Integrand size = 17, antiderivative size = 129 \[ \int (a+b x)^4 (c+d x)^{3/2} \, dx=\frac {2 (b c-a d)^4 (c+d x)^{5/2}}{5 d^5}-\frac {8 b (b c-a d)^3 (c+d x)^{7/2}}{7 d^5}+\frac {4 b^2 (b c-a d)^2 (c+d x)^{9/2}}{3 d^5}-\frac {8 b^3 (b c-a d) (c+d x)^{11/2}}{11 d^5}+\frac {2 b^4 (c+d x)^{13/2}}{13 d^5} \] Output:
2/5*(-a*d+b*c)^4*(d*x+c)^(5/2)/d^5-8/7*b*(-a*d+b*c)^3*(d*x+c)^(7/2)/d^5+4/ 3*b^2*(-a*d+b*c)^2*(d*x+c)^(9/2)/d^5-8/11*b^3*(-a*d+b*c)*(d*x+c)^(11/2)/d^ 5+2/13*b^4*(d*x+c)^(13/2)/d^5
Time = 0.08 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.19 \[ \int (a+b x)^4 (c+d x)^{3/2} \, dx=\frac {2 (c+d x)^{5/2} \left (3003 a^4 d^4+1716 a^3 b d^3 (-2 c+5 d x)+286 a^2 b^2 d^2 \left (8 c^2-20 c d x+35 d^2 x^2\right )+52 a b^3 d \left (-16 c^3+40 c^2 d x-70 c d^2 x^2+105 d^3 x^3\right )+b^4 \left (128 c^4-320 c^3 d x+560 c^2 d^2 x^2-840 c d^3 x^3+1155 d^4 x^4\right )\right )}{15015 d^5} \] Input:
Integrate[(a + b*x)^4*(c + d*x)^(3/2),x]
Output:
(2*(c + d*x)^(5/2)*(3003*a^4*d^4 + 1716*a^3*b*d^3*(-2*c + 5*d*x) + 286*a^2 *b^2*d^2*(8*c^2 - 20*c*d*x + 35*d^2*x^2) + 52*a*b^3*d*(-16*c^3 + 40*c^2*d* x - 70*c*d^2*x^2 + 105*d^3*x^3) + b^4*(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)))/(15015*d^5)
Time = 0.23 (sec) , antiderivative size = 129, 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.
| \(\displaystyle \int (a+b x)^4 (c+d x)^{3/2} \, dx\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \int \left (-\frac {4 b^3 (c+d x)^{9/2} (b c-a d)}{d^4}+\frac {6 b^2 (c+d x)^{7/2} (b c-a d)^2}{d^4}-\frac {4 b (c+d x)^{5/2} (b c-a d)^3}{d^4}+\frac {(c+d x)^{3/2} (a d-b c)^4}{d^4}+\frac {b^4 (c+d x)^{11/2}}{d^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {8 b^3 (c+d x)^{11/2} (b c-a d)}{11 d^5}+\frac {4 b^2 (c+d x)^{9/2} (b c-a d)^2}{3 d^5}-\frac {8 b (c+d x)^{7/2} (b c-a d)^3}{7 d^5}+\frac {2 (c+d x)^{5/2} (b c-a d)^4}{5 d^5}+\frac {2 b^4 (c+d x)^{13/2}}{13 d^5}\) |
Input:
Int[(a + b*x)^4*(c + d*x)^(3/2),x]
Output:
(2*(b*c - a*d)^4*(c + d*x)^(5/2))/(5*d^5) - (8*b*(b*c - a*d)^3*(c + d*x)^( 7/2))/(7*d^5) + (4*b^2*(b*c - a*d)^2*(c + d*x)^(9/2))/(3*d^5) - (8*b^3*(b* c - a*d)*(c + d*x)^(11/2))/(11*d^5) + (2*b^4*(c + d*x)^(13/2))/(13*d^5)
