\(\int \frac {(A+B x) (c^2-d^2 x^2)^{5/2}}{(c+d x)^9} \, dx\) [37]

Optimal result

Integrand size = 29, antiderivative size = 126 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{5/2}}{(c+d x)^9} \, dx=\frac {(B c-A d) \left (c^2-d^2 x^2\right )^{7/2}}{11 c d^2 (c+d x)^9}-\frac {(9 B c+2 A d) \left (c^2-d^2 x^2\right )^{7/2}}{99 c^2 d^2 (c+d x)^8}-\frac {(9 B c+2 A d) \left (c^2-d^2 x^2\right )^{7/2}}{693 c^3 d^2 (c+d x)^7} \] Output:

1/11*(-A*d+B*c)*(-d^2*x^2+c^2)^(7/2)/c/d^2/(d*x+c)^9-1/99*(2*A*d+9*B*c)*(- 
d^2*x^2+c^2)^(7/2)/c^2/d^2/(d*x+c)^8-1/693*(2*A*d+9*B*c)*(-d^2*x^2+c^2)^(7 
/2)/c^3/d^2/(d*x+c)^7
 

Mathematica [A] (verified)

Time = 1.27 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.67 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{5/2}}{(c+d x)^9} \, dx=-\frac {(c-d x)^3 \sqrt {c^2-d^2 x^2} \left (9 B c \left (c^2+9 c d x+d^2 x^2\right )+A d \left (79 c^2+18 c d x+2 d^2 x^2\right )\right )}{693 c^3 d^2 (c+d x)^6} \] Input:

Integrate[((A + B*x)*(c^2 - d^2*x^2)^(5/2))/(c + d*x)^9,x]
 

Output:

-1/693*((c - d*x)^3*Sqrt[c^2 - d^2*x^2]*(9*B*c*(c^2 + 9*c*d*x + d^2*x^2) + 
 A*d*(79*c^2 + 18*c*d*x + 2*d^2*x^2)))/(c^3*d^2*(c + d*x)^6)
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.02, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {671, 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.

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

\(\Big \downarrow \) 671

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

\(\Big \downarrow \) 461

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

\(\Big \downarrow \) 460

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

Input:

Int[((A + B*x)*(c^2 - d^2*x^2)^(5/2))/(c + d*x)^9,x]
 

Output:

((B*c - A*d)*(c^2 - d^2*x^2)^(7/2))/(11*c*d^2*(c + d*x)^9) + ((9*B*c + 2*A 
*d)*(-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*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 = 1.46 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.67

method	result	size
gosper	\(-\frac {\left (-d x +c \right ) \left (2 A \,d^{3} x^{2}+9 B c \,d^{2} x^{2}+18 A c \,d^{2} x +81 B \,c^{2} d x +79 A \,c^{2} d +9 B \,c^{3}\right ) \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{693 \left (d x +c \right )^{8} c^{3} d^{2}}\)	\(85\)
orering	\(-\frac {\left (-d x +c \right ) \left (2 A \,d^{3} x^{2}+9 B c \,d^{2} x^{2}+18 A c \,d^{2} x +81 B \,c^{2} d x +79 A \,c^{2} d +9 B \,c^{3}\right ) \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{693 \left (d x +c \right )^{8} c^{3} d^{2}}\)	\(85\)
trager	\(-\frac {\left (-2 A \,d^{6} x^{5}-9 B c \,d^{5} x^{5}-12 A c \,d^{5} x^{4}-54 B \,c^{2} d^{4} x^{4}-31 A \,c^{2} d^{4} x^{3}+207 B \,c^{3} d^{3} x^{3}+185 A \,c^{3} d^{3} x^{2}-207 B \,c^{4} d^{2} x^{2}-219 A \,c^{4} d^{2} x +54 B \,c^{5} d x +79 A \,c^{5} d +9 B \,c^{6}\right ) \sqrt {-d^{2} x^{2}+c^{2}}}{693 c^{3} \left (d x +c \right )^{6} d^{2}}\)	\(151\)
default	\(\frac {B \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {7}{2}}}{9 c d \left (x +\frac {c}{d}\right )^{8}}-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {7}{2}}}{63 c^{2} \left (x +\frac {c}{d}\right )^{7}}\right )}{d^{9}}+\frac {\left (A d -B c \right ) \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {7}{2}}}{11 c d \left (x +\frac {c}{d}\right )^{9}}+\frac {2 d \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {7}{2}}}{9 c d \left (x +\frac {c}{d}\right )^{8}}-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {7}{2}}}{63 c^{2} \left (x +\frac {c}{d}\right )^{7}}\right )}{11 c}\right )}{d^{10}}\)	\(247\)

Input:

int((B*x+A)*(-d^2*x^2+c^2)^(5/2)/(d*x+c)^9,x,method=_RETURNVERBOSE)
 

Output:

-1/693*(-d*x+c)*(2*A*d^3*x^2+9*B*c*d^2*x^2+18*A*c*d^2*x+81*B*c^2*d*x+79*A* 
c^2*d+9*B*c^3)*(-d^2*x^2+c^2)^(5/2)/(d*x+c)^8/c^3/d^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 358 vs. \(2 (114) = 228\).

