Integrand size = 17, antiderivative size = 158 \[ \int (a+b x)^5 (c+d x)^{3/2} \, dx=-\frac {2 (b c-a d)^5 (c+d x)^{5/2}}{5 d^6}+\frac {10 b (b c-a d)^4 (c+d x)^{7/2}}{7 d^6}-\frac {20 b^2 (b c-a d)^3 (c+d x)^{9/2}}{9 d^6}+\frac {20 b^3 (b c-a d)^2 (c+d x)^{11/2}}{11 d^6}-\frac {10 b^4 (b c-a d) (c+d x)^{13/2}}{13 d^6}+\frac {2 b^5 (c+d x)^{15/2}}{15 d^6} \] Output:
-2/5*(-a*d+b*c)^5*(d*x+c)^(5/2)/d^6+10/7*b*(-a*d+b*c)^4*(d*x+c)^(7/2)/d^6- 20/9*b^2*(-a*d+b*c)^3*(d*x+c)^(9/2)/d^6+20/11*b^3*(-a*d+b*c)^2*(d*x+c)^(11 /2)/d^6-10/13*b^4*(-a*d+b*c)*(d*x+c)^(13/2)/d^6+2/15*b^5*(d*x+c)^(15/2)/d^ 6
Time = 0.11 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.37 \[ \int (a+b x)^5 (c+d x)^{3/2} \, dx=\frac {2 (c+d x)^{5/2} \left (9009 a^5 d^5+6435 a^4 b d^4 (-2 c+5 d x)+1430 a^3 b^2 d^3 \left (8 c^2-20 c d x+35 d^2 x^2\right )+390 a^2 b^3 d^2 \left (-16 c^3+40 c^2 d x-70 c d^2 x^2+105 d^3 x^3\right )+15 a b^4 d \left (128 c^4-320 c^3 d x+560 c^2 d^2 x^2-840 c d^3 x^3+1155 d^4 x^4\right )+b^5 \left (-256 c^5+640 c^4 d x-1120 c^3 d^2 x^2+1680 c^2 d^3 x^3-2310 c d^4 x^4+3003 d^5 x^5\right )\right )}{45045 d^6} \] Input:
Integrate[(a + b*x)^5*(c + d*x)^(3/2),x]
Output:
(2*(c + d*x)^(5/2)*(9009*a^5*d^5 + 6435*a^4*b*d^4*(-2*c + 5*d*x) + 1430*a^ 3*b^2*d^3*(8*c^2 - 20*c*d*x + 35*d^2*x^2) + 390*a^2*b^3*d^2*(-16*c^3 + 40* c^2*d*x - 70*c*d^2*x^2 + 105*d^3*x^3) + 15*a*b^4*d*(128*c^4 - 320*c^3*d*x + 560*c^2*d^2*x^2 - 840*c*d^3*x^3 + 1155*d^4*x^4) + b^5*(-256*c^5 + 640*c^ 4*d*x - 1120*c^3*d^2*x^2 + 1680*c^2*d^3*x^3 - 2310*c*d^4*x^4 + 3003*d^5*x^ 5)))/(45045*d^6)
Time = 0.26 (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.
