\(\int \frac {(a+b x+c x^2)^p}{(b d+2 c d x)^5} \, dx\) [337]

Optimal result

Integrand size = 24, antiderivative size = 63 \[ \int \frac {\left (a+b x+c x^2\right )^p}{(b d+2 c d x)^5} \, dx=\frac {\left (a+b x+c x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (3,1+p,2+p,1-\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{\left (b^2-4 a c\right )^3 d^5 (1+p)} \] Output:

(c*x^2+b*x+a)^(p+1)*hypergeom([3, p+1],[2+p],1-(2*c*x+b)^2/(-4*a*c+b^2))/( 
-4*a*c+b^2)^3/d^5/(p+1)
 

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+b x+c x^2\right )^p}{(b d+2 c d x)^5} \, dx=\frac {(a+x (b+c x))^{1+p} \operatorname {Hypergeometric2F1}\left (3,1+p,2+p,\frac {4 c (a+x (b+c x))}{-b^2+4 a c}\right )}{\left (b^2-4 a c\right )^3 d^5 (1+p)} \] Input:

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

Output:

((a + x*(b + c*x))^(1 + p)*Hypergeometric2F1[3, 1 + p, 2 + p, (4*c*(a + x* 
(b + c*x)))/(-b^2 + 4*a*c)])/((b^2 - 4*a*c)^3*d^5*(1 + p))
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.54, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1118, 243, 75}

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.

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

\(\Big \downarrow \) 1118

\(\displaystyle \frac {\int \frac {\left (\frac {(b d+2 c x d)^2}{4 c d^2}+\frac {1}{4} \left (4 a-\frac {b^2}{c}\right )\right )^p}{(b d+2 c x d)^5}d(b d+2 c x d)}{2 c d}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {\int \frac {\left (\frac {(b d+2 c x d)^2}{4 c d^2}+\frac {1}{4} \left (4 a-\frac {b^2}{c}\right )\right )^p}{(b d+2 c x d)^6}d(b d+2 c x d)^2}{4 c d}\)

\(\Big \downarrow \) 75

\(\displaystyle \frac {\left (\frac {1}{4} \left (4 a-\frac {b^2}{c}\right )+\frac {(b d+2 c d x)^2}{4 c d^2}\right )^{p+1} \operatorname {Hypergeometric2F1}\left (3,p+1,p+2,1-\frac {(b d+2 c x d)^2}{\left (b^2-4 a c\right ) d^2}\right )}{d^5 (p+1) \left (b^2-4 a c\right )^3}\)

Input:

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

Output:

(((4*a - b^2/c)/4 + (b*d + 2*c*d*x)^2/(4*c*d^2))^(1 + p)*Hypergeometric2F1 
[3, 1 + p, 2 + p, 1 - (b*d + 2*c*d*x)^2/((b^2 - 4*a*c)*d^2)])/((b^2 - 4*a* 
c)^3*d^5*(1 + p))
 

Defintions of rubi rules used

rule 75

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x 
)^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))*Hypergeometric2F1[-m, n + 1, n + 2, 1 + 
 d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (IntegerQ[m] 
 || GtQ[-d/(b*c), 0])
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 1118

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S 
ymbol] :> Simp[1/e   Subst[Int[x^m*(a - b^2/(4*c) + (c*x^2)/e^2)^p, x], x, 
d + e*x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[2*c*d - b*e, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

integrate((c*x^2+b*x+a)^p/(2*c*d*x+b*d)^5,x, algorithm="fricas")
 

Output:

integral((c*x^2 + b*x + a)^p/(32*c^5*d^5*x^5 + 80*b*c^4*d^5*x^4 + 80*b^2*c 
^3*d^5*x^3 + 40*b^3*c^2*d^5*x^2 + 10*b^4*c*d^5*x + b^5*d^5), x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

integrate((c*x^2+b*x+a)^p/(2*c*d*x+b*d)^5,x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((c*x^2+b*x+a)^p/(2*c*d*x+b*d)^5,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - (a + b*x + c*x**2)**p*a + 32*int(((a + b*x + c*x**2)**p*x**2)/(8*a**2* 
b**5*c + 80*a**2*b**4*c**2*x + 320*a**2*b**3*c**3*x**2 + 640*a**2*b**2*c** 
4*x**3 + 640*a**2*b*c**5*x**4 + 256*a**2*c**6*x**5 - a*b**7*p - 10*a*b**6* 
c*p*x + 8*a*b**6*c*x - 40*a*b**5*c**2*p*x**2 + 88*a*b**5*c**2*x**2 - 80*a* 
b**4*c**3*p*x**3 + 400*a*b**4*c**3*x**3 - 80*a*b**3*c**4*p*x**4 + 960*a*b* 
*3*c**4*x**4 - 32*a*b**2*c**5*p*x**5 + 1280*a*b**2*c**5*x**5 + 896*a*b*c** 
6*x**6 + 256*a*c**7*x**7 - b**8*p*x - 11*b**7*c*p*x**2 - 50*b**6*c**2*p*x* 
*3 - 120*b**5*c**3*p*x**4 - 160*b**4*c**4*p*x**5 - 112*b**3*c**5*p*x**6 - 
32*b**2*c**6*p*x**7),x)*a**2*b**4*c**3*p + 256*int(((a + b*x + c*x**2)**p* 
x**2)/(8*a**2*b**5*c + 80*a**2*b**4*c**2*x + 320*a**2*b**3*c**3*x**2 + 640 
*a**2*b**2*c**4*x**3 + 640*a**2*b*c**5*x**4 + 256*a**2*c**6*x**5 - a*b**7* 
p - 10*a*b**6*c*p*x + 8*a*b**6*c*x - 40*a*b**5*c**2*p*x**2 + 88*a*b**5*c** 
2*x**2 - 80*a*b**4*c**3*p*x**3 + 400*a*b**4*c**3*x**3 - 80*a*b**3*c**4*p*x 
**4 + 960*a*b**3*c**4*x**4 - 32*a*b**2*c**5*p*x**5 + 1280*a*b**2*c**5*x**5 
 + 896*a*b*c**6*x**6 + 256*a*c**7*x**7 - b**8*p*x - 11*b**7*c*p*x**2 - 50* 
b**6*c**2*p*x**3 - 120*b**5*c**3*p*x**4 - 160*b**4*c**4*p*x**5 - 112*b**3* 
c**5*p*x**6 - 32*b**2*c**6*p*x**7),x)*a**2*b**3*c**4*p*x + 768*int(((a + b 
*x + c*x**2)**p*x**2)/(8*a**2*b**5*c + 80*a**2*b**4*c**2*x + 320*a**2*b**3 
*c**3*x**2 + 640*a**2*b**2*c**4*x**3 + 640*a**2*b*c**5*x**4 + 256*a**2*c** 
6*x**5 - a*b**7*p - 10*a*b**6*c*p*x + 8*a*b**6*c*x - 40*a*b**5*c**2*p*x...