\(\int \frac {(A+B x) (c^2-d^2 x^2)^{3/2}}{(c+d x)^9} \, dx\) [24]

Optimal result

Integrand size = 29, antiderivative size = 210 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^9} \, dx=\frac {(B c-A d) \left (c^2-d^2 x^2\right )^{5/2}}{13 c d^2 (c+d x)^9}-\frac {(9 B c+4 A d) \left (c^2-d^2 x^2\right )^{5/2}}{143 c^2 d^2 (c+d x)^8}-\frac {(9 B c+4 A d) \left (c^2-d^2 x^2\right )^{5/2}}{429 c^3 d^2 (c+d x)^7}-\frac {2 (9 B c+4 A d) \left (c^2-d^2 x^2\right )^{5/2}}{3003 c^4 d^2 (c+d x)^6}-\frac {2 (9 B c+4 A d) \left (c^2-d^2 x^2\right )^{5/2}}{15015 c^5 d^2 (c+d x)^5} \] Output:

1/13*(-A*d+B*c)*(-d^2*x^2+c^2)^(5/2)/c/d^2/(d*x+c)^9-1/143*(4*A*d+9*B*c)*( 
-d^2*x^2+c^2)^(5/2)/c^2/d^2/(d*x+c)^8-1/429*(4*A*d+9*B*c)*(-d^2*x^2+c^2)^( 
5/2)/c^3/d^2/(d*x+c)^7-2/3003*(4*A*d+9*B*c)*(-d^2*x^2+c^2)^(5/2)/c^4/d^2/( 
d*x+c)^6-2/15015*(4*A*d+9*B*c)*(-d^2*x^2+c^2)^(5/2)/c^5/d^2/(d*x+c)^5
 

Mathematica [A] (verified)

Time = 1.57 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.62 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^9} \, dx=-\frac {(c-d x)^2 \sqrt {c^2-d^2 x^2} \left (3 B c \left (71 c^4+639 c^3 d x+231 c^2 d^2 x^2+54 c d^3 x^3+6 d^4 x^4\right )+A d \left (1763 c^4+852 c^3 d x+308 c^2 d^2 x^2+72 c d^3 x^3+8 d^4 x^4\right )\right )}{15015 c^5 d^2 (c+d x)^7} \] Input:

Integrate[((A + B*x)*(c^2 - d^2*x^2)^(3/2))/(c + d*x)^9,x]
 

Output:

-1/15015*((c - d*x)^2*Sqrt[c^2 - d^2*x^2]*(3*B*c*(71*c^4 + 639*c^3*d*x + 2 
31*c^2*d^2*x^2 + 54*c*d^3*x^3 + 6*d^4*x^4) + A*d*(1763*c^4 + 852*c^3*d*x + 
 308*c^2*d^2*x^2 + 72*c*d^3*x^3 + 8*d^4*x^4)))/(c^5*d^2*(c + d*x)^7)
 

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.

\(\displaystyle \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^9} \, dx\)

\(\Big \downarrow \) 671

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

\(\Big \downarrow \) 461

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

\(\Big \downarrow \) 461

\(\displaystyle \frac {(4 A d+9 B c) \left (\frac {3 \left (\frac {2 \int \frac {\left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^6}dx}{9 c}-\frac {\left (c^2-d^2 x^2\right )^{5/2}}{9 c d (c+d x)^7}\right )}{11 c}-\frac {\left (c^2-d^2 x^2\right )^{5/2}}{11 c d (c+d x)^8}\right )}{13 c d}+\frac {\left (c^2-d^2 x^2\right )^{5/2} (B c-A d)}{13 c d^2 (c+d x)^9}\)

\(\Big \downarrow \) 461

\(\displaystyle \frac {(4 A d+9 B c) \left (\frac {3 \left (\frac {2 \left (\frac {\int \frac {\left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^5}dx}{7 c}-\frac {\left (c^2-d^2 x^2\right )^{5/2}}{7 c d (c+d x)^6}\right )}{9 c}-\frac {\left (c^2-d^2 x^2\right )^{5/2}}{9 c d (c+d x)^7}\right )}{11 c}-\frac {\left (c^2-d^2 x^2\right )^{5/2}}{11 c d (c+d x)^8}\right )}{13 c d}+\frac {\left (c^2-d^2 x^2\right )^{5/2} (B c-A d)}{13 c d^2 (c+d x)^9}\)

