Integrand size = 17, antiderivative size = 156 \[ \int (a+b x)^5 \sqrt {c+d x} \, dx=-\frac {2 (b c-a d)^5 (c+d x)^{3/2}}{3 d^6}+\frac {2 b (b c-a d)^4 (c+d x)^{5/2}}{d^6}-\frac {20 b^2 (b c-a d)^3 (c+d x)^{7/2}}{7 d^6}+\frac {20 b^3 (b c-a d)^2 (c+d x)^{9/2}}{9 d^6}-\frac {10 b^4 (b c-a d) (c+d x)^{11/2}}{11 d^6}+\frac {2 b^5 (c+d x)^{13/2}}{13 d^6} \]
-2/3*(-a*d+b*c)^5*(d*x+c)^(3/2)/d^6+2*b*(-a*d+b*c)^4*(d*x+c)^(5/2)/d^6-20/ 7*b^2*(-a*d+b*c)^3*(d*x+c)^(7/2)/d^6+20/9*b^3*(-a*d+b*c)^2*(d*x+c)^(9/2)/d ^6-10/11*b^4*(-a*d+b*c)*(d*x+c)^(11/2)/d^6+2/13*b^5*(d*x+c)^(13/2)/d^6
Time = 0.11 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.39 \[ \int (a+b x)^5 \sqrt {c+d x} \, dx=\frac {2 (c+d x)^{3/2} \left (3003 a^5 d^5+3003 a^4 b d^4 (-2 c+3 d x)+858 a^3 b^2 d^3 \left (8 c^2-12 c d x+15 d^2 x^2\right )+286 a^2 b^3 d^2 \left (-16 c^3+24 c^2 d x-30 c d^2 x^2+35 d^3 x^3\right )+13 a b^4 d \left (128 c^4-192 c^3 d x+240 c^2 d^2 x^2-280 c d^3 x^3+315 d^4 x^4\right )+b^5 \left (-256 c^5+384 c^4 d x-480 c^3 d^2 x^2+560 c^2 d^3 x^3-630 c d^4 x^4+693 d^5 x^5\right )\right )}{9009 d^6} \]
(2*(c + d*x)^(3/2)*(3003*a^5*d^5 + 3003*a^4*b*d^4*(-2*c + 3*d*x) + 858*a^3 *b^2*d^3*(8*c^2 - 12*c*d*x + 15*d^2*x^2) + 286*a^2*b^3*d^2*(-16*c^3 + 24*c ^2*d*x - 30*c*d^2*x^2 + 35*d^3*x^3) + 13*a*b^4*d*(128*c^4 - 192*c^3*d*x + 240*c^2*d^2*x^2 - 280*c*d^3*x^3 + 315*d^4*x^4) + b^5*(-256*c^5 + 384*c^4*d *x - 480*c^3*d^2*x^2 + 560*c^2*d^3*x^3 - 630*c*d^4*x^4 + 693*d^5*x^5)))/(9 009*d^6)
Time = 0.28 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {53, 2009}
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.
