\(\int x^3 (a+b x^2)^p (c+d x^2)^q (e+f x^2)^2 \, dx\) [76]

Optimal result

Integrand size = 31, antiderivative size = 576 \[ \int x^3 \left (a+b x^2\right )^p \left (c+d x^2\right )^q \left (e+f x^2\right )^2 \, dx=\frac {a f (b c f (2+p)-2 b d e (3+p+q)+a d f (4+p+2 q)) \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^{1+q}}{2 b^3 d^2 (2+p+q) (3+p+q)}-\frac {a f^2 \left (a+b x^2\right )^{2+p} \left (c+d x^2\right )^{1+q}}{2 b^3 d (3+p+q)}-\frac {f (b c f (3+p)-2 b d e (4+p+q)+a d f (5+p+2 q)) \left (a+b x^2\right )^{2+p} \left (c+d x^2\right )^{1+q}}{2 b^3 d^2 (3+p+q) (4+p+q)}+\frac {f^2 \left (a+b x^2\right )^{3+p} \left (c+d x^2\right )^{1+q}}{2 b^3 d (4+p+q)}+\frac {a \left (d (2+p+q) \left (a b c f^2 (2+p)+a^2 d f^2 (1+q)-b^2 d e^2 (3+p+q)\right )-f (b c (1+p)+a d (1+q)) (b c f (2+p)-2 b d e (3+p+q)+a d f (4+p+2 q))\right ) \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^{1+q} \operatorname {Hypergeometric2F1}\left (1,2+p+q,2+p,-\frac {d \left (a+b x^2\right )}{b c-a d}\right )}{2 b^3 d^2 (b c-a d) (1+p) (2+p+q) (3+p+q)}-\frac {\left (d (3+p+q) \left (a b c f^2 (3+p)+a^2 d f^2 (1+q)-b^2 d e^2 (4+p+q)\right )-f (b c (2+p)+a d (1+q)) (b c f (3+p)-2 b d e (4+p+q)+a d f (5+p+2 q))\right ) \left (a+b x^2\right )^{2+p} \left (c+d x^2\right )^{1+q} \operatorname {Hypergeometric2F1}\left (1,3+p+q,3+p,-\frac {d \left (a+b x^2\right )}{b c-a d}\right )}{2 b^3 d^2 (b c-a d) (2+p) (3+p+q) (4+p+q)} \] Output:

1/2*a*f*(b*c*f*(2+p)-2*b*d*e*(3+p+q)+a*d*f*(4+p+2*q))*(b*x^2+a)^(p+1)*(d*x 
^2+c)^(1+q)/b^3/d^2/(2+p+q)/(3+p+q)-1/2*a*f^2*(b*x^2+a)^(2+p)*(d*x^2+c)^(1 
+q)/b^3/d/(3+p+q)-1/2*f*(b*c*f*(3+p)-2*b*d*e*(4+p+q)+a*d*f*(5+p+2*q))*(b*x 
^2+a)^(2+p)*(d*x^2+c)^(1+q)/b^3/d^2/(3+p+q)/(4+p+q)+1/2*f^2*(b*x^2+a)^(3+p 
)*(d*x^2+c)^(1+q)/b^3/d/(4+p+q)+1/2*a*(d*(2+p+q)*(a*b*c*f^2*(2+p)+a^2*d*f^ 
2*(1+q)-b^2*d*e^2*(3+p+q))-f*(b*c*(p+1)+a*d*(1+q))*(b*c*f*(2+p)-2*b*d*e*(3 
+p+q)+a*d*f*(4+p+2*q)))*(b*x^2+a)^(p+1)*(d*x^2+c)^(1+q)*hypergeom([1, 2+p+ 
q],[2+p],-d*(b*x^2+a)/(-a*d+b*c))/b^3/d^2/(-a*d+b*c)/(p+1)/(2+p+q)/(3+p+q) 
-1/2*(d*(3+p+q)*(a*b*c*f^2*(3+p)+a^2*d*f^2*(1+q)-b^2*d*e^2*(4+p+q))-f*(b*c 
*(2+p)+a*d*(1+q))*(b*c*f*(3+p)-2*b*d*e*(4+p+q)+a*d*f*(5+p+2*q)))*(b*x^2+a) 
^(2+p)*(d*x^2+c)^(1+q)*hypergeom([1, 3+p+q],[3+p],-d*(b*x^2+a)/(-a*d+b*c)) 
/b^3/d^2/(-a*d+b*c)/(2+p)/(3+p+q)/(4+p+q)
 