Time = 0.19 (sec) , antiderivative size = 358, normalized size of antiderivative = 2.84 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{5/2}}{(c+d x)^9} \, dx=-\frac {9 \, B c^{7} + 79 \, A c^{6} d + {\left (9 \, B c d^{6} + 79 \, A d^{7}\right )} x^{6} + 6 \, {\left (9 \, B c^{2} d^{5} + 79 \, A c d^{6}\right )} x^{5} + 15 \, {\left (9 \, B c^{3} d^{4} + 79 \, A c^{2} d^{5}\right )} x^{4} + 20 \, {\left (9 \, B c^{4} d^{3} + 79 \, A c^{3} d^{4}\right )} x^{3} + 15 \, {\left (9 \, B c^{5} d^{2} + 79 \, A c^{4} d^{3}\right )} x^{2} + 6 \, {\left (9 \, B c^{6} d + 79 \, A c^{5} d^{2}\right )} x + {\left (9 \, B c^{6} + 79 \, A c^{5} d - {\left (9 \, B c d^{5} + 2 \, A d^{6}\right )} x^{5} - 6 \, {\left (9 \, B c^{2} d^{4} + 2 \, A c d^{5}\right )} x^{4} + {\left (207 \, B c^{3} d^{3} - 31 \, A c^{2} d^{4}\right )} x^{3} - {\left (207 \, B c^{4} d^{2} - 185 \, A c^{3} d^{3}\right )} x^{2} + 3 \, {\left (18 \, B c^{5} d - 73 \, A c^{4} d^{2}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}}}{693 \, {\left (c^{3} d^{8} x^{6} + 6 \, c^{4} d^{7} x^{5} + 15 \, c^{5} d^{6} x^{4} + 20 \, c^{6} d^{5} x^{3} + 15 \, c^{7} d^{4} x^{2} + 6 \, c^{8} d^{3} x + c^{9} d^{2}\right )}} \] Input:

integrate((B*x+A)*(-d^2*x^2+c^2)^(5/2)/(d*x+c)^9,x, algorithm="fricas")
 

Output:

-1/693*(9*B*c^7 + 79*A*c^6*d + (9*B*c*d^6 + 79*A*d^7)*x^6 + 6*(9*B*c^2*d^5 
 + 79*A*c*d^6)*x^5 + 15*(9*B*c^3*d^4 + 79*A*c^2*d^5)*x^4 + 20*(9*B*c^4*d^3 
 + 79*A*c^3*d^4)*x^3 + 15*(9*B*c^5*d^2 + 79*A*c^4*d^3)*x^2 + 6*(9*B*c^6*d 
+ 79*A*c^5*d^2)*x + (9*B*c^6 + 79*A*c^5*d - (9*B*c*d^5 + 2*A*d^6)*x^5 - 6* 
(9*B*c^2*d^4 + 2*A*c*d^5)*x^4 + (207*B*c^3*d^3 - 31*A*c^2*d^4)*x^3 - (207* 
B*c^4*d^2 - 185*A*c^3*d^3)*x^2 + 3*(18*B*c^5*d - 73*A*c^4*d^2)*x)*sqrt(-d^ 
2*x^2 + c^2))/(c^3*d^8*x^6 + 6*c^4*d^7*x^5 + 15*c^5*d^6*x^4 + 20*c^6*d^5*x 
^3 + 15*c^7*d^4*x^2 + 6*c^8*d^3*x + c^9*d^2)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{5/2}}{(c+d x)^9} \, dx=\text {Timed out} \] Input:

integrate((B*x+A)*(-d**2*x**2+c**2)**(5/2)/(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. 1439 vs. \(2 (114) = 228\).

