Integrand size = 31, antiderivative size = 158 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{5/2}}{\sqrt {c+d x}} \, dx=\frac {64 c^2 (B c-13 A d) \left (c^2-d^2 x^2\right )^{7/2}}{9009 d^2 (c+d x)^{7/2}}+\frac {16 c (B c-13 A d) \left (c^2-d^2 x^2\right )^{7/2}}{1287 d^2 (c+d x)^{5/2}}+\frac {2 (B c-13 A d) \left (c^2-d^2 x^2\right )^{7/2}}{143 d^2 (c+d x)^{3/2}}-\frac {2 B \left (c^2-d^2 x^2\right )^{7/2}}{13 d^2 \sqrt {c+d x}} \] Output:
64/9009*c^2*(-13*A*d+B*c)*(-d^2*x^2+c^2)^(7/2)/d^2/(d*x+c)^(7/2)+16/1287*c *(-13*A*d+B*c)*(-d^2*x^2+c^2)^(7/2)/d^2/(d*x+c)^(5/2)+2/143*(-13*A*d+B*c)* (-d^2*x^2+c^2)^(7/2)/d^2/(d*x+c)^(3/2)-2/13*B*(-d^2*x^2+c^2)^(7/2)/d^2/(d* x+c)^(1/2)
Time = 0.71 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.61 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{5/2}}{\sqrt {c+d x}} \, dx=-\frac {2 (c-d x)^3 \sqrt {c^2-d^2 x^2} \left (13 A d \left (151 c^2+182 c d x+63 d^2 x^2\right )+B \left (542 c^3+1897 c^2 d x+2016 c d^2 x^2+693 d^3 x^3\right )\right )}{9009 d^2 \sqrt {c+d x}} \] Input:
Integrate[((A + B*x)*(c^2 - d^2*x^2)^(5/2))/Sqrt[c + d*x],x]
Output:
(-2*(c - d*x)^3*Sqrt[c^2 - d^2*x^2]*(13*A*d*(151*c^2 + 182*c*d*x + 63*d^2* x^2) + B*(542*c^3 + 1897*c^2*d*x + 2016*c*d^2*x^2 + 693*d^3*x^3)))/(9009*d ^2*Sqrt[c + d*x])
Time = 0.43 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {672, 459, 459, 458}
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.
| \(\displaystyle \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{5/2}}{\sqrt {c+d x}} \, dx\) |
| \(\Big \downarrow \) 672 |
| \(\displaystyle -\frac {(B c-13 A d) \int \frac {\left (c^2-d^2 x^2\right )^{5/2}}{\sqrt {c+d x}}dx}{13 d}-\frac {2 B \left (c^2-d^2 x^2\right )^{7/2}}{13 d^2 \sqrt {c+d x}}\) |
| \(\Big \downarrow \) 459 |
| \(\displaystyle -\frac {(B c-13 A d) \left (\frac {8}{11} c \int \frac {\left (c^2-d^2 x^2\right )^{5/2}}{(c+d x)^{3/2}}dx-\frac {2 \left (c^2-d^2 x^2\right )^{7/2}}{11 d (c+d x)^{3/2}}\right )}{13 d}-\frac {2 B \left (c^2-d^2 x^2\right )^{7/2}}{13 d^2 \sqrt {c+d x}}\) |
| \(\Big \downarrow \) 459 |
| \(\displaystyle -\frac {(B c-13 A d) \left (\frac {8}{11} c \left (\frac {4}{9} c \int \frac {\left (c^2-d^2 x^2\right )^{5/2}}{(c+d x)^{5/2}}dx-\frac {2 \left (c^2-d^2 x^2\right )^{7/2}}{9 d (c+d x)^{5/2}}\right )-\frac {2 \left (c^2-d^2 x^2\right )^{7/2}}{11 d (c+d x)^{3/2}}\right )}{13 d}-\frac {2 B \left (c^2-d^2 x^2\right )^{7/2}}{13 d^2 \sqrt {c+d x}}\) |
| \(\Big \downarrow \) 458 |
| \(\displaystyle -\frac {\left (\frac {8}{11} c \left (-\frac {2 \left (c^2-d^2 x^2\right )^{7/2}}{9 d (c+d x)^{5/2}}-\frac {8 c \left (c^2-d^2 x^2\right )^{7/2}}{63 d (c+d x)^{7/2}}\right )-\frac {2 \left (c^2-d^2 x^2\right )^{7/2}}{11 d (c+d x)^{3/2}}\right ) (B c-13 A d)}{13 d}-\frac {2 B \left (c^2-d^2 x^2\right )^{7/2}}{13 d^2 \sqrt {c+d x}}\) |
