Integrand size = 17, antiderivative size = 158 \[ \int (a+b x)^5 (c+d x)^{5/2} \, dx=-\frac {2 (b c-a d)^5 (c+d x)^{7/2}}{7 d^6}+\frac {10 b (b c-a d)^4 (c+d x)^{9/2}}{9 d^6}-\frac {20 b^2 (b c-a d)^3 (c+d x)^{11/2}}{11 d^6}+\frac {20 b^3 (b c-a d)^2 (c+d x)^{13/2}}{13 d^6}-\frac {2 b^4 (b c-a d) (c+d x)^{15/2}}{3 d^6}+\frac {2 b^5 (c+d x)^{17/2}}{17 d^6} \]
-2/7*(-a*d+b*c)^5*(d*x+c)^(7/2)/d^6+10/9*b*(-a*d+b*c)^4*(d*x+c)^(9/2)/d^6- 20/11*b^2*(-a*d+b*c)^3*(d*x+c)^(11/2)/d^6+20/13*b^3*(-a*d+b*c)^2*(d*x+c)^( 13/2)/d^6-2/3*b^4*(-a*d+b*c)*(d*x+c)^(15/2)/d^6+2/17*b^5*(d*x+c)^(17/2)/d^ 6
Time = 0.13 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.37 \[ \int (a+b x)^5 (c+d x)^{5/2} \, dx=\frac {2 (c+d x)^{7/2} \left (21879 a^5 d^5+12155 a^4 b d^4 (-2 c+7 d x)+2210 a^3 b^2 d^3 \left (8 c^2-28 c d x+63 d^2 x^2\right )+510 a^2 b^3 d^2 \left (-16 c^3+56 c^2 d x-126 c d^2 x^2+231 d^3 x^3\right )+17 a b^4 d \left (128 c^4-448 c^3 d x+1008 c^2 d^2 x^2-1848 c d^3 x^3+3003 d^4 x^4\right )+b^5 \left (-256 c^5+896 c^4 d x-2016 c^3 d^2 x^2+3696 c^2 d^3 x^3-6006 c d^4 x^4+9009 d^5 x^5\right )\right )}{153153 d^6} \]
(2*(c + d*x)^(7/2)*(21879*a^5*d^5 + 12155*a^4*b*d^4*(-2*c + 7*d*x) + 2210* a^3*b^2*d^3*(8*c^2 - 28*c*d*x + 63*d^2*x^2) + 510*a^2*b^3*d^2*(-16*c^3 + 5 6*c^2*d*x - 126*c*d^2*x^2 + 231*d^3*x^3) + 17*a*b^4*d*(128*c^4 - 448*c^3*d *x + 1008*c^2*d^2*x^2 - 1848*c*d^3*x^3 + 3003*d^4*x^4) + b^5*(-256*c^5 + 8 96*c^4*d*x - 2016*c^3*d^2*x^2 + 3696*c^2*d^3*x^3 - 6006*c*d^4*x^4 + 9009*d ^5*x^5)))/(153153*d^6)
Time = 0.27 (sec) , antiderivative size = 158, 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.
