Integrand size = 17, antiderivative size = 129 \[ \int (a+b x)^4 (c+d x)^{5/2} \, dx=\frac {2 (b c-a d)^4 (c+d x)^{7/2}}{7 d^5}-\frac {8 b (b c-a d)^3 (c+d x)^{9/2}}{9 d^5}+\frac {12 b^2 (b c-a d)^2 (c+d x)^{11/2}}{11 d^5}-\frac {8 b^3 (b c-a d) (c+d x)^{13/2}}{13 d^5}+\frac {2 b^4 (c+d x)^{15/2}}{15 d^5} \]
2/7*(-a*d+b*c)^4*(d*x+c)^(7/2)/d^5-8/9*b*(-a*d+b*c)^3*(d*x+c)^(9/2)/d^5+12 /11*b^2*(-a*d+b*c)^2*(d*x+c)^(11/2)/d^5-8/13*b^3*(-a*d+b*c)*(d*x+c)^(13/2) /d^5+2/15*b^4*(d*x+c)^(15/2)/d^5
Time = 0.09 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.19 \[ \int (a+b x)^4 (c+d x)^{5/2} \, dx=\frac {2 (c+d x)^{7/2} \left (6435 a^4 d^4+2860 a^3 b d^3 (-2 c+7 d x)+390 a^2 b^2 d^2 \left (8 c^2-28 c d x+63 d^2 x^2\right )+60 a b^3 d \left (-16 c^3+56 c^2 d x-126 c d^2 x^2+231 d^3 x^3\right )+b^4 \left (128 c^4-448 c^3 d x+1008 c^2 d^2 x^2-1848 c d^3 x^3+3003 d^4 x^4\right )\right )}{45045 d^5} \]
(2*(c + d*x)^(7/2)*(6435*a^4*d^4 + 2860*a^3*b*d^3*(-2*c + 7*d*x) + 390*a^2 *b^2*d^2*(8*c^2 - 28*c*d*x + 63*d^2*x^2) + 60*a*b^3*d*(-16*c^3 + 56*c^2*d* x - 126*c*d^2*x^2 + 231*d^3*x^3) + b^4*(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)))/(45045*d^5)
Time = 0.24 (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.
| \(\displaystyle \int (a+b x)^4 (c+d x)^{5/2} \, dx\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \int \left (-\frac {4 b^3 (c+d x)^{11/2} (b c-a d)}{d^4}+\frac {6 b^2 (c+d x)^{9/2} (b c-a d)^2}{d^4}-\frac {4 b (c+d x)^{7/2} (b c-a d)^3}{d^4}+\frac {(c+d x)^{5/2} (a d-b c)^4}{d^4}+\frac {b^4 (c+d x)^{13/2}}{d^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {8 b^3 (c+d x)^{13/2} (b c-a d)}{13 d^5}+\frac {12 b^2 (c+d x)^{11/2} (b c-a d)^2}{11 d^5}-\frac {8 b (c+d x)^{9/2} (b c-a d)^3}{9 d^5}+\frac {2 (c+d x)^{7/2} (b c-a d)^4}{7 d^5}+\frac {2 b^4 (c+d x)^{15/2}}{15 d^5}\) |
(2*(b*c - a*d)^4*(c + d*x)^(7/2))/(7*d^5) - (8*b*(b*c - a*d)^3*(c + d*x)^( 9/2))/(9*d^5) + (12*b^2*(b*c - a*d)^2*(c + d*x)^(11/2))/(11*d^5) - (8*b^3* (b*c - a*d)*(c + d*x)^(13/2))/(13*d^5) + (2*b^4*(c + d*x)^(15/2))/(15*d^5)