| \(\displaystyle \int (a+b x)^5 \sqrt {c+d x} \, dx\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \int \left (-\frac {5 b^4 (c+d x)^{9/2} (b c-a d)}{d^5}+\frac {10 b^3 (c+d x)^{7/2} (b c-a d)^2}{d^5}-\frac {10 b^2 (c+d x)^{5/2} (b c-a d)^3}{d^5}+\frac {5 b (c+d x)^{3/2} (b c-a d)^4}{d^5}+\frac {\sqrt {c+d x} (a d-b c)^5}{d^5}+\frac {b^5 (c+d x)^{11/2}}{d^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {10 b^4 (c+d x)^{11/2} (b c-a d)}{11 d^6}+\frac {20 b^3 (c+d x)^{9/2} (b c-a d)^2}{9 d^6}-\frac {20 b^2 (c+d x)^{7/2} (b c-a d)^3}{7 d^6}+\frac {2 b (c+d x)^{5/2} (b c-a d)^4}{d^6}-\frac {2 (c+d x)^{3/2} (b c-a d)^5}{3 d^6}+\frac {2 b^5 (c+d x)^{13/2}}{13 d^6}\) |
(-2*(b*c - a*d)^5*(c + d*x)^(3/2))/(3*d^6) + (2*b*(b*c - a*d)^4*(c + d*x)^ (5/2))/d^6 - (20*b^2*(b*c - a*d)^3*(c + d*x)^(7/2))/(7*d^6) + (20*b^3*(b*c - a*d)^2*(c + d*x)^(9/2))/(9*d^6) - (10*b^4*(b*c - a*d)*(c + d*x)^(11/2)) /(11*d^6) + (2*b^5*(c + d*x)^(13/2))/(13*d^6)
3.14.75.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Time = 0.52 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.78
| method | result | size |
| derivativedivides | \(\frac {\frac {2 b^{5} \left (d x +c \right )^{\frac {13}{2}}}{13}+\frac {10 \left (a d -b c \right ) b^{4} \left (d x +c \right )^{\frac {11}{2}}}{11}+\frac {20 \left (a d -b c \right )^{2} b^{3} \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {20 \left (a d -b c \right )^{3} b^{2} \left (d x +c \right )^{\frac {7}{2}}}{7}+2 \left (a d -b c \right )^{4} b \left (d x +c \right )^{\frac {5}{2}}+\frac {2 \left (a d -b c \right )^{5} \left (d x +c \right )^{\frac {3}{2}}}{3}}{d^{6}}\) | \(121\) |
| default | \(\frac {\frac {2 b^{5} \left (d x +c \right )^{\frac {13}{2}}}{13}+\frac {10 \left (a d -b c \right ) b^{4} \left (d x +c \right )^{\frac {11}{2}}}{11}+\frac {20 \left (a d -b c \right )^{2} b^{3} \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {20 \left (a d -b c \right )^{3} b^{2} \left (d x +c \right )^{\frac {7}{2}}}{7}+2 \left (a d -b c \right )^{4} b \left (d x +c \right )^{\frac {5}{2}}+\frac {2 \left (a d -b c \right )^{5} \left (d x +c \right )^{\frac {3}{2}}}{3}}{d^{6}}\) | \(121\) |
| pseudoelliptic | \(\frac {2 \left (d x +c \right )^{\frac {3}{2}} \left (\left (\frac {3}{13} b^{5} x^{5}+\frac {15}{11} a \,b^{4} x^{4}+\frac {10}{3} a^{2} b^{3} x^{3}+\frac {30}{7} a^{3} b^{2} x^{2}+3 a^{4} b x +a^{5}\right ) d^{5}-2 \left (\frac {15}{143} b^{4} x^{4}+\frac {20}{33} a \,b^{3} x^{3}+\frac {10}{7} a^{2} b^{2} x^{2}+\frac {12}{7} a^{3} b x +a^{4}\right ) b c \,d^{4}+\frac {16 b^{2} \left (\frac {35}{429} b^{3} x^{3}+\frac {5}{11} a \,b^{2} x^{2}+a^{2} b x +a^{3}\right ) c^{2} d^{3}}{7}-\frac {32 b^{3} \left (\frac {15}{143} b^{2} x^{2}+\frac {6}{11} a b x +a^{2}\right ) c^{3} d^{2}}{21}+\frac {128 b^{4} \left (\frac {3 b x}{13}+a \right ) c^{4} d}{231}-\frac {256 b^{5} c^{5}}{3003}\right )}{3 d^{6}}\) | \(203\) |
