Integrand size = 29, antiderivative size = 168 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^8} \, dx=\frac {(B c-A d) \left (c^2-d^2 x^2\right )^{5/2}}{11 c d^2 (c+d x)^8}-\frac {(8 B c+3 A d) \left (c^2-d^2 x^2\right )^{5/2}}{99 c^2 d^2 (c+d x)^7}-\frac {2 (8 B c+3 A d) \left (c^2-d^2 x^2\right )^{5/2}}{693 c^3 d^2 (c+d x)^6}-\frac {2 (8 B c+3 A d) \left (c^2-d^2 x^2\right )^{5/2}}{3465 c^4 d^2 (c+d x)^5} \] Output:
1/11*(-A*d+B*c)*(-d^2*x^2+c^2)^(5/2)/c/d^2/(d*x+c)^8-1/99*(3*A*d+8*B*c)*(- d^2*x^2+c^2)^(5/2)/c^2/d^2/(d*x+c)^7-2/693*(3*A*d+8*B*c)*(-d^2*x^2+c^2)^(5 /2)/c^3/d^2/(d*x+c)^6-2/3465*(3*A*d+8*B*c)*(-d^2*x^2+c^2)^(5/2)/c^4/d^2/(d *x+c)^5
Time = 1.08 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.65 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^8} \, dx=-\frac {(c-d x)^2 \sqrt {c^2-d^2 x^2} \left (3 A d \left (152 c^3+61 c^2 d x+16 c d^2 x^2+2 d^3 x^3\right )+B c \left (61 c^3+488 c^2 d x+128 c d^2 x^2+16 d^3 x^3\right )\right )}{3465 c^4 d^2 (c+d x)^6} \] Input:
Integrate[((A + B*x)*(c^2 - d^2*x^2)^(3/2))/(c + d*x)^8,x]
Output:
-1/3465*((c - d*x)^2*Sqrt[c^2 - d^2*x^2]*(3*A*d*(152*c^3 + 61*c^2*d*x + 16 *c*d^2*x^2 + 2*d^3*x^3) + B*c*(61*c^3 + 488*c^2*d*x + 128*c*d^2*x^2 + 16*d ^3*x^3)))/(c^4*d^2*(c + d*x)^6)
Time = 0.45 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {671, 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.
| \(\displaystyle \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^8} \, dx\) |
| \(\Big \downarrow \) 671 |
| \(\displaystyle \frac {(3 A d+8 B c) \int \frac {\left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^7}dx}{11 c d}+\frac {\left (c^2-d^2 x^2\right )^{5/2} (B c-A d)}{11 c d^2 (c+d x)^8}\) |
| \(\Big \downarrow \) 461 |
| \(\displaystyle \frac {(3 A d+8 B c) \left (\frac {2 \int \frac {\left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^6}dx}{9 c}-\frac {\left (c^2-d^2 x^2\right )^{5/2}}{9 c d (c+d x)^7}\right )}{11 c d}+\frac {\left (c^2-d^2 x^2\right )^{5/2} (B c-A d)}{11 c d^2 (c+d x)^8}\) |
| \(\Big \downarrow \) 461 |
| \(\displaystyle \frac {(3 A d+8 B c) \left (\frac {2 \left (\frac {\int \frac {\left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^5}dx}{7 c}-\frac {\left (c^2-d^2 x^2\right )^{5/2}}{7 c d (c+d x)^6}\right )}{9 c}-\frac {\left (c^2-d^2 x^2\right )^{5/2}}{9 c d (c+d x)^7}\right )}{11 c d}+\frac {\left (c^2-d^2 x^2\right )^{5/2} (B c-A d)}{11 c d^2 (c+d x)^8}\) |
| \(\Big \downarrow \) 460 |
| \(\displaystyle \frac {\left (c^2-d^2 x^2\right )^{5/2} (B c-A d)}{11 c d^2 (c+d x)^8}+\frac {\left (\frac {2 \left (-\frac {\left (c^2-d^2 x^2\right )^{5/2}}{35 c^2 d (c+d x)^5}-\frac {\left (c^2-d^2 x^2\right )^{5/2}}{7 c d (c+d x)^6}\right )}{9 c}-\frac {\left (c^2-d^2 x^2\right )^{5/2}}{9 c d (c+d x)^7}\right ) (3 A d+8 B c)}{11 c d}\) |
