\(\int x^3 (a+b x^2)^p (c+d x^2)^q (e+f x^2) \, dx\) [67]

Optimal result

Integrand size = 29, antiderivative size = 264 \[ \int x^3 \left (a+b x^2\right )^p \left (c+d x^2\right )^q \left (e+f x^2\right ) \, dx=\frac {(b d e (3+p+q)-f (b c (2+p)+a d (4+p+2 q))) \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^{1+q}}{2 b^2 d^2 (2+p+q) (3+p+q)}+\frac {f \left (a+b x^2\right )^{2+p} \left (c+d x^2\right )^{1+q}}{2 b^2 d (3+p+q)}-\frac {(a d f (2+p+q) (b c (2+p)+a d (1+q))+(b c (1+p)+a d (1+q)) (b d e (3+p+q)-f (b c (2+p)+a d (4+p+2 q)))) \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^{1+q} \operatorname {Hypergeometric2F1}\left (1,2+p+q,2+p,-\frac {d \left (a+b x^2\right )}{b c-a d}\right )}{2 b^2 d^2 (b c-a d) (1+p) (2+p+q) (3+p+q)} \] Output:

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

Time = 0.34 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.74 \[ \int x^3 \left (a+b x^2\right )^p \left (c+d x^2\right )^q \left (e+f x^2\right ) \, dx=\frac {\left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^q \left (\frac {b \left (c+d x^2\right )}{b c-a d}\right )^{-q} \left ((b c-a d)^2 f \operatorname {Hypergeometric2F1}\left (1+p,-2-q,2+p,\frac {d \left (a+b x^2\right )}{-b c+a d}\right )+b \left (-\left ((b c-a d) (-d e+2 c f) \operatorname {Hypergeometric2F1}\left (1+p,-1-q,2+p,\frac {d \left (a+b x^2\right )}{-b c+a d}\right )\right )+b c (-d e+c f) \operatorname {Hypergeometric2F1}\left (1+p,-q,2+p,\frac {d \left (a+b x^2\right )}{-b c+a d}\right )\right )\right )}{2 b^3 d^2 (1+p)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.91, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {435, 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.

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

\(\Big \downarrow \) 435

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

\(\Big \downarrow \) 164

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

\(\Big \downarrow \) 80

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

\(\Big \downarrow \) 79

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

Input:

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

Output:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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