\(\int (d+e x)^2 (b x+c x^2)^p \, dx\) [249]

Optimal result

Integrand size = 19, antiderivative size = 165 \[ \int (d+e x)^2 \left (b x+c x^2\right )^p \, dx=-\frac {e (b e (2+p)-2 c d (3+2 p)) \left (b x+c x^2\right )^{1+p}}{2 c^2 (1+p) (3+2 p)}+\frac {e^2 x \left (b x+c x^2\right )^{1+p}}{c (3+2 p)}+\frac {\left (b^2 e^2 (2+p)+2 c^2 d^2 (3+2 p)-2 b c d e (3+2 p)\right ) \left (b x+c x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,2 (1+p),2+p,-\frac {c x}{b}\right )}{2 b c^2 (1+p) (3+2 p)} \] Output:

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

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.83 \[ \int (d+e x)^2 \left (b x+c x^2\right )^p \, dx=\frac {x (x (b+c x))^p \left (1+\frac {c x}{b}\right )^{-p} \left (-e (b+c x) \left (1+\frac {c x}{b}\right )^p (b e (2+p)-2 c (d (3+2 p)+e (1+p) x))+\left (b^2 e^2 (2+p)+2 c^2 d^2 (3+2 p)-2 b c d e (3+2 p)\right ) \operatorname {Hypergeometric2F1}\left (-p,1+p,2+p,-\frac {c x}{b}\right )\right )}{2 c^2 (1+p) (3+2 p)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.07, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {1166, 25, 1160, 1096}

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.

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

\(\Big \downarrow \) 1166

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1160

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

\(\Big \downarrow \) 1096

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 1096

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Rt[b^2 
 - 4*a*c, 2]}, Simp[(-(a + b*x + c*x^2)^(p + 1)/(q*(p + 1)*((q - b - 2*c*x) 
/(2*q))^(p + 1)))*Hypergeometric2F1[-p, p + 1, p + 2, (b + q + 2*c*x)/(2*q) 
], x]] /; FreeQ[{a, b, c, p}, x] &&  !IntegerQ[4*p] &&  !IntegerQ[3*p]
 

rule 1160

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol 
] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b 
*e)/(2*c)   Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] 
 && NeQ[p, -1]
 

rule 1166

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 
 1))), x] + Simp[1/(c*(m + 2*p + 1))   Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m 
+ 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]* 
(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && If[Ration 
alQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadrat 
icQ[a, b, c, d, e, m, p, x]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

Integral((x*(b + c*x))**p*(d + e*x)**2, x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

((b*x + c*x**2)**p*b**3*e**2*p**2 + 3*(b*x + c*x**2)**p*b**3*e**2*p + 2*(b 
*x + c*x**2)**p*b**3*e**2 - 4*(b*x + c*x**2)**p*b**2*c*d*e*p**2 - 10*(b*x 
+ c*x**2)**p*b**2*c*d*e*p - 6*(b*x + c*x**2)**p*b**2*c*d*e - 2*(b*x + c*x* 
*2)**p*b**2*c*e**2*p**2*x - 4*(b*x + c*x**2)**p*b**2*c*e**2*p*x + 4*(b*x + 
 c*x**2)**p*b*c**2*d**2*p**2 + 10*(b*x + c*x**2)**p*b*c**2*d**2*p + 6*(b*x 
 + c*x**2)**p*b*c**2*d**2 + 8*(b*x + c*x**2)**p*b*c**2*d*e*p**2*x + 12*(b* 
x + c*x**2)**p*b*c**2*d*e*p*x + 4*(b*x + c*x**2)**p*b*c**2*e**2*p**2*x**2 
+ 2*(b*x + c*x**2)**p*b*c**2*e**2*p*x**2 + 8*(b*x + c*x**2)**p*c**3*d**2*p 
**2*x + 20*(b*x + c*x**2)**p*c**3*d**2*p*x + 12*(b*x + c*x**2)**p*c**3*d** 
2*x + 16*(b*x + c*x**2)**p*c**3*d*e*p**2*x**2 + 32*(b*x + c*x**2)**p*c**3* 
d*e*p*x**2 + 12*(b*x + c*x**2)**p*c**3*d*e*x**2 + 8*(b*x + c*x**2)**p*c**3 
*e**2*p**2*x**3 + 12*(b*x + c*x**2)**p*c**3*e**2*p*x**3 + 4*(b*x + c*x**2) 
**p*c**3*e**2*x**3 - 4*int((b*x + c*x**2)**p/(4*b*p**2*x + 8*b*p*x + 3*b*x 
 + 4*c*p**2*x**2 + 8*c*p*x**2 + 3*c*x**2),x)*b**4*e**2*p**5 - 20*int((b*x 
+ c*x**2)**p/(4*b*p**2*x + 8*b*p*x + 3*b*x + 4*c*p**2*x**2 + 8*c*p*x**2 + 
3*c*x**2),x)*b**4*e**2*p**4 - 35*int((b*x + c*x**2)**p/(4*b*p**2*x + 8*b*p 
*x + 3*b*x + 4*c*p**2*x**2 + 8*c*p*x**2 + 3*c*x**2),x)*b**4*e**2*p**3 - 25 
*int((b*x + c*x**2)**p/(4*b*p**2*x + 8*b*p*x + 3*b*x + 4*c*p**2*x**2 + 8*c 
*p*x**2 + 3*c*x**2),x)*b**4*e**2*p**2 - 6*int((b*x + c*x**2)**p/(4*b*p**2* 
x + 8*b*p*x + 3*b*x + 4*c*p**2*x**2 + 8*c*p*x**2 + 3*c*x**2),x)*b**4*e*...