| gosper | \(\frac {2 \left (d x +c \right )^{\frac {3}{2}} \left (693 x^{5} b^{5} d^{5}+4095 x^{4} a \,b^{4} d^{5}-630 x^{4} b^{5} c \,d^{4}+10010 x^{3} a^{2} b^{3} d^{5}-3640 x^{3} a \,b^{4} c \,d^{4}+560 x^{3} b^{5} c^{2} d^{3}+12870 x^{2} a^{3} b^{2} d^{5}-8580 x^{2} a^{2} b^{3} c \,d^{4}+3120 x^{2} a \,b^{4} c^{2} d^{3}-480 x^{2} b^{5} c^{3} d^{2}+9009 x \,a^{4} b \,d^{5}-10296 x \,a^{3} b^{2} c \,d^{4}+6864 x \,a^{2} b^{3} c^{2} d^{3}-2496 x a \,b^{4} c^{3} d^{2}+384 x \,b^{5} c^{4} d +3003 a^{5} d^{5}-6006 a^{4} b c \,d^{4}+6864 a^{3} b^{2} c^{2} d^{3}-4576 a^{2} b^{3} c^{3} d^{2}+1664 a \,b^{4} c^{4} d -256 b^{5} c^{5}\right )}{9009 d^{6}}\) | \(273\) |
| trager | \(\frac {2 \left (693 b^{5} d^{6} x^{6}+4095 a \,b^{4} d^{6} x^{5}+63 b^{5} c \,d^{5} x^{5}+10010 a^{2} b^{3} d^{6} x^{4}+455 a \,b^{4} c \,d^{5} x^{4}-70 b^{5} c^{2} d^{4} x^{4}+12870 a^{3} b^{2} d^{6} x^{3}+1430 a^{2} b^{3} c \,d^{5} x^{3}-520 a \,b^{4} c^{2} d^{4} x^{3}+80 b^{5} c^{3} d^{3} x^{3}+9009 a^{4} b \,d^{6} x^{2}+2574 a^{3} b^{2} c \,d^{5} x^{2}-1716 a^{2} b^{3} c^{2} d^{4} x^{2}+624 a \,b^{4} c^{3} d^{3} x^{2}-96 b^{5} c^{4} d^{2} x^{2}+3003 a^{5} d^{6} x +3003 a^{4} b c \,d^{5} x -3432 a^{3} b^{2} c^{2} d^{4} x +2288 a^{2} b^{3} c^{3} d^{3} x -832 a \,b^{4} c^{4} d^{2} x +128 b^{5} c^{5} d x +3003 a^{5} c \,d^{5}-6006 a^{4} b \,c^{2} d^{4}+6864 a^{3} b^{2} c^{3} d^{3}-4576 a^{2} b^{3} c^{4} d^{2}+1664 a \,b^{4} c^{5} d -256 b^{5} c^{6}\right ) \sqrt {d x +c}}{9009 d^{6}}\) | \(361\) |
| risch | \(\frac {2 \left (693 b^{5} d^{6} x^{6}+4095 a \,b^{4} d^{6} x^{5}+63 b^{5} c \,d^{5} x^{5}+10010 a^{2} b^{3} d^{6} x^{4}+455 a \,b^{4} c \,d^{5} x^{4}-70 b^{5} c^{2} d^{4} x^{4}+12870 a^{3} b^{2} d^{6} x^{3}+1430 a^{2} b^{3} c \,d^{5} x^{3}-520 a \,b^{4} c^{2} d^{4} x^{3}+80 b^{5} c^{3} d^{3} x^{3}+9009 a^{4} b \,d^{6} x^{2}+2574 a^{3} b^{2} c \,d^{5} x^{2}-1716 a^{2} b^{3} c^{2} d^{4} x^{2}+624 a \,b^{4} c^{3} d^{3} x^{2}-96 b^{5} c^{4} d^{2} x^{2}+3003 a^{5} d^{6} x +3003 a^{4} b c \,d^{5} x -3432 a^{3} b^{2} c^{2} d^{4} x +2288 a^{2} b^{3} c^{3} d^{3} x -832 a \,b^{4} c^{4} d^{2} x +128 b^{5} c^{5} d x +3003 a^{5} c \,d^{5}-6006 a^{4} b \,c^{2} d^{4}+6864 a^{3} b^{2} c^{3} d^{3}-4576 a^{2} b^{3} c^{4} d^{2}+1664 a \,b^{4} c^{5} d -256 b^{5} c^{6}\right ) \sqrt {d x +c}}{9009 d^{6}}\) | \(361\) |
2/d^6*(1/13*b^5*(d*x+c)^(13/2)+5/11*(a*d-b*c)*b^4*(d*x+c)^(11/2)+10/9*(a*d -b*c)^2*b^3*(d*x+c)^(9/2)+10/7*(a*d-b*c)^3*b^2*(d*x+c)^(7/2)+(a*d-b*c)^4*b *(d*x+c)^(5/2)+1/3*(a*d-b*c)^5*(d*x+c)^(3/2))
Leaf count of result is larger than twice the leaf count of optimal. 338 vs. \(2 (134) = 268\).
