c x^2\right )^p \left (d+f x^2\right )^q \, dx+B \int x \left (a+c x^2\right )^p \left (d+f x^2\right )^q \, dx+C \int x^2 \left (a+c x^2\right )^p \left (d+f x^2\right )^q \, dx \\ & = \frac {1}{2} B \text {Subst}\left (\int (a+c x)^p (d+f x)^q \, dx,x,x^2\right )+\left (A \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p}\right ) \int \left (1+\frac {c x^2}{a}\right )^p \left (d+f x^2\right )^q \, dx+\left (C \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p}\right ) \int x^2 \left (1+\frac {c x^2}{a}\right )^p \left (d+f x^2\right )^q \, dx \\ & = \frac {1}{2} \left (B \left (d+f x^2\right )^q \left (\frac {c \left (d+f x^2\right )}{c d-a f}\right )^{-q}\right ) \text {Subst}\left (\int (a+c x)^p \left (\frac {c d}{c d-a f}+\frac {c f x}{c d-a f}\right )^q \, dx,x,x^2\right )+\left (A \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \left (d+f x^2\right )^q \left (1+\frac {f x^2}{d}\right )^{-q}\right ) \int \left (1+\frac {c x^2}{a}\right )^p \left (1+\frac {f x^2}{d}\right )^q \, dx+\left (C \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \left (d+f x^2\right )^q \left (1+\frac {f x^2}{d}\right )^{-q}\right ) \int x^2 \left (1+\frac {c x^2}{a}\right )^p \left (1+\frac {f x^2}{d}\right )^q \, dx \\ & = A x \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \left (d+f x^2\right )^q \left (1+\frac {f x^2}{d}\right )^{-q} F_1\left (\frac {1}{2};-p,-q;\frac {3}{2};-\frac {c x^2}{a},-\frac {f x^2}{d}\right )+\frac {1}{3} C x^3 \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \left (d+f x^2\right )^q \left (1+\frac {f x^2}{d}\right )^{-q} F_1\left (\frac {3}{2};-p,-q;\frac {5}{2};-\frac {c x^2}{a},-\frac {f x^2}{d}\right )+\frac {B \left (a+c x^2\right )^{1+p} \left (d+f x^2\right )^q \left (\frac {c \left (d+f x^2\right )}{c d-a f}\right )^{-q} \, _2F_1\left (1+p,-q;2+p;-\frac {f \left (a+c x^2\right )}{c d-a f}\right )}{2 c (1+p)} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 0.40 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.20 \[ \int \left (a+c x^2\right )^p \left (A+B x+C x^2\right ) \left (d+f x^2\right )^q \, dx=\frac {1}{6} x \left (a+c x^2\right )^p \left (d+f x^2\right )^q \left (3 B x \left (1+\frac {c x^2}{a}\right )^{-p} \left (1+\frac {f x^2}{d}\right )^{-q} \operatorname {AppellF1}\left (1,-p,-q,2,-\frac {c x^2}{a},-\frac {f x^2}{d}\right )+\frac {18 a A d \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {c x^2}{a},-\frac {f x^2}{d}\right )}{3 a d \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {c x^2}{a},-\frac {f x^2}{d}\right )+2 x^2 \left (c d p \operatorname {AppellF1}\left (\frac {3}{2},1-p,-q,\frac {5}{2},-\frac {c x^2}{a},-\frac {f x^2}{d}\right )+a f q \operatorname {AppellF1}\left (\frac {3}{2},-p,1-q,\frac {5}{2},-\frac {c x^2}{a},-\frac {f x^2}{d}\right )\right )}+2 C x^2 \left (1+\frac {c x^2}{a}\right )^{-p} \left (1+\frac {f x^2}{d}\right )^{-q} \operatorname {AppellF1}\left (\frac {3}{2},-p,-q,\frac {5}{2},-\frac {c x^2}{a},-\frac {f x^2}{d}\right )\right ) \]

[In]

Integrate[(a + c*x^2)^p*(A + B*x + C*x^2)*(d + f*x^2)^q,x]

[Out]

