\(\int \frac {(A+B x) (c^2-d^2 x^2)^{5/2}}{(c+d x)^{11}} \, dx\) [39]

Optimal result

Integrand size = 29, antiderivative size = 210 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{5/2}}{(c+d x)^{11}} \, dx=\frac {(B c-A d) \left (c^2-d^2 x^2\right )^{7/2}}{15 c d^2 (c+d x)^{11}}-\frac {(11 B c+4 A d) \left (c^2-d^2 x^2\right )^{7/2}}{195 c^2 d^2 (c+d x)^{10}}-\frac {(11 B c+4 A d) \left (c^2-d^2 x^2\right )^{7/2}}{715 c^3 d^2 (c+d x)^9}-\frac {2 (11 B c+4 A d) \left (c^2-d^2 x^2\right )^{7/2}}{6435 c^4 d^2 (c+d x)^8}-\frac {2 (11 B c+4 A d) \left (c^2-d^2 x^2\right )^{7/2}}{45045 c^5 d^2 (c+d x)^7} \] Output:

1/15*(-A*d+B*c)*(-d^2*x^2+c^2)^(7/2)/c/d^2/(d*x+c)^11-1/195*(4*A*d+11*B*c) 
*(-d^2*x^2+c^2)^(7/2)/c^2/d^2/(d*x+c)^10-1/715*(4*A*d+11*B*c)*(-d^2*x^2+c^ 
2)^(7/2)/c^3/d^2/(d*x+c)^9-2/6435*(4*A*d+11*B*c)*(-d^2*x^2+c^2)^(7/2)/c^4/ 
d^2/(d*x+c)^8-2/45045*(4*A*d+11*B*c)*(-d^2*x^2+c^2)^(7/2)/c^5/d^2/(d*x+c)^ 
7
 

Mathematica [A] (verified)

Time = 2.12 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.62 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{5/2}}{(c+d x)^{11}} \, dx=-\frac {(c-d x)^3 \sqrt {c^2-d^2 x^2} \left (11 B c \left (37 c^4+407 c^3 d x+117 c^2 d^2 x^2+22 c d^3 x^3+2 d^4 x^4\right )+A d \left (4243 c^4+1628 c^3 d x+468 c^2 d^2 x^2+88 c d^3 x^3+8 d^4 x^4\right )\right )}{45045 c^5 d^2 (c+d x)^8} \] Input:

Integrate[((A + B*x)*(c^2 - d^2*x^2)^(5/2))/(c + d*x)^11,x]
 

Output:

-1/45045*((c - d*x)^3*Sqrt[c^2 - d^2*x^2]*(11*B*c*(37*c^4 + 407*c^3*d*x + 
117*c^2*d^2*x^2 + 22*c*d^3*x^3 + 2*d^4*x^4) + A*d*(4243*c^4 + 1628*c^3*d*x 
 + 468*c^2*d^2*x^2 + 88*c*d^3*x^3 + 8*d^4*x^4)))/(c^5*d^2*(c + d*x)^8)
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {671, 461, 461, 461, 460}

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.

\(\displaystyle \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{5/2}}{(c+d x)^{11}} \, dx\)

\(\Big \downarrow \) 671

\(\displaystyle \frac {(4 A d+11 B c) \int \frac {\left (c^2-d^2 x^2\right )^{5/2}}{(c+d x)^{10}}dx}{15 c d}+\frac {\left (c^2-d^2 x^2\right )^{7/2} (B c-A d)}{15 c d^2 (c+d x)^{11}}\)

\(\Big \downarrow \) 461

\(\displaystyle \frac {(4 A d+11 B c) \left (\frac {3 \int \frac {\left (c^2-d^2 x^2\right )^{5/2}}{(c+d x)^9}dx}{13 c}-\frac {\left (c^2-d^2 x^2\right )^{7/2}}{13 c d (c+d x)^{10}}\right )}{15 c d}+\frac {\left (c^2-d^2 x^2\right )^{7/2} (B c-A d)}{15 c d^2 (c+d x)^{11}}\)

