\(\int x^3 (a+b x^2)^p (c+d x^2)^q \, dx\) [1636]

Optimal result

Integrand size = 22, antiderivative size = 134 \[ \int x^3 \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx=\frac {\left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^{1+q}}{2 b d (2+p+q)}-\frac {(b c (1+p)+a d (1+q)) \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^{1+q} \operatorname {Hypergeometric2F1}\left (1,2+p+q,2+p,-\frac {d \left (a+b x^2\right )}{b c-a d}\right )}{2 b d (b c-a d) (1+p) (2+p+q)} \] Output:

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

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.88 \[ \int x^3 \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx=\frac {\left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^q \left (b \left (c+d x^2\right )-\frac {(b c (1+p)+a d (1+q)) \left (\frac {b \left (c+d x^2\right )}{b c-a d}\right )^{-q} \operatorname {Hypergeometric2F1}\left (1+p,-q,2+p,\frac {d \left (a+b x^2\right )}{-b c+a d}\right )}{1+p}\right )}{2 b^2 d (2+p+q)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {354, 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.

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

\(\Big \downarrow \) 354

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

\(\Big \downarrow \) 90

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

\(\Big \downarrow \) 80

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

\(\Big \downarrow \) 79

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

Input:

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

Output:

(((a + b*x^2)^(1 + p)*(c + d*x^2)^(1 + q))/(b*d*(2 + p + q)) - ((b*c*(1 + 
p) + a*d*(1 + 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^2*d*(1 + p)*(2 + 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 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 354

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S 
ymbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x 
, x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ 
[(m - 1)/2]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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