\(\int (A+B x) (d+e x)^m (a^2+2 a b x+b^2 x^2)^p \, dx\) [512]

Optimal result

Integrand size = 31, antiderivative size = 167 \[ \int (A+B x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\frac {B (a+b x) (d+e x)^{1+m} \left (a^2+2 a b x+b^2 x^2\right )^p}{b e (2+m+2 p)}+\frac {(a B e (1+m)-A b e (2+m+2 p)+b B (d+2 d p)) (a+b x) (d+e x)^{1+m} \left (a^2+2 a b x+b^2 x^2\right )^p \operatorname {Hypergeometric2F1}\left (1,2+m+2 p,2+m,\frac {b (d+e x)}{b d-a e}\right )}{b e (b d-a e) (1+m) (2+m+2 p)} \] Output:

B*(b*x+a)*(e*x+d)^(1+m)*(b^2*x^2+2*a*b*x+a^2)^p/b/e/(2+m+2*p)+(a*B*e*(1+m) 
-A*b*e*(2+m+2*p)+b*B*(2*d*p+d))*(b*x+a)*(e*x+d)^(1+m)*(b^2*x^2+2*a*b*x+a^2 
)^p*hypergeom([1, 2+m+2*p],[2+m],b*(e*x+d)/(-a*e+b*d))/b/e/(-a*e+b*d)/(1+m 
)/(2+m+2*p)
 

Mathematica [A] (warning: unable to verify)

Time = 0.19 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.75 \[ \int (A+B x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\frac {\left ((a+b x)^2\right )^p (d+e x)^{1+m} \left (B e (a+b x)-\frac {(a B e (1+m)-A b e (2+m+2 p)+b B (d+2 d p)) \left (\frac {e (a+b x)}{-b d+a e}\right )^{-2 p} \operatorname {Hypergeometric2F1}\left (1+m,-2 p,2+m,\frac {b (d+e x)}{b d-a e}\right )}{1+m}\right )}{b e^2 (2+m+2 p)} \] Input:

Integrate[(A + B*x)*(d + e*x)^m*(a^2 + 2*a*b*x + b^2*x^2)^p,x]
 

Output:

(((a + b*x)^2)^p*(d + e*x)^(1 + m)*(B*e*(a + b*x) - ((a*B*e*(1 + m) - A*b* 
e*(2 + m + 2*p) + b*B*(d + 2*d*p))*Hypergeometric2F1[1 + m, -2*p, 2 + m, ( 
b*(d + e*x))/(b*d - a*e)])/((1 + m)*((e*(a + b*x))/(-(b*d) + a*e))^(2*p))) 
)/(b*e^2*(2 + m + 2*p))
 

Rubi [A] (warning: unable to verify)

Time = 0.61 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.12, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {1187, 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.

\(\displaystyle \int (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^p (d+e x)^m \, dx\)

\(\Big \downarrow \) 1187

\(\displaystyle \left (a b+b^2 x\right )^{-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p \int \left (x b^2+a b\right )^{2 p} (A+B x) (d+e x)^mdx\)

\(\Big \downarrow \) 90

\(\displaystyle \left (a b+b^2 x\right )^{-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p \left (\left (A-\frac {B (a e (m+1)+b (2 d p+d))}{b e (m+2 p+2)}\right ) \int \left (x b^2+a b\right )^{2 p} (d+e x)^mdx+\frac {B \left (a b+b^2 x\right )^{2 p+1} (d+e x)^{m+1}}{b^2 e (m+2 p+2)}\right )\)

\(\Big \downarrow \) 80

\(\displaystyle \left (a b+b^2 x\right )^{-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p \left (\left (a b+b^2 x\right )^{2 p} \left (-\frac {e (a+b x)}{b d-a e}\right )^{-2 p} \left (A-\frac {B (a e (m+1)+b (2 d p+d))}{b e (m+2 p+2)}\right ) \int (d+e x)^m \left (-\frac {b x e}{b d-a e}-\frac {a e}{b d-a e}\right )^{2 p}dx+\frac {B \left (a b+b^2 x\right )^{2 p+1} (d+e x)^{m+1}}{b^2 e (m+2 p+2)}\right )\)

\(\Big \downarrow \) 79

\(\displaystyle \left (a b+b^2 x\right )^{-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p \left (\frac {\left (a b+b^2 x\right )^{2 p} (d+e x)^{m+1} \left (-\frac {e (a+b x)}{b d-a e}\right )^{-2 p} \left (A-\frac {B (a e (m+1)+b (2 d p+d))}{b e (m+2 p+2)}\right ) \operatorname {Hypergeometric2F1}\left (m+1,-2 p,m+2,\frac {b (d+e x)}{b d-a e}\right )}{e (m+1)}+\frac {B \left (a b+b^2 x\right )^{2 p+1} (d+e x)^{m+1}}{b^2 e (m+2 p+2)}\right )\)

Input:

Int[(A + B*x)*(d + e*x)^m*(a^2 + 2*a*b*x + b^2*x^2)^p,x]
 

Output:

((a^2 + 2*a*b*x + b^2*x^2)^p*((B*(a*b + b^2*x)^(1 + 2*p)*(d + e*x)^(1 + m) 
)/(b^2*e*(2 + m + 2*p)) + ((A - (B*(a*e*(1 + m) + b*(d + 2*d*p)))/(b*e*(2 
+ m + 2*p)))*(a*b + b^2*x)^(2*p)*(d + e*x)^(1 + m)*Hypergeometric2F1[1 + m 
, -2*p, 2 + m, (b*(d + e*x))/(b*d - a*e)])/(e*(1 + m)*(-((e*(a + b*x))/(b* 
d - a*e)))^(2*p))))/(a*b + b^2*x)^(2*p)
 