| \(\displaystyle \int (a+b x)^5 (c+d x)^{3/2} \, dx\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \int \left (-\frac {5 b^4 (c+d x)^{11/2} (b c-a d)}{d^5}+\frac {10 b^3 (c+d x)^{9/2} (b c-a d)^2}{d^5}-\frac {10 b^2 (c+d x)^{7/2} (b c-a d)^3}{d^5}+\frac {5 b (c+d x)^{5/2} (b c-a d)^4}{d^5}+\frac {(c+d x)^{3/2} (a d-b c)^5}{d^5}+\frac {b^5 (c+d x)^{13/2}}{d^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {10 b^4 (c+d x)^{13/2} (b c-a d)}{13 d^6}+\frac {20 b^3 (c+d x)^{11/2} (b c-a d)^2}{11 d^6}-\frac {20 b^2 (c+d x)^{9/2} (b c-a d)^3}{9 d^6}+\frac {10 b (c+d x)^{7/2} (b c-a d)^4}{7 d^6}-\frac {2 (c+d x)^{5/2} (b c-a d)^5}{5 d^6}+\frac {2 b^5 (c+d x)^{15/2}}{15 d^6}\) |
Input:
Int[(a + b*x)^5*(c + d*x)^(3/2),x]
Output:
(-2*(b*c - a*d)^5*(c + d*x)^(5/2))/(5*d^6) + (10*b*(b*c - a*d)^4*(c + d*x) ^(7/2))/(7*d^6) - (20*b^2*(b*c - a*d)^3*(c + d*x)^(9/2))/(9*d^6) + (20*b^3 *(b*c - a*d)^2*(c + d*x)^(11/2))/(11*d^6) - (10*b^4*(b*c - a*d)*(c + d*x)^ (13/2))/(13*d^6) + (2*b^5*(c + d*x)^(15/2))/(15*d^6)
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 = 0.32 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.77
| method | result | size |
| derivativedivides | \(\frac {\frac {2 b^{5} \left (x d +c \right )^{\frac {15}{2}}}{15}+\frac {10 \left (a d -b c \right ) b^{4} \left (x d +c \right )^{\frac {13}{2}}}{13}+\frac {20 \left (a d -b c \right )^{2} b^{3} \left (x d +c \right )^{\frac {11}{2}}}{11}+\frac {20 \left (a d -b c \right )^{3} b^{2} \left (x d +c \right )^{\frac {9}{2}}}{9}+\frac {10 \left (a d -b c \right )^{4} b \left (x d +c \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a d -b c \right )^{5} \left (x d +c \right )^{\frac {5}{2}}}{5}}{d^{6}}\) | \(122\) |
| default | \(\frac {\frac {2 b^{5} \left (x d +c \right )^{\frac {15}{2}}}{15}+\frac {10 \left (a d -b c \right ) b^{4} \left (x d +c \right )^{\frac {13}{2}}}{13}+\frac {20 \left (a d -b c \right )^{2} b^{3} \left (x d +c \right )^{\frac {11}{2}}}{11}+\frac {20 \left (a d -b c \right )^{3} b^{2} \left (x d +c \right )^{\frac {9}{2}}}{9}+\frac {10 \left (a d -b c \right )^{4} b \left (x d +c \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a d -b c \right )^{5} \left (x d +c \right )^{\frac {5}{2}}}{5}}{d^{6}}\) | \(122\) |