\(\Big \downarrow \) 460

\(\displaystyle \frac {\left (c^2-d^2 x^2\right )^{5/2} (B c-A d)}{13 c d^2 (c+d x)^9}+\frac {\left (\frac {3 \left (\frac {2 \left (-\frac {\left (c^2-d^2 x^2\right )^{5/2}}{35 c^2 d (c+d x)^5}-\frac {\left (c^2-d^2 x^2\right )^{5/2}}{7 c d (c+d x)^6}\right )}{9 c}-\frac {\left (c^2-d^2 x^2\right )^{5/2}}{9 c d (c+d x)^7}\right )}{11 c}-\frac {\left (c^2-d^2 x^2\right )^{5/2}}{11 c d (c+d x)^8}\right ) (4 A d+9 B c)}{13 c d}\)

Input:

Int[((A + B*x)*(c^2 - d^2*x^2)^(3/2))/(c + d*x)^9,x]
 

Output:

((B*c - A*d)*(c^2 - d^2*x^2)^(5/2))/(13*c*d^2*(c + d*x)^9) + ((9*B*c + 4*A 
*d)*(-1/11*(c^2 - d^2*x^2)^(5/2)/(c*d*(c + d*x)^8) + (3*(-1/9*(c^2 - d^2*x 
^2)^(5/2)/(c*d*(c + d*x)^7) + (2*(-1/7*(c^2 - d^2*x^2)^(5/2)/(c*d*(c + d*x 
)^6) - (c^2 - d^2*x^2)^(5/2)/(35*c^2*d*(c + d*x)^5)))/(9*c)))/(11*c)))/(13 
*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 = 0.74 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.63

method	result	size
gosper	\(-\frac {\left (-d x +c \right ) \left (8 A \,d^{5} x^{4}+18 B c \,d^{4} x^{4}+72 A c \,d^{4} x^{3}+162 B \,c^{2} d^{3} x^{3}+308 A \,c^{2} d^{3} x^{2}+693 B \,c^{3} d^{2} x^{2}+852 A \,c^{3} d^{2} x +1917 B \,c^{4} d x +1763 A \,c^{4} d +213 B \,c^{5}\right ) \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{15015 \left (d x +c \right )^{8} c^{5} d^{2}}\)	\(133\)
orering	\(-\frac {\left (-d x +c \right ) \left (8 A \,d^{5} x^{4}+18 B c \,d^{4} x^{4}+72 A c \,d^{4} x^{3}+162 B \,c^{2} d^{3} x^{3}+308 A \,c^{2} d^{3} x^{2}+693 B \,c^{3} d^{2} x^{2}+852 A \,c^{3} d^{2} x +1917 B \,c^{4} d x +1763 A \,c^{4} d +213 B \,c^{5}\right ) \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{15015 \left (d x +c \right )^{8} c^{5} d^{2}}\)	\(133\)
trager	\(-\frac {\left (8 A \,d^{7} x^{6}+18 B c \,d^{6} x^{6}+56 A c \,d^{6} x^{5}+126 B \,c^{2} d^{5} x^{5}+172 A \,c^{2} d^{5} x^{4}+387 B \,c^{3} d^{4} x^{4}+308 A \,c^{3} d^{4} x^{3}+693 B \,c^{4} d^{3} x^{3}+367 c^{4} x^{2} A \,d^{3}-2928 c^{5} x^{2} B \,d^{2}-2674 A \,c^{5} d^{2} x +1491 B \,c^{6} d x +1763 A \,c^{6} d +213 B \,c^{7}\right ) \sqrt {-d^{2} x^{2}+c^{2}}}{15015 c^{5} \left (d x +c \right )^{7} d^{2}}\)	\(175\)
default	\(\frac {B \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {5}{2}}}{11 c d \left (x +\frac {c}{d}\right )^{8}}+\frac {3 d \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {5}{2}}}{9 c d \left (x +\frac {c}{d}\right )^{7}}+\frac {2 d \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {5}{2}}}{7 c d \left (x +\frac {c}{d}\right )^{6}}-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {5}{2}}}{35 c^{2} \left (x +\frac {c}{d}\right )^{5}}\right )}{9 c}\right )}{11 c}\right )}{d^{9}}+\frac {\left (A d -B c \right ) \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {5}{2}}}{13 c d \left (x +\frac {c}{d}\right )^{9}}+\frac {4 d \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {5}{2}}}{11 c d \left (x +\frac {c}{d}\right )^{8}}+\frac {3 d \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {5}{2}}}{9 c d \left (x +\frac {c}{d}\right )^{7}}+\frac {2 d \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {5}{2}}}{7 c d \left (x +\frac {c}{d}\right )^{6}}-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {5}{2}}}{35 c^{2} \left (x +\frac {c}{d}\right )^{5}}\right )}{9 c}\right )}{11 c}\right )}{13 c}\right )}{d^{10}}\)	\(455\)

