\(\int (a+c x^2)^p (A+C x^2) (d+f x^2)^q \, dx\) [398]

Optimal result

Integrand size = 26, antiderivative size = 166 \[ \int \left (a+c x^2\right )^p \left (A+C x^2\right ) \left (d+f x^2\right )^q \, dx=A x \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \left (d+f x^2\right )^q \left (1+\frac {f x^2}{d}\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {c x^2}{a},-\frac {f x^2}{d}\right )+\frac {1}{3} C x^3 \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \left (d+f x^2\right )^q \left (1+\frac {f x^2}{d}\right )^{-q} \operatorname {AppellF1}\left (\frac {3}{2},-p,-q,\frac {5}{2},-\frac {c x^2}{a},-\frac {f x^2}{d}\right ) \]

[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)

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {545, 441, 440, 525, 524} \[ \int \left (a+c x^2\right )^p \left (A+C x^2\right ) \left (d+f x^2\right )^q \, dx=A x \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} \left (d+f x^2\right )^q \left (\frac {f x^2}{d}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {c x^2}{a},-\frac {f x^2}{d}\right )+\frac {1}{3} C x^3 \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} \left (d+f x^2\right )^q \left (\frac {f x^2}{d}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {3}{2},-p,-q,\frac {5}{2},-\frac {c x^2}{a},-\frac {f x^2}{d}\right ) \]

[In]

Int[(a + c*x^2)^p*(A + 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)

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 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 545

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

Rubi steps \begin{align*} \text {integral}& = A \int \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 \\ & = \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 \\ & = \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 ) \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 0.32 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.46 \[ \int \left (a+c x^2\right )^p \left (A+C x^2\right ) \left (d+f x^2\right )^q \, dx=\frac {1}{3} x \left (a+c x^2\right )^p \left (d+f x^2\right )^q \left (\frac {9 a A d \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {c x^2}{a},-\frac {f x^2}{d}\right )}{3 a d \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {c x^2}{a},-\frac {f x^2}{d}\right )+2 x^2 \left (c d p \operatorname {AppellF1}\left (\frac {3}{2},1-p,-q,\frac {5}{2},-\frac {c x^2}{a},-\frac {f x^2}{d}\right )+a f q \operatorname {AppellF1}\left (\frac {3}{2},-p,1-q,\frac {5}{2},-\frac {c x^2}{a},-\frac {f x^2}{d}\right )\right )}+C x^2 \left (1+\frac {c x^2}{a}\right )^{-p} \left (1+\frac {f x^2}{d}\right )^{-q} \operatorname {AppellF1}\left (\frac {3}{2},-p,-q,\frac {5}{2},-\frac {c x^2}{a},-\frac {f x^2}{d}\right )\right ) \]

[In]

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

[Out]

(x*(a + c*x^2)^p*(d + f*x^2)^q*((9*a*A*d*AppellF1[1/2, -p, -q, 3/2, -((c*x^2)/a), -((f*x^2)/d)])/(3*a*d*Appell
F1[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)])) + (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)))/3

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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