Time = 0.07 (sec) , antiderivative size = 1439, normalized size of antiderivative = 11.42 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{5/2}}{(c+d x)^9} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)*(-d^2*x^2+c^2)^(5/2)/(d*x+c)^9,x, algorithm="maxima")
 

Output:

1/3*(-d^2*x^2 + c^2)^(5/2)*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) - 5/12*(-d^2*x^2 + c^2)^(3/2)*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) + 5/22*sqrt(-d^2*x^2 + c^2)*B*c^3/(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) - 1/3*(-d^2*x^2 + c^2)^(5/2)*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) - 1/2*(-d^2*x^2 + c^2)^(5/2)*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) + 5/12*(-d^2*x^2 + c^2)^(3/2)*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) + 5/6*(-d^2*x^2 + c^2)^(3/2)*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) - 5/22*sqrt(-d^2*x^2 + c^2)*A*c^2/(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) - 25/44*sqrt(- 
d^2*x^2 + c^2)*B*c^2/(d^7*x^5 + 5*c*d^6*x^4 + 10*c^2*d^5*x^3 + 10*c^3*d^4* 
x^2 + 5*c^4*d^3*x + c^5*d^2) + 5/396*sqrt(-d^2*x^2 + c^2)*A*c/(d^6*x^5 + 5 
*c*d^5*x^4 + 10*c^2*d^4*x^3 + 10*c^3*d^3*x^2 + 5*c^4*d^2*x + c^5*d) + 5/15 
4*sqrt(-d^2*x^2 + c^2)*B*c/(d^6*x^4 + 4*c*d^5*x^3 + 6*c^2*d^4*x^2 + 4*c...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 666 vs. \(2 (114) = 228\).

Time = 0.17 (sec) , antiderivative size = 666, normalized size of antiderivative = 5.29 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{5/2}}{(c+d x)^9} \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)*(-d^2*x^2+c^2)^(5/2)/(d*x+c)^9,x, algorithm="giac")
 

Output:

2/693*(9*B*c + 79*A*d + 99*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))*B*c/(d^2*x) 
 + 176*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))*A/(d*x) - 198*(c*d + sqrt(-d^2* 
x^2 + c^2)*abs(d))^2*B*c/(d^4*x^2) + 2959*(c*d + sqrt(-d^2*x^2 + c^2)*abs( 
d))^2*A/(d^3*x^2) + 2178*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^3*B*c/(d^6*x^ 
3) + 4950*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^3*A/(d^5*x^3) - 1188*(c*d + 
sqrt(-d^2*x^2 + c^2)*abs(d))^4*B*c/(d^8*x^4) + 15444*(c*d + sqrt(-d^2*x^2 
+ c^2)*abs(d))^4*A/(d^7*x^4) + 5544*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^5* 
B*c/(d^10*x^5) + 15246*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^5*A/(d^9*x^5) - 
 1386*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^6*B*c/(d^12*x^6) + 21252*(c*d + 
sqrt(-d^2*x^2 + c^2)*abs(d))^6*A/(d^11*x^6) + 4158*(c*d + sqrt(-d^2*x^2 + 
c^2)*abs(d))^7*B*c/(d^14*x^7) + 10626*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^ 
7*A/(d^13*x^7) - 693*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^8*B*c/(d^16*x^8) 
+ 8085*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^8*A/(d^15*x^8) + 693*(c*d + sqr 
t(-d^2*x^2 + c^2)*abs(d))^9*B*c/(d^18*x^9) + 1386*(c*d + sqrt(-d^2*x^2 + c 
^2)*abs(d))^9*A/(d^17*x^9) + 693*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^10*A/ 
(d^19*x^10))/(c^3*d*((c*d + sqrt(-d^2*x^2 + c^2)*abs(d))/(d^2*x) + 1)^11*a 
bs(d))
 

Mupad [B] (verification not implemented)