Time = 0.22 (sec) , antiderivative size = 338, normalized size of antiderivative = 2.17 \[ \int (a+b x)^5 \sqrt {c+d x} \, dx=\frac {2 \, {\left (693 \, b^{5} d^{6} x^{6} - 256 \, b^{5} c^{6} + 1664 \, a b^{4} c^{5} d - 4576 \, a^{2} b^{3} c^{4} d^{2} + 6864 \, a^{3} b^{2} c^{3} d^{3} - 6006 \, a^{4} b c^{2} d^{4} + 3003 \, a^{5} c d^{5} + 63 \, {\left (b^{5} c d^{5} + 65 \, a b^{4} d^{6}\right )} x^{5} - 35 \, {\left (2 \, b^{5} c^{2} d^{4} - 13 \, a b^{4} c d^{5} - 286 \, a^{2} b^{3} d^{6}\right )} x^{4} + 10 \, {\left (8 \, b^{5} c^{3} d^{3} - 52 \, a b^{4} c^{2} d^{4} + 143 \, a^{2} b^{3} c d^{5} + 1287 \, a^{3} b^{2} d^{6}\right )} x^{3} - 3 \, {\left (32 \, b^{5} c^{4} d^{2} - 208 \, a b^{4} c^{3} d^{3} + 572 \, a^{2} b^{3} c^{2} d^{4} - 858 \, a^{3} b^{2} c d^{5} - 3003 \, a^{4} b d^{6}\right )} x^{2} + {\left (128 \, b^{5} c^{5} d - 832 \, a b^{4} c^{4} d^{2} + 2288 \, a^{2} b^{3} c^{3} d^{3} - 3432 \, a^{3} b^{2} c^{2} d^{4} + 3003 \, a^{4} b c d^{5} + 3003 \, a^{5} d^{6}\right )} x\right )} \sqrt {d x + c}}{9009 \, d^{6}} \]
2/9009*(693*b^5*d^6*x^6 - 256*b^5*c^6 + 1664*a*b^4*c^5*d - 4576*a^2*b^3*c^ 4*d^2 + 6864*a^3*b^2*c^3*d^3 - 6006*a^4*b*c^2*d^4 + 3003*a^5*c*d^5 + 63*(b ^5*c*d^5 + 65*a*b^4*d^6)*x^5 - 35*(2*b^5*c^2*d^4 - 13*a*b^4*c*d^5 - 286*a^ 2*b^3*d^6)*x^4 + 10*(8*b^5*c^3*d^3 - 52*a*b^4*c^2*d^4 + 143*a^2*b^3*c*d^5 + 1287*a^3*b^2*d^6)*x^3 - 3*(32*b^5*c^4*d^2 - 208*a*b^4*c^3*d^3 + 572*a^2* b^3*c^2*d^4 - 858*a^3*b^2*c*d^5 - 3003*a^4*b*d^6)*x^2 + (128*b^5*c^5*d - 8 32*a*b^4*c^4*d^2 + 2288*a^2*b^3*c^3*d^3 - 3432*a^3*b^2*c^2*d^4 + 3003*a^4* b*c*d^5 + 3003*a^5*d^6)*x)*sqrt(d*x + c)/d^6
Leaf count of result is larger than twice the leaf count of optimal. 335 vs. \(2 (144) = 288\).