\(\Big \downarrow \) 461

\(\displaystyle \frac {(4 A d+11 B c) \left (\frac {3 \left (\frac {2 \int \frac {\left (c^2-d^2 x^2\right )^{5/2}}{(c+d x)^8}dx}{11 c}-\frac {\left (c^2-d^2 x^2\right )^{7/2}}{11 c d (c+d x)^9}\right )}{13 c}-\frac {\left (c^2-d^2 x^2\right )^{7/2}}{13 c d (c+d x)^{10}}\right )}{15 c d}+\frac {\left (c^2-d^2 x^2\right )^{7/2} (B c-A d)}{15 c d^2 (c+d x)^{11}}\)

\(\Big \downarrow \) 461

\(\displaystyle \frac {(4 A d+11 B c) \left (\frac {3 \left (\frac {2 \left (\frac {\int \frac {\left (c^2-d^2 x^2\right )^{5/2}}{(c+d x)^7}dx}{9 c}-\frac {\left (c^2-d^2 x^2\right )^{7/2}}{9 c d (c+d x)^8}\right )}{11 c}-\frac {\left (c^2-d^2 x^2\right )^{7/2}}{11 c d (c+d x)^9}\right )}{13 c}-\frac {\left (c^2-d^2 x^2\right )^{7/2}}{13 c d (c+d x)^{10}}\right )}{15 c d}+\frac {\left (c^2-d^2 x^2\right )^{7/2} (B c-A d)}{15 c d^2 (c+d x)^{11}}\)

\(\Big \downarrow \) 460

\(\displaystyle \frac {\left (c^2-d^2 x^2\right )^{7/2} (B c-A d)}{15 c d^2 (c+d x)^{11}}+\frac {\left (\frac {3 \left (\frac {2 \left (-\frac {\left (c^2-d^2 x^2\right )^{7/2}}{63 c^2 d (c+d x)^7}-\frac {\left (c^2-d^2 x^2\right )^{7/2}}{9 c d (c+d x)^8}\right )}{11 c}-\frac {\left (c^2-d^2 x^2\right )^{7/2}}{11 c d (c+d x)^9}\right )}{13 c}-\frac {\left (c^2-d^2 x^2\right )^{7/2}}{13 c d (c+d x)^{10}}\right ) (4 A d+11 B c)}{15 c d}\)

Input:

Int[((A + B*x)*(c^2 - d^2*x^2)^(5/2))/(c + d*x)^11,x]
 

Output:

((B*c - A*d)*(c^2 - d^2*x^2)^(7/2))/(15*c*d^2*(c + d*x)^11) + ((11*B*c + 4 
*A*d)*(-1/13*(c^2 - d^2*x^2)^(7/2)/(c*d*(c + d*x)^10) + (3*(-1/11*(c^2 - d 
^2*x^2)^(7/2)/(c*d*(c + d*x)^9) + (2*(-1/9*(c^2 - d^2*x^2)^(7/2)/(c*d*(c + 
 d*x)^8) - (c^2 - d^2*x^2)^(7/2)/(63*c^2*d*(c + d*x)^7)))/(11*c)))/(13*c)) 
)/(15*c*d)
 

Defintions of rubi rules used

rule 460

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
(-d)*(c + d*x)^n*((a + b*x^2)^(p + 1)/(b*c*n)), x] /; FreeQ[{a, b, c, d, n, 
 p}, x] && EqQ[b*c^2 + a*d^2, 0] && EqQ[n + 2*p + 2, 0]
 

rule 461

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
(-d)*(c + d*x)^n*((a + b*x^2)^(p + 1)/(2*b*c*(n + p + 1))), x] + Simp[Simpl 
ify[n + 2*p + 2]/(2*c*(n + p + 1))   Int[(c + d*x)^(n + 1)*(a + b*x^2)^p, x 
], x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && ILtQ[Simp 
lify[n + 2*p + 2], 0] && (LtQ[n, -1] || GtQ[n + p, 0])
 

rule 671

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_ 
), x_Symbol] :> Simp[(d*g - e*f)*(d + e*x)^m*((a + c*x^2)^(p + 1)/(2*c*d*(m 
 + p + 1))), x] + Simp[(m*(g*c*d + c*e*f) + 2*e*c*f*(p + 1))/(e*(2*c*d)*(m 
+ p + 1))   Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, 
e, f, g, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && ((LtQ[m, -1] &&  !IGtQ[m + p 
 + 1, 0]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p 
 + 1, 0]
 