Input:
Int[((A + B*x)*(c^2 - d^2*x^2)^(3/2))/(c + d*x)^8,x]
Output:
((B*c - A*d)*(c^2 - d^2*x^2)^(5/2))/(11*c*d^2*(c + d*x)^8) + ((8*B*c + 3*A *d)*(-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*d)
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]
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])
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]
Time = 0.64 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.65
| method | result | size |
| gosper | \(-\frac {\left (-d x +c \right ) \left (6 A \,d^{4} x^{3}+16 B c \,d^{3} x^{3}+48 A c \,d^{3} x^{2}+128 x^{2} c^{2} B \,d^{2}+183 A \,c^{2} d^{2} x +488 B \,c^{3} d x +456 A \,c^{3} d +61 B \,c^{4}\right ) \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{3465 \left (d x +c \right )^{7} c^{4} d^{2}}\) | \(109\) |
| orering | \(-\frac {\left (-d x +c \right ) \left (6 A \,d^{4} x^{3}+16 B c \,d^{3} x^{3}+48 A c \,d^{3} x^{2}+128 x^{2} c^{2} B \,d^{2}+183 A \,c^{2} d^{2} x +488 B \,c^{3} d x +456 A \,c^{3} d +61 B \,c^{4}\right ) \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{3465 \left (d x +c \right )^{7} c^{4} d^{2}}\) | \(109\) |
| trager | \(-\frac {\left (6 A \,d^{6} x^{5}+16 B c \,d^{5} x^{5}+36 A c \,d^{5} x^{4}+96 B \,c^{2} d^{4} x^{4}+93 A \,c^{2} d^{4} x^{3}+248 B \,c^{3} d^{3} x^{3}+138 A \,c^{3} d^{3} x^{2}-787 B \,c^{4} d^{2} x^{2}-729 A \,c^{4} d^{2} x +366 B \,c^{5} d x +456 A \,c^{5} d +61 B \,c^{6}\right ) \sqrt {-d^{2} x^{2}+c^{2}}}{3465 c^{4} \left (d x +c \right )^{6} d^{2}}\) | \(151\) |
| default | \(\frac {B \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {5}{2}}}{9 c d \left (x +\frac {c}{d}\right )^{7}}+\frac {2 d \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {5}{2}}}{7 c d \left (x +\frac {c}{d}\right )^{6}}-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {5}{2}}}{35 c^{2} \left (x +\frac {c}{d}\right )^{5}}\right )}{9 c}\right )}{d^{8}}+\frac {\left (A d -B c \right ) \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {5}{2}}}{11 c d \left (x +\frac {c}{d}\right )^{8}}+\frac {3 d \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {5}{2}}}{9 c d \left (x +\frac {c}{d}\right )^{7}}+\frac {2 d \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {5}{2}}}{7 c d \left (x +\frac {c}{d}\right )^{6}}-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {5}{2}}}{35 c^{2} \left (x +\frac {c}{d}\right )^{5}}\right )}{9 c}\right )}{11 c}\right )}{d^{9}}\) | \(351\) |
Input:
int((B*x+A)*(-d^2*x^2+c^2)^(3/2)/(d*x+c)^8,x,method=_RETURNVERBOSE)
Output:
-1/3465*(-d*x+c)*(6*A*d^4*x^3+16*B*c*d^3*x^3+48*A*c*d^3*x^2+128*B*c^2*d^2* x^2+183*A*c^2*d^2*x+488*B*c^3*d*x+456*A*c^3*d+61*B*c^4)*(-d^2*x^2+c^2)^(3/ 2)/(d*x+c)^7/c^4/d^2
Leaf count of result is larger than twice the leaf count of optimal. 359 vs. \(2 (152) = 304\).
