\(\int \frac {(c+d x)^n (A+B x+C x^2+D x^3)}{(a+b x)^{11/2}} \, dx\) [189]

Optimal result

Integrand size = 32, antiderivative size = 415 \[ \int \frac {(c+d x)^n \left (A+B x+C x^2+D x^3\right )}{(a+b x)^{11/2}} \, dx=-\frac {2 \left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) (c+d x)^{1+n}}{9 b^3 (b c-a d) (a+b x)^{9/2}}+\frac {2 (5 b c D+2 a d D (2-3 n)-b C d (3-2 n)) (c+d x)^{1+n}}{b^3 d^2 (3-2 n) (5-2 n) (a+b x)^{7/2}}-\frac {2 D (c+d x)^{1+n}}{b^3 d (3-2 n) (a+b x)^{5/2}}-\frac {2 \left (b^3 d (9 B c-A d (7-2 n)) (3-2 n)-4 a^3 d^2 D \left (6+5 n-n^2\right )+\frac {9 (b c-a d) (5 b c D+2 a d D (2-3 n)-b C d (3-2 n)) (7 b c-2 a d (1+n))}{d (5-2 n)}+2 a^2 b d \left (45 c D+C d \left (3+n-2 n^2\right )\right )-a b^2 \left (45 c^2 D+9 c C d (3-2 n)+2 B d^2 \left (3+n-2 n^2\right )\right )\right ) (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},-n,-\frac {5}{2},-\frac {d (a+b x)}{b c-a d}\right )}{63 b^4 d (b c-a d) (3-2 n) (a+b x)^{7/2}} \] Output:

-2/9*(A*b^3-a*(B*b^2-C*a*b+D*a^2))*(d*x+c)^(1+n)/b^3/(-a*d+b*c)/(b*x+a)^(9 
/2)+2*(5*D*b*c+2*a*d*D*(2-3*n)-b*C*d*(3-2*n))*(d*x+c)^(1+n)/b^3/d^2/(3-2*n 
)/(5-2*n)/(b*x+a)^(7/2)-2*D*(d*x+c)^(1+n)/b^3/d/(3-2*n)/(b*x+a)^(5/2)-2/63 
*(b^3*d*(9*B*c-A*d*(7-2*n))*(3-2*n)-4*a^3*d^2*D*(-n^2+5*n+6)+9*(-a*d+b*c)* 
(5*D*b*c+2*a*d*D*(2-3*n)-b*C*d*(3-2*n))*(7*b*c-2*a*d*(1+n))/d/(5-2*n)+2*a^ 
2*b*d*(45*D*c+C*d*(-2*n^2+n+3))-a*b^2*(45*D*c^2+9*c*C*d*(3-2*n)+2*B*d^2*(- 
2*n^2+n+3)))*(d*x+c)^n*hypergeom([-7/2, -n],[-5/2],-d*(b*x+a)/(-a*d+b*c))/ 
b^4/d/(-a*d+b*c)/(3-2*n)/(b*x+a)^(7/2)/((b*(d*x+c)/(-a*d+b*c))^n)
 

Mathematica [A] (verified)

Time = 3.34 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.56 \[ \int \frac {(c+d x)^n \left (A+B x+C x^2+D x^3\right )}{(a+b x)^{11/2}} \, dx=-\frac {2 (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \left (35 \left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) \operatorname {Hypergeometric2F1}\left (-\frac {9}{2},-n,-\frac {7}{2},\frac {d (a+b x)}{-b c+a d}\right )+3 (a+b x) \left (15 \left (b^2 B-2 a b C+3 a^2 D\right ) \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},-n,-\frac {5}{2},\frac {d (a+b x)}{-b c+a d}\right )-7 (a+b x) \left ((-3 b C+9 a D) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},-n,-\frac {3}{2},\frac {d (a+b x)}{-b c+a d}\right )-5 D (a+b x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-n,-\frac {1}{2},\frac {d (a+b x)}{-b c+a d}\right )\right )\right )\right )}{315 b^4 (a+b x)^{9/2}} \] Input:

Integrate[((c + d*x)^n*(A + B*x + C*x^2 + D*x^3))/(a + b*x)^(11/2),x]
 

Output:

(-2*(c + d*x)^n*(35*(A*b^3 - a*(b^2*B - a*b*C + a^2*D))*Hypergeometric2F1[ 
-9/2, -n, -7/2, (d*(a + b*x))/(-(b*c) + a*d)] + 3*(a + b*x)*(15*(b^2*B - 2 
*a*b*C + 3*a^2*D)*Hypergeometric2F1[-7/2, -n, -5/2, (d*(a + b*x))/(-(b*c) 
+ a*d)] - 7*(a + b*x)*((-3*b*C + 9*a*D)*Hypergeometric2F1[-5/2, -n, -3/2, 
(d*(a + b*x))/(-(b*c) + a*d)] - 5*D*(a + b*x)*Hypergeometric2F1[-3/2, -n, 
-1/2, (d*(a + b*x))/(-(b*c) + a*d)]))))/(315*b^4*(a + b*x)^(9/2)*((b*(c + 
d*x))/(b*c - a*d))^n)
 

Rubi [A] (warning: unable to verify)

Time = 1.00 (sec) , antiderivative size = 589, normalized size of antiderivative = 1.42, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {2124, 27, 1193, 27, 87, 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.

\(\displaystyle \int \frac {(c+d x)^n \left (A+B x+C x^2+D x^3\right )}{(a+b x)^{11/2}} \, dx\)

\(\Big \downarrow \) 2124

\(\displaystyle -\frac {2 \int -\frac {(c+d x)^n \left (9 \left (c-\frac {a d}{b}\right ) D x^2+\frac {9 (b c-a d) (b C-a D) x}{b^2}+\frac {-2 d D (n+1) a^3+b (9 c D+2 C d (n+1)) a^2-b^2 (9 c C+2 B d (n+1)) a+b^3 (9 B c-A d (7-2 n))}{b^3}\right )}{2 (a+b x)^{9/2}}dx}{9 (b c-a d)}-\frac {2 (c+d x)^{n+1} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{9 b^3 (a+b x)^{9/2} (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {(c+d x)^n \left (-\frac {2 d D (n+1) a^3}{b^3}+\frac {(9 c D+2 C d (n+1)) a^2}{b^2}-\frac {(9 c C+2 B d (n+1)) a}{b}+9 \left (c-\frac {a d}{b}\right ) D x^2+9 B c-A d (7-2 n)+\frac {9 (b c-a d) (b C-a D) x}{b^2}\right )}{(a+b x)^{9/2}}dx}{9 (b c-a d)}-\frac {2 (c+d x)^{n+1} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{9 b^3 (a+b x)^{9/2} (b c-a d)}\)

\(\Big \downarrow \) 1193

\(\displaystyle \frac {-\frac {2 \int \frac {(c+d x)^n \left (2 d^2 D \left (2 n^2+15 n+13\right ) a^3-b d \left (9 c D (6 n+13)+4 C d \left (n^2+3 n+2\right )\right ) a^2+2 b^2 \left (63 D c^2+18 C d (n+1) c-B d^2 \left (-2 n^2+3 n+5\right )\right ) a-b^3 \left (63 C c^2-9 B d (5-2 n) c+A d^2 \left (4 n^2-24 n+35\right )\right )-63 b (b c-a d)^2 D x\right )}{2 b^3 (a+b x)^{7/2}}dx}{7 (b c-a d)}-\frac {2 (c+d x)^{n+1} \left (-2 a^3 d D (n+10)+a^2 b (27 c D+C d (2 n+11))-2 a b^2 (B d (n+1)+9 c C)+b^3 (9 B c-A d (7-2 n))\right )}{7 b^3 (a+b x)^{7/2} (b c-a d)}}{9 (b c-a d)}-\frac {2 (c+d x)^{n+1} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{9 b^3 (a+b x)^{9/2} (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {\int \frac {(c+d x)^n \left (2 d^2 D \left (2 n^2+15 n+13\right ) a^3-b d \left (9 c D (6 n+13)+4 C d \left (n^2+3 n+2\right )\right ) a^2+2 b^2 \left (63 D c^2+18 C d (n+1) c-B d^2 \left (-2 n^2+3 n+5\right )\right ) a-b^3 \left (63 C c^2-9 B d (5-2 n) c+A d^2 \left (4 n^2-24 n+35\right )\right )-63 b (b c-a d)^2 D x\right )}{(a+b x)^{7/2}}dx}{7 b^3 (b c-a d)}-\frac {2 (c+d x)^{n+1} \left (-2 a^3 d D (n+10)+a^2 b (27 c D+C d (2 n+11))-2 a b^2 (B d (n+1)+9 c C)+b^3 (9 B c-A d (7-2 n))\right )}{7 b^3 (a+b x)^{7/2} (b c-a d)}}{9 (b c-a d)}-\frac {2 (c+d x)^{n+1} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{9 b^3 (a+b x)^{9/2} (b c-a d)}\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {-\frac {-\frac {\left (2 d \left (\frac {3}{2}-n\right ) \left (2 a^3 d^2 D \left (2 n^2+15 n+13\right )-a^2 b d \left (9 c D (6 n+13)+4 C d \left (n^2+3 n+2\right )\right )+2 a b^2 \left (-B d^2 \left (-2 n^2+3 n+5\right )+63 c^2 D+18 c C d (n+1)\right )-b^3 \left (A d^2 \left (4 n^2-24 n+35\right )-9 B c d (5-2 n)+63 c^2 C\right )\right )+126 D (b c-a d)^2 \left (\frac {5 b c}{2}-a d (n+1)\right )\right ) \int \frac {(c+d x)^n}{(a+b x)^{5/2}}dx}{5 (b c-a d)}-\frac {2 (c+d x)^{n+1} \left (a^3 d^2 D \left (4 n^2+30 n+89\right )-a^2 b d \left (27 c D (2 n+9)+4 C d \left (n^2+3 n+2\right )\right )+a b^2 \left (-2 B d^2 \left (-2 n^2+3 n+5\right )+189 c^2 D+36 c C d (n+1)\right )-b^3 \left (A d^2 \left (4 n^2-24 n+35\right )-9 B c d (5-2 n)+63 c^2 C\right )\right )}{5 (a+b x)^{5/2} (b c-a d)}}{7 b^3 (b c-a d)}-\frac {2 (c+d x)^{n+1} \left (-2 a^3 d D (n+10)+a^2 b (27 c D+C d (2 n+11))-2 a b^2 (B d (n+1)+9 c C)+b^3 (9 B c-A d (7-2 n))\right )}{7 b^3 (a+b x)^{7/2} (b c-a d)}}{9 (b c-a d)}-\frac {2 (c+d x)^{n+1} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{9 b^3 (a+b x)^{9/2} (b c-a d)}\)

