\(\int x^2 (c+d x)^n (a+b x^2)^p \, dx\) [1937]

Optimal result

Integrand size = 20, antiderivative size = 383 \[ \int x^2 (c+d x)^n \left (a+b x^2\right )^p \, dx=\frac {(c+d x)^{1+n} \left (a+b x^2\right )^{1+p}}{b d (3+n+2 p)}-\frac {\left (a d^2 (1+n)-2 b c^2 (1+p)\right ) (c+d x)^{1+n} \left (a+b x^2\right )^p \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \operatorname {AppellF1}\left (1+n,-p,-p,2+n,\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{b d^3 (1+n) (3+n+2 p)}-\frac {2 c (1+p) (c+d x)^{2+n} \left (a+b x^2\right )^p \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \operatorname {AppellF1}\left (2+n,-p,-p,3+n,\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{d^3 (2+n) (3+n+2 p)} \] Output:

(d*x+c)^(1+n)*(b*x^2+a)^(p+1)/b/d/(3+n+2*p)-(a*d^2*(1+n)-2*b*c^2*(p+1))*(d 
*x+c)^(1+n)*(b*x^2+a)^p*AppellF1(1+n,-p,-p,2+n,(d*x+c)/(c-(-a)^(1/2)*d/b^( 
1/2)),(d*x+c)/(c+(-a)^(1/2)*d/b^(1/2)))/b/d^3/(1+n)/(3+n+2*p)/((1-(d*x+c)/ 
(c-(-a)^(1/2)*d/b^(1/2)))^p)/((1-(d*x+c)/(c+(-a)^(1/2)*d/b^(1/2)))^p)-2*c* 
(p+1)*(d*x+c)^(2+n)*(b*x^2+a)^p*AppellF1(2+n,-p,-p,3+n,(d*x+c)/(c-(-a)^(1/ 
2)*d/b^(1/2)),(d*x+c)/(c+(-a)^(1/2)*d/b^(1/2)))/d^3/(2+n)/(3+n+2*p)/((1-(d 
*x+c)/(c-(-a)^(1/2)*d/b^(1/2)))^p)/((1-(d*x+c)/(c+(-a)^(1/2)*d/b^(1/2)))^p 
)
 

Mathematica [F]

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

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 1.07 (sec) , antiderivative size = 625, normalized size of antiderivative = 1.63, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {624, 624, 514, 150}

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.

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

\(\Big \downarrow \) 624

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

\(\Big \downarrow \) 624

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

\(\Big \downarrow \) 514

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

\(\Big \downarrow \) 150

\(\displaystyle \frac {\frac {\left (a+b x^2\right )^p (c+d x)^{n+3} \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{\frac {\sqrt {-a} d}{\sqrt {b}}+c}\right )^{-p} \operatorname {AppellF1}\left (n+3,-p,-p,n+4,\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{d^2 (n+3)}-\frac {c \left (a+b x^2\right )^p (c+d x)^{n+2} \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{\frac {\sqrt {-a} d}{\sqrt {b}}+c}\right )^{-p} \operatorname {AppellF1}\left (n+2,-p,-p,n+3,\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{d^2 (n+2)}}{d}-\frac {c \left (\frac {\left (a+b x^2\right )^p (c+d x)^{n+2} \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{\frac {\sqrt {-a} d}{\sqrt {b}}+c}\right )^{-p} \operatorname {AppellF1}\left (n+2,-p,-p,n+3,\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{d^2 (n+2)}-\frac {c \left (a+b x^2\right )^p (c+d x)^{n+1} \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{\frac {\sqrt {-a} d}{\sqrt {b}}+c}\right )^{-p} \operatorname {AppellF1}\left (n+1,-p,-p,n+2,\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{d^2 (n+1)}\right )}{d}\)

Input:

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

Output:

-((c*(-((c*(c + d*x)^(1 + n)*(a + b*x^2)^p*AppellF1[1 + n, -p, -p, 2 + n, 
(c + d*x)/(c - (Sqrt[-a]*d)/Sqrt[b]), (c + d*x)/(c + (Sqrt[-a]*d)/Sqrt[b]) 
])/(d^2*(1 + n)*(1 - (c + d*x)/(c - (Sqrt[-a]*d)/Sqrt[b]))^p*(1 - (c + d*x 
)/(c + (Sqrt[-a]*d)/Sqrt[b]))^p)) + ((c + d*x)^(2 + n)*(a + b*x^2)^p*Appel 
lF1[2 + n, -p, -p, 3 + n, (c + d*x)/(c - (Sqrt[-a]*d)/Sqrt[b]), (c + d*x)/ 
(c + (Sqrt[-a]*d)/Sqrt[b])])/(d^2*(2 + n)*(1 - (c + d*x)/(c - (Sqrt[-a]*d) 
/Sqrt[b]))^p*(1 - (c + d*x)/(c + (Sqrt[-a]*d)/Sqrt[b]))^p)))/d) + (-((c*(c 
 + d*x)^(2 + n)*(a + b*x^2)^p*AppellF1[2 + n, -p, -p, 3 + n, (c + d*x)/(c 
- (Sqrt[-a]*d)/Sqrt[b]), (c + d*x)/(c + (Sqrt[-a]*d)/Sqrt[b])])/(d^2*(2 + 
n)*(1 - (c + d*x)/(c - (Sqrt[-a]*d)/Sqrt[b]))^p*(1 - (c + d*x)/(c + (Sqrt[ 
-a]*d)/Sqrt[b]))^p)) + ((c + d*x)^(3 + n)*(a + b*x^2)^p*AppellF1[3 + n, -p 
, -p, 4 + n, (c + d*x)/(c - (Sqrt[-a]*d)/Sqrt[b]), (c + d*x)/(c + (Sqrt[-a 
]*d)/Sqrt[b])])/(d^2*(3 + n)*(1 - (c + d*x)/(c - (Sqrt[-a]*d)/Sqrt[b]))^p* 
(1 - (c + d*x)/(c + (Sqrt[-a]*d)/Sqrt[b]))^p))/d
 