(x*(a + c*x^2)^p*(d + f*x^2)^q*((3*B*x*AppellF1[1, -p, -q, 2, -((c*x^2)/a), -((f*x^2)/d)])/((1 + (c*x^2)/a)^p*
(1 + (f*x^2)/d)^q) + (18*a*A*d*AppellF1[1/2, -p, -q, 3/2, -((c*x^2)/a), -((f*x^2)/d)])/(3*a*d*AppellF1[1/2, -p
, -q, 3/2, -((c*x^2)/a), -((f*x^2)/d)] + 2*x^2*(c*d*p*AppellF1[3/2, 1 - p, -q, 5/2, -((c*x^2)/a), -((f*x^2)/d)
] + a*f*q*AppellF1[3/2, -p, 1 - q, 5/2, -((c*x^2)/a), -((f*x^2)/d)])) + (2*C*x^2*AppellF1[3/2, -p, -q, 5/2, -(
(c*x^2)/a), -((f*x^2)/d)])/((1 + (c*x^2)/a)^p*(1 + (f*x^2)/d)^q)))/6

Maple [F]

\[\int \left (c \,x^{2}+a \right )^{p} \left (C \,x^{2}+B x +A \right ) \left (f \,x^{2}+d \right )^{q}d x\]

[In]

int((c*x^2+a)^p*(C*x^2+B*x+A)*(f*x^2+d)^q,x)

[Out]

int((c*x^2+a)^p*(C*x^2+B*x+A)*(f*x^2+d)^q,x)

Fricas [F]

\[ \int \left (a+c x^2\right )^p \left (A+B x+C x^2\right ) \left (d+f x^2\right )^q \, dx=\int { {\left (C x^{2} + B x + A\right )} {\left (c x^{2} + a\right )}^{p} {\left (f x^{2} + d\right )}^{q} \,d x } \]

[In]

integrate((c*x^2+a)^p*(C*x^2+B*x+A)*(f*x^2+d)^q,x, algorithm="fricas")

[Out]

integral((C*x^2 + B*x + A)*(c*x^2 + a)^p*(f*x^2 + d)^q, x)

Sympy [F(-1)]

Timed out. \[ \int \left (a+c x^2\right )^p \left (A+B x+C x^2\right ) \left (d+f x^2\right )^q \, dx=\text {Timed out} \]

[In]

integrate((c*x**2+a)**p*(C*x**2+B*x+A)*(f*x**2+d)**q,x)

[Out]

Timed out

Maxima [F]

\[ \int \left (a+c x^2\right )^p \left (A+B x+C x^2\right ) \left (d+f x^2\right )^q \, dx=\int { {\left (C x^{2} + B x + A\right )} {\left (c x^{2} + a\right )}^{p} {\left (f x^{2} + d\right )}^{q} \,d x } \]

[In]

integrate((c*x^2+a)^p*(C*x^2+B*x+A)*(f*x^2+d)^q,x, algorithm="maxima")

[Out]

integrate((C*x^2 + B*x + A)*(c*x^2 + a)^p*(f*x^2 + d)^q, x)

Giac [F]

\[ \int \left (a+c x^2\right )^p \left (A+B x+C x^2\right ) \left (d+f x^2\right )^q \, dx=\int { {\left (C x^{2} + B x + A\right )} {\left (c x^{2} + a\right )}^{p} {\left (f x^{2} + d\right )}^{q} \,d x } \]

[In]

integrate((c*x^2+a)^p*(C*x^2+B*x+A)*(f*x^2+d)^q,x, algorithm="giac")

[Out]

integrate((C*x^2 + B*x + A)*(c*x^2 + a)^p*(f*x^2 + d)^q, x)

Mupad [F(-1)]

Timed out. \[ \int \left (a+c x^2\right )^p \left (A+B x+C x^2\right ) \left (d+f x^2\right )^q \, dx=\int {\left (c\,x^2+a\right )}^p\,{\left (f\,x^2+d\right )}^q\,\left (C\,x^2+B\,x+A\right ) \,d x \]

[In]

int((a + c*x^2)^p*(d + f*x^2)^q*(A + B*x + C*x^2),x)

[Out]

int((a + c*x^2)^p*(d + f*x^2)^q*(A + B*x + C*x^2), x)