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 = 0.30 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.78
| method | result | size |
| derivativedivides | \(\frac {\frac {2 b^{4} \left (x d +c \right )^{\frac {13}{2}}}{13}+\frac {8 b^{3} \left (a d -b c \right ) \left (x d +c \right )^{\frac {11}{2}}}{11}+\frac {4 \left (a d -b c \right )^{2} b^{2} \left (x d +c \right )^{\frac {9}{2}}}{3}+\frac {8 \left (a d -b c \right )^{3} b \left (x d +c \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a d -b c \right )^{4} \left (x d +c \right )^{\frac {5}{2}}}{5}}{d^{5}}\) | \(100\) |
| default | \(\frac {\frac {2 b^{4} \left (x d +c \right )^{\frac {13}{2}}}{13}+\frac {8 b^{3} \left (a d -b c \right ) \left (x d +c \right )^{\frac {11}{2}}}{11}+\frac {4 \left (a d -b c \right )^{2} b^{2} \left (x d +c \right )^{\frac {9}{2}}}{3}+\frac {8 \left (a d -b c \right )^{3} b \left (x d +c \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a d -b c \right )^{4} \left (x d +c \right )^{\frac {5}{2}}}{5}}{d^{5}}\) | \(100\) |
| pseudoelliptic | \(\frac {2 \left (x d +c \right )^{\frac {5}{2}} \left (\left (\frac {5}{13} b^{4} x^{4}+\frac {20}{11} a \,x^{3} b^{3}+\frac {10}{3} a^{2} b^{2} x^{2}+\frac {20}{7} a^{3} b x +a^{4}\right ) d^{4}-\frac {8 c \left (\frac {35}{143} b^{3} x^{3}+\frac {35}{33} a \,b^{2} x^{2}+\frac {5}{3} a^{2} b x +a^{3}\right ) b \,d^{3}}{7}+\frac {16 c^{2} b^{2} \left (\frac {35}{143} b^{2} x^{2}+\frac {10}{11} a b x +a^{2}\right ) d^{2}}{21}-\frac {64 \left (\frac {5 b x}{13}+a \right ) c^{3} b^{3} d}{231}+\frac {128 c^{4} b^{4}}{3003}\right )}{5 d^{5}}\) | \(143\) |
| gosper | \(\frac {2 \left (x d +c \right )^{\frac {5}{2}} \left (1155 d^{4} x^{4} b^{4}+5460 a \,b^{3} d^{4} x^{3}-840 b^{4} c \,d^{3} x^{3}+10010 a^{2} b^{2} d^{4} x^{2}-3640 a \,b^{3} c \,d^{3} x^{2}+560 b^{4} c^{2} d^{2} x^{2}+8580 a^{3} b \,d^{4} x -5720 a^{2} b^{2} c \,d^{3} x +2080 a \,b^{3} c^{2} d^{2} x -320 b^{4} c^{3} d x +3003 d^{4} a^{4}-3432 a^{3} b c \,d^{3}+2288 a^{2} b^{2} c^{2} d^{2}-832 a \,b^{3} c^{3} d +128 c^{4} b^{4}\right )}{15015 d^{5}}\) | \(186\) |