3.14.100.3.1 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 = 0.30 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.78
| method | result | size |
| derivativedivides | \(\frac {\frac {2 b^{4} \left (d x +c \right )^{\frac {15}{2}}}{15}+\frac {8 \left (a d -b c \right ) b^{3} \left (d x +c \right )^{\frac {13}{2}}}{13}+\frac {12 \left (a d -b c \right )^{2} b^{2} \left (d x +c \right )^{\frac {11}{2}}}{11}+\frac {8 \left (a d -b c \right )^{3} b \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {2 \left (a d -b c \right )^{4} \left (d x +c \right )^{\frac {7}{2}}}{7}}{d^{5}}\) | \(100\) |
| default | \(\frac {\frac {2 b^{4} \left (d x +c \right )^{\frac {15}{2}}}{15}+\frac {8 \left (a d -b c \right ) b^{3} \left (d x +c \right )^{\frac {13}{2}}}{13}+\frac {12 \left (a d -b c \right )^{2} b^{2} \left (d x +c \right )^{\frac {11}{2}}}{11}+\frac {8 \left (a d -b c \right )^{3} b \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {2 \left (a d -b c \right )^{4} \left (d x +c \right )^{\frac {7}{2}}}{7}}{d^{5}}\) | \(100\) |
| pseudoelliptic | \(\frac {2 \left (d x +c \right )^{\frac {7}{2}} \left (\left (\frac {7}{15} d^{4} x^{4}-\frac {56}{195} c \,d^{3} x^{3}+\frac {112}{715} c^{2} d^{2} x^{2}-\frac {448}{6435} c^{3} d x +\frac {128}{6435} c^{4}\right ) b^{4}-\frac {64 d \left (-\frac {231}{16} d^{3} x^{3}+\frac {63}{8} c \,d^{2} x^{2}-\frac {7}{2} c^{2} d x +c^{3}\right ) a \,b^{3}}{429}+\frac {16 \left (\frac {63}{8} d^{2} x^{2}-\frac {7}{2} c d x +c^{2}\right ) d^{2} a^{2} b^{2}}{33}-\frac {8 \left (-\frac {7 d x}{2}+c \right ) d^{3} a^{3} b}{9}+a^{4} d^{4}\right )}{7 d^{5}}\) | \(144\) |
| gosper | \(\frac {2 \left (d x +c \right )^{\frac {7}{2}} \left (3003 d^{4} x^{4} b^{4}+13860 a \,b^{3} d^{4} x^{3}-1848 b^{4} c \,d^{3} x^{3}+24570 a^{2} b^{2} d^{4} x^{2}-7560 a \,b^{3} c \,d^{3} x^{2}+1008 b^{4} c^{2} d^{2} x^{2}+20020 a^{3} b \,d^{4} x -10920 a^{2} b^{2} c \,d^{3} x +3360 a \,b^{3} c^{2} d^{2} x -448 b^{4} c^{3} d x +6435 a^{4} d^{4}-5720 a^{3} b c \,d^{3}+3120 a^{2} b^{2} c^{2} d^{2}-960 a \,b^{3} c^{3} d +128 b^{4} c^{4}\right )}{45045 d^{5}}\) | \(186\) |