Input:

int((B*x+A)*(-d^2*x^2+c^2)^(3/2)/(d*x+c)^9,x,method=_RETURNVERBOSE)
 

Output:

-1/15015*(-d*x+c)*(8*A*d^5*x^4+18*B*c*d^4*x^4+72*A*c*d^4*x^3+162*B*c^2*d^3 
*x^3+308*A*c^2*d^3*x^2+693*B*c^3*d^2*x^2+852*A*c^3*d^2*x+1917*B*c^4*d*x+17 
63*A*c^4*d+213*B*c^5)*(-d^2*x^2+c^2)^(3/2)/(d*x+c)^8/c^5/d^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 418 vs. \(2 (190) = 380\).

Time = 0.29 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.99 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^9} \, dx=-\frac {213 \, B c^{8} + 1763 \, A c^{7} d + {\left (213 \, B c d^{7} + 1763 \, A d^{8}\right )} x^{7} + 7 \, {\left (213 \, B c^{2} d^{6} + 1763 \, A c d^{7}\right )} x^{6} + 21 \, {\left (213 \, B c^{3} d^{5} + 1763 \, A c^{2} d^{6}\right )} x^{5} + 35 \, {\left (213 \, B c^{4} d^{4} + 1763 \, A c^{3} d^{5}\right )} x^{4} + 35 \, {\left (213 \, B c^{5} d^{3} + 1763 \, A c^{4} d^{4}\right )} x^{3} + 21 \, {\left (213 \, B c^{6} d^{2} + 1763 \, A c^{5} d^{3}\right )} x^{2} + 7 \, {\left (213 \, B c^{7} d + 1763 \, A c^{6} d^{2}\right )} x + {\left (213 \, B c^{7} + 1763 \, A c^{6} d + 2 \, {\left (9 \, B c d^{6} + 4 \, A d^{7}\right )} x^{6} + 14 \, {\left (9 \, B c^{2} d^{5} + 4 \, A c d^{6}\right )} x^{5} + 43 \, {\left (9 \, B c^{3} d^{4} + 4 \, A c^{2} d^{5}\right )} x^{4} + 77 \, {\left (9 \, B c^{4} d^{3} + 4 \, A c^{3} d^{4}\right )} x^{3} - {\left (2928 \, B c^{5} d^{2} - 367 \, A c^{4} d^{3}\right )} x^{2} + 7 \, {\left (213 \, B c^{6} d - 382 \, A c^{5} d^{2}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}}}{15015 \, {\left (c^{5} d^{9} x^{7} + 7 \, c^{6} d^{8} x^{6} + 21 \, c^{7} d^{7} x^{5} + 35 \, c^{8} d^{6} x^{4} + 35 \, c^{9} d^{5} x^{3} + 21 \, c^{10} d^{4} x^{2} + 7 \, c^{11} d^{3} x + c^{12} d^{2}\right )}} \] Input:

integrate((B*x+A)*(-d^2*x^2+c^2)^(3/2)/(d*x+c)^9,x, algorithm="fricas")
 

Output:

-1/15015*(213*B*c^8 + 1763*A*c^7*d + (213*B*c*d^7 + 1763*A*d^8)*x^7 + 7*(2 
13*B*c^2*d^6 + 1763*A*c*d^7)*x^6 + 21*(213*B*c^3*d^5 + 1763*A*c^2*d^6)*x^5 
 + 35*(213*B*c^4*d^4 + 1763*A*c^3*d^5)*x^4 + 35*(213*B*c^5*d^3 + 1763*A*c^ 
4*d^4)*x^3 + 21*(213*B*c^6*d^2 + 1763*A*c^5*d^3)*x^2 + 7*(213*B*c^7*d + 17 
63*A*c^6*d^2)*x + (213*B*c^7 + 1763*A*c^6*d + 2*(9*B*c*d^6 + 4*A*d^7)*x^6 
+ 14*(9*B*c^2*d^5 + 4*A*c*d^6)*x^5 + 43*(9*B*c^3*d^4 + 4*A*c^2*d^5)*x^4 + 
77*(9*B*c^4*d^3 + 4*A*c^3*d^4)*x^3 - (2928*B*c^5*d^2 - 367*A*c^4*d^3)*x^2 
+ 7*(213*B*c^6*d - 382*A*c^5*d^2)*x)*sqrt(-d^2*x^2 + c^2))/(c^5*d^9*x^7 + 
7*c^6*d^8*x^6 + 21*c^7*d^7*x^5 + 35*c^8*d^6*x^4 + 35*c^9*d^5*x^3 + 21*c^10 
*d^4*x^2 + 7*c^11*d^3*x + c^12*d^2)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^9} \, dx=\text {Timed out} \] Input:

integrate((B*x+A)*(-d**2*x**2+c**2)**(3/2)/(d*x+c)**9,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1512 vs. \(2 (190) = 380\).

Time = 0.07 (sec) , antiderivative size = 1512, normalized size of antiderivative = 7.20 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^9} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)*(-d^2*x^2+c^2)^(3/2)/(d*x+c)^9,x, algorithm="maxima")
 

Output:

1/5*(-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) - 6/65*sqrt(-d^2*x^2 + c^2)*B*c^2/(d^9*x^7 + 7*c*d^8*x^6 
+ 21*c^2*d^7*x^5 + 35*c^3*d^6*x^4 + 35*c^4*d^5*x^3 + 21*c^5*d^4*x^2 + 7*c^ 
6*d^3*x + c^7*d^2) - 1/5*(-d^2*x^2 + c^2)^(3/2)*A/(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) - 1/4*(-d^2*x^2 + c^2)^(3/2)*B/(d^9*x^7 + 
 7*c*d^8*x^6 + 21*c^2*d^7*x^5 + 35*c^3*d^6*x^4 + 35*c^4*d^5*x^3 + 21*c^5*d 
^4*x^2 + 7*c^6*d^3*x + c^7*d^2) + 6/65*sqrt(-d^2*x^2 + c^2)*A*c/(d^8*x^7 + 
 7*c*d^7*x^6 + 21*c^2*d^6*x^5 + 35*c^3*d^5*x^4 + 35*c^4*d^4*x^3 + 21*c^5*d 
^3*x^2 + 7*c^6*d^2*x + c^7*d) + 201/1430*sqrt(-d^2*x^2 + c^2)*B*c/(d^8*x^6 
 + 6*c*d^7*x^5 + 15*c^2*d^6*x^4 + 20*c^3*d^5*x^3 + 15*c^4*d^4*x^2 + 6*c^5* 
d^3*x + c^6*d^2) + 1/429*sqrt(-d^2*x^2 + c^2)*B*c/(c*d^7*x^5 + 5*c^2*d^6*x 
^4 + 10*c^3*d^5*x^3 + 10*c^4*d^4*x^2 + 5*c^5*d^3*x + c^6*d^2) + 4/3003*sqr 
t(-d^2*x^2 + c^2)*B*c/(c^2*d^6*x^4 + 4*c^3*d^5*x^3 + 6*c^4*d^4*x^2 + 4*c^5 
*d^3*x + c^6*d^2) + 4/5005*sqrt(-d^2*x^2 + c^2)*B*c/(c^3*d^5*x^3 + 3*c^4*d 
^4*x^2 + 3*c^5*d^3*x + c^6*d^2) + 8/15015*sqrt(-d^2*x^2 + c^2)*B*c/(c^4*d^ 
4*x^2 + 2*c^5*d^3*x + c^6*d^2) + 8/15015*sqrt(-d^2*x^2 + c^2)*B*c/(c^5*d^3 
*x + c^6*d^2) - 3/715*sqrt(-d^2*x^2 + c^2)*A/(d^7*x^6 + 6*c*d^6*x^5 + 15*c 
^2*d^5*x^4 + 20*c^3*d^4*x^3 + 15*c^4*d^3*x^2 + 6*c^5*d^2*x + c^6*d) - 1...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 796 vs. \(2 (190) = 380\).