| orering | \(\frac {2 \left (x d +c \right )^{\frac {5}{2}} \left (1155 d^{4} x^{4} b^{4}+5460 a \,b^{3} d^{4} x^{3}-840 b^{4} c \,d^{3} x^{3}+10010 a^{2} b^{2} d^{4} x^{2}-3640 a \,b^{3} c \,d^{3} x^{2}+560 b^{4} c^{2} d^{2} x^{2}+8580 a^{3} b \,d^{4} x -5720 a^{2} b^{2} c \,d^{3} x +2080 a \,b^{3} c^{2} d^{2} x -320 b^{4} c^{3} d x +3003 d^{4} a^{4}-3432 a^{3} b c \,d^{3}+2288 a^{2} b^{2} c^{2} d^{2}-832 a \,b^{3} c^{3} d +128 c^{4} b^{4}\right )}{15015 d^{5}}\) | \(186\) |
| trager | \(\frac {2 \left (1155 b^{4} d^{6} x^{6}+5460 a \,b^{3} d^{6} x^{5}+1470 b^{4} c \,d^{5} x^{5}+10010 a^{2} b^{2} d^{6} x^{4}+7280 a \,b^{3} c \,d^{5} x^{4}+35 b^{4} c^{2} d^{4} x^{4}+8580 a^{3} b \,d^{6} x^{3}+14300 a^{2} b^{2} c \,d^{5} x^{3}+260 a \,b^{3} c^{2} d^{4} x^{3}-40 b^{4} c^{3} d^{3} x^{3}+3003 a^{4} d^{6} x^{2}+13728 a^{3} b c \,d^{5} x^{2}+858 a^{2} b^{2} c^{2} d^{4} x^{2}-312 a \,b^{3} c^{3} d^{3} x^{2}+48 b^{4} c^{4} d^{2} x^{2}+6006 a^{4} c \,d^{5} x +1716 a^{3} b \,c^{2} d^{4} x -1144 a^{2} b^{2} c^{3} d^{3} x +416 a \,b^{3} c^{4} d^{2} x -64 b^{4} c^{5} d x +3003 a^{4} c^{2} d^{4}-3432 a^{3} b \,c^{3} d^{3}+2288 a^{2} b^{2} c^{4} d^{2}-832 a \,b^{3} c^{5} d +128 b^{4} c^{6}\right ) \sqrt {x d +c}}{15015 d^{5}}\) | \(332\) |
| risch | \(\frac {2 \left (1155 b^{4} d^{6} x^{6}+5460 a \,b^{3} d^{6} x^{5}+1470 b^{4} c \,d^{5} x^{5}+10010 a^{2} b^{2} d^{6} x^{4}+7280 a \,b^{3} c \,d^{5} x^{4}+35 b^{4} c^{2} d^{4} x^{4}+8580 a^{3} b \,d^{6} x^{3}+14300 a^{2} b^{2} c \,d^{5} x^{3}+260 a \,b^{3} c^{2} d^{4} x^{3}-40 b^{4} c^{3} d^{3} x^{3}+3003 a^{4} d^{6} x^{2}+13728 a^{3} b c \,d^{5} x^{2}+858 a^{2} b^{2} c^{2} d^{4} x^{2}-312 a \,b^{3} c^{3} d^{3} x^{2}+48 b^{4} c^{4} d^{2} x^{2}+6006 a^{4} c \,d^{5} x +1716 a^{3} b \,c^{2} d^{4} x -1144 a^{2} b^{2} c^{3} d^{3} x +416 a \,b^{3} c^{4} d^{2} x -64 b^{4} c^{5} d x +3003 a^{4} c^{2} d^{4}-3432 a^{3} b \,c^{3} d^{3}+2288 a^{2} b^{2} c^{4} d^{2}-832 a \,b^{3} c^{5} d +128 b^{4} c^{6}\right ) \sqrt {x d +c}}{15015 d^{5}}\) | \(332\) |
Input:
int((b*x+a)^4*(d*x+c)^(3/2),x,method=_RETURNVERBOSE)
Output:
2/d^5*(1/13*b^4*(d*x+c)^(13/2)+4/11*b^3*(a*d-b*c)*(d*x+c)^(11/2)+2/3*(a*d- b*c)^2*b^2*(d*x+c)^(9/2)+4/7*(a*d-b*c)^3*b*(d*x+c)^(7/2)+1/5*(a*d-b*c)^4*( d*x+c)^(5/2))
Leaf count of result is larger than twice the leaf count of optimal. 311 vs. \(2 (109) = 218\).