Mathematica [A] (warning: unable to verify)

Time = 2.77 (sec) , antiderivative size = 535, normalized size of antiderivative = 0.93 \[ \int x^3 \left (a+b x^2\right )^p \left (c+d x^2\right )^q \left (e+f x^2\right )^2 \, dx=\frac {\left (a+b x^2\right )^p \left (c+d x^2\right )^q \left (\frac {b \left (c+d x^2\right ) \left (d^2 f (1+q) (b d e (5+p+q)-f (b c (3+p)+a d (2+q))) \left (a+b x^2\right )^2+b d^3 f (1+q) (3+p+q) \left (a+b x^2\right )^2 \left (e+f x^2\right )+(-b c+a d) \left (a^2 d^2 f^2 \left (2+3 q+q^2\right )-2 a b d f (1+q) (-c f (2+p)+d e (4+p+q))+b^2 \left (c^2 f^2 \left (6+5 p+p^2\right )-2 c d e f (2+p) (4+p+q)+d^2 e^2 \left (12+p^2+7 q+q^2+p (7+2 q)\right )\right )\right ) \left (\frac {d \left (a+b x^2\right )}{-b c+a d}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-1-p,1+q,2+q,\frac {b \left (c+d x^2\right )}{b c-a d}\right )\right )}{(1+q) (4+p+q)}-\frac {a d^2 \left (a+b x^2\right ) \left (b f (1+p) (b d e (4+p+q)-f (b c (2+p)+a d (2+q))) \left (c+d x^2\right )+b^2 d f (1+p) (2+p+q) \left (c+d x^2\right ) \left (e+f x^2\right )+\left (a^2 d^2 f^2 \left (2+3 q+q^2\right )-2 a b d f (1+q) (-c f (1+p)+d e (3+p+q))+b^2 \left (c^2 f^2 \left (2+3 p+p^2\right )-2 c d e f (1+p) (3+p+q)+d^2 e^2 \left (6+p^2+5 q+q^2+p (5+2 q)\right )\right )\right ) \left (\frac {b \left (c+d x^2\right )}{b c-a d}\right )^{-q} \operatorname {Hypergeometric2F1}\left (1+p,-q,2+p,\frac {d \left (a+b x^2\right )}{-b c+a d}\right )\right )}{(1+p) (2+p+q)}\right )}{2 b^4 d^4 (3+p+q)} \] Input:

Integrate[x^3*(a + b*x^2)^p*(c + d*x^2)^q*(e + f*x^2)^2,x]
 

Output:

((a + b*x^2)^p*(c + d*x^2)^q*((b*(c + d*x^2)*(d^2*f*(1 + q)*(b*d*e*(5 + p 
+ q) - f*(b*c*(3 + p) + a*d*(2 + q)))*(a + b*x^2)^2 + b*d^3*f*(1 + q)*(3 + 
 p + q)*(a + b*x^2)^2*(e + f*x^2) + ((-(b*c) + a*d)*(a^2*d^2*f^2*(2 + 3*q 
+ q^2) - 2*a*b*d*f*(1 + q)*(-(c*f*(2 + p)) + d*e*(4 + p + q)) + b^2*(c^2*f 
^2*(6 + 5*p + p^2) - 2*c*d*e*f*(2 + p)*(4 + p + q) + d^2*e^2*(12 + p^2 + 7 
*q + q^2 + p*(7 + 2*q))))*Hypergeometric2F1[-1 - p, 1 + q, 2 + q, (b*(c + 
d*x^2))/(b*c - a*d)])/((d*(a + b*x^2))/(-(b*c) + a*d))^p))/((1 + q)*(4 + p 
 + q)) - (a*d^2*(a + b*x^2)*(b*f*(1 + p)*(b*d*e*(4 + p + q) - f*(b*c*(2 + 
p) + a*d*(2 + q)))*(c + d*x^2) + b^2*d*f*(1 + p)*(2 + p + q)*(c + d*x^2)*( 
e + f*x^2) + ((a^2*d^2*f^2*(2 + 3*q + q^2) - 2*a*b*d*f*(1 + q)*(-(c*f*(1 + 
 p)) + d*e*(3 + p + q)) + b^2*(c^2*f^2*(2 + 3*p + p^2) - 2*c*d*e*f*(1 + p) 
*(3 + p + q) + d^2*e^2*(6 + p^2 + 5*q + q^2 + p*(5 + 2*q))))*Hypergeometri 
c2F1[1 + p, -q, 2 + p, (d*(a + b*x^2))/(-(b*c) + a*d)])/((b*(c + d*x^2))/( 
b*c - a*d))^q))/((1 + p)*(2 + p + q))))/(2*b^4*d^4*(3 + p + q))
 

Rubi [A] (warning: unable to verify)

Time = 1.34 (sec) , antiderivative size = 556, normalized size of antiderivative = 0.97, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {435, 170, 25, 164, 80, 79}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x^3 \left (e+f x^2\right )^2 \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx\)

\(\Big \downarrow \) 435

\(\displaystyle \frac {1}{2} \int x^2 \left (b x^2+a\right )^p \left (d x^2+c\right )^q \left (f x^2+e\right )^2dx^2\)

\(\Big \downarrow \) 170

\(\displaystyle \frac {1}{2} \left (\frac {\int -\left (b x^2+a\right )^p \left (d x^2+c\right )^q \left (f x^2+e\right ) \left (-\left ((2 b d e-f (b c (p+3)+a d (q+3))) x^2\right )+2 a c f+e (b c (p+1)+a d (q+1))\right )dx^2}{b d (p+q+4)}+\frac {\left (e+f x^2\right )^2 \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{b d (p+q+4)}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{2} \left (\frac {\left (e+f x^2\right )^2 \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{b d (p+q+4)}-\frac {\int \left (b x^2+a\right )^p \left (d x^2+c\right )^q \left (f x^2+e\right ) \left (-\left ((2 b d e-b c f (p+3)-a d f (q+3)) x^2\right )+2 a c f+b c e (p+1)+a d e (q+1)\right )dx^2}{b d (p+q+4)}\right )\)

\(\Big \downarrow \) 164

\(\displaystyle \frac {1}{2} \left (\frac {\left (e+f x^2\right )^2 \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{b d (p+q+4)}-\frac {\frac {\left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1} \left (-b d f x^2 (p+q+2) (-a d f (q+3)-b c f (p+3)+2 b d e)+b c f (p+2) (-a d f (q+3)-b c f (p+3)+2 b d e)+a d f (q+2) (-a d f (q+3)-b c f (p+3)+2 b d e)-2 b d (p+q+3) (b e (d e-c f (p+2))-a f (c f+d e (q+2)))\right )}{b^2 d^2 (p+q+2) (p+q+3)}-\frac {\left (a^2 d^2 f (q+1) (q+2) (-a d f (q+3)-b c f (p+3)+2 b d e)-\left (b^2 \left (c^2 (-f) (p+1) (p+2) (-a d f (q+3)-b c f (p+3)+2 b d e)+d^2 e (p+q+2) (p+q+3) (2 a c f+a d e (q+1)+b c e (p+1))+2 c d (p+1) (p+q+3) (b e (d e-c f (p+2))-a f (c f+d e (q+2)))\right )\right )+a b d (q+1) (2 c f (p+1) (-a d f (q+3)-b c f (p+3)+2 b d e)-2 d (p+q+3) (b e (d e-c f (p+2))-a f (c f+d e (q+2))))\right ) \int \left (b x^2+a\right )^p \left (d x^2+c\right )^qdx^2}{b^2 d^2 (p+q+2) (p+q+3)}}{b d (p+q+4)}\right )\)