| trager | \(\frac {2 \left (3003 b^{4} d^{7} x^{7}+13860 a \,b^{3} d^{7} x^{6}+7161 b^{4} c \,d^{6} x^{6}+24570 a^{2} b^{2} d^{7} x^{5}+34020 a \,b^{3} c \,d^{6} x^{5}+4473 b^{4} c^{2} d^{5} x^{5}+20020 a^{3} b \,d^{7} x^{4}+62790 a^{2} b^{2} c \,d^{6} x^{4}+22260 a \,b^{3} c^{2} d^{5} x^{4}+35 b^{4} c^{3} d^{4} x^{4}+6435 a^{4} d^{7} x^{3}+54340 a^{3} b c \,d^{6} x^{3}+44070 a^{2} b^{2} c^{2} d^{5} x^{3}+300 a \,b^{3} c^{3} d^{4} x^{3}-40 b^{4} c^{4} d^{3} x^{3}+19305 a^{4} c \,d^{6} x^{2}+42900 a^{3} b \,c^{2} d^{5} x^{2}+1170 a^{2} b^{2} c^{3} d^{4} x^{2}-360 a \,b^{3} c^{4} d^{3} x^{2}+48 b^{4} c^{5} d^{2} x^{2}+19305 a^{4} c^{2} d^{5} x +2860 a^{3} b \,c^{3} d^{4} x -1560 a^{2} b^{2} c^{4} d^{3} x +480 a \,b^{3} c^{5} d^{2} x -64 b^{4} c^{6} d x +6435 a^{4} c^{3} d^{4}-5720 a^{3} b \,c^{4} d^{3}+3120 a^{2} b^{2} c^{5} d^{2}-960 a \,b^{3} c^{6} d +128 b^{4} c^{7}\right ) \sqrt {d x +c}}{45045 d^{5}}\) | \(407\) |
| risch | \(\frac {2 \left (3003 b^{4} d^{7} x^{7}+13860 a \,b^{3} d^{7} x^{6}+7161 b^{4} c \,d^{6} x^{6}+24570 a^{2} b^{2} d^{7} x^{5}+34020 a \,b^{3} c \,d^{6} x^{5}+4473 b^{4} c^{2} d^{5} x^{5}+20020 a^{3} b \,d^{7} x^{4}+62790 a^{2} b^{2} c \,d^{6} x^{4}+22260 a \,b^{3} c^{2} d^{5} x^{4}+35 b^{4} c^{3} d^{4} x^{4}+6435 a^{4} d^{7} x^{3}+54340 a^{3} b c \,d^{6} x^{3}+44070 a^{2} b^{2} c^{2} d^{5} x^{3}+300 a \,b^{3} c^{3} d^{4} x^{3}-40 b^{4} c^{4} d^{3} x^{3}+19305 a^{4} c \,d^{6} x^{2}+42900 a^{3} b \,c^{2} d^{5} x^{2}+1170 a^{2} b^{2} c^{3} d^{4} x^{2}-360 a \,b^{3} c^{4} d^{3} x^{2}+48 b^{4} c^{5} d^{2} x^{2}+19305 a^{4} c^{2} d^{5} x +2860 a^{3} b \,c^{3} d^{4} x -1560 a^{2} b^{2} c^{4} d^{3} x +480 a \,b^{3} c^{5} d^{2} x -64 b^{4} c^{6} d x +6435 a^{4} c^{3} d^{4}-5720 a^{3} b \,c^{4} d^{3}+3120 a^{2} b^{2} c^{5} d^{2}-960 a \,b^{3} c^{6} d +128 b^{4} c^{7}\right ) \sqrt {d x +c}}{45045 d^{5}}\) | \(407\) |
2/d^5*(1/15*b^4*(d*x+c)^(15/2)+4/13*(a*d-b*c)*b^3*(d*x+c)^(13/2)+6/11*(a*d -b*c)^2*b^2*(d*x+c)^(11/2)+4/9*(a*d-b*c)^3*b*(d*x+c)^(9/2)+1/7*(a*d-b*c)^4 *(d*x+c)^(7/2))
Leaf count of result is larger than twice the leaf count of optimal. 377 vs. \(2 (109) = 218\).