Input:
Int[((A + B*x)*(c^2 - d^2*x^2)^(5/2))/Sqrt[c + d*x],x]
Output:
(-2*B*(c^2 - d^2*x^2)^(7/2))/(13*d^2*Sqrt[c + d*x]) - ((B*c - 13*A*d)*((-2 *(c^2 - d^2*x^2)^(7/2))/(11*d*(c + d*x)^(3/2)) + (8*c*((-8*c*(c^2 - d^2*x^ 2)^(7/2))/(63*d*(c + d*x)^(7/2)) - (2*(c^2 - d^2*x^2)^(7/2))/(9*d*(c + d*x )^(5/2))))/11))/(13*d)
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ d*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*(p + 1))), x] /; FreeQ[{a, b, c , d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && EqQ[n + p, 0]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ d*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*(n + 2*p + 1))), x] + Simp[2*c* (Simplify[n + p]/(n + 2*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] && IGtQ[Simplif y[n + p], 0]
Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_ ), x_Symbol] :> Simp[g*(d + e*x)^m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Simp[(m*(d*g + e*f) + 2*e*f*(p + 1))/(e*(m + 2*p + 2)) Int[(d + e*x )^m*(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] && NeQ[m + 2*p + 2, 0]
Time = 0.41 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.58
| method | result | size |
| gosper | \(-\frac {2 \left (-d x +c \right ) \left (693 B \,d^{3} x^{3}+819 A \,d^{3} x^{2}+2016 B c \,d^{2} x^{2}+2366 A c \,d^{2} x +1897 B \,c^{2} d x +1963 A \,c^{2} d +542 B \,c^{3}\right ) \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{9009 d^{2} \left (d x +c \right )^{\frac {5}{2}}}\) | \(91\) |
| orering | \(-\frac {2 \left (-d x +c \right ) \left (693 B \,d^{3} x^{3}+819 A \,d^{3} x^{2}+2016 B c \,d^{2} x^{2}+2366 A c \,d^{2} x +1897 B \,c^{2} d x +1963 A \,c^{2} d +542 B \,c^{3}\right ) \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{9009 d^{2} \left (d x +c \right )^{\frac {5}{2}}}\) | \(91\) |
| default | \(-\frac {2 \sqrt {-d^{2} x^{2}+c^{2}}\, \left (-d x +c \right )^{3} \left (693 B \,d^{3} x^{3}+819 A \,d^{3} x^{2}+2016 B c \,d^{2} x^{2}+2366 A c \,d^{2} x +1897 B \,c^{2} d x +1963 A \,c^{2} d +542 B \,c^{3}\right )}{9009 \sqrt {d x +c}\, d^{2}}\) | \(93\) |
| risch | \(-\frac {2 \sqrt {\frac {-d^{2} x^{2}+c^{2}}{d x +c}}\, \sqrt {d x +c}\, \left (-693 B \,d^{6} x^{6}-819 A \,d^{6} x^{5}+63 B c \,d^{5} x^{5}+91 A c \,d^{5} x^{4}+2072 B \,c^{2} d^{4} x^{4}+2678 A \,c^{2} d^{4} x^{3}-206 B \,c^{3} d^{3} x^{3}-390 A \,c^{3} d^{3} x^{2}-2049 B \,c^{4} d^{2} x^{2}-3523 A \,c^{4} d^{2} x +271 B \,c^{5} d x +1963 A \,c^{5} d +542 B \,c^{6}\right ) \sqrt {-d x +c}}{9009 \sqrt {-d^{2} x^{2}+c^{2}}\, d^{2}}\) | \(187\) |