Time = 0.22 (sec) , antiderivative size = 359, normalized size of antiderivative = 2.14 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^8} \, dx=-\frac {61 \, B c^{7} + 456 \, A c^{6} d + {\left (61 \, B c d^{6} + 456 \, A d^{7}\right )} x^{6} + 6 \, {\left (61 \, B c^{2} d^{5} + 456 \, A c d^{6}\right )} x^{5} + 15 \, {\left (61 \, B c^{3} d^{4} + 456 \, A c^{2} d^{5}\right )} x^{4} + 20 \, {\left (61 \, B c^{4} d^{3} + 456 \, A c^{3} d^{4}\right )} x^{3} + 15 \, {\left (61 \, B c^{5} d^{2} + 456 \, A c^{4} d^{3}\right )} x^{2} + 6 \, {\left (61 \, B c^{6} d + 456 \, A c^{5} d^{2}\right )} x + {\left (61 \, B c^{6} + 456 \, A c^{5} d + 2 \, {\left (8 \, B c d^{5} + 3 \, A d^{6}\right )} x^{5} + 12 \, {\left (8 \, B c^{2} d^{4} + 3 \, A c d^{5}\right )} x^{4} + 31 \, {\left (8 \, B c^{3} d^{3} + 3 \, A c^{2} d^{4}\right )} x^{3} - {\left (787 \, B c^{4} d^{2} - 138 \, A c^{3} d^{3}\right )} x^{2} + 3 \, {\left (122 \, B c^{5} d - 243 \, A c^{4} d^{2}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}}}{3465 \, {\left (c^{4} d^{8} x^{6} + 6 \, c^{5} d^{7} x^{5} + 15 \, c^{6} d^{6} x^{4} + 20 \, c^{7} d^{5} x^{3} + 15 \, c^{8} d^{4} x^{2} + 6 \, c^{9} d^{3} x + c^{10} d^{2}\right )}} \] Input:
integrate((B*x+A)*(-d^2*x^2+c^2)^(3/2)/(d*x+c)^8,x, algorithm="fricas")
Output:
-1/3465*(61*B*c^7 + 456*A*c^6*d + (61*B*c*d^6 + 456*A*d^7)*x^6 + 6*(61*B*c ^2*d^5 + 456*A*c*d^6)*x^5 + 15*(61*B*c^3*d^4 + 456*A*c^2*d^5)*x^4 + 20*(61 *B*c^4*d^3 + 456*A*c^3*d^4)*x^3 + 15*(61*B*c^5*d^2 + 456*A*c^4*d^3)*x^2 + 6*(61*B*c^6*d + 456*A*c^5*d^2)*x + (61*B*c^6 + 456*A*c^5*d + 2*(8*B*c*d^5 + 3*A*d^6)*x^5 + 12*(8*B*c^2*d^4 + 3*A*c*d^5)*x^4 + 31*(8*B*c^3*d^3 + 3*A* c^2*d^4)*x^3 - (787*B*c^4*d^2 - 138*A*c^3*d^3)*x^2 + 3*(122*B*c^5*d - 243* A*c^4*d^2)*x)*sqrt(-d^2*x^2 + c^2))/(c^4*d^8*x^6 + 6*c^5*d^7*x^5 + 15*c^6* d^6*x^4 + 20*c^7*d^5*x^3 + 15*c^8*d^4*x^2 + 6*c^9*d^3*x + c^10*d^2)
Timed out. \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^8} \, dx=\text {Timed out} \] Input:
integrate((B*x+A)*(-d**2*x**2+c**2)**(3/2)/(d*x+c)**8,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 1199 vs. \(2 (152) = 304\).
Time = 0.06 (sec) , antiderivative size = 1199, normalized size of antiderivative = 7.14 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^8} \, dx=\text {Too large to display} \] Input:
integrate((B*x+A)*(-d^2*x^2+c^2)^(3/2)/(d*x+c)^8,x, algorithm="maxima")
Output:
1/4*(-d^2*x^2 + c^2)^(3/2)*B*c/(d^9*x^7 + 7*c*d^8*x^6 + 21*c^2*d^7*x^5 + 3 5*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) - 3/22*sqrt(-d^2*x^2 + c^2)*B*c^2/(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) - 1/4*(-d^2*x^2 + c^2)^(3/2)*A/(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) - 1/3*(-d^2*x^2 + c ^2)^(3/2)*B/(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) + 3/22*sqrt(-d^2*x^2 + c^2)*A*c/(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) + 91/396*sqrt(-d^2*x^2 + c^2)*B*c/(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) + 1/231*sqrt(- d^2*x^2 + c^2)*B*c/(c*d^6*x^4 + 4*c^2*d^5*x^3 + 6*c^3*d^4*x^2 + 4*c^4*d^3* x + c^5*d^2) + 1/385*sqrt(-d^2*x^2 + c^2)*B*c/(c^2*d^5*x^3 + 3*c^3*d^4*x^2 + 3*c^4*d^3*x + c^5*d^2) + 2/1155*sqrt(-d^2*x^2 + c^2)*B*c/(c^3*d^4*x^2 + 2*c^4*d^3*x + c^5*d^2) + 2/1155*sqrt(-d^2*x^2 + c^2)*B*c/(c^4*d^3*x + c^5 *d^2) - 1/132*sqrt(-d^2*x^2 + c^2)*A/(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) - 1/231*sqrt(-d^2*x^2 + c^2)*A/ (c*d^5*x^4 + 4*c^2*d^4*x^3 + 6*c^3*d^3*x^2 + 4*c^4*d^2*x + c^5*d) - 1/385* sqrt(-d^2*x^2 + c^2)*A/(c^2*d^4*x^3 + 3*c^3*d^3*x^2 + 3*c^4*d^2*x + c^5*d) - 2/1155*sqrt(-d^2*x^2 + c^2)*A/(c^3*d^3*x^2 + 2*c^4*d^2*x + c^5*d) - ...