\(\Big \downarrow \) 80

\(\displaystyle \frac {-\frac {-\frac {(c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \left (2 d \left (\frac {3}{2}-n\right ) \left (2 a^3 d^2 D \left (2 n^2+15 n+13\right )-a^2 b d \left (9 c D (6 n+13)+4 C d \left (n^2+3 n+2\right )\right )+2 a b^2 \left (-B d^2 \left (-2 n^2+3 n+5\right )+63 c^2 D+18 c C d (n+1)\right )-b^3 \left (A d^2 \left (4 n^2-24 n+35\right )-9 B c d (5-2 n)+63 c^2 C\right )\right )+126 D (b c-a d)^2 \left (\frac {5 b c}{2}-a d (n+1)\right )\right ) \int \frac {\left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^n}{(a+b x)^{5/2}}dx}{5 (b c-a d)}-\frac {2 (c+d x)^{n+1} \left (a^3 d^2 D \left (4 n^2+30 n+89\right )-a^2 b d \left (27 c D (2 n+9)+4 C d \left (n^2+3 n+2\right )\right )+a b^2 \left (-2 B d^2 \left (-2 n^2+3 n+5\right )+189 c^2 D+36 c C d (n+1)\right )-b^3 \left (A d^2 \left (4 n^2-24 n+35\right )-9 B c d (5-2 n)+63 c^2 C\right )\right )}{5 (a+b x)^{5/2} (b c-a d)}}{7 b^3 (b c-a d)}-\frac {2 (c+d x)^{n+1} \left (-2 a^3 d D (n+10)+a^2 b (27 c D+C d (2 n+11))-2 a b^2 (B d (n+1)+9 c C)+b^3 (9 B c-A d (7-2 n))\right )}{7 b^3 (a+b x)^{7/2} (b c-a d)}}{9 (b c-a d)}-\frac {2 (c+d x)^{n+1} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{9 b^3 (a+b x)^{9/2} (b c-a d)}\)