| \(\displaystyle \int (a+b x)^5 (c+d x)^{5/2} \, dx\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \int \left (-\frac {5 b^4 (c+d x)^{13/2} (b c-a d)}{d^5}+\frac {10 b^3 (c+d x)^{11/2} (b c-a d)^2}{d^5}-\frac {10 b^2 (c+d x)^{9/2} (b c-a d)^3}{d^5}+\frac {5 b (c+d x)^{7/2} (b c-a d)^4}{d^5}+\frac {(c+d x)^{5/2} (a d-b c)^5}{d^5}+\frac {b^5 (c+d x)^{15/2}}{d^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 b^4 (c+d x)^{15/2} (b c-a d)}{3 d^6}+\frac {20 b^3 (c+d x)^{13/2} (b c-a d)^2}{13 d^6}-\frac {20 b^2 (c+d x)^{11/2} (b c-a d)^3}{11 d^6}+\frac {10 b (c+d x)^{9/2} (b c-a d)^4}{9 d^6}-\frac {2 (c+d x)^{7/2} (b c-a d)^5}{7 d^6}+\frac {2 b^5 (c+d x)^{17/2}}{17 d^6}\) |
(-2*(b*c - a*d)^5*(c + d*x)^(7/2))/(7*d^6) + (10*b*(b*c - a*d)^4*(c + d*x) ^(9/2))/(9*d^6) - (20*b^2*(b*c - a*d)^3*(c + d*x)^(11/2))/(11*d^6) + (20*b ^3*(b*c - a*d)^2*(c + d*x)^(13/2))/(13*d^6) - (2*b^4*(b*c - a*d)*(c + d*x) ^(15/2))/(3*d^6) + (2*b^5*(c + d*x)^(17/2))/(17*d^6)
3.14.99.3.1 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 = 0.82 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.77
| method | result | size |
| derivativedivides | \(\frac {\frac {2 b^{5} \left (d x +c \right )^{\frac {17}{2}}}{17}+\frac {2 \left (a d -b c \right ) b^{4} \left (d x +c \right )^{\frac {15}{2}}}{3}+\frac {20 \left (a d -b c \right )^{2} b^{3} \left (d x +c \right )^{\frac {13}{2}}}{13}+\frac {20 \left (a d -b c \right )^{3} b^{2} \left (d x +c \right )^{\frac {11}{2}}}{11}+\frac {10 \left (a d -b c \right )^{4} b \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {2 \left (a d -b c \right )^{5} \left (d x +c \right )^{\frac {7}{2}}}{7}}{d^{6}}\) | \(122\) |
| default | \(\frac {\frac {2 b^{5} \left (d x +c \right )^{\frac {17}{2}}}{17}+\frac {2 \left (a d -b c \right ) b^{4} \left (d x +c \right )^{\frac {15}{2}}}{3}+\frac {20 \left (a d -b c \right )^{2} b^{3} \left (d x +c \right )^{\frac {13}{2}}}{13}+\frac {20 \left (a d -b c \right )^{3} b^{2} \left (d x +c \right )^{\frac {11}{2}}}{11}+\frac {10 \left (a d -b c \right )^{4} b \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {2 \left (a d -b c \right )^{5} \left (d x +c \right )^{\frac {7}{2}}}{7}}{d^{6}}\) | \(122\) |
| pseudoelliptic | \(\frac {2 \left (\left (\frac {7}{17} b^{5} x^{5}+\frac {7}{3} a \,b^{4} x^{4}+\frac {70}{13} a^{2} b^{3} x^{3}+\frac {70}{11} a^{3} b^{2} x^{2}+\frac {35}{9} a^{4} b x +a^{5}\right ) d^{5}-\frac {10 b \left (\frac {21}{85} b^{4} x^{4}+\frac {84}{65} a \,b^{3} x^{3}+\frac {378}{143} a^{2} b^{2} x^{2}+\frac {28}{11} a^{3} b x +a^{4}\right ) c \,d^{4}}{9}+\frac {80 b^{2} \left (\frac {231}{1105} b^{3} x^{3}+\frac {63}{65} a \,b^{2} x^{2}+\frac {21}{13} a^{2} b x +a^{3}\right ) c^{2} d^{3}}{99}-\frac {160 \left (\frac {21}{85} b^{2} x^{2}+\frac {14}{15} a b x +a^{2}\right ) b^{3} c^{3} d^{2}}{429}+\frac {128 b^{4} \left (\frac {7 b x}{17}+a \right ) c^{4} d}{1287}-\frac {256 b^{5} c^{5}}{21879}\right ) \left (d x +c \right )^{\frac {7}{2}}}{7 d^{6}}\) | \(204\) |