\(\Big \downarrow \) 80

\(\displaystyle \frac {1}{2} \left (\frac {\left (e+f x^2\right )^2 \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{b d (p+q+4)}-\frac {\frac {\left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1} \left (-b d f x^2 (p+q+2) (-a d f (q+3)-b c f (p+3)+2 b d e)+b c f (p+2) (-a d f (q+3)-b c f (p+3)+2 b d e)+a d f (q+2) (-a d f (q+3)-b c f (p+3)+2 b d e)-2 b d (p+q+3) (b e (d e-c f (p+2))-a f (c f+d e (q+2)))\right )}{b^2 d^2 (p+q+2) (p+q+3)}-\frac {\left (c+d x^2\right )^q \left (\frac {b \left (c+d x^2\right )}{b c-a d}\right )^{-q} \left (a^2 d^2 f (q+1) (q+2) (-a d f (q+3)-b c f (p+3)+2 b d e)-\left (b^2 \left (c^2 (-f) (p+1) (p+2) (-a d f (q+3)-b c f (p+3)+2 b d e)+d^2 e (p+q+2) (p+q+3) (2 a c f+a d e (q+1)+b c e (p+1))+2 c d (p+1) (p+q+3) (b e (d e-c f (p+2))-a f (c f+d e (q+2)))\right )\right )+a b d (q+1) (2 c f (p+1) (-a d f (q+3)-b c f (p+3)+2 b d e)-2 d (p+q+3) (b e (d e-c f (p+2))-a f (c f+d e (q+2))))\right ) \int \left (b x^2+a\right )^p \left (\frac {b d x^2}{b c-a d}+\frac {b c}{b c-a d}\right )^qdx^2}{b^2 d^2 (p+q+2) (p+q+3)}}{b d (p+q+4)}\right )\)

\(\Big \downarrow \) 79

\(\displaystyle \frac {1}{2} \left (\frac {\left (e+f x^2\right )^2 \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{b d (p+q+4)}-\frac {\frac {\left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1} \left (-b d f x^2 (p+q+2) (-a d f (q+3)-b c f (p+3)+2 b d e)+b c f (p+2) (-a d f (q+3)-b c f (p+3)+2 b d e)+a d f (q+2) (-a d f (q+3)-b c f (p+3)+2 b d e)-2 b d (p+q+3) (b e (d e-c f (p+2))-a f (c f+d e (q+2)))\right )}{b^2 d^2 (p+q+2) (p+q+3)}-\frac {\left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^q \left (\frac {b \left (c+d x^2\right )}{b c-a d}\right )^{-q} \left (a^2 d^2 f (q+1) (q+2) (-a d f (q+3)-b c f (p+3)+2 b d e)-\left (b^2 \left (c^2 (-f) (p+1) (p+2) (-a d f (q+3)-b c f (p+3)+2 b d e)+d^2 e (p+q+2) (p+q+3) (2 a c f+a d e (q+1)+b c e (p+1))+2 c d (p+1) (p+q+3) (b e (d e-c f (p+2))-a f (c f+d e (q+2)))\right )\right )+a b d (q+1) (2 c f (p+1) (-a d f (q+3)-b c f (p+3)+2 b d e)-2 d (p+q+3) (b e (d e-c f (p+2))-a f (c f+d e (q+2))))\right ) \operatorname {Hypergeometric2F1}\left (p+1,-q,p+2,-\frac {d \left (b x^2+a\right )}{b c-a d}\right )}{b^3 d^2 (p+1) (p+q+2) (p+q+3)}}{b d (p+q+4)}\right )\)