Time = 0.08 (sec) , antiderivative size = 311, normalized size of antiderivative = 2.41 \[ \int (a+b x)^4 (c+d x)^{3/2} \, dx=\frac {2 \, {\left (1155 \, b^{4} d^{6} x^{6} + 128 \, b^{4} c^{6} - 832 \, a b^{3} c^{5} d + 2288 \, a^{2} b^{2} c^{4} d^{2} - 3432 \, a^{3} b c^{3} d^{3} + 3003 \, a^{4} c^{2} d^{4} + 210 \, {\left (7 \, b^{4} c d^{5} + 26 \, a b^{3} d^{6}\right )} x^{5} + 35 \, {\left (b^{4} c^{2} d^{4} + 208 \, a b^{3} c d^{5} + 286 \, a^{2} b^{2} d^{6}\right )} x^{4} - 20 \, {\left (2 \, b^{4} c^{3} d^{3} - 13 \, a b^{3} c^{2} d^{4} - 715 \, a^{2} b^{2} c d^{5} - 429 \, a^{3} b d^{6}\right )} x^{3} + 3 \, {\left (16 \, b^{4} c^{4} d^{2} - 104 \, a b^{3} c^{3} d^{3} + 286 \, a^{2} b^{2} c^{2} d^{4} + 4576 \, a^{3} b c d^{5} + 1001 \, a^{4} d^{6}\right )} x^{2} - 2 \, {\left (32 \, b^{4} c^{5} d - 208 \, a b^{3} c^{4} d^{2} + 572 \, a^{2} b^{2} c^{3} d^{3} - 858 \, a^{3} b c^{2} d^{4} - 3003 \, a^{4} c d^{5}\right )} x\right )} \sqrt {d x + c}}{15015 \, d^{5}} \] Input:
integrate((b*x+a)^4*(d*x+c)^(3/2),x, algorithm="fricas")
Output:
2/15015*(1155*b^4*d^6*x^6 + 128*b^4*c^6 - 832*a*b^3*c^5*d + 2288*a^2*b^2*c ^4*d^2 - 3432*a^3*b*c^3*d^3 + 3003*a^4*c^2*d^4 + 210*(7*b^4*c*d^5 + 26*a*b ^3*d^6)*x^5 + 35*(b^4*c^2*d^4 + 208*a*b^3*c*d^5 + 286*a^2*b^2*d^6)*x^4 - 2 0*(2*b^4*c^3*d^3 - 13*a*b^3*c^2*d^4 - 715*a^2*b^2*c*d^5 - 429*a^3*b*d^6)*x ^3 + 3*(16*b^4*c^4*d^2 - 104*a*b^3*c^3*d^3 + 286*a^2*b^2*c^2*d^4 + 4576*a^ 3*b*c*d^5 + 1001*a^4*d^6)*x^2 - 2*(32*b^4*c^5*d - 208*a*b^3*c^4*d^2 + 572* a^2*b^2*c^3*d^3 - 858*a^3*b*c^2*d^4 - 3003*a^4*c*d^5)*x)*sqrt(d*x + c)/d^5
Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (119) = 238\).