Leaf count of result is larger than twice the leaf count of optimal. 666 vs. \(2 (152) = 304\).
Time = 0.15 (sec) , antiderivative size = 666, normalized size of antiderivative = 3.96 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^8} \, dx =\text {Too large to display} \] Input:
integrate((B*x+A)*(-d^2*x^2+c^2)^(3/2)/(d*x+c)^8,x, algorithm="giac")
Output:
2/3465*(61*B*c + 456*A*d + 671*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))*B*c/(d^ 2*x) + 1551*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))*A/(d*x) - 110*(c*d + sqrt( -d^2*x^2 + c^2)*abs(d))^2*B*c/(d^4*x^2) + 14685*(c*d + sqrt(-d^2*x^2 + c^2 )*abs(d))^2*A/(d^3*x^2) + 8910*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^3*B*c/( d^6*x^3) + 33660*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^3*A/(d^5*x^3) + 3960* (c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^4*B*c/(d^8*x^4) + 81180*(c*d + sqrt(-d ^2*x^2 + c^2)*abs(d))^4*A/(d^7*x^4) + 22176*(c*d + sqrt(-d^2*x^2 + c^2)*ab s(d))^5*B*c/(d^10*x^5) + 98406*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^5*A/(d^ 9*x^5) + 6006*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^6*B*c/(d^12*x^6) + 11226 6*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^6*A/(d^11*x^6) + 16170*(c*d + sqrt(- d^2*x^2 + c^2)*abs(d))^7*B*c/(d^14*x^7) + 69300*(c*d + sqrt(-d^2*x^2 + c^2 )*abs(d))^7*A/(d^13*x^7) + 1155*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^8*B*c/ (d^16*x^8) + 41580*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^8*A/(d^15*x^8) + 34 65*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^9*B*c/(d^18*x^9) + 10395*(c*d + sqr t(-d^2*x^2 + c^2)*abs(d))^9*A/(d^17*x^9) + 3465*(c*d + sqrt(-d^2*x^2 + c^2 )*abs(d))^10*A/(d^19*x^10))/(c^4*d*((c*d + sqrt(-d^2*x^2 + c^2)*abs(d))/(d ^2*x) + 1)^11*abs(d))
Time = 11.44 (sec) , antiderivative size = 912, normalized size of antiderivative = 5.43 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^8} \, dx=\frac {223\,B\,\sqrt {c^2-d^2\,x^2}}{693\,\left (c^4\,d^2+4\,c^3\,d^3\,x+6\,c^2\,d^4\,x^2+4\,c\,d^5\,x^3+d^6\,x^4\right )}-\frac {236\,B\,\sqrt {c^2-d^2\,x^2}}{10395\,\left (c^4\,d^2+x\,c^3\,d^3\right )}-\frac {A\,\sqrt {c^2-d^2\,x^2}}{385\,\left (c^5\,d+3\,c^4\,d^2\,x+3\,c^3\,d^3\,x^2+c^2\,d^4\,x^3\right )}-\frac {8\,B\,\sqrt {c^2-d^2\,x^2}}{1155\,\left (c^4\,d^2+3\,c^3\,d^3\,x+3\,c^2\,d^4\,x^2+c\,d^5\,x^3\right )}-\frac {2\,A\,\sqrt {c^2-d^2\,x^2}}{315\,\left (c^5\,d+2\,c^4\,d^2\,x+c^3\,d^3\,x^2\right )}+\frac {8\,A\,\sqrt {c^2-d^2\,x^2}}{33\,\left (c^5\,d+5\,c^4\,d^2\,x+10\,c^3\,d^3\,x^2+10\,c^2\,d^4\,x^3+5\,c\,d^5\,x^4+d^6\,x^5\right )}-\frac {A\,\sqrt {c^2-d^2\,x^2}}{231\,\left (c^5\,d+4\,c^4\,d^2\,x+6\,c^3\,d^3\,x^2+4\,c^2\,d^4\,x^3+c\,d^5\,x^4\right )}+\frac {41\,A\,\sqrt {c^2-d^2\,x^2}}{10395\,\left (c^5\,d+x\,c^4\,d^2\right )}+\frac {B\,\sqrt {c^2-d^2\,x^2}}{495\,\left (c^4\,d^2+2\,c^3\,d^3\,x+c^2\,d^4\,x^2\right )}+\frac {16\,A\,d\,\sqrt {c^2-d^2\,x^2}}{3465\,\left (c^5\,d^2+2\,c^4\,d^3\,x+c^3\,d^4\,x^2\right )}-\frac {23\,B\,c\,\sqrt {c^2-d^2\,x^2}}{3465\,\left (c^5\,d^2+2\,c^4\,d^3\,x+c^3\,d^4\,x^2\right )}-\frac {68\,B\,c\,\sqrt {c^2-d^2\,x^2}}{99\,\left (c^5\,d^2+5\,c^4\,d^3\,x+10\,c^3\,d^4\,x^2+10\,c^2\,d^5\,x^3+5\,c\,d^6\,x^4+d^7\,x^5\right )}+\frac {4\,B\,c^2\,\sqrt {c^2-d^2\,x^2}}{11\,\left (c^6\,d^2+6\,c^5\,d^3\,x+15\,c^4\,d^4\,x^2+20\,c^3\,d^5\,x^3+15\,c^2\,d^6\,x^4+6\,c\,d^7\,x^5+d^8\,x^6\right )}-\frac {59\,A\,d\,\sqrt {c^2-d^2\,x^2}}{10395\,\left (c^5\,d^2+x\,c^4\,d^3\right )}+\frac {188\,B\,c\,\sqrt {c^2-d^2\,x^2}}{10395\,\left (c^5\,d^2+x\,c^4\,d^3\right )}-\frac {4\,A\,c\,\sqrt {c^2-d^2\,x^2}}{11\,\left (c^6\,d+6\,c^5\,d^2\,x+15\,c^4\,d^3\,x^2+20\,c^3\,d^4\,x^3+15\,c^2\,d^5\,x^4+6\,c\,d^6\,x^5+d^7\,x^6\right )} \] Input:
int(((c^2 - d^2*x^2)^(3/2)*(A + B*x))/(c + d*x)^8,x)
Output:
(223*B*(c^2 - d^2*x^2)^(1/2))/(693*(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)) - (236*B*(c^2 - d^2*x^2)^(1/2))/(10395*(c^4*d^2 + c^3*d^3*x)) - (A*(c^2 - d^2*x^2)^(1/2))/(385*(c^5*d + 3*c^4*d^2*x + 3*c^ 3*d^3*x^2 + c^2*d^4*x^3)) - (8*B*(c^2 - d^2*x^2)^(1/2))/(1155*(c^4*d^2 + 3 *c^3*d^3*x + c*d^5*x^3 + 3*c^2*d^4*x^2)) - (2*A*(c^2 - d^2*x^2)^(1/2))/(31 5*(c^5*d + 2*c^4*d^2*x + c^3*d^3*x^2)) + (8*A*(c^2 - d^2*x^2)^(1/2))/(33*( 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)) - (A*(c^2 - d^2*x^2)^(1/2))/(231*(c^5*d + 4*c^4*d^2*x + c*d^5*x^4 + 6*c^3*d^3*x^2 + 4*c^2*d^4*x^3)) + (41*A*(c^2 - d^2*x^2)^(1/2))/(10395*(c^5 *d + c^4*d^2*x)) + (B*(c^2 - d^2*x^2)^(1/2))/(495*(c^4*d^2 + 2*c^3*d^3*x + c^2*d^4*x^2)) + (16*A*d*(c^2 - d^2*x^2)^(1/2))/(3465*(c^5*d^2 + 2*c^4*d^3 *x + c^3*d^4*x^2)) - (23*B*c*(c^2 - d^2*x^2)^(1/2))/(3465*(c^5*d^2 + 2*c^4 *d^3*x + c^3*d^4*x^2)) - (68*B*c*(c^2 - d^2*x^2)^(1/2))/(99*(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* B*c^2*(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)) - (59*A*d*(c^2 - d^2*x^2)^(1/2))/(10395*(c^5*d^2 + c^4*d^3*x)) + (188*B*c*(c^2 - d^2*x^2) ^(1/2))/(10395*(c^5*d^2 + c^4*d^3*x)) - (4*A*c*(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))
