\(\int x^5 (a+b x^2)^p (c+d x^2)^q \, dx\) [1635]

Optimal result

Integrand size = 22, antiderivative size = 230 \[ \int x^5 \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx=-\frac {(b c (2+p)+a d (4+p+2 q)) \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^{1+q}}{2 b^2 d^2 (2+p+q) (3+p+q)}+\frac {\left (a+b x^2\right )^{2+p} \left (c+d x^2\right )^{1+q}}{2 b^2 d (3+p+q)}+\frac {\left (b^2 c^2 \left (2+3 p+p^2\right )+2 a b c d (1+p) (1+q)+a^2 d^2 \left (2+3 q+q^2\right )\right ) \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^{1+q} \operatorname {Hypergeometric2F1}\left (1,2+p+q,2+p,-\frac {d \left (a+b x^2\right )}{b c-a d}\right )}{2 b^2 d^2 (b c-a d) (1+p) (2+p+q) (3+p+q)} \] Output:

-1/2*(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*(b*x^2+a)^(2+p)*(d*x^2+c)^(1+q)/b^2/d/(3+p+q)+1/2*(b^2*c^ 
2*(p^2+3*p+2)+2*a*b*c*d*(p+1)*(1+q)+a^2*d^2*(q^2+3*q+2))*(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 = 0.34 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.85 \[ \int x^5 \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx=\frac {\left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^q \left (-\frac {(b c (2+p)+a d (2+q)) \left (c+d x^2\right )}{b d (2+p+q)}+x^2 \left (c+d x^2\right )+\frac {\left (b^2 c^2 \left (2+3 p+p^2\right )+2 a b c d (1+p) (1+q)+a^2 d^2 \left (2+3 q+q^2\right )\right ) \left (\frac {b \left (c+d x^2\right )}{b c-a d}\right )^{-q} \operatorname {Hypergeometric2F1}\left (1+p,-q,2+p,\frac {d \left (a+b x^2\right )}{-b c+a d}\right )}{b^2 d (1+p) (2+p+q)}\right )}{2 b d (3+p+q)} \] Input:

Integrate[x^5*(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*(2 + p) + a*d*(2 + q))*(c + d* 
x^2))/(b*d*(2 + p + q))) + x^2*(c + d*x^2) + ((b^2*c^2*(2 + 3*p + p^2) + 2 
*a*b*c*d*(1 + p)*(1 + q) + a^2*d^2*(2 + 3*q + q^2))*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 = 0.37 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {354, 101, 25, 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.

\(\displaystyle \int x^5 \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx\)

\(\Big \downarrow \) 354

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

\(\Big \downarrow \) 101

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 90

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

\(\Big \downarrow \) 80

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

\(\Big \downarrow \) 79

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

Input:

Int[x^5*(a + b*x^2)^p*(c + d*x^2)^q,x]
 

Output:

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

\[\int x^{5} \left (b \,x^{2}+a \right )^{p} \left (x^{2} d +c \right )^{q}d x\]

Input:

int(x^5*(b*x^2+a)^p*(d*x^2+c)^q,x)
 

Output:

int(x^5*(b*x^2+a)^p*(d*x^2+c)^q,x)
 

Fricas [F]

\[ \int x^5 \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx=\int { {\left (b x^{2} + a\right )}^{p} {\left (d x^{2} + c\right )}^{q} x^{5} \,d x } \] Input:

integrate(x^5*(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^5, x)
 

Sympy [F(-1)]

Timed out. \[ \int x^5 \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx=\text {Timed out} \] Input:

integrate(x**5*(b*x**2+a)**p*(d*x**2+c)**q,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int x^5 \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx=\int { {\left (b x^{2} + a\right )}^{p} {\left (d x^{2} + c\right )}^{q} x^{5} \,d x } \] Input:

integrate(x^5*(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^5, x)
 

Giac [F]

\[ \int x^5 \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx=\int { {\left (b x^{2} + a\right )}^{p} {\left (d x^{2} + c\right )}^{q} x^{5} \,d x } \] Input:

integrate(x^5*(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^5, x)
 

Mupad [F(-1)]

Timed out. \[ \int x^5 \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx=\int x^5\,{\left (b\,x^2+a\right )}^p\,{\left (d\,x^2+c\right )}^q \,d x \] Input:

int(x^5*(a + b*x^2)^p*(c + d*x^2)^q,x)
 

Output:

int(x^5*(a + b*x^2)^p*(c + d*x^2)^q, x)
 

Reduce [F]

\[ \int x^5 \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx=\text {too large to display} \] Input:

int(x^5*(b*x^2+a)^p*(d*x^2+c)^q,x)
                                                                                    
                                                                                    
 

Output:

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