Input:
int((B*x+A)*(-d^2*x^2+c^2)^(5/2)/(d*x+c)^(1/2),x,method=_RETURNVERBOSE)
Output:
-2/9009*(-d*x+c)*(693*B*d^3*x^3+819*A*d^3*x^2+2016*B*c*d^2*x^2+2366*A*c*d^ 2*x+1897*B*c^2*d*x+1963*A*c^2*d+542*B*c^3)*(-d^2*x^2+c^2)^(5/2)/d^2/(d*x+c )^(5/2)
Time = 0.08 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.04 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{5/2}}{\sqrt {c+d x}} \, dx=\frac {2 \, {\left (693 \, B d^{6} x^{6} - 542 \, B c^{6} - 1963 \, A c^{5} d - 63 \, {\left (B c d^{5} - 13 \, A d^{6}\right )} x^{5} - 7 \, {\left (296 \, B c^{2} d^{4} + 13 \, A c d^{5}\right )} x^{4} + 206 \, {\left (B c^{3} d^{3} - 13 \, A c^{2} d^{4}\right )} x^{3} + 3 \, {\left (683 \, B c^{4} d^{2} + 130 \, A c^{3} d^{3}\right )} x^{2} - 271 \, {\left (B c^{5} d - 13 \, A c^{4} d^{2}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}} \sqrt {d x + c}}{9009 \, {\left (d^{3} x + c d^{2}\right )}} \] Input:
integrate((B*x+A)*(-d^2*x^2+c^2)^(5/2)/(d*x+c)^(1/2),x, algorithm="fricas" )
Output:
2/9009*(693*B*d^6*x^6 - 542*B*c^6 - 1963*A*c^5*d - 63*(B*c*d^5 - 13*A*d^6) *x^5 - 7*(296*B*c^2*d^4 + 13*A*c*d^5)*x^4 + 206*(B*c^3*d^3 - 13*A*c^2*d^4) *x^3 + 3*(683*B*c^4*d^2 + 130*A*c^3*d^3)*x^2 - 271*(B*c^5*d - 13*A*c^4*d^2 )*x)*sqrt(-d^2*x^2 + c^2)*sqrt(d*x + c)/(d^3*x + c*d^2)
\[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{5/2}}{\sqrt {c+d x}} \, dx=\int \frac {\left (- \left (- c + d x\right ) \left (c + d x\right )\right )^{\frac {5}{2}} \left (A + B x\right )}{\sqrt {c + d x}}\, dx \] Input:
integrate((B*x+A)*(-d**2*x**2+c**2)**(5/2)/(d*x+c)**(1/2),x)
Output:
Integral((-(-c + d*x)*(c + d*x))**(5/2)*(A + B*x)/sqrt(c + d*x), x)
Time = 0.05 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.91 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{5/2}}{\sqrt {c+d x}} \, dx=\frac {2 \, {\left (63 \, d^{5} x^{5} - 7 \, c d^{4} x^{4} - 206 \, c^{2} d^{3} x^{3} + 30 \, c^{3} d^{2} x^{2} + 271 \, c^{4} d x - 151 \, c^{5}\right )} \sqrt {-d x + c} A}{693 \, d} + \frac {2 \, {\left (693 \, d^{6} x^{6} - 63 \, c d^{5} x^{5} - 2072 \, c^{2} d^{4} x^{4} + 206 \, c^{3} d^{3} x^{3} + 2049 \, c^{4} d^{2} x^{2} - 271 \, c^{5} d x - 542 \, c^{6}\right )} \sqrt {-d x + c} B}{9009 \, d^{2}} \] Input:
integrate((B*x+A)*(-d^2*x^2+c^2)^(5/2)/(d*x+c)^(1/2),x, algorithm="maxima" )
Output:
2/693*(63*d^5*x^5 - 7*c*d^4*x^4 - 206*c^2*d^3*x^3 + 30*c^3*d^2*x^2 + 271*c ^4*d*x - 151*c^5)*sqrt(-d*x + c)*A/d + 2/9009*(693*d^6*x^6 - 63*c*d^5*x^5 - 2072*c^2*d^4*x^4 + 206*c^3*d^3*x^3 + 2049*c^4*d^2*x^2 - 271*c^5*d*x - 54 2*c^6)*sqrt(-d*x + c)*B/d^2
Leaf count of result is larger than twice the leaf count of optimal. 422 vs. \(2 (134) = 268\).
