\(\int x (a+b x^2)^p (c+d x^2)^q (e+f x^2)^2 \, dx\) [77]

Optimal result

Integrand size = 29, antiderivative size = 287 \[ \int x \left (a+b x^2\right )^p \left (c+d x^2\right )^q \left (e+f x^2\right )^2 \, dx=-\frac {f (b c f (2+p)-2 b d e (3+p+q)+a d f (4+p+2 q)) \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^{1+q}}{2 b^2 d^2 (2+p+q) (3+p+q)}+\frac {f^2 \left (a+b x^2\right )^{2+p} \left (c+d x^2\right )^{1+q}}{2 b^2 d (3+p+q)}-\frac {\left (d (2+p+q) \left (a b c f^2 (2+p)+a^2 d f^2 (1+q)-b^2 d e^2 (3+p+q)\right )-f (b c (1+p)+a d (1+q)) (b c f (2+p)-2 b d e (3+p+q)+a d f (4+p+2 q))\right ) \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^{1+q} \operatorname {Hypergeometric2F1}\left (1,2+p+q,2+p,-\frac {d \left (a+b x^2\right )}{b c-a d}\right )}{2 b^2 d^2 (b c-a d) (1+p) (2+p+q) (3+p+q)} \] Output:

-1/2*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^2/d^2/(2+p+q)/(3+p+q)+1/2*f^2*(b*x^2+a)^(2+p)*(d*x^2+c)^(1+q) 
/b^2/d/(3+p+q)-1/2*(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^2/d^2/(-a*d+b*c)/(p+1)/(2+p+q)/(3+p+q)
 

Mathematica [A] (warning: unable to verify)

Time = 0.50 (sec) , antiderivative size = 270, normalized size of antiderivative = 0.94 \[ \int x \left (a+b x^2\right )^p \left (c+d x^2\right )^q \left (e+f x^2\right )^2 \, dx=\frac {\left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^q \left (\frac {f (b d e (4+p+q)-f (b c (2+p)+a d (2+q))) \left (c+d x^2\right )}{b d (2+p+q)}+f \left (c+d x^2\right ) \left (e+f x^2\right )+\frac {\left (a^2 d^2 f^2 \left (2+3 q+q^2\right )-2 a b d f (1+q) (-c f (1+p)+d e (3+p+q))+b^2 \left (c^2 f^2 \left (2+3 p+p^2\right )-2 c d e f (1+p) (3+p+q)+d^2 e^2 \left (6+p^2+5 q+q^2+p (5+2 q)\right )\right )\right ) \left (\frac {b \left (c+d x^2\right )}{b c-a d}\right )^{-q} \operatorname {Hypergeometric2F1}\left (1+p,-q,2+p,\frac {d \left (a+b x^2\right )}{-b c+a d}\right )}{b^2 d (1+p) (2+p+q)}\right )}{2 b d (3+p+q)} \] Input:

Integrate[x*(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)^q*((f*(b*d*e*(4 + p + q) - f*(b*c*(2 + p) 
 + a*d*(2 + q)))*(c + d*x^2))/(b*d*(2 + p + q)) + f*(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))))*Hypergeometric2F1[1 + 
p, -q, 2 + p, (d*(a + b*x^2))/(-(b*c) + a*d)])/(b^2*d*(1 + p)*(2 + p + q)* 
((b*(c + d*x^2))/(b*c - a*d))^q)))/(2*b*d*(3 + p + q))
 

Rubi [A] (warning: unable to verify)

Time = 0.82 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {435, 101, 90, 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.

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

\(\Big \downarrow \) 435

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

\(\Big \downarrow \) 101

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

\(\Big \downarrow \) 90

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

\(\Big \downarrow \) 80

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

\(\Big \downarrow \) 79

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

Input:

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

Output:

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

Defintions of rubi rules used

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 90

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

rule 101

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + 
 p + 3))), x] + Simp[1/(d*f*(n + p + 3))   Int[(c + d*x)^n*(e + f*x)^p*Simp 
[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f 
*(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, 
 c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

integrate(x*(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, x)
 

Giac [F]

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

integrate(x*(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, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

((c + d*x**2)**q*(a + b*x**2)**p*a**3*c*d**2*f**2*p*q + 2*(c + d*x**2)**q* 
(a + b*x**2)**p*a**3*c*d**2*f**2*p - (c + d*x**2)**q*(a + b*x**2)**p*a**3* 
d**3*f**2*p*q**2*x**2 - 2*(c + d*x**2)**q*(a + b*x**2)**p*a**3*d**3*f**2*p 
*q*x**2 - 2*(c + d*x**2)**q*(a + b*x**2)**p*a**2*b*c**2*d*f**2*p*q - 2*(c 
+ d*x**2)**q*(a + b*x**2)**p*a**2*b*c*d**2*e*f*p**2 - 2*(c + d*x**2)**q*(a 
 + b*x**2)**p*a**2*b*c*d**2*e*f*p*q - 6*(c + d*x**2)**q*(a + b*x**2)**p*a* 
*2*b*c*d**2*e*f*p - (c + d*x**2)**q*(a + b*x**2)**p*a**2*b*c*d**2*f**2*p** 
2*q*x**2 - 2*(c + d*x**2)**q*(a + b*x**2)**p*a**2*b*c*d**2*f**2*p**2*x**2 
+ 2*(c + d*x**2)**q*(a + b*x**2)**p*a**2*b*c*d**2*f**2*p*q**2*x**2 + 2*(c 
+ d*x**2)**q*(a + b*x**2)**p*a**2*b*d**3*e*f*p**2*q*x**2 + 2*(c + d*x**2)* 
*q*(a + b*x**2)**p*a**2*b*d**3*e*f*p*q**2*x**2 + 6*(c + d*x**2)**q*(a + b* 
x**2)**p*a**2*b*d**3*e*f*p*q*x**2 + (c + d*x**2)**q*(a + b*x**2)**p*a**2*b 
*d**3*f**2*p**2*q*x**4 + (c + d*x**2)**q*(a + b*x**2)**p*a**2*b*d**3*f**2* 
p*q**2*x**4 + (c + d*x**2)**q*(a + b*x**2)**p*a**2*b*d**3*f**2*p*q*x**4 + 
(c + d*x**2)**q*(a + b*x**2)**p*a*b**2*c**3*f**2*p*q + 2*(c + d*x**2)**q*( 
a + b*x**2)**p*a*b**2*c**3*f**2*q - 2*(c + d*x**2)**q*(a + b*x**2)**p*a*b* 
*2*c**2*d*e*f*p*q - 2*(c + d*x**2)**q*(a + b*x**2)**p*a*b**2*c**2*d*e*f*q* 
*2 - 6*(c + d*x**2)**q*(a + b*x**2)**p*a*b**2*c**2*d*e*f*q + 2*(c + d*x**2 
)**q*(a + b*x**2)**p*a*b**2*c**2*d*f**2*p**2*q*x**2 - (c + d*x**2)**q*(a + 
 b*x**2)**p*a*b**2*c**2*d*f**2*p*q**2*x**2 - 2*(c + d*x**2)**q*(a + b*x...