Input:

Int[x^3*(a + b*x^2)^p*(c + d*x^2)^q*(e + f*x^2)^2,x]
 

Output:

(((a + b*x^2)^(1 + p)*(c + d*x^2)^(1 + q)*(e + f*x^2)^2)/(b*d*(4 + p + q)) 
 - (((a + b*x^2)^(1 + p)*(c + d*x^2)^(1 + q)*(b*c*f*(2 + p)*(2*b*d*e - b*c 
*f*(3 + p) - a*d*f*(3 + q)) + a*d*f*(2 + q)*(2*b*d*e - b*c*f*(3 + p) - a*d 
*f*(3 + q)) - 2*b*d*(3 + p + q)*(b*e*(d*e - c*f*(2 + p)) - a*f*(c*f + d*e* 
(2 + q))) - b*d*f*(2 + p + q)*(2*b*d*e - b*c*f*(3 + p) - a*d*f*(3 + q))*x^ 
2))/(b^2*d^2*(2 + p + q)*(3 + p + q)) - ((a^2*d^2*f*(1 + q)*(2 + q)*(2*b*d 
*e - b*c*f*(3 + p) - a*d*f*(3 + q)) + a*b*d*(1 + q)*(2*c*f*(1 + p)*(2*b*d* 
e - b*c*f*(3 + p) - a*d*f*(3 + q)) - 2*d*(3 + p + q)*(b*e*(d*e - c*f*(2 + 
p)) - a*f*(c*f + d*e*(2 + q)))) - b^2*(d^2*e*(2 + p + q)*(3 + p + q)*(2*a* 
c*f + b*c*e*(1 + p) + a*d*e*(1 + q)) - c^2*f*(1 + p)*(2 + p)*(2*b*d*e - b* 
c*f*(3 + p) - a*d*f*(3 + q)) + 2*c*d*(1 + p)*(3 + p + q)*(b*e*(d*e - c*f*( 
2 + p)) - a*f*(c*f + d*e*(2 + q)))))*(a + b*x^2)^(1 + p)*(c + d*x^2)^q*Hyp 
ergeometric2F1[1 + p, -q, 2 + p, -((d*(a + b*x^2))/(b*c - a*d))])/(b^3*d^2 
*(1 + p)*(2 + p + q)*(3 + p + q)*((b*(c + d*x^2))/(b*c - a*d))^q))/(b*d*(4 
 + p + q)))/2
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 79

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] 
&&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] 
 ||  !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
 

rule 80

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(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] &&  !IntegerQ[m] &&  !Integ 
erQ[n] && (RationalQ[m] ||  !SimplerQ[n + 1, m + 1])
 

rule 164

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ 
))*((g_.) + (h_.)*(x_)), x_] :> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - 
 b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( 
c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h 
*(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 
3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + 
d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3))   Int[( 
a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] 
&& NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]
 

rule 170

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( 
e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) 
  Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 
) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) 
+ h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre 
eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] 
 && IntegerQ[m]
 

rule 435

Int[(x_)^(m_.)*((a_.) + (b_.)*(x_)^2)^(p_.)*((c_.) + (d_.)*(x_)^2)^(q_.)*(( 
e_.) + (f_.)*(x_)^2)^(r_.), x_Symbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2) 
*(a + b*x)^p*(c + d*x)^q*(e + f*x)^r, x], x, x^2], x] /; FreeQ[{a, b, c, d, 
 e, f, p, q, r}, x] && IntegerQ[(m - 1)/2]
 