Time = 0.94 (sec) , antiderivative size = 335, normalized size of antiderivative = 2.15 \[ \int (a+b x)^5 \sqrt {c+d x} \, dx=\begin {cases} \frac {2 \left (\frac {b^{5} \left (c + d x\right )^{\frac {13}{2}}}{13 d^{5}} + \frac {\left (c + d x\right )^{\frac {11}{2}} \cdot \left (5 a b^{4} d - 5 b^{5} c\right )}{11 d^{5}} + \frac {\left (c + d x\right )^{\frac {9}{2}} \cdot \left (10 a^{2} b^{3} d^{2} - 20 a b^{4} c d + 10 b^{5} c^{2}\right )}{9 d^{5}} + \frac {\left (c + d x\right )^{\frac {7}{2}} \cdot \left (10 a^{3} b^{2} d^{3} - 30 a^{2} b^{3} c d^{2} + 30 a b^{4} c^{2} d - 10 b^{5} c^{3}\right )}{7 d^{5}} + \frac {\left (c + d x\right )^{\frac {5}{2}} \cdot \left (5 a^{4} b d^{4} - 20 a^{3} b^{2} c d^{3} + 30 a^{2} b^{3} c^{2} d^{2} - 20 a b^{4} c^{3} d + 5 b^{5} c^{4}\right )}{5 d^{5}} + \frac {\left (c + d x\right )^{\frac {3}{2}} \left (a^{5} d^{5} - 5 a^{4} b c d^{4} + 10 a^{3} b^{2} c^{2} d^{3} - 10 a^{2} b^{3} c^{3} d^{2} + 5 a b^{4} c^{4} d - b^{5} c^{5}\right )}{3 d^{5}}\right )}{d} & \text {for}\: d \neq 0 \\\sqrt {c} \left (\begin {cases} a^{5} x & \text {for}\: b = 0 \\\frac {\left (a + b x\right )^{6}}{6 b} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]
Piecewise((2*(b**5*(c + d*x)**(13/2)/(13*d**5) + (c + d*x)**(11/2)*(5*a*b* *4*d - 5*b**5*c)/(11*d**5) + (c + d*x)**(9/2)*(10*a**2*b**3*d**2 - 20*a*b* *4*c*d + 10*b**5*c**2)/(9*d**5) + (c + d*x)**(7/2)*(10*a**3*b**2*d**3 - 30 *a**2*b**3*c*d**2 + 30*a*b**4*c**2*d - 10*b**5*c**3)/(7*d**5) + (c + d*x)* *(5/2)*(5*a**4*b*d**4 - 20*a**3*b**2*c*d**3 + 30*a**2*b**3*c**2*d**2 - 20* a*b**4*c**3*d + 5*b**5*c**4)/(5*d**5) + (c + d*x)**(3/2)*(a**5*d**5 - 5*a* *4*b*c*d**4 + 10*a**3*b**2*c**2*d**3 - 10*a**2*b**3*c**3*d**2 + 5*a*b**4*c **4*d - b**5*c**5)/(3*d**5))/d, Ne(d, 0)), (sqrt(c)*Piecewise((a**5*x, Eq( b, 0)), ((a + b*x)**6/(6*b), True)), True))
Time = 0.20 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.66 \[ \int (a+b x)^5 \sqrt {c+d x} \, dx=\frac {2 \, {\left (693 \, {\left (d x + c\right )}^{\frac {13}{2}} b^{5} - 4095 \, {\left (b^{5} c - a b^{4} d\right )} {\left (d x + c\right )}^{\frac {11}{2}} + 10010 \, {\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} {\left (d x + c\right )}^{\frac {9}{2}} - 12870 \, {\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} {\left (d x + c\right )}^{\frac {7}{2}} + 9009 \, {\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )} {\left (d x + c\right )}^{\frac {5}{2}} - 3003 \, {\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )} {\left (d x + c\right )}^{\frac {3}{2}}\right )}}{9009 \, d^{6}} \]
2/9009*(693*(d*x + c)^(13/2)*b^5 - 4095*(b^5*c - a*b^4*d)*(d*x + c)^(11/2) + 10010*(b^5*c^2 - 2*a*b^4*c*d + a^2*b^3*d^2)*(d*x + c)^(9/2) - 12870*(b^ 5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*(d*x + c)^(7/2) + 9 009*(b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b *d^4)*(d*x + c)^(5/2) - 3003*(b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)*(d*x + c)^(3/2))/d^6
Leaf count of result is larger than twice the leaf count of optimal. 641 vs. \(2 (134) = 268\).