| pseudoelliptic | \(\frac {2 \left (\left (\frac {1}{3} b^{5} x^{5}+\frac {25}{13} a \,b^{4} x^{4}+\frac {50}{11} a^{2} b^{3} x^{3}+\frac {50}{9} a^{3} b^{2} x^{2}+\frac {25}{7} a^{4} b x +a^{5}\right ) d^{5}-\frac {10 \left (\frac {7}{39} b^{4} x^{4}+\frac {140}{143} a \,x^{3} b^{3}+\frac {70}{33} a^{2} b^{2} x^{2}+\frac {20}{9} a^{3} b x +a^{4}\right ) c b \,d^{4}}{7}+\frac {80 c^{2} \left (\frac {21}{143} b^{3} x^{3}+\frac {105}{143} a \,b^{2} x^{2}+\frac {15}{11} a^{2} b x +a^{3}\right ) b^{2} d^{3}}{63}-\frac {160 \left (\frac {7}{39} b^{2} x^{2}+\frac {10}{13} a b x +a^{2}\right ) c^{3} b^{3} d^{2}}{231}+\frac {640 c^{4} \left (\frac {b x}{3}+a \right ) b^{4} d}{3003}-\frac {256 c^{5} b^{5}}{9009}\right ) \left (x d +c \right )^{\frac {5}{2}}}{5 d^{6}}\) | \(204\) |
| gosper | \(\frac {2 \left (x d +c \right )^{\frac {5}{2}} \left (3003 x^{5} b^{5} d^{5}+17325 x^{4} a \,b^{4} d^{5}-2310 x^{4} b^{5} c \,d^{4}+40950 x^{3} a^{2} b^{3} d^{5}-12600 x^{3} a \,b^{4} c \,d^{4}+1680 x^{3} b^{5} c^{2} d^{3}+50050 x^{2} a^{3} b^{2} d^{5}-27300 x^{2} a^{2} b^{3} c \,d^{4}+8400 x^{2} a \,b^{4} c^{2} d^{3}-1120 x^{2} b^{5} c^{3} d^{2}+32175 x \,a^{4} b \,d^{5}-28600 x \,a^{3} b^{2} c \,d^{4}+15600 x \,a^{2} b^{3} c^{2} d^{3}-4800 x a \,b^{4} c^{3} d^{2}+640 x \,b^{5} c^{4} d +9009 a^{5} d^{5}-12870 a^{4} b c \,d^{4}+11440 a^{3} b^{2} c^{2} d^{3}-6240 a^{2} b^{3} c^{3} d^{2}+1920 a \,b^{4} c^{4} d -256 c^{5} b^{5}\right )}{45045 d^{6}}\) | \(273\) |
| orering | \(\frac {2 \left (x d +c \right )^{\frac {5}{2}} \left (3003 x^{5} b^{5} d^{5}+17325 x^{4} a \,b^{4} d^{5}-2310 x^{4} b^{5} c \,d^{4}+40950 x^{3} a^{2} b^{3} d^{5}-12600 x^{3} a \,b^{4} c \,d^{4}+1680 x^{3} b^{5} c^{2} d^{3}+50050 x^{2} a^{3} b^{2} d^{5}-27300 x^{2} a^{2} b^{3} c \,d^{4}+8400 x^{2} a \,b^{4} c^{2} d^{3}-1120 x^{2} b^{5} c^{3} d^{2}+32175 x \,a^{4} b \,d^{5}-28600 x \,a^{3} b^{2} c \,d^{4}+15600 x \,a^{2} b^{3} c^{2} d^{3}-4800 x a \,b^{4} c^{3} d^{2}+640 x \,b^{5} c^{4} d +9009 a^{5} d^{5}-12870 a^{4} b c \,d^{4}+11440 a^{3} b^{2} c^{2} d^{3}-6240 a^{2} b^{3} c^{3} d^{2}+1920 a \,b^{4} c^{4} d -256 c^{5} b^{5}\right )}{45045 d^{6}}\) | \(273\) |
| trager | \(\frac {2 \left (3003 b^{5} d^{7} x^{7}+17325 a \,b^{4} d^{7} x^{6}+3696 b^{5} c \,d^{6} x^{6}+40950 a^{2} b^{3} d^{7} x^{5}+22050 a \,b^{4} c \,d^{6} x^{5}+63 b^{5} c^{2} d^{5} x^{5}+50050 a^{3} b^{2} d^{7} x^{4}+54600 a^{2} b^{3} c \,d^{6} x^{4}+525 a \,b^{4} c^{2} d^{5} x^{4}-70 b^{5} c^{3} d^{4} x^{4}+32175 a^{4} b \,d^{7} x^{3}+71500 a^{3} b^{2} c \,d^{6} x^{3}+1950 a^{2} b^{3} c^{2} d^{5} x^{3}-600 a \,b^{4} c^{3} d^{4} x^{3}+80 b^{5} c^{4} d^{3} x^{3}+9009 a^{5} d^{7} x^{2}+51480 a^{4} b c \,d^{6} x^{2}+4290 a^{3} b^{2} c^{2} d^{5} x^{2}-2340 a^{2} b^{3} c^{3} d^{4} x^{2}+720 a \,b^{4} c^{4} d^{3} x^{2}-96 b^{5} c^{5} d^{2} x^{2}+18018 a^{5} c \,d^{6} x +6435 a^{4} b \,c^{2} d^{5} x -5720 a^{3} b^{2} c^{3} d^{4} x +3120 a^{2} b^{3} c^{4} d^{3} x -960 a \,b^{4} c^{5} d^{2} x +128 b^{5} c^{6} d x +9009 a^{5} c^{2} d^{5}-12870 a^{4} b \,c^{3} d^{4}+11440 a^{3} b^{2} c^{4} d^{3}-6240 a^{2} b^{3} c^{5} d^{2}+1920 a \,b^{4} c^{6} d -256 b^{5} c^{7}\right ) \sqrt {x d +c}}{45045 d^{6}}\) | \(453\) |