Time = 0.77 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.88 \[ \int (a+b x)^4 (c+d x)^{3/2} \, dx=\begin {cases} \frac {2 \left (\frac {b^{4} \left (c + d x\right )^{\frac {13}{2}}}{13 d^{4}} + \frac {\left (c + d x\right )^{\frac {11}{2}} \cdot \left (4 a b^{3} d - 4 b^{4} c\right )}{11 d^{4}} + \frac {\left (c + d x\right )^{\frac {9}{2}} \cdot \left (6 a^{2} b^{2} d^{2} - 12 a b^{3} c d + 6 b^{4} c^{2}\right )}{9 d^{4}} + \frac {\left (c + d x\right )^{\frac {7}{2}} \cdot \left (4 a^{3} b d^{3} - 12 a^{2} b^{2} c d^{2} + 12 a b^{3} c^{2} d - 4 b^{4} c^{3}\right )}{7 d^{4}} + \frac {\left (c + d x\right )^{\frac {5}{2}} \left (a^{4} d^{4} - 4 a^{3} b c d^{3} + 6 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d + b^{4} c^{4}\right )}{5 d^{4}}\right )}{d} & \text {for}\: d \neq 0 \\c^{\frac {3}{2}} \left (\begin {cases} a^{4} x & \text {for}\: b = 0 \\\frac {\left (a + b x\right )^{5}}{5 b} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((b*x+a)**4*(d*x+c)**(3/2),x)
Output:
Piecewise((2*(b**4*(c + d*x)**(13/2)/(13*d**4) + (c + d*x)**(11/2)*(4*a*b* *3*d - 4*b**4*c)/(11*d**4) + (c + d*x)**(9/2)*(6*a**2*b**2*d**2 - 12*a*b** 3*c*d + 6*b**4*c**2)/(9*d**4) + (c + d*x)**(7/2)*(4*a**3*b*d**3 - 12*a**2* b**2*c*d**2 + 12*a*b**3*c**2*d - 4*b**4*c**3)/(7*d**4) + (c + d*x)**(5/2)* (a**4*d**4 - 4*a**3*b*c*d**3 + 6*a**2*b**2*c**2*d**2 - 4*a*b**3*c**3*d + b **4*c**4)/(5*d**4))/d, Ne(d, 0)), (c**(3/2)*Piecewise((a**4*x, Eq(b, 0)), ((a + b*x)**5/(5*b), True)), True))
Time = 0.03 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.40 \[ \int (a+b x)^4 (c+d x)^{3/2} \, dx=\frac {2 \, {\left (1155 \, {\left (d x + c\right )}^{\frac {13}{2}} b^{4} - 5460 \, {\left (b^{4} c - a b^{3} d\right )} {\left (d x + c\right )}^{\frac {11}{2}} + 10010 \, {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} {\left (d x + c\right )}^{\frac {9}{2}} - 8580 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} {\left (d x + c\right )}^{\frac {7}{2}} + 3003 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} {\left (d x + c\right )}^{\frac {5}{2}}\right )}}{15015 \, d^{5}} \] Input:
integrate((b*x+a)^4*(d*x+c)^(3/2),x, algorithm="maxima")
Output:
2/15015*(1155*(d*x + c)^(13/2)*b^4 - 5460*(b^4*c - a*b^3*d)*(d*x + c)^(11/ 2) + 10010*(b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*(d*x + c)^(9/2) - 8580*(b ^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*(d*x + c)^(7/2) + 30 03*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4) *(d*x + c)^(5/2))/d^5
Leaf count of result is larger than twice the leaf count of optimal. 807 vs. \(2 (109) = 218\).
Time = 0.13 (sec) , antiderivative size = 807, normalized size of antiderivative = 6.26 \[ \int (a+b x)^4 (c+d x)^{3/2} \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^4*(d*x+c)^(3/2),x, algorithm="giac")
Output:
2/45045*(45045*sqrt(d*x + c)*a^4*c^2 + 30030*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)*a^4*c + 60060*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)*a^3*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^ 4 + 18018*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2 )*a^2*b^2*c^2/d^2 + 24024*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*s qrt(d*x + c)*c^2)*a^3*b*c/d + 5148*(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*b^3*c^2/d^3 + 15444* (5*(d*x + c)^(7/2) - 21*(d*x + c)^(5/2)*c + 35*(d*x + c)^(3/2)*c^2 - 35*sq rt(d*x + c)*c^3)*a^2*b^2*c/d^2 + 5148*(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/d + 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)*b^4*c^2/d^4 + 1144*(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^3*c/d^3 + 858*(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^2/d^2 + 130*(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^4*c/d^4 + 260*(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)*a*b^3/d^...