Maple [A] (verified)

Time = 2.73 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.63

method	result	size
gosper	\(-\frac {\left (-d x +c \right ) \left (8 A \,d^{5} x^{4}+22 B c \,d^{4} x^{4}+88 A c \,d^{4} x^{3}+242 B \,c^{2} d^{3} x^{3}+468 A \,c^{2} d^{3} x^{2}+1287 B \,c^{3} d^{2} x^{2}+1628 A \,c^{3} d^{2} x +4477 B \,c^{4} d x +4243 A \,c^{4} d +407 B \,c^{5}\right ) \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{45045 \left (d x +c \right )^{10} c^{5} d^{2}}\)	\(133\)
orering	\(-\frac {\left (-d x +c \right ) \left (8 A \,d^{5} x^{4}+22 B c \,d^{4} x^{4}+88 A c \,d^{4} x^{3}+242 B \,c^{2} d^{3} x^{3}+468 A \,c^{2} d^{3} x^{2}+1287 B \,c^{3} d^{2} x^{2}+1628 A \,c^{3} d^{2} x +4477 B \,c^{4} d x +4243 A \,c^{4} d +407 B \,c^{5}\right ) \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{45045 \left (d x +c \right )^{10} c^{5} d^{2}}\)	\(133\)
trager	\(-\frac {\left (-8 A \,d^{8} x^{7}-22 B c \,d^{7} x^{7}-64 A c \,d^{7} x^{6}-176 B \,c^{2} d^{6} x^{6}-228 A \,c^{2} d^{6} x^{5}-627 B \,c^{3} d^{5} x^{5}-480 A \,c^{3} d^{5} x^{4}-1320 B \,c^{4} d^{4} x^{4}-675 A \,c^{4} d^{4} x^{3}+9405 B \,c^{5} d^{3} x^{3}+8313 c^{5} x^{2} A \,d^{3}-10923 c^{6} x^{2} B \,d^{2}-11101 A \,c^{6} d^{2} x +3256 B \,c^{7} d x +4243 A \,c^{7} d +407 B \,c^{8}\right ) \sqrt {-d^{2} x^{2}+c^{2}}}{45045 c^{5} \left (d x +c \right )^{8} d^{2}}\)	\(199\)
default	\(\frac {B \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {7}{2}}}{13 c d \left (x +\frac {c}{d}\right )^{10}}+\frac {3 d \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {7}{2}}}{11 c d \left (x +\frac {c}{d}\right )^{9}}+\frac {2 d \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {7}{2}}}{9 c d \left (x +\frac {c}{d}\right )^{8}}-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {7}{2}}}{63 c^{2} \left (x +\frac {c}{d}\right )^{7}}\right )}{11 c}\right )}{13 c}\right )}{d^{11}}+\frac {\left (A d -B c \right ) \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {7}{2}}}{15 c d \left (x +\frac {c}{d}\right )^{11}}+\frac {4 d \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {7}{2}}}{13 c d \left (x +\frac {c}{d}\right )^{10}}+\frac {3 d \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {7}{2}}}{11 c d \left (x +\frac {c}{d}\right )^{9}}+\frac {2 d \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {7}{2}}}{9 c d \left (x +\frac {c}{d}\right )^{8}}-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {7}{2}}}{63 c^{2} \left (x +\frac {c}{d}\right )^{7}}\right )}{11 c}\right )}{13 c}\right )}{15 c}\right )}{d^{12}}\)	\(455\)

Input:

int((B*x+A)*(-d^2*x^2+c^2)^(5/2)/(d*x+c)^11,x,method=_RETURNVERBOSE)
 

Output:

-1/45045*(-d*x+c)*(8*A*d^5*x^4+22*B*c*d^4*x^4+88*A*c*d^4*x^3+242*B*c^2*d^3 
*x^3+468*A*c^2*d^3*x^2+1287*B*c^3*d^2*x^2+1628*A*c^3*d^2*x+4477*B*c^4*d*x+ 
4243*A*c^4*d+407*B*c^5)*(-d^2*x^2+c^2)^(5/2)/(d*x+c)^10/c^5/d^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 476 vs. \(2 (190) = 380\).