| risch | \(\frac {2 \left (3003 b^{5} d^{7} x^{7}+17325 a \,b^{4} d^{7} x^{6}+3696 b^{5} c \,d^{6} x^{6}+40950 a^{2} b^{3} d^{7} x^{5}+22050 a \,b^{4} c \,d^{6} x^{5}+63 b^{5} c^{2} d^{5} x^{5}+50050 a^{3} b^{2} d^{7} x^{4}+54600 a^{2} b^{3} c \,d^{6} x^{4}+525 a \,b^{4} c^{2} d^{5} x^{4}-70 b^{5} c^{3} d^{4} x^{4}+32175 a^{4} b \,d^{7} x^{3}+71500 a^{3} b^{2} c \,d^{6} x^{3}+1950 a^{2} b^{3} c^{2} d^{5} x^{3}-600 a \,b^{4} c^{3} d^{4} x^{3}+80 b^{5} c^{4} d^{3} x^{3}+9009 a^{5} d^{7} x^{2}+51480 a^{4} b c \,d^{6} x^{2}+4290 a^{3} b^{2} c^{2} d^{5} x^{2}-2340 a^{2} b^{3} c^{3} d^{4} x^{2}+720 a \,b^{4} c^{4} d^{3} x^{2}-96 b^{5} c^{5} d^{2} x^{2}+18018 a^{5} c \,d^{6} x +6435 a^{4} b \,c^{2} d^{5} x -5720 a^{3} b^{2} c^{3} d^{4} x +3120 a^{2} b^{3} c^{4} d^{3} x -960 a \,b^{4} c^{5} d^{2} x +128 b^{5} c^{6} d x +9009 a^{5} c^{2} d^{5}-12870 a^{4} b \,c^{3} d^{4}+11440 a^{3} b^{2} c^{4} d^{3}-6240 a^{2} b^{3} c^{5} d^{2}+1920 a \,b^{4} c^{6} d -256 b^{5} c^{7}\right ) \sqrt {x d +c}}{45045 d^{6}}\) | \(453\) |
Input:
int((b*x+a)^5*(d*x+c)^(3/2),x,method=_RETURNVERBOSE)
Output:
2/d^6*(1/15*b^5*(d*x+c)^(15/2)+5/13*(a*d-b*c)*b^4*(d*x+c)^(13/2)+10/11*(a* d-b*c)^2*b^3*(d*x+c)^(11/2)+10/9*(a*d-b*c)^3*b^2*(d*x+c)^(9/2)+5/7*(a*d-b* c)^4*b*(d*x+c)^(7/2)+1/5*(a*d-b*c)^5*(d*x+c)^(5/2))
Leaf count of result is larger than twice the leaf count of optimal. 418 vs. \(2 (134) = 268\).
Time = 0.08 (sec) , antiderivative size = 418, normalized size of antiderivative = 2.65 \[ \int (a+b x)^5 (c+d x)^{3/2} \, dx=\frac {2 \, {\left (3003 \, b^{5} d^{7} x^{7} - 256 \, b^{5} c^{7} + 1920 \, a b^{4} c^{6} d - 6240 \, a^{2} b^{3} c^{5} d^{2} + 11440 \, a^{3} b^{2} c^{4} d^{3} - 12870 \, a^{4} b c^{3} d^{4} + 9009 \, a^{5} c^{2} d^{5} + 231 \, {\left (16 \, b^{5} c d^{6} + 75 \, a b^{4} d^{7}\right )} x^{6} + 63 \, {\left (b^{5} c^{2} d^{5} + 350 \, a b^{4} c d^{6} + 650 \, a^{2} b^{3} d^{7}\right )} x^{5} - 35 \, {\left (2 \, b^{5} c^{3} d^{4} - 15 \, a b^{4} c^{2} d^{5} - 1560 \, a^{2} b^{3} c d^{6} - 1430 \, a^{3} b^{2} d^{7}\right )} x^{4} + 5 \, {\left (16 \, b^{5} c^{4} d^{3} - 120 \, a b^{4} c^{3} d^{4} + 390 \, a^{2} b^{3} c^{2} d^{5} + 14300 \, a^{3} b^{2} c d^{6} + 6435 \, a^{4} b d^{7}\right )} x^{3} - 3 \, {\left (32 \, b^{5} c^{5} d^{2} - 240 \, a b^{4} c^{4} d^{3} + 780 \, a^{2} b^{3} c^{3} d^{4} - 1430 \, a^{3} b^{2} c^{2} d^{5} - 17160 \, a^{4} b c d^{6} - 3003 \, a^{5} d^{7}\right )} x^{2} + {\left (128 \, b^{5} c^{6} d - 960 \, a b^{4} c^{5} d^{2} + 3120 \, a^{2} b^{3} c^{4} d^{3} - 5720 \, a^{3} b^{2} c^{3} d^{4} + 6435 \, a^{4} b c^{2} d^{5} + 18018 \, a^{5} c d^{6}\right )} x\right )} \sqrt {d x + c}}{45045 \, d^{6}} \] Input:
integrate((b*x+a)^5*(d*x+c)^(3/2),x, algorithm="fricas")
Output:
2/45045*(3003*b^5*d^7*x^7 - 256*b^5*c^7 + 1920*a*b^4*c^6*d - 6240*a^2*b^3* c^5*d^2 + 11440*a^3*b^2*c^4*d^3 - 12870*a^4*b*c^3*d^4 + 9009*a^5*c^2*d^5 + 231*(16*b^5*c*d^6 + 75*a*b^4*d^7)*x^6 + 63*(b^5*c^2*d^5 + 350*a*b^4*c*d^6 + 650*a^2*b^3*d^7)*x^5 - 35*(2*b^5*c^3*d^4 - 15*a*b^4*c^2*d^5 - 1560*a^2* b^3*c*d^6 - 1430*a^3*b^2*d^7)*x^4 + 5*(16*b^5*c^4*d^3 - 120*a*b^4*c^3*d^4 + 390*a^2*b^3*c^2*d^5 + 14300*a^3*b^2*c*d^6 + 6435*a^4*b*d^7)*x^3 - 3*(32* b^5*c^5*d^2 - 240*a*b^4*c^4*d^3 + 780*a^2*b^3*c^3*d^4 - 1430*a^3*b^2*c^2*d ^5 - 17160*a^4*b*c*d^6 - 3003*a^5*d^7)*x^2 + (128*b^5*c^6*d - 960*a*b^4*c^ 5*d^2 + 3120*a^2*b^3*c^4*d^3 - 5720*a^3*b^2*c^3*d^4 + 6435*a^4*b*c^2*d^5 + 18018*a^5*c*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 = 0.86 (sec) , antiderivative size = 335, normalized size of antiderivative = 2.12 \[ \int (a+b x)^5 (c+d x)^{3/2} \, dx=\begin {cases} \frac {2 \left (\frac {b^{5} \left (c + d x\right )^{\frac {15}{2}}}{15 d^{5}} + \frac {\left (c + d x\right )^{\frac {13}{2}} \cdot \left (5 a b^{4} d - 5 b^{5} c\right )}{13 d^{5}} + \frac {\left (c + d x\right )^{\frac {11}{2}} \cdot \left (10 a^{2} b^{3} d^{2} - 20 a b^{4} c d + 10 b^{5} c^{2}\right )}{11 d^{5}} + \frac {\left (c + d x\right )^{\frac {9}{2}} \cdot \left (10 a^{3} b^{2} d^{3} - 30 a^{2} b^{3} c d^{2} + 30 a b^{4} c^{2} d - 10 b^{5} c^{3}\right )}{9 d^{5}} + \frac {\left (c + d x\right )^{\frac {7}{2}} \cdot \left (5 a^{4} b d^{4} - 20 a^{3} b^{2} c d^{3} + 30 a^{2} b^{3} c^{2} d^{2} - 20 a b^{4} c^{3} d + 5 b^{5} c^{4}\right )}{7 d^{5}} + \frac {\left (c + d x\right )^{\frac {5}{2}} \left (a^{5} d^{5} - 5 a^{4} b c d^{4} + 10 a^{3} b^{2} c^{2} d^{3} - 10 a^{2} b^{3} c^{3} d^{2} + 5 a b^{4} c^{4} d - b^{5} c^{5}\right )}{5 d^{5}}\right )}{d} & \text {for}\: d \neq 0 \\c^{\frac {3}{2}} \left (\begin {cases} a^{5} x & \text {for}\: b = 0 \\\frac {\left (a + b x\right )^{6}}{6 b} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((b*x+a)**5*(d*x+c)**(3/2),x)
Output:
Piecewise((2*(b**5*(c + d*x)**(15/2)/(15*d**5) + (c + d*x)**(13/2)*(5*a*b* *4*d - 5*b**5*c)/(13*d**5) + (c + d*x)**(11/2)*(10*a**2*b**3*d**2 - 20*a*b **4*c*d + 10*b**5*c**2)/(11*d**5) + (c + d*x)**(9/2)*(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)/(9*d**5) + (c + d*x )**(7/2)*(5*a**4*b*d**4 - 20*a**3*b**2*c*d**3 + 30*a**2*b**3*c**2*d**2 - 2 0*a*b**4*c**3*d + 5*b**5*c**4)/(7*d**5) + (c + d*x)**(5/2)*(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)/(5*d**5))/d, Ne(d, 0)), (c**(3/2)*Piecewise((a**5*x, Eq(b, 0)), ((a + b*x)**6/(6*b), True)), True))