\(\Big \downarrow \) 79

\(\displaystyle \frac {-\frac {\frac {2 (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-n,-\frac {1}{2},-\frac {d (a+b x)}{b c-a d}\right ) \left (2 d \left (\frac {3}{2}-n\right ) \left (2 a^3 d^2 D \left (2 n^2+15 n+13\right )-a^2 b d \left (9 c D (6 n+13)+4 C d \left (n^2+3 n+2\right )\right )+2 a b^2 \left (-B d^2 \left (-2 n^2+3 n+5\right )+63 c^2 D+18 c C d (n+1)\right )-b^3 \left (A d^2 \left (4 n^2-24 n+35\right )-9 B c d (5-2 n)+63 c^2 C\right )\right )+126 D (b c-a d)^2 \left (\frac {5 b c}{2}-a d (n+1)\right )\right )}{15 b (a+b x)^{3/2} (b c-a d)}-\frac {2 (c+d x)^{n+1} \left (a^3 d^2 D \left (4 n^2+30 n+89\right )-a^2 b d \left (27 c D (2 n+9)+4 C d \left (n^2+3 n+2\right )\right )+a b^2 \left (-2 B d^2 \left (-2 n^2+3 n+5\right )+189 c^2 D+36 c C d (n+1)\right )-b^3 \left (A d^2 \left (4 n^2-24 n+35\right )-9 B c d (5-2 n)+63 c^2 C\right )\right )}{5 (a+b x)^{5/2} (b c-a d)}}{7 b^3 (b c-a d)}-\frac {2 (c+d x)^{n+1} \left (-2 a^3 d D (n+10)+a^2 b (27 c D+C d (2 n+11))-2 a b^2 (B d (n+1)+9 c C)+b^3 (9 B c-A d (7-2 n))\right )}{7 b^3 (a+b x)^{7/2} (b c-a d)}}{9 (b c-a d)}-\frac {2 (c+d x)^{n+1} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{9 b^3 (a+b x)^{9/2} (b c-a d)}\)

Input:

Int[((c + d*x)^n*(A + B*x + C*x^2 + D*x^3))/(a + b*x)^(11/2),x]
 

Output:

(-2*(A*b^3 - a*(b^2*B - a*b*C + a^2*D))*(c + d*x)^(1 + n))/(9*b^3*(b*c - a 
*d)*(a + b*x)^(9/2)) + ((-2*(b^3*(9*B*c - A*d*(7 - 2*n)) - 2*a^3*d*D*(10 + 
 n) - 2*a*b^2*(9*c*C + B*d*(1 + n)) + a^2*b*(27*c*D + C*d*(11 + 2*n)))*(c 
+ d*x)^(1 + n))/(7*b^3*(b*c - a*d)*(a + b*x)^(7/2)) - ((-2*(a^3*d^2*D*(89 
+ 30*n + 4*n^2) + a*b^2*(189*c^2*D + 36*c*C*d*(1 + n) - 2*B*d^2*(5 + 3*n - 
 2*n^2)) - a^2*b*d*(27*c*D*(9 + 2*n) + 4*C*d*(2 + 3*n + n^2)) - b^3*(63*c^ 
2*C - 9*B*c*d*(5 - 2*n) + A*d^2*(35 - 24*n + 4*n^2)))*(c + d*x)^(1 + n))/( 
5*(b*c - a*d)*(a + b*x)^(5/2)) + (2*(126*(b*c - a*d)^2*D*((5*b*c)/2 - a*d* 
(1 + n)) + 2*d*(3/2 - n)*(2*a^3*d^2*D*(13 + 15*n + 2*n^2) + 2*a*b^2*(63*c^ 
2*D + 18*c*C*d*(1 + n) - B*d^2*(5 + 3*n - 2*n^2)) - a^2*b*d*(9*c*D*(13 + 6 
*n) + 4*C*d*(2 + 3*n + n^2)) - b^3*(63*c^2*C - 9*B*c*d*(5 - 2*n) + A*d^2*( 
35 - 24*n + 4*n^2))))*(c + d*x)^n*Hypergeometric2F1[-3/2, -n, -1/2, -((d*( 
a + b*x))/(b*c - a*d))])/(15*b*(b*c - a*d)*(a + b*x)^(3/2)*((b*(c + d*x))/ 
(b*c - a*d))^n))/(7*b^3*(b*c - a*d)))/(9*(b*c - a*d))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

rule 1193

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) 
 + (c_.)*(x_)^2)^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x 
 + c*x^2)^p, d + e*x, x], R = PolynomialRemainder[(a + b*x + c*x^2)^p, d + 
e*x, x]}, Simp[R*(d + e*x)^(m + 1)*((f + g*x)^(n + 1)/((m + 1)*(e*f - d*g)) 
), x] + Simp[1/((m + 1)*(e*f - d*g))   Int[(d + e*x)^(m + 1)*(f + g*x)^n*Ex 
pandToSum[(m + 1)*(e*f - d*g)*Qx - g*R*(m + n + 2), x], x], x]] /; FreeQ[{a 
, b, c, d, e, f, g, n}, x] && IGtQ[p, 0] && ILtQ[2*m, -2] &&  !IntegerQ[n] 
&&  !(EqQ[m, -2] && EqQ[p, 1] && EqQ[2*c*d - b*e, 0])
 

rule 2124

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] : 
> With[{Qx = PolynomialQuotient[Px, a + b*x, x], R = PolynomialRemainder[Px 
, a + b*x, x]}, Simp[R*(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((m + 1)*(b*c - 
 a*d))), x] + Simp[1/((m + 1)*(b*c - a*d))   Int[(a + b*x)^(m + 1)*(c + d*x 
)^n*ExpandToSum[(m + 1)*(b*c - a*d)*Qx - d*R*(m + n + 2), x], x], x]] /; Fr 
eeQ[{a, b, c, d, n}, x] && PolyQ[Px, x] && LtQ[m, -1] && (IntegerQ[m] ||  ! 
ILtQ[n, -1])
 