Time = 0.23 (sec) , antiderivative size = 377, normalized size of antiderivative = 2.92 \[ \int (a+b x)^4 (c+d x)^{5/2} \, dx=\frac {2 \, {\left (3003 \, b^{4} d^{7} x^{7} + 128 \, b^{4} c^{7} - 960 \, a b^{3} c^{6} d + 3120 \, a^{2} b^{2} c^{5} d^{2} - 5720 \, a^{3} b c^{4} d^{3} + 6435 \, a^{4} c^{3} d^{4} + 231 \, {\left (31 \, b^{4} c d^{6} + 60 \, a b^{3} d^{7}\right )} x^{6} + 63 \, {\left (71 \, b^{4} c^{2} d^{5} + 540 \, a b^{3} c d^{6} + 390 \, a^{2} b^{2} d^{7}\right )} x^{5} + 35 \, {\left (b^{4} c^{3} d^{4} + 636 \, a b^{3} c^{2} d^{5} + 1794 \, a^{2} b^{2} c d^{6} + 572 \, a^{3} b d^{7}\right )} x^{4} - 5 \, {\left (8 \, b^{4} c^{4} d^{3} - 60 \, a b^{3} c^{3} d^{4} - 8814 \, a^{2} b^{2} c^{2} d^{5} - 10868 \, a^{3} b c d^{6} - 1287 \, a^{4} d^{7}\right )} x^{3} + 3 \, {\left (16 \, b^{4} c^{5} d^{2} - 120 \, a b^{3} c^{4} d^{3} + 390 \, a^{2} b^{2} c^{3} d^{4} + 14300 \, a^{3} b c^{2} d^{5} + 6435 \, a^{4} c d^{6}\right )} x^{2} - {\left (64 \, b^{4} c^{6} d - 480 \, a b^{3} c^{5} d^{2} + 1560 \, a^{2} b^{2} c^{4} d^{3} - 2860 \, a^{3} b c^{3} d^{4} - 19305 \, a^{4} c^{2} d^{5}\right )} x\right )} \sqrt {d x + c}}{45045 \, d^{5}} \]
2/45045*(3003*b^4*d^7*x^7 + 128*b^4*c^7 - 960*a*b^3*c^6*d + 3120*a^2*b^2*c ^5*d^2 - 5720*a^3*b*c^4*d^3 + 6435*a^4*c^3*d^4 + 231*(31*b^4*c*d^6 + 60*a* b^3*d^7)*x^6 + 63*(71*b^4*c^2*d^5 + 540*a*b^3*c*d^6 + 390*a^2*b^2*d^7)*x^5 + 35*(b^4*c^3*d^4 + 636*a*b^3*c^2*d^5 + 1794*a^2*b^2*c*d^6 + 572*a^3*b*d^ 7)*x^4 - 5*(8*b^4*c^4*d^3 - 60*a*b^3*c^3*d^4 - 8814*a^2*b^2*c^2*d^5 - 1086 8*a^3*b*c*d^6 - 1287*a^4*d^7)*x^3 + 3*(16*b^4*c^5*d^2 - 120*a*b^3*c^4*d^3 + 390*a^2*b^2*c^3*d^4 + 14300*a^3*b*c^2*d^5 + 6435*a^4*c*d^6)*x^2 - (64*b^ 4*c^6*d - 480*a*b^3*c^5*d^2 + 1560*a^2*b^2*c^4*d^3 - 2860*a^3*b*c^3*d^4 - 19305*a^4*c^2*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 = 1.10 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.88 \[ \int (a+b x)^4 (c+d x)^{5/2} \, dx=\begin {cases} \frac {2 \left (\frac {b^{4} \left (c + d x\right )^{\frac {15}{2}}}{15 d^{4}} + \frac {\left (c + d x\right )^{\frac {13}{2}} \cdot \left (4 a b^{3} d - 4 b^{4} c\right )}{13 d^{4}} + \frac {\left (c + d x\right )^{\frac {11}{2}} \cdot \left (6 a^{2} b^{2} d^{2} - 12 a b^{3} c d + 6 b^{4} c^{2}\right )}{11 d^{4}} + \frac {\left (c + d x\right )^{\frac {9}{2}} \cdot \left (4 a^{3} b d^{3} - 12 a^{2} b^{2} c d^{2} + 12 a b^{3} c^{2} d - 4 b^{4} c^{3}\right )}{9 d^{4}} + \frac {\left (c + d x\right )^{\frac {7}{2}} \left (a^{4} d^{4} - 4 a^{3} b c d^{3} + 6 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d + b^{4} c^{4}\right )}{7 d^{4}}\right )}{d} & \text {for}\: d \neq 0 \\c^{\frac {5}{2}} \left (\begin {cases} a^{4} x & \text {for}\: b = 0 \\\frac {\left (a + b x\right )^{5}}{5 b} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]