Time = 0.03 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.64 \[ \int (a+b x)^5 (c+d x)^{3/2} \, dx=\frac {2 \, {\left (3003 \, {\left (d x + c\right )}^{\frac {15}{2}} b^{5} - 17325 \, {\left (b^{5} c - a b^{4} d\right )} {\left (d x + c\right )}^{\frac {13}{2}} + 40950 \, {\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} {\left (d x + c\right )}^{\frac {11}{2}} - 50050 \, {\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} {\left (d x + c\right )}^{\frac {9}{2}} + 32175 \, {\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )} {\left (d x + c\right )}^{\frac {7}{2}} - 9009 \, {\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )} {\left (d x + c\right )}^{\frac {5}{2}}\right )}}{45045 \, d^{6}} \] Input:
integrate((b*x+a)^5*(d*x+c)^(3/2),x, algorithm="maxima")
Output:
2/45045*(3003*(d*x + c)^(15/2)*b^5 - 17325*(b^5*c - a*b^4*d)*(d*x + c)^(13 /2) + 40950*(b^5*c^2 - 2*a*b^4*c*d + a^2*b^3*d^2)*(d*x + c)^(11/2) - 50050 *(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)^(9/2) + 32175*(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)^(7/2) - 9009*(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)^(5/2))/d^6
Leaf count of result is larger than twice the leaf count of optimal. 1084 vs. \(2 (134) = 268\).
Time = 0.14 (sec) , antiderivative size = 1084, normalized size of antiderivative = 6.86 \[ \int (a+b x)^5 (c+d x)^{3/2} \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^5*(d*x+c)^(3/2),x, algorithm="giac")
Output:
2/45045*(45045*sqrt(d*x + c)*a^5*c^2 + 30030*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)*a^5*c + 75075*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)*a^4*b*c^2/d + 3003*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*a^ 5 + 30030*(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^2/d^2 + 30030*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*s qrt(d*x + c)*c^2)*a^4*b*c/d + 12870*(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^2/d^3 + 257 40*(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/d^2 + 6435*(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/d + 715*(3 5*(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^2/d^4 + 2860*(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/d^3 + 1430*(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^3*b^2/d^2 + 65*(63*(d*x + c)^(11/2) - 3 85*(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^2/d^5 + 650*(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)...