Time = 0.14 (sec) , antiderivative size = 422, normalized size of antiderivative = 2.67 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{5/2}}{\sqrt {c+d x}} \, dx=-\frac {2 \, {\left (15015 \, {\left (-d x + c\right )}^{\frac {3}{2}} A c^{4} d - 3003 \, {\left (3 \, {\left (d x - c\right )}^{2} \sqrt {-d x + c} - 5 \, {\left (-d x + c\right )}^{\frac {3}{2}} c\right )} B c^{4} + 858 \, {\left (15 \, {\left (d x - c\right )}^{3} \sqrt {-d x + c} + 42 \, {\left (d x - c\right )}^{2} \sqrt {-d x + c} c - 35 \, {\left (-d x + c\right )}^{\frac {3}{2}} c^{2}\right )} A c^{2} d + 286 \, {\left (35 \, {\left (d x - c\right )}^{4} \sqrt {-d x + c} + 135 \, {\left (d x - c\right )}^{3} \sqrt {-d x + c} c + 189 \, {\left (d x - c\right )}^{2} \sqrt {-d x + c} c^{2} - 105 \, {\left (-d x + c\right )}^{\frac {3}{2}} c^{3}\right )} B c^{2} - 13 \, {\left (315 \, {\left (d x - c\right )}^{5} \sqrt {-d x + c} + 1540 \, {\left (d x - c\right )}^{4} \sqrt {-d x + c} c + 2970 \, {\left (d x - c\right )}^{3} \sqrt {-d x + c} c^{2} + 2772 \, {\left (d x - c\right )}^{2} \sqrt {-d x + c} c^{3} - 1155 \, {\left (-d x + c\right )}^{\frac {3}{2}} c^{4}\right )} A d - 5 \, {\left (693 \, {\left (d x - c\right )}^{6} \sqrt {-d x + c} + 4095 \, {\left (d x - c\right )}^{5} \sqrt {-d x + c} c + 10010 \, {\left (d x - c\right )}^{4} \sqrt {-d x + c} c^{2} + 12870 \, {\left (d x - c\right )}^{3} \sqrt {-d x + c} c^{3} + 9009 \, {\left (d x - c\right )}^{2} \sqrt {-d x + c} c^{4} - 3003 \, {\left (-d x + c\right )}^{\frac {3}{2}} c^{5}\right )} B\right )}}{45045 \, d^{2}} \] Input:
integrate((B*x+A)*(-d^2*x^2+c^2)^(5/2)/(d*x+c)^(1/2),x, algorithm="giac")
Output:
-2/45045*(15015*(-d*x + c)^(3/2)*A*c^4*d - 3003*(3*(d*x - c)^2*sqrt(-d*x + c) - 5*(-d*x + c)^(3/2)*c)*B*c^4 + 858*(15*(d*x - c)^3*sqrt(-d*x + c) + 4 2*(d*x - c)^2*sqrt(-d*x + c)*c - 35*(-d*x + c)^(3/2)*c^2)*A*c^2*d + 286*(3 5*(d*x - c)^4*sqrt(-d*x + c) + 135*(d*x - c)^3*sqrt(-d*x + c)*c + 189*(d*x - c)^2*sqrt(-d*x + c)*c^2 - 105*(-d*x + c)^(3/2)*c^3)*B*c^2 - 13*(315*(d* x - c)^5*sqrt(-d*x + c) + 1540*(d*x - c)^4*sqrt(-d*x + c)*c + 2970*(d*x - c)^3*sqrt(-d*x + c)*c^2 + 2772*(d*x - c)^2*sqrt(-d*x + c)*c^3 - 1155*(-d*x + c)^(3/2)*c^4)*A*d - 5*(693*(d*x - c)^6*sqrt(-d*x + c) + 4095*(d*x - c)^ 5*sqrt(-d*x + c)*c + 10010*(d*x - c)^4*sqrt(-d*x + c)*c^2 + 12870*(d*x - c )^3*sqrt(-d*x + c)*c^3 + 9009*(d*x - c)^2*sqrt(-d*x + c)*c^4 - 3003*(-d*x + c)^(3/2)*c^5)*B)/d^2