| gosper | \(\frac {2 \left (d x +c \right )^{\frac {7}{2}} \left (9009 x^{5} b^{5} d^{5}+51051 x^{4} a \,b^{4} d^{5}-6006 x^{4} b^{5} c \,d^{4}+117810 x^{3} a^{2} b^{3} d^{5}-31416 x^{3} a \,b^{4} c \,d^{4}+3696 x^{3} b^{5} c^{2} d^{3}+139230 x^{2} a^{3} b^{2} d^{5}-64260 x^{2} a^{2} b^{3} c \,d^{4}+17136 x^{2} a \,b^{4} c^{2} d^{3}-2016 x^{2} b^{5} c^{3} d^{2}+85085 x \,a^{4} b \,d^{5}-61880 x \,a^{3} b^{2} c \,d^{4}+28560 x \,a^{2} b^{3} c^{2} d^{3}-7616 x a \,b^{4} c^{3} d^{2}+896 x \,b^{5} c^{4} d +21879 a^{5} d^{5}-24310 a^{4} b c \,d^{4}+17680 a^{3} b^{2} c^{2} d^{3}-8160 a^{2} b^{3} c^{3} d^{2}+2176 a \,b^{4} c^{4} d -256 b^{5} c^{5}\right )}{153153 d^{6}}\) | \(273\) |
| trager | \(\frac {2 \left (9009 b^{5} d^{8} x^{8}+51051 a \,b^{4} d^{8} x^{7}+21021 b^{5} c \,d^{7} x^{7}+117810 a^{2} b^{3} d^{8} x^{6}+121737 a \,b^{4} c \,d^{7} x^{6}+12705 b^{5} c^{2} d^{6} x^{6}+139230 a^{3} b^{2} d^{8} x^{5}+289170 a^{2} b^{3} c \,d^{7} x^{5}+76041 a \,b^{4} c^{2} d^{6} x^{5}+63 b^{5} c^{3} d^{5} x^{5}+85085 a^{4} b \,d^{8} x^{4}+355810 a^{3} b^{2} c \,d^{7} x^{4}+189210 a^{2} b^{3} c^{2} d^{6} x^{4}+595 a \,b^{4} c^{3} d^{5} x^{4}-70 b^{5} c^{4} d^{4} x^{4}+21879 a^{5} d^{8} x^{3}+230945 a^{4} b c \,d^{7} x^{3}+249730 a^{3} b^{2} c^{2} d^{6} x^{3}+2550 a^{2} b^{3} c^{3} d^{5} x^{3}-680 a \,b^{4} c^{4} d^{4} x^{3}+80 b^{5} c^{5} d^{3} x^{3}+65637 a^{5} c \,d^{7} x^{2}+182325 a^{4} b \,c^{2} d^{6} x^{2}+6630 a^{3} b^{2} c^{3} d^{5} x^{2}-3060 a^{2} b^{3} c^{4} d^{4} x^{2}+816 a \,b^{4} c^{5} d^{3} x^{2}-96 b^{5} c^{6} d^{2} x^{2}+65637 a^{5} c^{2} d^{6} x +12155 a^{4} b \,c^{3} d^{5} x -8840 a^{3} b^{2} c^{4} d^{4} x +4080 a^{2} b^{3} c^{5} d^{3} x -1088 a \,b^{4} c^{6} d^{2} x +128 b^{5} c^{7} d x +21879 a^{5} c^{3} d^{5}-24310 a^{4} b \,c^{4} d^{4}+17680 a^{3} b^{2} c^{5} d^{3}-8160 a^{2} b^{3} c^{6} d^{2}+2176 a \,b^{4} c^{7} d -256 b^{5} c^{8}\right ) \sqrt {d x +c}}{153153 d^{6}}\) | \(545\) |
| risch | \(\frac {2 \left (9009 b^{5} d^{8} x^{8}+51051 a \,b^{4} d^{8} x^{7}+21021 b^{5} c \,d^{7} x^{7}+117810 a^{2} b^{3} d^{8} x^{6}+121737 a \,b^{4} c \,d^{7} x^{6}+12705 b^{5} c^{2} d^{6} x^{6}+139230 a^{3} b^{2} d^{8} x^{5}+289170 a^{2} b^{3} c \,d^{7} x^{5}+76041 a \,b^{4} c^{2} d^{6} x^{5}+63 b^{5} c^{3} d^{5} x^{5}+85085 a^{4} b \,d^{8} x^{4}+355810 a^{3} b^{2} c \,d^{7} x^{4}+189210 a^{2} b^{3} c^{2} d^{6} x^{4}+595 a \,b^{4} c^{3} d^{5} x^{4}-70 b^{5} c^{4} d^{4} x^{4}+21879 a^{5} d^{8} x^{3}+230945 