\(\int (c+d x)^2 (a+b (c+d x)^2)^p \, dx\) [49]

Optimal result

Integrand size = 21, antiderivative size = 55 \[ \int (c+d x)^2 \left (a+b (c+d x)^2\right )^p \, dx=\frac {(c+d x)^3 \left (a+b (c+d x)^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,\frac {5}{2}+p,\frac {5}{2},-\frac {b (c+d x)^2}{a}\right )}{3 a d} \] Output:

1/3*(d*x+c)^3*(a+b*(d*x+c)^2)^(p+1)*hypergeom([1, 5/2+p],[5/2],-b*(d*x+c)^ 
2/a)/a/d
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.24 \[ \int (c+d x)^2 \left (a+b (c+d x)^2\right )^p \, dx=\frac {(c+d x)^3 \left (a+b (c+d x)^2\right )^p \left (1+\frac {b (c+d x)^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-p,\frac {5}{2},-\frac {b (c+d x)^2}{a}\right )}{3 d} \] Input:

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

Output:

((c + d*x)^3*(a + b*(c + d*x)^2)^p*Hypergeometric2F1[3/2, -p, 5/2, -((b*(c 
 + d*x)^2)/a)])/(3*d*(1 + (b*(c + d*x)^2)/a)^p)
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.24, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {895, 279, 278}

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.

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

\(\Big \downarrow \) 895

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

\(\Big \downarrow \) 279

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

\(\Big \downarrow \) 278

\(\displaystyle \frac {(c+d x)^3 \left (a+b (c+d x)^2\right )^p \left (\frac {b (c+d x)^2}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-p,\frac {5}{2},-\frac {b (c+d x)^2}{a}\right )}{3 d}\)

Input:

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

Output:

((c + d*x)^3*(a + b*(c + d*x)^2)^p*Hypergeometric2F1[3/2, -p, 5/2, -((b*(c 
 + d*x)^2)/a)])/(3*d*(1 + (b*(c + d*x)^2)/a)^p)
 

Defintions of rubi rules used

rule 278

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( 
c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( 
-b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && (ILtQ[p, 0 
] || GtQ[a, 0])
 

rule 279

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^IntP 
art[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^FracPart[p])   Int[(c*x)^m* 
(1 + b*(x^2/a))^p, x], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && 
!(ILtQ[p, 0] || GtQ[a, 0])
 

rule 895

Int[(u_)^(m_.)*((a_) + (b_.)*(v_)^(n_))^(p_.), x_Symbol] :> Simp[u^m/(Coeff 
icient[v, x, 1]*v^m)   Subst[Int[x^m*(a + b*x^n)^p, x], x, v], x] /; FreeQ[ 
{a, b, m, n, p}, x] && LinearPairQ[u, v, x]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (c+d x)^2 \left (a+b (c+d x)^2\right )^p \, dx=\frac {-\left (b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a \right )^{p} a^{2}+2 \left (b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a \right )^{p} a b \,c^{2} p +2 \left (b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a \right )^{p} a b c d p x +2 \left (b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a \right )^{p} b^{2} c^{4} p +\left (b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a \right )^{p} b^{2} c^{4}+6 \left (b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a \right )^{p} b^{2} c^{3} d p x +3 \left (b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a \right )^{p} b^{2} c^{3} d x +6 \left (b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a \right )^{p} b^{2} c^{2} d^{2} p \,x^{2}+3 \left (b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a \right )^{p} b^{2} c^{2} d^{2} x^{2}+2 \left (b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a \right )^{p} b^{2} c \,d^{3} p \,x^{3}+\left (b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a \right )^{p} b^{2} c \,d^{3} x^{3}+8 \left (\int \frac {\left (b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a \right )^{p} x}{4 b \,d^{2} p^{2} x^{2}+8 b c d \,p^{2} x +8 b \,d^{2} p \,x^{2}+4 b \,c^{2} p^{2}+16 b c d p x +3 b \,d^{2} x^{2}+8 b \,c^{2} p +6 b c d x +4 a \,p^{2}+3 b \,c^{2}+8 a p +3 a}d x \right ) a^{2} b \,d^{2} p^{3}+16 \left (\int \frac {\left (b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a \right )^{p} x}{4 b \,d^{2} p^{2} x^{2}+8 b c d \,p^{2} x +8 b \,d^{2} p \,x^{2}+4 b \,c^{2} p^{2}+16 b c d p x +3 b \,d^{2} x^{2}+8 b \,c^{2} p +6 b c d x +4 a \,p^{2}+3 b \,c^{2}+8 a p +3 a}d x \right ) a^{2} b \,d^{2} p^{2}+6 \left (\int \frac {\left (b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a \right )^{p} x}{4 b \,d^{2} p^{2} x^{2}+8 b c d \,p^{2} x +8 b \,d^{2} p \,x^{2}+4 b \,c^{2} p^{2}+16 b c d p x +3 b \,d^{2} x^{2}+8 b \,c^{2} p +6 b c d x +4 a \,p^{2}+3 b \,c^{2}+8 a p +3 a}d x \right ) a^{2} b \,d^{2} p}{b^{2} c d \left (4 p^{2}+8 p +3\right )} \] Input:

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

Output:

( - (a + b*c**2 + 2*b*c*d*x + b*d**2*x**2)**p*a**2 + 2*(a + b*c**2 + 2*b*c 
*d*x + b*d**2*x**2)**p*a*b*c**2*p + 2*(a + b*c**2 + 2*b*c*d*x + b*d**2*x** 
2)**p*a*b*c*d*p*x + 2*(a + b*c**2 + 2*b*c*d*x + b*d**2*x**2)**p*b**2*c**4* 
p + (a + b*c**2 + 2*b*c*d*x + b*d**2*x**2)**p*b**2*c**4 + 6*(a + b*c**2 + 
2*b*c*d*x + b*d**2*x**2)**p*b**2*c**3*d*p*x + 3*(a + b*c**2 + 2*b*c*d*x + 
b*d**2*x**2)**p*b**2*c**3*d*x + 6*(a + b*c**2 + 2*b*c*d*x + b*d**2*x**2)** 
p*b**2*c**2*d**2*p*x**2 + 3*(a + b*c**2 + 2*b*c*d*x + b*d**2*x**2)**p*b**2 
*c**2*d**2*x**2 + 2*(a + b*c**2 + 2*b*c*d*x + b*d**2*x**2)**p*b**2*c*d**3* 
p*x**3 + (a + b*c**2 + 2*b*c*d*x + b*d**2*x**2)**p*b**2*c*d**3*x**3 + 8*in 
t(((a + b*c**2 + 2*b*c*d*x + b*d**2*x**2)**p*x)/(4*a*p**2 + 8*a*p + 3*a + 
4*b*c**2*p**2 + 8*b*c**2*p + 3*b*c**2 + 8*b*c*d*p**2*x + 16*b*c*d*p*x + 6* 
b*c*d*x + 4*b*d**2*p**2*x**2 + 8*b*d**2*p*x**2 + 3*b*d**2*x**2),x)*a**2*b* 
d**2*p**3 + 16*int(((a + b*c**2 + 2*b*c*d*x + b*d**2*x**2)**p*x)/(4*a*p**2 
 + 8*a*p + 3*a + 4*b*c**2*p**2 + 8*b*c**2*p + 3*b*c**2 + 8*b*c*d*p**2*x + 
16*b*c*d*p*x + 6*b*c*d*x + 4*b*d**2*p**2*x**2 + 8*b*d**2*p*x**2 + 3*b*d**2 
*x**2),x)*a**2*b*d**2*p**2 + 6*int(((a + b*c**2 + 2*b*c*d*x + b*d**2*x**2) 
**p*x)/(4*a*p**2 + 8*a*p + 3*a + 4*b*c**2*p**2 + 8*b*c**2*p + 3*b*c**2 + 8 
*b*c*d*p**2*x + 16*b*c*d*p*x + 6*b*c*d*x + 4*b*d**2*p**2*x**2 + 8*b*d**2*p 
*x**2 + 3*b*d**2*x**2),x)*a**2*b*d**2*p)/(b**2*c*d*(4*p**2 + 8*p + 3))