Time = 9.86 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.91 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{5/2}}{\sqrt {c+d x}} \, dx=\frac {\sqrt {c^2-d^2\,x^2}\,\left (\frac {2\,B\,d^4\,x^6}{13}-\frac {1084\,B\,c^6+3926\,A\,d\,c^5}{9009\,d^2}+\frac {2\,c^3\,x^2\,\left (130\,A\,d+683\,B\,c\right )}{3003}+\frac {x^5\,\left (1638\,A\,d^6-126\,B\,c\,d^5\right )}{9009\,d^2}-\frac {412\,c^2\,d\,x^3\,\left (13\,A\,d-B\,c\right )}{9009}+\frac {542\,c^4\,x\,\left (13\,A\,d-B\,c\right )}{9009\,d}-\frac {2\,c\,d^2\,x^4\,\left (13\,A\,d+296\,B\,c\right )}{1287}\right )}{\sqrt {c+d\,x}} \] Input:
int(((c^2 - d^2*x^2)^(5/2)*(A + B*x))/(c + d*x)^(1/2),x)
Output:
((c^2 - d^2*x^2)^(1/2)*((2*B*d^4*x^6)/13 - (1084*B*c^6 + 3926*A*c^5*d)/(90 09*d^2) + (2*c^3*x^2*(130*A*d + 683*B*c))/3003 + (x^5*(1638*A*d^6 - 126*B* c*d^5))/(9009*d^2) - (412*c^2*d*x^3*(13*A*d - B*c))/9009 + (542*c^4*x*(13* A*d - B*c))/(9009*d) - (2*c*d^2*x^4*(13*A*d + 296*B*c))/1287))/(c + d*x)^( 1/2)
Time = 0.23 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.90 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{5/2}}{\sqrt {c+d x}} \, dx=\frac {2 \sqrt {-d x +c}\, \left (693 b \,d^{6} x^{6}+819 a \,d^{6} x^{5}-63 b c \,d^{5} x^{5}-91 a c \,d^{5} x^{4}-2072 b \,c^{2} d^{4} x^{4}-2678 a \,c^{2} d^{4} x^{3}+206 b \,c^{3} d^{3} x^{3}+390 a \,c^{3} d^{3} x^{2}+2049 b \,c^{4} d^{2} x^{2}+3523 a \,c^{4} d^{2} x -271 b \,c^{5} d x -1963 a \,c^{5} d -542 b \,c^{6}\right )}{9009 d^{2}} \] Input:
int((B*x+A)*(-d^2*x^2+c^2)^(5/2)/(d*x+c)^(1/2),x)
Output:
(2*sqrt(c - d*x)*( - 1963*a*c**5*d + 3523*a*c**4*d**2*x + 390*a*c**3*d**3* x**2 - 2678*a*c**2*d**4*x**3 - 91*a*c*d**5*x**4 + 819*a*d**6*x**5 - 542*b* c**6 - 271*b*c**5*d*x + 2049*b*c**4*d**2*x**2 + 206*b*c**3*d**3*x**3 - 207 2*b*c**2*d**4*x**4 - 63*b*c*d**5*x**5 + 693*b*d**6*x**6))/(9009*d**2)