Maple [F]

\[\int \frac {\left (x d +c \right )^{n} \left (D x^{3}+C \,x^{2}+B x +A \right )}{\left (b x +a \right )^{\frac {11}{2}}}d x\]

Input:

int((d*x+c)^n*(D*x^3+C*x^2+B*x+A)/(b*x+a)^(11/2),x)
 

Output:

int((d*x+c)^n*(D*x^3+C*x^2+B*x+A)/(b*x+a)^(11/2),x)
 

Fricas [F]

\[ \int \frac {(c+d x)^n \left (A+B x+C x^2+D x^3\right )}{(a+b x)^{11/2}} \, dx=\int { \frac {{\left (D x^{3} + C x^{2} + B x + A\right )} {\left (d x + c\right )}^{n}}{{\left (b x + a\right )}^{\frac {11}{2}}} \,d x } \] Input:

integrate((d*x+c)^n*(D*x^3+C*x^2+B*x+A)/(b*x+a)^(11/2),x, algorithm="frica 
s")
 

Output:

integral((D*x^3 + C*x^2 + B*x + A)*sqrt(b*x + a)*(d*x + c)^n/(b^6*x^6 + 6* 
a*b^5*x^5 + 15*a^2*b^4*x^4 + 20*a^3*b^3*x^3 + 15*a^4*b^2*x^2 + 6*a^5*b*x + 
 a^6), x)
 

Sympy [F(-2)]

Exception generated. \[ \int \frac {(c+d x)^n \left (A+B x+C x^2+D x^3\right )}{(a+b x)^{11/2}} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

integrate((d*x+c)**n*(D*x**3+C*x**2+B*x+A)/(b*x+a)**(11/2),x)
 

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

\[ \int \frac {(c+d x)^n \left (A+B x+C x^2+D x^3\right )}{(a+b x)^{11/2}} \, dx=\int { \frac {{\left (D x^{3} + C x^{2} + B x + A\right )} {\left (d x + c\right )}^{n}}{{\left (b x + a\right )}^{\frac {11}{2}}} \,d x } \] Input:

integrate((d*x+c)^n*(D*x^3+C*x^2+B*x+A)/(b*x+a)^(11/2),x, algorithm="maxim 
a")
 

Output:

integrate((D*x^3 + C*x^2 + B*x + A)*(d*x + c)^n/(b*x + a)^(11/2), x)
 

Giac [F]

\[ \int \frac {(c+d x)^n \left (A+B x+C x^2+D x^3\right )}{(a+b x)^{11/2}} \, dx=\int { \frac {{\left (D x^{3} + C x^{2} + B x + A\right )} {\left (d x + c\right )}^{n}}{{\left (b x + a\right )}^{\frac {11}{2}}} \,d x } \] Input:

integrate((d*x+c)^n*(D*x^3+C*x^2+B*x+A)/(b*x+a)^(11/2),x, algorithm="giac" 
)
 

Output:

integrate((D*x^3 + C*x^2 + B*x + A)*(d*x + c)^n/(b*x + a)^(11/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^n \left (A+B x+C x^2+D x^3\right )}{(a+b x)^{11/2}} \, dx=\int \frac {{\left (c+d\,x\right )}^n\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (a+b\,x\right )}^{11/2}} \,d x \] Input:

int(((c + d*x)^n*(A + B*x + C*x^2 + x^3*D))/(a + b*x)^(11/2),x)
 

Output:

int(((c + d*x)^n*(A + B*x + C*x^2 + x^3*D))/(a + b*x)^(11/2), x)
 

Reduce [F]

\[ \int \frac {(c+d x)^n \left (A+B x+C x^2+D x^3\right )}{(a+b x)^{11/2}} \, dx=\int \frac {\left (d x +c \right )^{n} \left (D x^{3}+C \,x^{2}+B x +A \right )}{\left (b x +a \right )^{\frac {11}{2}}}d x \] Input:

int((d*x+c)^n*(D*x^3+C*x^2+B*x+A)/(b*x+a)^(11/2),x)
                                                                                    
                                                                                    
 

Output:

int((d*x+c)^n*(D*x^3+C*x^2+B*x+A)/(b*x+a)^(11/2),x)