Time = 0.15 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.87 \[ \int (a+b x)^5 (c+d x)^{3/2} \, dx=\frac {2\,b^5\,{\left (c+d\,x\right )}^{15/2}}{15\,d^6}-\frac {\left (10\,b^5\,c-10\,a\,b^4\,d\right )\,{\left (c+d\,x\right )}^{13/2}}{13\,d^6}+\frac {2\,{\left (a\,d-b\,c\right )}^5\,{\left (c+d\,x\right )}^{5/2}}{5\,d^6}+\frac {20\,b^2\,{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{9/2}}{9\,d^6}+\frac {20\,b^3\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{11/2}}{11\,d^6}+\frac {10\,b\,{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{7/2}}{7\,d^6} \] Input:
int((a + b*x)^5*(c + d*x)^(3/2),x)
Output:
(2*b^5*(c + d*x)^(15/2))/(15*d^6) - ((10*b^5*c - 10*a*b^4*d)*(c + d*x)^(13 /2))/(13*d^6) + (2*(a*d - b*c)^5*(c + d*x)^(5/2))/(5*d^6) + (20*b^2*(a*d - b*c)^3*(c + d*x)^(9/2))/(9*d^6) + (20*b^3*(a*d - b*c)^2*(c + d*x)^(11/2)) /(11*d^6) + (10*b*(a*d - b*c)^4*(c + d*x)^(7/2))/(7*d^6)
Time = 0.16 (sec) , antiderivative size = 451, normalized size of antiderivative = 2.85 \[ \int (a+b x)^5 (c+d x)^{3/2} \, dx=\frac {2 \sqrt {d x +c}\, \left (3003 b^{5} d^{7} x^{7}+17325 a \,b^{4} d^{7} x^{6}+3696 b^{5} c \,d^{6} x^{6}+40950 a^{2} b^{3} d^{7} x^{5}+22050 a \,b^{4} c \,d^{6} x^{5}+63 b^{5} c^{2} d^{5} x^{5}+50050 a^{3} b^{2} d^{7} x^{4}+54600 a^{2} b^{3} c \,d^{6} x^{4}+525 a \,b^{4} c^{2} d^{5} x^{4}-70 b^{5} c^{3} d^{4} x^{4}+32175 a^{4} b \,d^{7} x^{3}+71500 a^{3} b^{2} c \,d^{6} x^{3}+1950 a^{2} b^{3} c^{2} d^{5} x^{3}-600 a \,b^{4} c^{3} d^{4} x^{3}+80 b^{5} c^{4} d^{3} x^{3}+9009 a^{5} d^{7} x^{2}+51480 a^{4} b c \,d^{6} x^{2}+4290 a^{3} b^{2} c^{2} d^{5} x^{2}-2340 a^{2} b^{3} c^{3} d^{4} x^{2}+720 a \,b^{4} c^{4} d^{3} x^{2}-96 b^{5} c^{5} d^{2} x^{2}+18018 a^{5} c \,d^{6} x +6435 a^{4} b \,c^{2} d^{5} x -5720 a^{3} b^{2} c^{3} d^{4} x +3120 a^{2} b^{3} c^{4} d^{3} x -960 a \,b^{4} c^{5} d^{2} x +128 b^{5} c^{6} d x +9009 a^{5} c^{2} d^{5}-12870 a^{4} b \,c^{3} d^{4}+11440 a^{3} b^{2} c^{4} d^{3}-6240 a^{2} b^{3} c^{5} d^{2}+1920 a \,b^{4} c^{6} d -256 b^{5} c^{7}\right )}{45045 d^{6}} \] Input:
int((b*x+a)^5*(d*x+c)^(3/2),x)
Output:
(2*sqrt(c + d*x)*(9009*a**5*c**2*d**5 + 18018*a**5*c*d**6*x + 9009*a**5*d* *7*x**2 - 12870*a**4*b*c**3*d**4 + 6435*a**4*b*c**2*d**5*x + 51480*a**4*b* c*d**6*x**2 + 32175*a**4*b*d**7*x**3 + 11440*a**3*b**2*c**4*d**3 - 5720*a* *3*b**2*c**3*d**4*x + 4290*a**3*b**2*c**2*d**5*x**2 + 71500*a**3*b**2*c*d* *6*x**3 + 50050*a**3*b**2*d**7*x**4 - 6240*a**2*b**3*c**5*d**2 + 3120*a**2 *b**3*c**4*d**3*x - 2340*a**2*b**3*c**3*d**4*x**2 + 1950*a**2*b**3*c**2*d* *5*x**3 + 54600*a**2*b**3*c*d**6*x**4 + 40950*a**2*b**3*d**7*x**5 + 1920*a *b**4*c**6*d - 960*a*b**4*c**5*d**2*x + 720*a*b**4*c**4*d**3*x**2 - 600*a* b**4*c**3*d**4*x**3 + 525*a*b**4*c**2*d**5*x**4 + 22050*a*b**4*c*d**6*x**5 + 17325*a*b**4*d**7*x**6 - 256*b**5*c**7 + 128*b**5*c**6*d*x - 96*b**5*c* *5*d**2*x**2 + 80*b**5*c**4*d**3*x**3 - 70*b**5*c**3*d**4*x**4 + 63*b**5*c **2*d**5*x**5 + 3696*b**5*c*d**6*x**6 + 3003*b**5*d**7*x**7))/(45045*d**6)