Time = 0.30 (sec) , antiderivative size = 641, normalized size of antiderivative = 4.11 \[ \int (a+b x)^5 \sqrt {c+d x} \, dx =\text {Too large to display} \]
2/9009*(9009*sqrt(d*x + c)*a^5*c + 3003*((d*x + c)^(3/2) - 3*sqrt(d*x + c) *c)*a^5 + 15015*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)*a^4*b*c/d + 6006*(3* (d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*a^3*b^2*c/d ^2 + 3003*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2 )*a^4*b/d + 2574*(5*(d*x + c)^(7/2) - 21*(d*x + c)^(5/2)*c + 35*(d*x + c)^ (3/2)*c^2 - 35*sqrt(d*x + c)*c^3)*a^2*b^3*c/d^3 + 2574*(5*(d*x + c)^(7/2) - 21*(d*x + c)^(5/2)*c + 35*(d*x + c)^(3/2)*c^2 - 35*sqrt(d*x + c)*c^3)*a^ 3*b^2/d^2 + 143*(35*(d*x + c)^(9/2) - 180*(d*x + c)^(7/2)*c + 378*(d*x + c )^(5/2)*c^2 - 420*(d*x + c)^(3/2)*c^3 + 315*sqrt(d*x + c)*c^4)*a*b^4*c/d^4 + 286*(35*(d*x + c)^(9/2) - 180*(d*x + c)^(7/2)*c + 378*(d*x + c)^(5/2)*c ^2 - 420*(d*x + c)^(3/2)*c^3 + 315*sqrt(d*x + c)*c^4)*a^2*b^3/d^3 + 13*(63 *(d*x + c)^(11/2) - 385*(d*x + c)^(9/2)*c + 990*(d*x + c)^(7/2)*c^2 - 1386 *(d*x + c)^(5/2)*c^3 + 1155*(d*x + c)^(3/2)*c^4 - 693*sqrt(d*x + c)*c^5)*b ^5*c/d^5 + 65*(63*(d*x + c)^(11/2) - 385*(d*x + c)^(9/2)*c + 990*(d*x + c) ^(7/2)*c^2 - 1386*(d*x + c)^(5/2)*c^3 + 1155*(d*x + c)^(3/2)*c^4 - 693*sqr t(d*x + c)*c^5)*a*b^4/d^4 + 3*(231*(d*x + c)^(13/2) - 1638*(d*x + c)^(11/2 )*c + 5005*(d*x + c)^(9/2)*c^2 - 8580*(d*x + c)^(7/2)*c^3 + 9009*(d*x + c) ^(5/2)*c^4 - 6006*(d*x + c)^(3/2)*c^5 + 3003*sqrt(d*x + c)*c^6)*b^5/d^5)/d
Time = 0.09 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.88 \[ \int (a+b x)^5 \sqrt {c+d x} \, dx=\frac {2\,b^5\,{\left (c+d\,x\right )}^{13/2}}{13\,d^6}-\frac {\left (10\,b^5\,c-10\,a\,b^4\,d\right )\,{\left (c+d\,x\right )}^{11/2}}{11\,d^6}+\frac {2\,{\left (a\,d-b\,c\right )}^5\,{\left (c+d\,x\right )}^{3/2}}{3\,d^6}+\frac {20\,b^2\,{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{7/2}}{7\,d^6}+\frac {20\,b^3\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{9/2}}{9\,d^6}+\frac {2\,b\,{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{5/2}}{d^6} \]