Time = 0.63 (sec) , antiderivative size = 476, normalized size of antiderivative = 2.27 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{5/2}}{(c+d x)^{11}} \, dx=-\frac {407 \, B c^{9} + 4243 \, A c^{8} d + {\left (407 \, B c d^{8} + 4243 \, A d^{9}\right )} x^{8} + 8 \, {\left (407 \, B c^{2} d^{7} + 4243 \, A c d^{8}\right )} x^{7} + 28 \, {\left (407 \, B c^{3} d^{6} + 4243 \, A c^{2} d^{7}\right )} x^{6} + 56 \, {\left (407 \, B c^{4} d^{5} + 4243 \, A c^{3} d^{6}\right )} x^{5} + 70 \, {\left (407 \, B c^{5} d^{4} + 4243 \, A c^{4} d^{5}\right )} x^{4} + 56 \, {\left (407 \, B c^{6} d^{3} + 4243 \, A c^{5} d^{4}\right )} x^{3} + 28 \, {\left (407 \, B c^{7} d^{2} + 4243 \, A c^{6} d^{3}\right )} x^{2} + 8 \, {\left (407 \, B c^{8} d + 4243 \, A c^{7} d^{2}\right )} x + {\left (407 \, B c^{8} + 4243 \, A c^{7} d - 2 \, {\left (11 \, B c d^{7} + 4 \, A d^{8}\right )} x^{7} - 16 \, {\left (11 \, B c^{2} d^{6} + 4 \, A c d^{7}\right )} x^{6} - 57 \, {\left (11 \, B c^{3} d^{5} + 4 \, A c^{2} d^{6}\right )} x^{5} - 120 \, {\left (11 \, B c^{4} d^{4} + 4 \, A c^{3} d^{5}\right )} x^{4} + 45 \, {\left (209 \, B c^{5} d^{3} - 15 \, A c^{4} d^{4}\right )} x^{3} - 3 \, {\left (3641 \, B c^{6} d^{2} - 2771 \, A c^{5} d^{3}\right )} x^{2} + {\left (3256 \, B c^{7} d - 11101 \, A c^{6} d^{2}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}}}{45045 \, {\left (c^{5} d^{10} x^{8} + 8 \, c^{6} d^{9} x^{7} + 28 \, c^{7} d^{8} x^{6} + 56 \, c^{8} d^{7} x^{5} + 70 \, c^{9} d^{6} x^{4} + 56 \, c^{10} d^{5} x^{3} + 28 \, c^{11} d^{4} x^{2} + 8 \, c^{12} d^{3} x + c^{13} d^{2}\right )}} \] Input:

integrate((B*x+A)*(-d^2*x^2+c^2)^(5/2)/(d*x+c)^11,x, algorithm="fricas")
 

Output:

-1/45045*(407*B*c^9 + 4243*A*c^8*d + (407*B*c*d^8 + 4243*A*d^9)*x^8 + 8*(4 
07*B*c^2*d^7 + 4243*A*c*d^8)*x^7 + 28*(407*B*c^3*d^6 + 4243*A*c^2*d^7)*x^6 
 + 56*(407*B*c^4*d^5 + 4243*A*c^3*d^6)*x^5 + 70*(407*B*c^5*d^4 + 4243*A*c^ 
4*d^5)*x^4 + 56*(407*B*c^6*d^3 + 4243*A*c^5*d^4)*x^3 + 28*(407*B*c^7*d^2 + 
 4243*A*c^6*d^3)*x^2 + 8*(407*B*c^8*d + 4243*A*c^7*d^2)*x + (407*B*c^8 + 4 
243*A*c^7*d - 2*(11*B*c*d^7 + 4*A*d^8)*x^7 - 16*(11*B*c^2*d^6 + 4*A*c*d^7) 
*x^6 - 57*(11*B*c^3*d^5 + 4*A*c^2*d^6)*x^5 - 120*(11*B*c^4*d^4 + 4*A*c^3*d 
^5)*x^4 + 45*(209*B*c^5*d^3 - 15*A*c^4*d^4)*x^3 - 3*(3641*B*c^6*d^2 - 2771 
*A*c^5*d^3)*x^2 + (3256*B*c^7*d - 11101*A*c^6*d^2)*x)*sqrt(-d^2*x^2 + c^2) 
)/(c^5*d^10*x^8 + 8*c^6*d^9*x^7 + 28*c^7*d^8*x^6 + 56*c^8*d^7*x^5 + 70*c^9 
*d^6*x^4 + 56*c^10*d^5*x^3 + 28*c^11*d^4*x^2 + 8*c^12*d^3*x + c^13*d^2)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{5/2}}{(c+d x)^{11}} \, dx=\text {Timed out} \] Input:

integrate((B*x+A)*(-d**2*x**2+c**2)**(5/2)/(d*x+c)**11,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2142 vs. \(2 (190) = 380\).

Time = 0.08 (sec) , antiderivative size = 2142, normalized size of antiderivative = 10.20 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{5/2}}{(c+d x)^{11}} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)*(-d^2*x^2+c^2)^(5/2)/(d*x+c)^11,x, algorithm="maxima")
 

Output:

1/5*(-d^2*x^2 + c^2)^(5/2)*B*c/(d^12*x^10 + 10*c*d^11*x^9 + 45*c^2*d^10*x^ 
8 + 120*c^3*d^9*x^7 + 210*c^4*d^8*x^6 + 252*c^5*d^7*x^5 + 210*c^6*d^6*x^4 
+ 120*c^7*d^5*x^3 + 45*c^8*d^4*x^2 + 10*c^9*d^3*x + c^10*d^2) - 1/6*(-d^2* 
x^2 + c^2)^(3/2)*B*c^2/(d^11*x^9 + 9*c*d^10*x^8 + 36*c^2*d^9*x^7 + 84*c^3* 
d^8*x^6 + 126*c^4*d^7*x^5 + 126*c^5*d^6*x^4 + 84*c^6*d^5*x^3 + 36*c^7*d^4* 
x^2 + 9*c^8*d^3*x + c^9*d^2) + 1/15*sqrt(-d^2*x^2 + c^2)*B*c^3/(d^10*x^8 + 
 8*c*d^9*x^7 + 28*c^2*d^8*x^6 + 56*c^3*d^7*x^5 + 70*c^4*d^6*x^4 + 56*c^5*d 
^5*x^3 + 28*c^6*d^4*x^2 + 8*c^7*d^3*x + c^8*d^2) - 1/5*(-d^2*x^2 + c^2)^(5 
/2)*A/(d^11*x^10 + 10*c*d^10*x^9 + 45*c^2*d^9*x^8 + 120*c^3*d^8*x^7 + 210* 
c^4*d^7*x^6 + 252*c^5*d^6*x^5 + 210*c^6*d^5*x^4 + 120*c^7*d^4*x^3 + 45*c^8 
*d^3*x^2 + 10*c^9*d^2*x + c^10*d) - 1/4*(-d^2*x^2 + c^2)^(5/2)*B/(d^11*x^9 
 + 9*c*d^10*x^8 + 36*c^2*d^9*x^7 + 84*c^3*d^8*x^6 + 126*c^4*d^7*x^5 + 126* 
c^5*d^6*x^4 + 84*c^6*d^5*x^3 + 36*c^7*d^4*x^2 + 9*c^8*d^3*x + c^9*d^2) + 1 
/6*(-d^2*x^2 + c^2)^(3/2)*A*c/(d^10*x^9 + 9*c*d^9*x^8 + 36*c^2*d^8*x^7 + 8 
4*c^3*d^7*x^6 + 126*c^4*d^6*x^5 + 126*c^5*d^5*x^4 + 84*c^6*d^4*x^3 + 36*c^ 
7*d^3*x^2 + 9*c^8*d^2*x + c^9*d) + 1/4*(-d^2*x^2 + c^2)^(3/2)*B*c/(d^10*x^ 
8 + 8*c*d^9*x^7 + 28*c^2*d^8*x^6 + 56*c^3*d^7*x^5 + 70*c^4*d^6*x^4 + 56*c^ 
5*d^5*x^3 + 28*c^6*d^4*x^2 + 8*c^7*d^3*x + c^8*d^2) - 1/15*sqrt(-d^2*x^2 + 
 c^2)*A*c^2/(d^9*x^8 + 8*c*d^8*x^7 + 28*c^2*d^7*x^6 + 56*c^3*d^6*x^5 + 70* 
c^4*d^5*x^4 + 56*c^5*d^4*x^3 + 28*c^6*d^3*x^2 + 8*c^7*d^2*x + c^8*d) - ...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 926 vs. \(2 (190) = 380\).