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 1187

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ 
) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x + c*x^2)^FracPart[p]/(c^ 
IntPart[p]*(b/2 + c*x)^(2*FracPart[p]))   Int[(d + e*x)^m*(f + g*x)^n*(b/2 
+ c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[b^2 
 - 4*a*c, 0] &&  !IntegerQ[p]
 

Maple [F]

\[\int \left (B x +A \right ) \left (e x +d \right )^{m} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p}d x\]

Input:

int((B*x+A)*(e*x+d)^m*(b^2*x^2+2*a*b*x+a^2)^p,x)
 

Output:

int((B*x+A)*(e*x+d)^m*(b^2*x^2+2*a*b*x+a^2)^p,x)
 

Fricas [F]

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

integrate((B*x+A)*(e*x+d)^m*(b^2*x^2+2*a*b*x+a^2)^p,x, algorithm="fricas")
 

Output:

integral((B*x + A)*(b^2*x^2 + 2*a*b*x + a^2)^p*(e*x + d)^m, x)
 

Sympy [F(-2)]

Exception generated. \[ \int (A+B x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

integrate((B*x+A)*(e*x+d)**m*(b**2*x**2+2*a*b*x+a**2)**p,x)
 

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

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

integrate((B*x+A)*(e*x+d)^m*(b^2*x^2+2*a*b*x+a^2)^p,x, algorithm="maxima")
 

Output:

integrate((B*x + A)*(b^2*x^2 + 2*a*b*x + a^2)^p*(e*x + d)^m, x)
 

Giac [F]

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

integrate((B*x+A)*(e*x+d)^m*(b^2*x^2+2*a*b*x+a^2)^p,x, algorithm="giac")
 

Output:

integrate((B*x + A)*(b^2*x^2 + 2*a*b*x + a^2)^p*(e*x + d)^m, x)
 

Mupad [F(-1)]

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

int((A + B*x)*(d + e*x)^m*(a^2 + b^2*x^2 + 2*a*b*x)^p,x)
 

Output:

int((A + B*x)*(d + e*x)^m*(a^2 + b^2*x^2 + 2*a*b*x)^p, x)
 

Reduce [F]

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

int((B*x+A)*(e*x+d)^m*(b^2*x^2+2*a*b*x+a^2)^p,x)
 

Output:

((d + e*x)**m*(a**2 + 2*a*b*x + b**2*x**2)**p*a**2*d*e*m**2 + 4*(d + e*x)* 
*m*(a**2 + 2*a*b*x + b**2*x**2)**p*a**2*d*e*m*p + 2*(d + e*x)**m*(a**2 + 2 
*a*b*x + b**2*x**2)**p*a**2*d*e*m + 4*(d + e*x)**m*(a**2 + 2*a*b*x + b**2* 
x**2)**p*a**2*d*e*p**2 + 2*(d + e*x)**m*(a**2 + 2*a*b*x + b**2*x**2)**p*a* 
*2*d*e*p + (d + e*x)**m*(a**2 + 2*a*b*x + b**2*x**2)**p*a**2*e**2*m**2*x + 
 4*(d + e*x)**m*(a**2 + 2*a*b*x + b**2*x**2)**p*a**2*e**2*m*p*x + 2*(d + e 
*x)**m*(a**2 + 2*a*b*x + b**2*x**2)**p*a**2*e**2*m*x - (d + e*x)**m*(a**2 
+ 2*a*b*x + b**2*x**2)**p*a*b*d**2*m + (d + e*x)**m*(a**2 + 2*a*b*x + b**2 
*x**2)**p*a*b*d*e*m**2*x + 2*(d + e*x)**m*(a**2 + 2*a*b*x + b**2*x**2)**p* 
a*b*d*e*m*p*x + 8*(d + e*x)**m*(a**2 + 2*a*b*x + b**2*x**2)**p*a*b*d*e*p** 
2*x + 4*(d + e*x)**m*(a**2 + 2*a*b*x + b**2*x**2)**p*a*b*d*e*p*x + (d + e* 
x)**m*(a**2 + 2*a*b*x + b**2*x**2)**p*a*b*e**2*m**2*x**2 + 2*(d + e*x)**m* 
(a**2 + 2*a*b*x + b**2*x**2)**p*a*b*e**2*m*p*x**2 + (d + e*x)**m*(a**2 + 2 
*a*b*x + b**2*x**2)**p*a*b*e**2*m*x**2 + 2*(d + e*x)**m*(a**2 + 2*a*b*x + 
b**2*x**2)**p*b**2*d**2*m*p*x + 2*(d + e*x)**m*(a**2 + 2*a*b*x + b**2*x**2 
)**p*b**2*d*e*m*p*x**2 + 4*(d + e*x)**m*(a**2 + 2*a*b*x + b**2*x**2)**p*b* 
*2*d*e*p**2*x**2 + 2*(d + e*x)**m*(a**2 + 2*a*b*x + b**2*x**2)**p*b**2*d*e 
*p*x**2 + 4*int(((d + e*x)**m*(a**2 + 2*a*b*x + b**2*x**2)**p*x)/(a**2*d*e 
*m**3 + 4*a**2*d*e*m**2*p + 3*a**2*d*e*m**2 + 4*a**2*d*e*m*p**2 + 6*a**2*d 
*e*m*p + 2*a**2*d*e*m + a**2*e**2*m**3*x + 4*a**2*e**2*m**2*p*x + 3*a**...