Defintions of rubi rules used

rule 150

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ 
] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 
, (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] &&  !In 
tegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
 

rule 514

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ 
{q = Rt[-a/b, 2]}, Simp[(a + b*x^2)^p/(d*(1 - (c + d*x)/(c - d*q))^p*(1 - ( 
c + d*x)/(c + d*q))^p)   Subst[Int[x^n*Simp[1 - x/(c + d*q), x]^p*Simp[1 - 
x/(c - d*q), x]^p, x], x, c + d*x], x]] /; FreeQ[{a, b, c, d, n, p}, x] && 
NeQ[b*c^2 + a*d^2, 0]
 

rule 624

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> Simp[1/d   Int[x^(m - 1)*(c + d*x)^(n + 1)*(a + b*x^2)^p, x], x] - Si 
mp[c/d   Int[x^(m - 1)*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, 
 d, n, p}, x] && IGtQ[m, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - (c + d*x)**n*(a + b*x**2)**p*a**2*d**3*n**2 - 2*(c + d*x)**n*(a + b*x* 
*2)**p*a**2*d**3*n*p - 3*(c + d*x)**n*(a + b*x**2)**p*a**2*d**3*n - 2*(c + 
 d*x)**n*(a + b*x**2)**p*a**2*d**3*p - 2*(c + d*x)**n*(a + b*x**2)**p*a**2 
*d**3 + (c + d*x)**n*(a + b*x**2)**p*a*b*c**2*d*n**2 - (c + d*x)**n*(a + b 
*x**2)**p*a*b*c**2*d*n + 2*(c + d*x)**n*(a + b*x**2)**p*a*b*c*d**2*n*p*x + 
 4*(c + d*x)**n*(a + b*x**2)**p*a*b*c*d**2*p**2*x + 4*(c + d*x)**n*(a + b* 
x**2)**p*a*b*c*d**2*p*x - 2*(c + d*x)**n*(a + b*x**2)**p*b**2*c**3*n*p*x - 
 2*(c + d*x)**n*(a + b*x**2)**p*b**2*c**3*n*x + (c + d*x)**n*(a + b*x**2)* 
*p*b**2*c**2*d*n**2*x**2 + 2*(c + d*x)**n*(a + b*x**2)**p*b**2*c**2*d*n*p* 
x**2 + (c + d*x)**n*(a + b*x**2)**p*b**2*c**2*d*n*x**2 + (c + d*x)**n*(a + 
 b*x**2)**p*b**2*c*d**2*n**2*x**3 + 4*(c + d*x)**n*(a + b*x**2)**p*b**2*c* 
d**2*n*p*x**3 + 3*(c + d*x)**n*(a + b*x**2)**p*b**2*c*d**2*n*x**3 + 4*(c + 
 d*x)**n*(a + b*x**2)**p*b**2*c*d**2*p**2*x**3 + 6*(c + d*x)**n*(a + b*x** 
2)**p*b**2*c*d**2*p*x**3 + 2*(c + d*x)**n*(a + b*x**2)**p*b**2*c*d**2*x**3 
 + int(((c + d*x)**n*(a + b*x**2)**p*x**2)/(a*c*n**3 + 6*a*c*n**2*p + 6*a* 
c*n**2 + 12*a*c*n*p**2 + 24*a*c*n*p + 11*a*c*n + 8*a*c*p**3 + 24*a*c*p**2 
+ 22*a*c*p + 6*a*c + a*d*n**3*x + 6*a*d*n**2*p*x + 6*a*d*n**2*x + 12*a*d*n 
*p**2*x + 24*a*d*n*p*x + 11*a*d*n*x + 8*a*d*p**3*x + 24*a*d*p**2*x + 22*a* 
d*p*x + 6*a*d*x + b*c*n**3*x**2 + 6*b*c*n**2*p*x**2 + 6*b*c*n**2*x**2 + 12 
*b*c*n*p**2*x**2 + 24*b*c*n*p*x**2 + 11*b*c*n*x**2 + 8*b*c*p**3*x**2 + ...