(2*b^5*(c + d*x)^(13/2))/(13*d^6) - ((10*b^5*c - 10*a*b^4*d)*(c + d*x)^(11 /2))/(11*d^6) + (2*(a*d - b*c)^5*(c + d*x)^(3/2))/(3*d^6) + (20*b^2*(a*d - b*c)^3*(c + d*x)^(7/2))/(7*d^6) + (20*b^3*(a*d - b*c)^2*(c + d*x)^(9/2))/ (9*d^6) + (2*b*(a*d - b*c)^4*(c + d*x)^(5/2))/d^6
Time = 0.00 (sec) , antiderivative size = 359, normalized size of antiderivative = 2.30 \[ \int (a+b x)^5 \sqrt {c+d x} \, dx=\frac {2 \sqrt {d x +c}\, \left (693 b^{5} d^{6} x^{6}+4095 a \,b^{4} d^{6} x^{5}+63 b^{5} c \,d^{5} x^{5}+10010 a^{2} b^{3} d^{6} x^{4}+455 a \,b^{4} c \,d^{5} x^{4}-70 b^{5} c^{2} d^{4} x^{4}+12870 a^{3} b^{2} d^{6} x^{3}+1430 a^{2} b^{3} c \,d^{5} x^{3}-520 a \,b^{4} c^{2} d^{4} x^{3}+80 b^{5} c^{3} d^{3} x^{3}+9009 a^{4} b \,d^{6} x^{2}+2574 a^{3} b^{2} c \,d^{5} x^{2}-1716 a^{2} b^{3} c^{2} d^{4} x^{2}+624 a \,b^{4} c^{3} d^{3} x^{2}-96 b^{5} c^{4} d^{2} x^{2}+3003 a^{5} d^{6} x +3003 a^{4} b c \,d^{5} x -3432 a^{3} b^{2} c^{2} d^{4} x +2288 a^{2} b^{3} c^{3} d^{3} x -832 a \,b^{4} c^{4} d^{2} x +128 b^{5} c^{5} d x +3003 a^{5} c \,d^{5}-6006 a^{4} b \,c^{2} d^{4}+6864 a^{3} b^{2} c^{3} d^{3}-4576 a^{2} b^{3} c^{4} d^{2}+1664 a \,b^{4} c^{5} d -256 b^{5} c^{6}\right )}{9009 d^{6}} \]
int(sqrt(c + d*x)*(a**5 + 5*a**4*b*x + 10*a**3*b**2*x**2 + 10*a**2*b**3*x* *3 + 5*a*b**4*x**4 + b**5*x**5),x)
(2*sqrt(c + d*x)*(3003*a**5*c*d**5 + 3003*a**5*d**6*x - 6006*a**4*b*c**2*d **4 + 3003*a**4*b*c*d**5*x + 9009*a**4*b*d**6*x**2 + 6864*a**3*b**2*c**3*d **3 - 3432*a**3*b**2*c**2*d**4*x + 2574*a**3*b**2*c*d**5*x**2 + 12870*a**3 *b**2*d**6*x**3 - 4576*a**2*b**3*c**4*d**2 + 2288*a**2*b**3*c**3*d**3*x - 1716*a**2*b**3*c**2*d**4*x**2 + 1430*a**2*b**3*c*d**5*x**3 + 10010*a**2*b* *3*d**6*x**4 + 1664*a*b**4*c**5*d - 832*a*b**4*c**4*d**2*x + 624*a*b**4*c* *3*d**3*x**2 - 520*a*b**4*c**2*d**4*x**3 + 455*a*b**4*c*d**5*x**4 + 4095*a *b**4*d**6*x**5 - 256*b**5*c**6 + 128*b**5*c**5*d*x - 96*b**5*c**4*d**2*x* *2 + 80*b**5*c**3*d**3*x**3 - 70*b**5*c**2*d**4*x**4 + 63*b**5*c*d**5*x**5 + 693*b**5*d**6*x**6))/(9009*d**6)