Time = 0.14 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.87 \[ \int (a+b x)^4 (c+d x)^{3/2} \, dx=\frac {2\,b^4\,{\left (c+d\,x\right )}^{13/2}}{13\,d^5}-\frac {\left (8\,b^4\,c-8\,a\,b^3\,d\right )\,{\left (c+d\,x\right )}^{11/2}}{11\,d^5}+\frac {2\,{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{5/2}}{5\,d^5}+\frac {4\,b^2\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{9/2}}{3\,d^5}+\frac {8\,b\,{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{7/2}}{7\,d^5} \] Input:
int((a + b*x)^4*(c + d*x)^(3/2),x)
Output:
(2*b^4*(c + d*x)^(13/2))/(13*d^5) - ((8*b^4*c - 8*a*b^3*d)*(c + d*x)^(11/2 ))/(11*d^5) + (2*(a*d - b*c)^4*(c + d*x)^(5/2))/(5*d^5) + (4*b^2*(a*d - b* c)^2*(c + d*x)^(9/2))/(3*d^5) + (8*b*(a*d - b*c)^3*(c + d*x)^(7/2))/(7*d^5 )
Time = 0.16 (sec) , antiderivative size = 330, normalized size of antiderivative = 2.56 \[ \int (a+b x)^4 (c+d x)^{3/2} \, dx=\frac {2 \sqrt {d x +c}\, \left (1155 b^{4} d^{6} x^{6}+5460 a \,b^{3} d^{6} x^{5}+1470 b^{4} c \,d^{5} x^{5}+10010 a^{2} b^{2} d^{6} x^{4}+7280 a \,b^{3} c \,d^{5} x^{4}+35 b^{4} c^{2} d^{4} x^{4}+8580 a^{3} b \,d^{6} x^{3}+14300 a^{2} b^{2} c \,d^{5} x^{3}+260 a \,b^{3} c^{2} d^{4} x^{3}-40 b^{4} c^{3} d^{3} x^{3}+3003 a^{4} d^{6} x^{2}+13728 a^{3} b c \,d^{5} x^{2}+858 a^{2} b^{2} c^{2} d^{4} x^{2}-312 a \,b^{3} c^{3} d^{3} x^{2}+48 b^{4} c^{4} d^{2} x^{2}+6006 a^{4} c \,d^{5} x +1716 a^{3} b \,c^{2} d^{4} x -1144 a^{2} b^{2} c^{3} d^{3} x +416 a \,b^{3} c^{4} d^{2} x -64 b^{4} c^{5} d x +3003 a^{4} c^{2} d^{4}-3432 a^{3} b \,c^{3} d^{3}+2288 a^{2} b^{2} c^{4} d^{2}-832 a \,b^{3} c^{5} d +128 b^{4} c^{6}\right )}{15015 d^{5}} \] Input:
int((b*x+a)^4*(d*x+c)^(3/2),x)
Output:
(2*sqrt(c + d*x)*(3003*a**4*c**2*d**4 + 6006*a**4*c*d**5*x + 3003*a**4*d** 6*x**2 - 3432*a**3*b*c**3*d**3 + 1716*a**3*b*c**2*d**4*x + 13728*a**3*b*c* d**5*x**2 + 8580*a**3*b*d**6*x**3 + 2288*a**2*b**2*c**4*d**2 - 1144*a**2*b **2*c**3*d**3*x + 858*a**2*b**2*c**2*d**4*x**2 + 14300*a**2*b**2*c*d**5*x* *3 + 10010*a**2*b**2*d**6*x**4 - 832*a*b**3*c**5*d + 416*a*b**3*c**4*d**2* x - 312*a*b**3*c**3*d**3*x**2 + 260*a*b**3*c**2*d**4*x**3 + 7280*a*b**3*c* d**5*x**4 + 5460*a*b**3*d**6*x**5 + 128*b**4*c**6 - 64*b**4*c**5*d*x + 48* b**4*c**4*d**2*x**2 - 40*b**4*c**3*d**3*x**3 + 35*b**4*c**2*d**4*x**4 + 14 70*b**4*c*d**5*x**5 + 1155*b**4*d**6*x**6))/(15015*d**5)