Piecewise((2*(b**4*(c + d*x)**(15/2)/(15*d**4) + (c + d*x)**(13/2)*(4*a*b* *3*d - 4*b**4*c)/(13*d**4) + (c + d*x)**(11/2)*(6*a**2*b**2*d**2 - 12*a*b* *3*c*d + 6*b**4*c**2)/(11*d**4) + (c + d*x)**(9/2)*(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)/(9*d**4) + (c + d*x)**(7/2 )*(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)/(7*d**4))/d, Ne(d, 0)), (c**(5/2)*Piecewise((a**4*x, Eq(b, 0)) , ((a + b*x)**5/(5*b), True)), True))
Time = 0.21 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.40 \[ \int (a+b x)^4 (c+d x)^{5/2} \, dx=\frac {2 \, {\left (3003 \, {\left (d x + c\right )}^{\frac {15}{2}} b^{4} - 13860 \, {\left (b^{4} c - a b^{3} d\right )} {\left (d x + c\right )}^{\frac {13}{2}} + 24570 \, {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} {\left (d x + c\right )}^{\frac {11}{2}} - 20020 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} {\left (d x + c\right )}^{\frac {9}{2}} + 6435 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} {\left (d x + c\right )}^{\frac {7}{2}}\right )}}{45045 \, d^{5}} \]
2/45045*(3003*(d*x + c)^(15/2)*b^4 - 13860*(b^4*c - a*b^3*d)*(d*x + c)^(13 /2) + 24570*(b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*(d*x + c)^(11/2) - 20020 *(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)^(9/2) + 6435*(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)^(7/2))/d^5
Leaf count of result is larger than twice the leaf count of optimal. 1204 vs. \(2 (109) = 218\).
Time = 0.31 (sec) , antiderivative size = 1204, normalized size of antiderivative = 9.33 \[ \int (a+b x)^4 (c+d x)^{5/2} \, dx=\text {Too large to display} \]
2/45045*(45045*sqrt(d*x + c)*a^4*c^3 + 45045*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)*a^4*c^2 + 60060*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)*a^3*b*c^3/d + 9009*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)* a^4*c + 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^3/d^2 + 36036*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*a^3*b*c^2/d + 1287*(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 + 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^3/d^3 + 23166*(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^2*c^2/d^2 + 15444 *(5*(d*x + c)^(7/2) - 21*(d*x + c)^(5/2)*c + 35*(d*x + c)^(3/2)*c^2 - 35*s qrt(d*x + c)*c^3)*a^3*b*c/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^3/d^4 + 1716*(35*(d*x + c)^(9/2) - 180*(d*x + c)^(7/2)*c + 37 8*(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^2/d^3 + 2574*(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 *c/d^2 + 572*(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/d + 195* (63*(d*x + c)^(11/2) - 385*(d*x + c)^(9/2)*c + 990*(d*x + c)^(7/2)*c^2 ...
Time = 0.24 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.87 \[ \int (a+b x)^4 (c+d x)^{5/2} \, dx=\frac {2\,b^4\,{\left (c+d\,x\right )}^{15/2}}{15\,d^5}-\frac {\left (8\,b^4\,c-8\,a\,b^3\,d\right )\,{\left (c+d\,x\right )}^{13/2}}{13\,d^5}+\frac {2\,{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{7/2}}{7\,d^5}+\frac {12\,b^2\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{11/2}}{11\,d^5}+\frac {8\,b\,{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{9/2}}{9\,d^5} \]