Time = 0.22 (sec) , antiderivative size = 584, normalized size of antiderivative = 3.48 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^8} \, dx=\frac {630 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{5}+141 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{4} d x +1878 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{3} d^{2} x^{2}+1833 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{2} d^{3} x^{3}+906 \sqrt {-d^{2} x^{2}+c^{2}}\, a c \,d^{4} x^{4}+180 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,d^{5} x^{5}+61 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{5} x -1397 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{4} d \,x^{2}-362 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{3} d^{2} x^{3}-209 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{2} d^{3} x^{4}-45 \sqrt {-d^{2} x^{2}+c^{2}}\, b c \,d^{4} x^{5}-630 a \,c^{6}+141 a \,c^{5} d x -3477 a \,c^{4} d^{2} x^{2}-3435 a \,c^{3} d^{3} x^{3}-2553 a \,c^{2} d^{4} x^{4}-1014 a c \,d^{5} x^{5}-168 a \,d^{6} x^{6}+61 b \,c^{6} x +2068 b \,c^{5} d \,x^{2}+185 b \,c^{4} d^{2} x^{3}+1067 b \,c^{3} d^{3} x^{4}+446 b \,c^{2} d^{4} x^{5}+77 b c \,d^{5} x^{6}}{3465 c^{4} d \left (\sqrt {-d^{2} x^{2}+c^{2}}\, c^{5}+5 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{4} d x +10 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{3} d^{2} x^{2}+10 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{2} d^{3} x^{3}+5 \sqrt {-d^{2} x^{2}+c^{2}}\, c \,d^{4} x^{4}+\sqrt {-d^{2} x^{2}+c^{2}}\, d^{5} x^{5}-c^{6}-6 c^{5} d x -15 c^{4} d^{2} x^{2}-20 c^{3} d^{3} x^{3}-15 c^{2} d^{4} x^{4}-6 c \,d^{5} x^{5}-d^{6} x^{6}\right )} \] Input:
int((B*x+A)*(-d^2*x^2+c^2)^(3/2)/(d*x+c)^8,x)
Output:
(630*sqrt(c**2 - d**2*x**2)*a*c**5 + 141*sqrt(c**2 - d**2*x**2)*a*c**4*d*x + 1878*sqrt(c**2 - d**2*x**2)*a*c**3*d**2*x**2 + 1833*sqrt(c**2 - d**2*x* *2)*a*c**2*d**3*x**3 + 906*sqrt(c**2 - d**2*x**2)*a*c*d**4*x**4 + 180*sqrt (c**2 - d**2*x**2)*a*d**5*x**5 + 61*sqrt(c**2 - d**2*x**2)*b*c**5*x - 1397 *sqrt(c**2 - d**2*x**2)*b*c**4*d*x**2 - 362*sqrt(c**2 - d**2*x**2)*b*c**3* d**2*x**3 - 209*sqrt(c**2 - d**2*x**2)*b*c**2*d**3*x**4 - 45*sqrt(c**2 - d **2*x**2)*b*c*d**4*x**5 - 630*a*c**6 + 141*a*c**5*d*x - 3477*a*c**4*d**2*x **2 - 3435*a*c**3*d**3*x**3 - 2553*a*c**2*d**4*x**4 - 1014*a*c*d**5*x**5 - 168*a*d**6*x**6 + 61*b*c**6*x + 2068*b*c**5*d*x**2 + 185*b*c**4*d**2*x**3 + 1067*b*c**3*d**3*x**4 + 446*b*c**2*d**4*x**5 + 77*b*c*d**5*x**6)/(3465* c**4*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))