a^{4} b c \,d^{7} x^{3}+249730 a^{3} b^{2} c^{2} d^{6} x^{3}+2550 a^{2} b^{3} c^{3} d^{5} x^{3}-680 a \,b^{4} c^{4} d^{4} x^{3}+80 b^{5} c^{5} d^{3} x^{3}+65637 a^{5} c \,d^{7} x^{2}+182325 a^{4} b \,c^{2} d^{6} x^{2}+6630 a^{3} b^{2} c^{3} d^{5} x^{2}-3060 a^{2} b^{3} c^{4} d^{4} x^{2}+816 a \,b^{4} c^{5} d^{3} x^{2}-96 b^{5} c^{6} d^{2} x^{2}+65637 a^{5} c^{2} d^{6} x +12155 a^{4} b \,c^{3} d^{5} x -8840 a^{3} b^{2} c^{4} d^{4} x +4080 a^{2} b^{3} c^{5} d^{3} x -1088 a \,b^{4} c^{6} d^{2} x +128 b^{5} c^{7} d x +21879 a^{5} c^{3} d^{5}-24310 a^{4} b \,c^{4} d^{4}+17680 a^{3} b^{2} c^{5} d^{3}-8160 a^{2} b^{3} c^{6} d^{2}+2176 a \,b^{4} c^{7} d -256 b^{5} c^{8}\right ) \sqrt {d x +c}}{153153 d^{6}}\) | \(545\) |
2/d^6*(1/17*b^5*(d*x+c)^(17/2)+1/3*(a*d-b*c)*b^4*(d*x+c)^(15/2)+10/13*(a*d -b*c)^2*b^3*(d*x+c)^(13/2)+10/11*(a*d-b*c)^3*b^2*(d*x+c)^(11/2)+5/9*(a*d-b *c)^4*b*(d*x+c)^(9/2)+1/7*(a*d-b*c)^5*(d*x+c)^(7/2))
Leaf count of result is larger than twice the leaf count of optimal. 497 vs. \(2 (134) = 268\).
Time = 0.23 (sec) , antiderivative size = 497, normalized size of antiderivative = 3.15 \[ \int (a+b x)^5 (c+d x)^{5/2} \, dx=\frac {2 \, {\left (9009 \, b^{5} d^{8} x^{8} - 256 \, b^{5} c^{8} + 2176 \, a b^{4} c^{7} d - 8160 \, a^{2} b^{3} c^{6} d^{2} + 17680 \, a^{3} b^{2} c^{5} d^{3} - 24310 \, a^{4} b c^{4} d^{4} + 21879 \, a^{5} c^{3} d^{5} + 3003 \, {\left (7 \, b^{5} c d^{7} + 17 \, a b^{4} d^{8}\right )} x^{7} + 231 \, {\left (55 \, b^{5} c^{2} d^{6} + 527 \, a b^{4} c d^{7} + 510 \, a^{2} b^{3} d^{8}\right )} x^{6} + 63 \, {\left (b^{5} c^{3} d^{5} + 1207 \, a b^{4} c^{2} d^{6} + 4590 \, a^{2} b^{3} c d^{7} + 2210 \, a^{3} b^{2} d^{8}\right )} x^{5} - 35 \, {\left (2 \, b^{5} c^{4} d^{4} - 17 \, a b^{4} c^{3} d^{5} - 5406 \, a^{2} b^{3} c^{2} d^{6} - 10166 \, a^{3} b^{2} c d^{7} - 2431 \, a^{4} b d^{8}\right )} x^{4} + {\left (80 \, b^{5} c^{5} d^{3} - 680 \, a b^{4} c^{4} d^{4} + 2550 \, a^{2} b^{3} c^{3} d^{5} + 249730 \, a^{3} b^{2} c^{2} d^{6} + 230945 \, a^{4} b c d^{7} + 21879 \, a^{5} d^{8}\right )} x^{3} - 3 \, {\left (32 \, b^{5} c^{6} d^{2} - 272 \, a b^{4} c^{5} d^{3} + 1020 \, a^{2} b^{3} c^{4} d^{4} - 2210 \, a^{3} b^{2} c^{3} d^{5} - 60775 \, a^{4} b c^{2} d^{6} - 21879 \, a^{5} c d^{7}\right )} x^{2} + {\left (128 \, b^{5} c^{7} d - 1088 \, a b^{4} c^{6} d^{2} + 4080 \, a^{2} b^{3} c^{5} d^{3} - 8840 \, a^{3} b^{2} c^{4} d^{4} + 12155 \, a^{4} b c^{3} d^{5} + 65637 \, a^{5} c^{2} d^{6}\right )} x\right )} \sqrt {d x + c}}{153153 \, d^{6}} \]