Time = 11.64 (sec) , antiderivative size = 808, normalized size of antiderivative = 6.41 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{5/2}}{(c+d x)^9} \, dx=\frac {4\,B\,\sqrt {c^2-d^2\,x^2}}{315\,\left (c^3\,d^2+x\,c^2\,d^3\right )}-\frac {37\,B\,\sqrt {c^2-d^2\,x^2}}{77\,\left (c^3\,d^2+3\,c^2\,d^3\,x+3\,c\,d^4\,x^2+d^5\,x^3\right )}+\frac {2\,A\,\sqrt {c^2-d^2\,x^2}}{693\,\left (c^4\,d+2\,c^3\,d^2\,x+c^2\,d^3\,x^2\right )}+\frac {B\,\sqrt {c^2-d^2\,x^2}}{77\,\left (c^3\,d^2+2\,c^2\,d^3\,x+c\,d^4\,x^2\right )}-\frac {29\,A\,\sqrt {c^2-d^2\,x^2}}{3465\,\left (c^4\,d+x\,c^3\,d^2\right )}-\frac {226\,A\,\sqrt {c^2-d^2\,x^2}}{693\,\left (c^4\,d+4\,c^3\,d^2\,x+6\,c^2\,d^3\,x^2+4\,c\,d^4\,x^3+d^5\,x^4\right )}+\frac {A\,\sqrt {c^2-d^2\,x^2}}{231\,\left (c^4\,d+3\,c^3\,d^2\,x+3\,c^2\,d^3\,x^2+c\,d^4\,x^3\right )}-\frac {8\,A\,c^2\,\sqrt {c^2-d^2\,x^2}}{11\,\left (c^6\,d+6\,c^5\,d^2\,x+15\,c^4\,d^3\,x^2+20\,c^3\,d^4\,x^3+15\,c^2\,d^5\,x^4+6\,c\,d^6\,x^5+d^7\,x^6\right )}+\frac {8\,B\,c^3\,\sqrt {c^2-d^2\,x^2}}{11\,\left (c^6\,d^2+6\,c^5\,d^3\,x+15\,c^4\,d^4\,x^2+20\,c^3\,d^5\,x^3+15\,c^2\,d^6\,x^4+6\,c\,d^7\,x^5+d^8\,x^6\right )}+\frac {13\,A\,d\,\sqrt {c^2-d^2\,x^2}}{1155\,\left (c^4\,d^2+x\,c^3\,d^3\right )}+\frac {B\,c\,\sqrt {c^2-d^2\,x^2}}{3465\,\left (c^4\,d^2+x\,c^3\,d^3\right )}+\frac {118\,B\,c\,\sqrt {c^2-d^2\,x^2}}{77\,\left (c^4\,d^2+4\,c^3\,d^3\,x+6\,c^2\,d^4\,x^2+4\,c\,d^5\,x^3+d^6\,x^4\right )}-\frac {20\,B\,c^2\,\sqrt {c^2-d^2\,x^2}}{11\,\left (c^5\,d^2+5\,c^4\,d^3\,x+10\,c^3\,d^4\,x^2+10\,c^2\,d^5\,x^3+5\,c\,d^6\,x^4+d^7\,x^5\right )}+\frac {92\,A\,c\,\sqrt {c^2-d^2\,x^2}}{99\,\left (c^5\,d+5\,c^4\,d^2\,x+10\,c^3\,d^3\,x^2+10\,c^2\,d^4\,x^3+5\,c\,d^5\,x^4+d^6\,x^5\right )} \] Input:

int(((c^2 - d^2*x^2)^(5/2)*(A + B*x))/(c + d*x)^9,x)
 

Output:

(4*B*(c^2 - d^2*x^2)^(1/2))/(315*(c^3*d^2 + c^2*d^3*x)) - (37*B*(c^2 - d^2 
*x^2)^(1/2))/(77*(c^3*d^2 + d^5*x^3 + 3*c^2*d^3*x + 3*c*d^4*x^2)) + (2*A*( 
c^2 - d^2*x^2)^(1/2))/(693*(c^4*d + 2*c^3*d^2*x + c^2*d^3*x^2)) + (B*(c^2 
- d^2*x^2)^(1/2))/(77*(c^3*d^2 + 2*c^2*d^3*x + c*d^4*x^2)) - (29*A*(c^2 - 
d^2*x^2)^(1/2))/(3465*(c^4*d + c^3*d^2*x)) - (226*A*(c^2 - d^2*x^2)^(1/2)) 
/(693*(c^4*d + d^5*x^4 + 4*c^3*d^2*x + 4*c*d^4*x^3 + 6*c^2*d^3*x^2)) + (A* 
(c^2 - d^2*x^2)^(1/2))/(231*(c^4*d + 3*c^3*d^2*x + c*d^4*x^3 + 3*c^2*d^3*x 
^2)) - (8*A*c^2*(c^2 - d^2*x^2)^(1/2))/(11*(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)) + (8*B* 
c^3*(c^2 - d^2*x^2)^(1/2))/(11*(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)) + (13*A*d*(c^2 - 
d^2*x^2)^(1/2))/(1155*(c^4*d^2 + c^3*d^3*x)) + (B*c*(c^2 - d^2*x^2)^(1/2)) 
/(3465*(c^4*d^2 + c^3*d^3*x)) + (118*B*c*(c^2 - d^2*x^2)^(1/2))/(77*(c^4*d 
^2 + d^6*x^4 + 4*c^3*d^3*x + 4*c*d^5*x^3 + 6*c^2*d^4*x^2)) - (20*B*c^2*(c^ 
2 - d^2*x^2)^(1/2))/(11*(c^5*d^2 + d^7*x^5 + 5*c^4*d^3*x + 5*c*d^6*x^4 + 1 
0*c^3*d^4*x^2 + 10*c^2*d^5*x^3)) + (92*A*c*(c^2 - d^2*x^2)^(1/2))/(99*(c^5 
*d + d^6*x^5 + 5*c^4*d^2*x + 5*c*d^5*x^4 + 10*c^3*d^3*x^2 + 10*c^2*d^4*x^3 
))
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 574, normalized size of antiderivative = 4.56 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{5/2}}{(c+d x)^9} \, dx=\frac {126 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{5}+16 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{4} d x +655 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{3} d^{2} x^{2}+439 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{2} d^{3} x^{3}+223 \sqrt {-d^{2} x^{2}+c^{2}}\, a c \,d^{4} x^{4}+45 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,d^{5} x^{5}+9 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{5} x -297 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{4} d \,x^{2}+117 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{3} d^{2} x^{3}-99 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{2} d^{3} x^{4}-18 \sqrt {-d^{2} x^{2}+c^{2}}\, b c \,d^{4} x^{5}-126 a \,c^{6}+16 a \,c^{5} d x -1109 a \,c^{4} d^{2} x^{2}-724 a \,c^{3} d^{3} x^{3}-724 a \,c^{2} d^{4} x^{4}-292 a c \,d^{5} x^{5}-49 a \,d^{6} x^{6}+9 b \,c^{6} x +396 b \,c^{5} d \,x^{2}-234 b \,c^{4} d^{2} x^{3}+396 b \,c^{3} d^{3} x^{4}+9 b \,c^{2} d^{4} x^{5}}{693 c^{3} d \left (\sqrt {-d^{2} x^{2}+c^{2}}\, c^{5}+5 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{4} d x +10 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{3} d^{2} x^{2}+10 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{2} d^{3} x^{3}+5 \sqrt {-d^{2} x^{2}+c^{2}}\, c \,d^{4} x^{4}+\sqrt {-d^{2} x^{2}+c^{2}}\, d^{5} x^{5}-c^{6}-6 c^{5} d x -15 c^{4} d^{2} x^{2}-20 c^{3} d^{3} x^{3}-15 c^{2} d^{4} x^{4}-6 c \,d^{5} x^{5}-d^{6} x^{6}\right )} \] Input:

int((B*x+A)*(-d^2*x^2+c^2)^(5/2)/(d*x+c)^9,x)
 

Output:

(126*sqrt(c**2 - d**2*x**2)*a*c**5 + 16*sqrt(c**2 - d**2*x**2)*a*c**4*d*x 
+ 655*sqrt(c**2 - d**2*x**2)*a*c**3*d**2*x**2 + 439*sqrt(c**2 - d**2*x**2) 
*a*c**2*d**3*x**3 + 223*sqrt(c**2 - d**2*x**2)*a*c*d**4*x**4 + 45*sqrt(c** 
2 - d**2*x**2)*a*d**5*x**5 + 9*sqrt(c**2 - d**2*x**2)*b*c**5*x - 297*sqrt( 
c**2 - d**2*x**2)*b*c**4*d*x**2 + 117*sqrt(c**2 - d**2*x**2)*b*c**3*d**2*x 
**3 - 99*sqrt(c**2 - d**2*x**2)*b*c**2*d**3*x**4 - 18*sqrt(c**2 - d**2*x** 
2)*b*c*d**4*x**5 - 126*a*c**6 + 16*a*c**5*d*x - 1109*a*c**4*d**2*x**2 - 72 
4*a*c**3*d**3*x**3 - 724*a*c**2*d**4*x**4 - 292*a*c*d**5*x**5 - 49*a*d**6* 
x**6 + 9*b*c**6*x + 396*b*c**5*d*x**2 - 234*b*c**4*d**2*x**3 + 396*b*c**3* 
d**3*x**4 + 9*b*c**2*d**4*x**5)/(693*c**3*d*(sqrt(c**2 - d**2*x**2)*c**5 + 
 5*sqrt(c**2 - d**2*x**2)*c**4*d*x + 10*sqrt(c**2 - d**2*x**2)*c**3*d**2*x 
**2 + 10*sqrt(c**2 - d**2*x**2)*c**2*d**3*x**3 + 5*sqrt(c**2 - d**2*x**2)* 
c*d**4*x**4 + sqrt(c**2 - d**2*x**2)*d**5*x**5 - c**6 - 6*c**5*d*x - 15*c* 
*4*d**2*x**2 - 20*c**3*d**3*x**3 - 15*c**2*d**4*x**4 - 6*c*d**5*x**5 - d** 
6*x**6))