Time = 0.15 (sec) , antiderivative size = 926, normalized size of antiderivative = 4.41 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{5/2}}{(c+d x)^{11}} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)*(-d^2*x^2+c^2)^(5/2)/(d*x+c)^11,x, algorithm="giac")
 

Output:

2/45045*(407*B*c + 4243*A*d + 6105*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))*B*c 
/(d^2*x) + 18600*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))*A/(d*x) - 2310*(c*d + 
 sqrt(-d^2*x^2 + c^2)*abs(d))^2*B*c/(d^4*x^2) + 265335*(c*d + sqrt(-d^2*x^ 
2 + c^2)*abs(d))^2*A/(d^3*x^2) + 170170*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d) 
)^3*B*c/(d^6*x^3) + 864500*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^3*A/(d^5*x^ 
3) + 105105*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^4*B*c/(d^8*x^4) + 3088995* 
(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^4*A/(d^7*x^4) + 915915*(c*d + sqrt(-d^ 
2*x^2 + c^2)*abs(d))^5*B*c/(d^10*x^5) + 6066060*(c*d + sqrt(-d^2*x^2 + c^2 
)*abs(d))^5*A/(d^9*x^5) + 580580*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^6*B*c 
/(d^12*x^6) + 11026015*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^6*A/(d^11*x^6) 
+ 1827540*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^7*B*c/(d^14*x^7) + 13178880* 
(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^7*A/(d^13*x^7) + 791505*(c*d + sqrt(-d 
^2*x^2 + c^2)*abs(d))^8*B*c/(d^16*x^8) + 14124825*(c*d + sqrt(-d^2*x^2 + c 
^2)*abs(d))^8*A/(d^15*x^8) + 1456455*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^9 
*B*c/(d^18*x^9) + 10210200*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^9*A/(d^17*x 
^9) + 306306*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^10*B*c/(d^20*x^10) + 6675 
669*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^10*A/(d^19*x^10) + 450450*(c*d + s 
qrt(-d^2*x^2 + c^2)*abs(d))^11*B*c/(d^22*x^11) + 2702700*(c*d + sqrt(-d^2* 
x^2 + c^2)*abs(d))^11*A/(d^21*x^11) + 15015*(c*d + sqrt(-d^2*x^2 + c^2)*ab 
s(d))^12*B*c/(d^24*x^12) + 1066065*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^...
 

Mupad [B] (verification not implemented)

Time = 13.91 (sec) , antiderivative size = 1457, normalized size of antiderivative = 6.94 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{5/2}}{(c+d x)^{11}} \, dx =\text {Too large to display} \] Input:

int(((c^2 - d^2*x^2)^(5/2)*(A + B*x))/(c + d*x)^11,x)
 

Output:

(1138*B*(c^2 - d^2*x^2)^(1/2))/(96525*(c^5*d^2 + c^4*d^3*x)) - (463*A*(c^2 
 - d^2*x^2)^(1/2))/(225225*(c^6*d + 3*c^5*d^2*x + 3*c^4*d^3*x^2 + c^3*d^4* 
x^3)) - (142*A*(c^2 - d^2*x^2)^(1/2))/(715*(c^6*d + d^7*x^6 + 6*c^5*d^2*x 
+ 6*c*d^6*x^5 + 15*c^4*d^3*x^2 + 20*c^3*d^4*x^3 + 15*c^2*d^5*x^4)) + (A*(c 
^2 - d^2*x^2)^(1/2))/(1287*(c^6*d + 5*c^5*d^2*x + c*d^6*x^5 + 10*c^4*d^3*x 
^2 + 10*c^3*d^4*x^3 + 5*c^2*d^5*x^4)) + (2102*A*(c^2 - d^2*x^2)^(1/2))/(67 
5675*(c^6*d + 2*c^5*d^2*x + c^4*d^3*x^2)) + (58*B*(c^2 - d^2*x^2)^(1/2))/( 
20475*(c^5*d^2 + 3*c^4*d^3*x + 3*c^3*d^4*x^2 + c^2*d^5*x^3)) - (419*A*(c^2 
 - d^2*x^2)^(1/2))/(225225*(c^6*d + c^5*d^2*x)) - (139*B*(c^2 - d^2*x^2)^( 
1/2))/(20475*(c^5*d^2 + 2*c^4*d^3*x + c^3*d^4*x^2)) - (29*B*(c^2 - d^2*x^2 
)^(1/2))/(117*(c^5*d^2 + d^7*x^5 + 5*c^4*d^3*x + 5*c*d^6*x^4 + 10*c^3*d^4* 
x^2 + 10*c^2*d^5*x^3)) + (4*A*(c^2 - d^2*x^2)^(1/2))/(9009*(c^6*d + 4*c^5* 
d^2*x + 6*c^4*d^3*x^2 + 4*c^3*d^4*x^3 + c^2*d^5*x^4)) + (B*(c^2 - d^2*x^2) 
^(1/2))/(819*(c^5*d^2 + 4*c^4*d^3*x + c*d^6*x^4 + 6*c^3*d^4*x^2 + 4*c^2*d^ 
5*x^3)) + (62*B*c*(c^2 - d^2*x^2)^(1/2))/(65*(c^6*d^2 + d^8*x^6 + 6*c^5*d^ 
3*x + 6*c*d^7*x^5 + 15*c^4*d^4*x^2 + 20*c^3*d^5*x^3 + 15*c^2*d^6*x^4)) - ( 
244*B*c^2*(c^2 - d^2*x^2)^(1/2))/(195*(c^7*d^2 + d^9*x^7 + 7*c^6*d^3*x + 7 
*c*d^8*x^6 + 21*c^5*d^4*x^2 + 35*c^4*d^5*x^3 + 35*c^3*d^6*x^4 + 21*c^2*d^7 
*x^5)) - (1982*A*d*(c^2 - d^2*x^2)^(1/2))/(675675*(c^6*d^2 + 2*c^5*d^3*x + 
 c^4*d^4*x^2)) + (149*B*c*(c^2 - d^2*x^2)^(1/2))/(20475*(c^6*d^2 + 2*c^...
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 802, normalized size of antiderivative = 3.82 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{5/2}}{(c+d x)^{11}} \, dx=\frac {6006 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{7}-1771 a \,d^{8} x^{8}+407 b \,c^{8} x +1755 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,d^{7} x^{7}+407 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{7} x +1240 a \,c^{7} d x -68778 a \,c^{6} d^{2} x^{2}-89740 a \,c^{5} d^{3} x^{3}-123605 a \,c^{4} d^{4} x^{4}-98980 a \,c^{3} d^{5} x^{5}-49528 a \,c^{2} d^{6} x^{6}-14160 a c \,d^{7} x^{7}+25575 b \,c^{7} d \,x^{2}+2464 b \,c^{6} d^{2} x^{3}+39215 b \,c^{5} d^{3} x^{4}+22099 b \,c^{4} d^{4} x^{5}+10945 b \,c^{3} d^{5} x^{6}+3102 b \,c^{2} d^{6} x^{7}+385 b c \,d^{7} x^{8}+1240 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{6} d x +45336 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{5} d^{2} x^{2}+61030 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{4} d^{3} x^{3}+61225 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{3} d^{4} x^{4}+36795 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{2} d^{5} x^{5}+12277 \sqrt {-d^{2} x^{2}+c^{2}}\, a c \,d^{6} x^{6}-19470 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{6} d \,x^{2}-4840 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{5} d^{2} x^{3}-15565 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{4} d^{3} x^{4}-9174 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{3} d^{4} x^{5}-3025 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{2} d^{5} x^{6}-429 \sqrt {-d^{2} x^{2}+c^{2}}\, b c \,d^{6} x^{7}-6006 a \,c^{8}}{45045 c^{5} d \left (\sqrt {-d^{2} x^{2}+c^{2}}\, c^{7}+7 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{6} d x +21 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{5} d^{2} x^{2}+35 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{4} d^{3} x^{3}+35 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{3} d^{4} x^{4}+21 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{2} d^{5} x^{5}+7 \sqrt {-d^{2} x^{2}+c^{2}}\, c \,d^{6} x^{6}+\sqrt {-d^{2} x^{2}+c^{2}}\, d^{7} x^{7}-c^{8}-8 c^{7} d x -28 c^{6} d^{2} x^{2}-56 c^{5} d^{3} x^{3}-70 c^{4} d^{4} x^{4}-56 c^{3} d^{5} x^{5}-28 c^{2} d^{6} x^{6}-8 c \,d^{7} x^{7}-d^{8} x^{8}\right )} \] Input:

int((B*x+A)*(-d^2*x^2+c^2)^(5/2)/(d*x+c)^11,x)
 

Output:

(6006*sqrt(c**2 - d**2*x**2)*a*c**7 + 1240*sqrt(c**2 - d**2*x**2)*a*c**6*d 
*x + 45336*sqrt(c**2 - d**2*x**2)*a*c**5*d**2*x**2 + 61030*sqrt(c**2 - d** 
2*x**2)*a*c**4*d**3*x**3 + 61225*sqrt(c**2 - d**2*x**2)*a*c**3*d**4*x**4 + 
 36795*sqrt(c**2 - d**2*x**2)*a*c**2*d**5*x**5 + 12277*sqrt(c**2 - d**2*x* 
*2)*a*c*d**6*x**6 + 1755*sqrt(c**2 - d**2*x**2)*a*d**7*x**7 + 407*sqrt(c** 
2 - d**2*x**2)*b*c**7*x - 19470*sqrt(c**2 - d**2*x**2)*b*c**6*d*x**2 - 484 
0*sqrt(c**2 - d**2*x**2)*b*c**5*d**2*x**3 - 15565*sqrt(c**2 - d**2*x**2)*b 
*c**4*d**3*x**4 - 9174*sqrt(c**2 - d**2*x**2)*b*c**3*d**4*x**5 - 3025*sqrt 
(c**2 - d**2*x**2)*b*c**2*d**5*x**6 - 429*sqrt(c**2 - d**2*x**2)*b*c*d**6* 
x**7 - 6006*a*c**8 + 1240*a*c**7*d*x - 68778*a*c**6*d**2*x**2 - 89740*a*c* 
*5*d**3*x**3 - 123605*a*c**4*d**4*x**4 - 98980*a*c**3*d**5*x**5 - 49528*a* 
c**2*d**6*x**6 - 14160*a*c*d**7*x**7 - 1771*a*d**8*x**8 + 407*b*c**8*x + 2 
5575*b*c**7*d*x**2 + 2464*b*c**6*d**2*x**3 + 39215*b*c**5*d**3*x**4 + 2209 
9*b*c**4*d**4*x**5 + 10945*b*c**3*d**5*x**6 + 3102*b*c**2*d**6*x**7 + 385* 
b*c*d**7*x**8)/(45045*c**5*d*(sqrt(c**2 - d**2*x**2)*c**7 + 7*sqrt(c**2 - 
d**2*x**2)*c**6*d*x + 21*sqrt(c**2 - d**2*x**2)*c**5*d**2*x**2 + 35*sqrt(c 
**2 - d**2*x**2)*c**4*d**3*x**3 + 35*sqrt(c**2 - d**2*x**2)*c**3*d**4*x**4 
 + 21*sqrt(c**2 - d**2*x**2)*c**2*d**5*x**5 + 7*sqrt(c**2 - d**2*x**2)*c*d 
**6*x**6 + sqrt(c**2 - d**2*x**2)*d**7*x**7 - c**8 - 8*c**7*d*x - 28*c**6* 
d**2*x**2 - 56*c**5*d**3*x**3 - 70*c**4*d**4*x**4 - 56*c**3*d**5*x**5 -...