2/153153*(9009*b^5*d^8*x^8 - 256*b^5*c^8 + 2176*a*b^4*c^7*d - 8160*a^2*b^3 *c^6*d^2 + 17680*a^3*b^2*c^5*d^3 - 24310*a^4*b*c^4*d^4 + 21879*a^5*c^3*d^5 + 3003*(7*b^5*c*d^7 + 17*a*b^4*d^8)*x^7 + 231*(55*b^5*c^2*d^6 + 527*a*b^4 *c*d^7 + 510*a^2*b^3*d^8)*x^6 + 63*(b^5*c^3*d^5 + 1207*a*b^4*c^2*d^6 + 459 0*a^2*b^3*c*d^7 + 2210*a^3*b^2*d^8)*x^5 - 35*(2*b^5*c^4*d^4 - 17*a*b^4*c^3 *d^5 - 5406*a^2*b^3*c^2*d^6 - 10166*a^3*b^2*c*d^7 - 2431*a^4*b*d^8)*x^4 + (80*b^5*c^5*d^3 - 680*a*b^4*c^4*d^4 + 2550*a^2*b^3*c^3*d^5 + 249730*a^3*b^ 2*c^2*d^6 + 230945*a^4*b*c*d^7 + 21879*a^5*d^8)*x^3 - 3*(32*b^5*c^6*d^2 - 272*a*b^4*c^5*d^3 + 1020*a^2*b^3*c^4*d^4 - 2210*a^3*b^2*c^3*d^5 - 60775*a^ 4*b*c^2*d^6 - 21879*a^5*c*d^7)*x^2 + (128*b^5*c^7*d - 1088*a*b^4*c^6*d^2 + 4080*a^2*b^3*c^5*d^3 - 8840*a^3*b^2*c^4*d^4 + 12155*a^4*b*c^3*d^5 + 65637 *a^5*c^2*d^6)*x)*sqrt(d*x + c)/d^6
Leaf count of result is larger than twice the leaf count of optimal. 335 vs. \(2 (146) = 292\).
Time = 1.16 (sec) , antiderivative size = 335, normalized size of antiderivative = 2.12 \[ \int (a+b x)^5 (c+d x)^{5/2} \, dx=\begin {cases} \frac {2 \left (\frac {b^{5} \left (c + d x\right )^{\frac {17}{2}}}{17 d^{5}} + \frac {\left (c + d x\right )^{\frac {15}{2}} \cdot \left (5 a b^{4} d - 5 b^{5} c\right )}{15 d^{5}} + \frac {\left (c + d x\right )^{\frac {13}{2}} \cdot \left (10 a^{2} b^{3} d^{2} - 20 a b^{4} c d + 10 b^{5} c^{2}\right )}{13 d^{5}} + \frac {\left (c + d x\right )^{\frac {11}{2}} \cdot \left (10 a^{3} b^{2} d^{3} - 30 a^{2} b^{3} c d^{2} + 30 a b^{4} c^{2} d - 10 b^{5} c^{3}\right )}{11 d^{5}} + \frac {\left (c + d x\right )^{\frac {9}{2}} \cdot \left (5 a^{4} b d^{4} - 20 a^{3} b^{2} c d^{3} + 30 a^{2} b^{3} c^{2} d^{2} - 20 a b^{4} c^{3} d + 5 b^{5} c^{4}\right )}{9 d^{5}} + \frac {\left (c + d x\right )^{\frac {7}{2}} \left (a^{5} d^{5} - 5 a^{4} b c d^{4} + 10 a^{3} b^{2} c^{2} d^{3} - 10 a^{2} b^{3} c^{3} d^{2} + 5 a b^{4} c^{4} d - b^{5} c^{5}\right )}{7 d^{5}}\right )}{d} & \text {for}\: d \neq 0 \\c^{\frac {5}{2}} \left (\begin {cases} a^{5} x & \text {for}\: b = 0 \\\frac {\left (a + b x\right )^{6}}{6 b} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]
Piecewise((2*(b**5*(c + d*x)**(17/2)/(17*d**5) + (c + d*x)**(15/2)*(5*a*b* *4*d - 5*b**5*c)/(15*d**5) + (c + d*x)**(13/2)*(10*a**2*b**3*d**2 - 20*a*b **4*c*d + 10*b**5*c**2)/(13*d**5) + (c + d*x)**(11/2)*(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)/(11*d**5) + (c + d *x)**(9/2)*(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)/(9*d**5) + (c + d*x)**(7/2)*(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)/(7*d**5))/d, Ne(d, 0)), (c**(5/2)*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 = 1.64 \[ \int (a+b x)^5 (c+d x)^{5/2} \, dx=\frac {2 \, {\left (9009 \, {\left (d x + c\right )}^{\frac {17}{2}} b^{5} - 51051 \, {\left (b^{5} c - a b^{4} d\right )} {\left (d x + c\right )}^{\frac {15}{2}} + 117810 \, {\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} {\left (d x + c\right )}^{\frac {13}{2}} - 139230 \, {\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} {\left (d x + c\right )}^{\frac {11}{2}} + 85085 \, {\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )} {\left (d x + c\right )}^{\frac {9}{2}} - 21879 \, {\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )} {\left (d x + c\right )}^{\frac {7}{2}}\right )}}{153153 \, d^{6}} \]
2/153153*(9009*(d*x + c)^(17/2)*b^5 - 51051*(b^5*c - a*b^4*d)*(d*x + c)^(1 5/2) + 117810*(b^5*c^2 - 2*a*b^4*c*d + a^2*b^3*d^2)*(d*x + c)^(13/2) - 139 230*(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)^(1 1/2) + 85085*(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)^(9/2) - 21879*(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)^(7/2) )/d^6
Leaf count of result is larger than twice the leaf count of optimal. 1599 vs. \(2 (134) = 268\).
Time = 0.33 (sec) , antiderivative size = 1599, normalized size of antiderivative = 10.12 \[ \int (a+b x)^5 (c+d x)^{5/2} \, dx=\text {Too large to display} \]
2/765765*(765765*sqrt(d*x + c)*a^5*c^3 + 765765*((d*x + c)^(3/2) - 3*sqrt( d*x + c)*c)*a^5*c^2 + 1276275*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)*a^4*b* c^3/d + 153153*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c )*c^2)*a^5*c + 510510*(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^3/d^2 + 765765*(3*(d*x + c)^(5/2) - 10*(d*x + c)^( 3/2)*c + 15*sqrt(d*x + c)*c^2)*a^4*b*c^2/d + 21879*(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^5 + 218790*(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^3/d^3 + 656370*(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* c^2/d^2 + 328185*(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^4*b*c/d + 12155*(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^3/d^4 + 72930*(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*c^2/d^3 + 72930*(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*sq rt(d*x + c)*c^4)*a^3*b^2*c/d^2 + 12155*(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^4*b/d + 1105*(63*(d*x + c)^(11/2) - 385*(d*x + c)^(9/2)*c...
Time = 0.27 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.87 \[ \int (a+b x)^5 (c+d x)^{5/2} \, dx=\frac {2\,b^5\,{\left (c+d\,x\right )}^{17/2}}{17\,d^6}-\frac {\left (10\,b^5\,c-10\,a\,b^4\,d\right )\,{\left (c+d\,x\right )}^{15/2}}{15\,d^6}+\frac {2\,{\left (a\,d-b\,c\right )}^5\,{\left (c+d\,x\right )}^{7/2}}{7\,d^6}+\frac {20\,b^2\,{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{11/2}}{11\,d^6}+\frac {20\,b^3\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{13/2}}{13\,d^6}+\frac {10\,b\,{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{9/2}}{9\,d^6} \]