Maple [F]

\[\int x^{3} \left (b \,x^{2}+a \right )^{p} \left (d \,x^{2}+c \right )^{q} \left (f \,x^{2}+e \right )^{2}d x\]

Input:

int(x^3*(b*x^2+a)^p*(d*x^2+c)^q*(f*x^2+e)^2,x)
 

Output:

int(x^3*(b*x^2+a)^p*(d*x^2+c)^q*(f*x^2+e)^2,x)
 

Fricas [F]

\[ \int x^3 \left (a+b x^2\right )^p \left (c+d x^2\right )^q \left (e+f x^2\right )^2 \, dx=\int { {\left (f x^{2} + e\right )}^{2} {\left (b x^{2} + a\right )}^{p} {\left (d x^{2} + c\right )}^{q} x^{3} \,d x } \] Input:

integrate(x^3*(b*x^2+a)^p*(d*x^2+c)^q*(f*x^2+e)^2,x, algorithm="fricas")
 

Output:

integral((f^2*x^7 + 2*e*f*x^5 + e^2*x^3)*(b*x^2 + a)^p*(d*x^2 + c)^q, x)
 

Sympy [F(-1)]

Timed out. \[ \int x^3 \left (a+b x^2\right )^p \left (c+d x^2\right )^q \left (e+f x^2\right )^2 \, dx=\text {Timed out} \] Input:

integrate(x**3*(b*x**2+a)**p*(d*x**2+c)**q*(f*x**2+e)**2,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int x^3 \left (a+b x^2\right )^p \left (c+d x^2\right )^q \left (e+f x^2\right )^2 \, dx=\int { {\left (f x^{2} + e\right )}^{2} {\left (b x^{2} + a\right )}^{p} {\left (d x^{2} + c\right )}^{q} x^{3} \,d x } \] Input:

integrate(x^3*(b*x^2+a)^p*(d*x^2+c)^q*(f*x^2+e)^2,x, algorithm="maxima")
 

Output:

integrate((f*x^2 + e)^2*(b*x^2 + a)^p*(d*x^2 + c)^q*x^3, x)
 

Giac [F]

\[ \int x^3 \left (a+b x^2\right )^p \left (c+d x^2\right )^q \left (e+f x^2\right )^2 \, dx=\int { {\left (f x^{2} + e\right )}^{2} {\left (b x^{2} + a\right )}^{p} {\left (d x^{2} + c\right )}^{q} x^{3} \,d x } \] Input:

integrate(x^3*(b*x^2+a)^p*(d*x^2+c)^q*(f*x^2+e)^2,x, algorithm="giac")
 

Output:

integrate((f*x^2 + e)^2*(b*x^2 + a)^p*(d*x^2 + c)^q*x^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int x^3 \left (a+b x^2\right )^p \left (c+d x^2\right )^q \left (e+f x^2\right )^2 \, dx=\int x^3\,{\left (b\,x^2+a\right )}^p\,{\left (d\,x^2+c\right )}^q\,{\left (f\,x^2+e\right )}^2 \,d x \] Input:

int(x^3*(a + b*x^2)^p*(c + d*x^2)^q*(e + f*x^2)^2,x)
 

Output:

int(x^3*(a + b*x^2)^p*(c + d*x^2)^q*(e + f*x^2)^2, x)
 

Reduce [F]

\[ \int x^3 \left (a+b x^2\right )^p \left (c+d x^2\right )^q \left (e+f x^2\right )^2 \, dx=\int x^{3} \left (b \,x^{2}+a \right )^{p} \left (d \,x^{2}+c \right )^{q} \left (f \,x^{2}+e \right )^{2}d x \] Input:

int(x^3*(b*x^2+a)^p*(d*x^2+c)^q*(f*x^2+e)^2,x)
                                                                                    
                                                                                    
 

Output:

int(x^3*(b*x^2+a)^p*(d*x^2+c)^q*(f*x^2+e)^2,x)