Time = 0.15 (sec) , antiderivative size = 796, normalized size of antiderivative = 3.79 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^9} \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)*(-d^2*x^2+c^2)^(3/2)/(d*x+c)^9,x, algorithm="giac")
 

Output:

2/15015*(213*B*c + 1763*A*d + 2769*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))*B*c 
/(d^2*x) + 7904*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))*A/(d*x) + 1599*(c*d + 
sqrt(-d^2*x^2 + c^2)*abs(d))^2*B*c/(d^4*x^2) + 77454*(c*d + sqrt(-d^2*x^2 
+ c^2)*abs(d))^2*A/(d^3*x^2) + 45903*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^3 
*B*c/(d^6*x^3) + 233948*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^3*A/(d^5*x^3) 
+ 47190*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^4*B*c/(d^8*x^4) + 659945*(c*d 
+ sqrt(-d^2*x^2 + c^2)*abs(d))^4*A/(d^7*x^4) + 181038*(c*d + sqrt(-d^2*x^2 
 + c^2)*abs(d))^5*B*c/(d^10*x^5) + 1094808*(c*d + sqrt(-d^2*x^2 + c^2)*abs 
(d))^5*A/(d^9*x^5) + 131274*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^6*B*c/(d^1 
2*x^6) + 1559844*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^6*A/(d^11*x^6) + 2342 
34*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^7*B*c/(d^14*x^7) + 1465464*(c*d + s 
qrt(-d^2*x^2 + c^2)*abs(d))^7*A/(d^13*x^7) + 93093*(c*d + sqrt(-d^2*x^2 + 
c^2)*abs(d))^8*B*c/(d^16*x^8) + 1174173*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d) 
)^8*A/(d^15*x^8) + 105105*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^9*B*c/(d^18* 
x^9) + 600600*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^9*A/(d^17*x^9) + 15015*( 
c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^10*B*c/(d^20*x^10) + 270270*(c*d + sqrt 
(-d^2*x^2 + c^2)*abs(d))^10*A/(d^19*x^10) + 15015*(c*d + sqrt(-d^2*x^2 + c 
^2)*abs(d))^11*B*c/(d^22*x^11) + 60060*(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)*abs(d))^12*A/(d^23*x 
^12))/(c^5*d*((c*d + sqrt(-d^2*x^2 + c^2)*abs(d))/(d^2*x) + 1)^13*abs(d...
 

Mupad [B] (verification not implemented)

Time = 11.97 (sec) , antiderivative size = 1237, normalized size of antiderivative = 5.89 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^9} \, dx =\text {Too large to display} \] Input:

int(((c^2 - d^2*x^2)^(3/2)*(A + B*x))/(c + d*x)^9,x)
 

Output:

(838*B*(c^2 - d^2*x^2)^(1/2))/(135135*(c^5*d^2 + c^4*d^3*x)) - (53*A*(c^2 
- d^2*x^2)^(1/2))/(15015*(c^6*d + 3*c^5*d^2*x + 3*c^4*d^3*x^2 + c^3*d^4*x^ 
3)) + (28*A*(c^2 - d^2*x^2)^(1/2))/(143*(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))/(429*(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)) + (10*A*(c^2 - d^2*x^2)^(1/2))/(9009*(c^ 
6*d + 2*c^5*d^2*x + c^4*d^3*x^2)) + (2*B*(c^2 - d^2*x^2)^(1/2))/(1365*(c^5 
*d^2 + 3*c^4*d^3*x + 3*c^3*d^4*x^2 + c^2*d^5*x^3)) - (23*A*(c^2 - d^2*x^2) 
^(1/2))/(45045*(c^6*d + c^5*d^2*x)) - (49*B*(c^2 - d^2*x^2)^(1/2))/(6435*( 
c^5*d^2 + 2*c^4*d^3*x + c^3*d^4*x^2)) + (35*B*(c^2 - d^2*x^2)^(1/2))/(143* 
(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))/(3003*(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)) - (3*B*(c^2 - d^2*x^2)^(1/2))/(100 
1*(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)) - ( 
80*B*c*(c^2 - d^2*x^2)^(1/2))/(143*(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)) + (4*B*c^2*(c 
^2 - d^2*x^2)^(1/2))/(13*(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)) - (74* 
A*d*(c^2 - d^2*x^2)^(1/2))/(45045*(c^6*d^2 + 2*c^5*d^3*x + c^4*d^4*x^2)) + 
 (289*B*c*(c^2 - d^2*x^2)^(1/2))/(45045*(c^6*d^2 + 2*c^5*d^3*x + c^4*d^...
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 693, normalized size of antiderivative = 3.30 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^9} \, dx=\frac {-2310 a \,c^{7}+555 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,d^{6} x^{6}+213 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{6} x +608 a \,c^{6} d x -14528 a \,c^{5} d^{2} x^{2}-19086 a \,c^{4} d^{3} x^{3}-19009 a \,c^{3} d^{4} x^{4}-11371 a \,c^{2} d^{5} x^{5}-3781 a c \,d^{6} x^{6}+8892 b \,c^{6} d \,x^{2}+3834 b \,c^{5} d^{2} x^{3}+7761 b \,c^{4} d^{3} x^{4}+4734 b \,c^{3} d^{4} x^{5}+1599 b \,c^{2} d^{5} x^{6}+231 b c \,d^{6} x^{7}+608 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{5} d x +8572 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{4} d^{2} x^{2}+11248 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{3} d^{3} x^{3}+8377 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{2} d^{4} x^{4}+3338 \sqrt {-d^{2} x^{2}+c^{2}}\, a c \,d^{5} x^{5}-6123 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{5} d \,x^{2}-3567 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{4} d^{2} x^{3}-2808 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{3} d^{3} x^{4}-1152 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{2} d^{4} x^{5}-195 \sqrt {-d^{2} x^{2}+c^{2}}\, b c \,d^{5} x^{6}+2310 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{6}-539 a \,d^{7} x^{7}+213 b \,c^{7} x}{15015 c^{5} d \left (\sqrt {-d^{2} x^{2}+c^{2}}\, c^{6}+6 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{5} d x +15 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{4} d^{2} x^{2}+20 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{3} d^{3} x^{3}+15 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{2} d^{4} x^{4}+6 \sqrt {-d^{2} x^{2}+c^{2}}\, c \,d^{5} x^{5}+\sqrt {-d^{2} x^{2}+c^{2}}\, d^{6} x^{6}-c^{7}-7 c^{6} d x -21 c^{5} d^{2} x^{2}-35 c^{4} d^{3} x^{3}-35 c^{3} d^{4} x^{4}-21 c^{2} d^{5} x^{5}-7 c \,d^{6} x^{6}-d^{7} x^{7}\right )} \] Input:

int((B*x+A)*(-d^2*x^2+c^2)^(3/2)/(d*x+c)^9,x)
 

Output:

(2310*sqrt(c**2 - d**2*x**2)*a*c**6 + 608*sqrt(c**2 - d**2*x**2)*a*c**5*d* 
x + 8572*sqrt(c**2 - d**2*x**2)*a*c**4*d**2*x**2 + 11248*sqrt(c**2 - d**2* 
x**2)*a*c**3*d**3*x**3 + 8377*sqrt(c**2 - d**2*x**2)*a*c**2*d**4*x**4 + 33 
38*sqrt(c**2 - d**2*x**2)*a*c*d**5*x**5 + 555*sqrt(c**2 - d**2*x**2)*a*d** 
6*x**6 + 213*sqrt(c**2 - d**2*x**2)*b*c**6*x - 6123*sqrt(c**2 - d**2*x**2) 
*b*c**5*d*x**2 - 3567*sqrt(c**2 - d**2*x**2)*b*c**4*d**2*x**3 - 2808*sqrt( 
c**2 - d**2*x**2)*b*c**3*d**3*x**4 - 1152*sqrt(c**2 - d**2*x**2)*b*c**2*d* 
*4*x**5 - 195*sqrt(c**2 - d**2*x**2)*b*c*d**5*x**6 - 2310*a*c**7 + 608*a*c 
**6*d*x - 14528*a*c**5*d**2*x**2 - 19086*a*c**4*d**3*x**3 - 19009*a*c**3*d 
**4*x**4 - 11371*a*c**2*d**5*x**5 - 3781*a*c*d**6*x**6 - 539*a*d**7*x**7 + 
 213*b*c**7*x + 8892*b*c**6*d*x**2 + 3834*b*c**5*d**2*x**3 + 7761*b*c**4*d 
**3*x**4 + 4734*b*c**3*d**4*x**5 + 1599*b*c**2*d**5*x**6 + 231*b*c*d**6*x* 
*7)/(15015*c**5*d*(sqrt(c**2 - d**2*x**2)*c**6 + 6*sqrt(c**2 - d**2*x**2)* 
c**5*d*x + 15*sqrt(c**2 - d**2*x**2)*c**4*d**2*x**2 + 20*sqrt(c**2 - d**2* 
x**2)*c**3*d**3*x**3 + 15*sqrt(c**2 - d**2*x**2)*c**2*d**4*x**4 + 6*sqrt(c 
**2 - d**2*x**2)*c*d**5*x**5 + sqrt(c**2 - d**2*x**2)*d**6*x**6 - c**7 - 7 
*c**6*d*x - 21*c**5*d**2*x**2 - 35*c**4*d**3*x**3 - 35*c**3*d**4*x**4 - 21 
*c**2*d**5*x**5 - 7*c*d**6*x**6 - d**7*x**7))