Integrand size = 21, antiderivative size = 147 \[ \int (d+e x)^{3/2} \left (b x+c x^2\right )^2 \, dx=\frac {2 d^2 (c d-b e)^2 (d+e x)^{5/2}}{5 e^5}-\frac {4 d (c d-b e) (2 c d-b e) (d+e x)^{7/2}}{7 e^5}+\frac {2 \left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^{9/2}}{9 e^5}-\frac {4 c (2 c d-b e) (d+e x)^{11/2}}{11 e^5}+\frac {2 c^2 (d+e x)^{13/2}}{13 e^5} \] Output:
2/5*d^2*(-b*e+c*d)^2*(e*x+d)^(5/2)/e^5-4/7*d*(-b*e+c*d)*(-b*e+2*c*d)*(e*x+ d)^(7/2)/e^5+2/9*(b^2*e^2-6*b*c*d*e+6*c^2*d^2)*(e*x+d)^(9/2)/e^5-4/11*c*(- b*e+2*c*d)*(e*x+d)^(11/2)/e^5+2/13*c^2*(e*x+d)^(13/2)/e^5
Time = 0.08 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.85 \[ \int (d+e x)^{3/2} \left (b x+c x^2\right )^2 \, dx=\frac {2 (d+e x)^{5/2} \left (143 b^2 e^2 \left (8 d^2-20 d e x+35 e^2 x^2\right )+78 b c e \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )+3 c^2 \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )\right )}{45045 e^5} \] Input:
Integrate[(d + e*x)^(3/2)*(b*x + c*x^2)^2,x]
Output:
(2*(d + e*x)^(5/2)*(143*b^2*e^2*(8*d^2 - 20*d*e*x + 35*e^2*x^2) + 78*b*c*e *(-16*d^3 + 40*d^2*e*x - 70*d*e^2*x^2 + 105*e^3*x^3) + 3*c^2*(128*d^4 - 32 0*d^3*e*x + 560*d^2*e^2*x^2 - 840*d*e^3*x^3 + 1155*e^4*x^4)))/(45045*e^5)
Time = 0.45 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {1140, 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.
| \(\displaystyle \int \left (b x+c x^2\right )^2 (d+e x)^{3/2} \, dx\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle \int \left (\frac {(d+e x)^{7/2} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{e^4}+\frac {d^2 (d+e x)^{3/2} (c d-b e)^2}{e^4}-\frac {2 c (d+e x)^{9/2} (2 c d-b e)}{e^4}+\frac {2 d (d+e x)^{5/2} (c d-b e) (b e-2 c d)}{e^4}+\frac {c^2 (d+e x)^{11/2}}{e^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 (d+e x)^{9/2} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{9 e^5}+\frac {2 d^2 (d+e x)^{5/2} (c d-b e)^2}{5 e^5}-\frac {4 c (d+e x)^{11/2} (2 c d-b e)}{11 e^5}-\frac {4 d (d+e x)^{7/2} (c d-b e) (2 c d-b e)}{7 e^5}+\frac {2 c^2 (d+e x)^{13/2}}{13 e^5}\) |
Input:
Int[(d + e*x)^(3/2)*(b*x + c*x^2)^2,x]
Output:
(2*d^2*(c*d - b*e)^2*(d + e*x)^(5/2))/(5*e^5) - (4*d*(c*d - b*e)*(2*c*d - b*e)*(d + e*x)^(7/2))/(7*e^5) + (2*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2)*(d + e*x)^(9/2))/(9*e^5) - (4*c*(2*c*d - b*e)*(d + e*x)^(11/2))/(11*e^5) + (2*c ^2*(d + e*x)^(13/2))/(13*e^5)
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x _Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0]
Time = 0.72 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.73
| method | result | size |
| pseudoelliptic | \(\frac {16 \left (e x +d \right )^{\frac {5}{2}} \left (\frac {35 x^{2} \left (\frac {9}{13} c^{2} x^{2}+\frac {18}{11} c b x +b^{2}\right ) e^{4}}{8}-\frac {5 \left (\frac {126}{143} c^{2} x^{2}+\frac {21}{11} c b x +b^{2}\right ) x d \,e^{3}}{2}+d^{2} \left (\frac {210}{143} c^{2} x^{2}+\frac {30}{11} c b x +b^{2}\right ) e^{2}-\frac {12 \left (\frac {10 c x}{13}+b \right ) c \,d^{3} e}{11}+\frac {48 c^{2} d^{4}}{143}\right )}{315 e^{5}}\) | \(108\) |
| gosper | \(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (3465 c^{2} x^{4} e^{4}+8190 x^{3} b c \,e^{4}-2520 d \,c^{2} x^{3} e^{3}+5005 x^{2} b^{2} e^{4}-5460 x^{2} b c d \,e^{3}+1680 x^{2} c^{2} d^{2} e^{2}-2860 x \,b^{2} d \,e^{3}+3120 x b c \,d^{2} e^{2}-960 x \,c^{2} d^{3} e +1144 d^{2} e^{2} b^{2}-1248 d^{3} e b c +384 c^{2} d^{4}\right )}{45045 e^{5}}\) | \(141\) |
| derivativedivides | \(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (-2 c^{2} d +2 c \left (b e -c d \right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (c^{2} d^{2}-4 d c \left (b e -c d \right )+\left (b e -c d \right )^{2}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (2 d^{2} c \left (b e -c d \right )-2 d \left (b e -c d \right )^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 d^{2} \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{5}}\) | \(144\) |
| default | \(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (-2 c^{2} d +2 c \left (b e -c d \right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (c^{2} d^{2}-4 d c \left (b e -c d \right )+\left (b e -c d \right )^{2}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (2 d^{2} c \left (b e -c d \right )-2 d \left (b e -c d \right )^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 d^{2} \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{5}}\) | \(144\) |
| orering | \(\frac {2 \left (3465 c^{2} x^{4} e^{4}+8190 x^{3} b c \,e^{4}-2520 d \,c^{2} x^{3} e^{3}+5005 x^{2} b^{2} e^{4}-5460 x^{2} b c d \,e^{3}+1680 x^{2} c^{2} d^{2} e^{2}-2860 x \,b^{2} d \,e^{3}+3120 x b c \,d^{2} e^{2}-960 x \,c^{2} d^{3} e +1144 d^{2} e^{2} b^{2}-1248 d^{3} e b c +384 c^{2} d^{4}\right ) \left (e x +d \right )^{\frac {5}{2}} \left (c \,x^{2}+b x \right )^{2}}{45045 e^{5} \left (c x +b \right )^{2} x^{2}}\) | \(162\) |
| trager | \(\frac {2 \left (3465 e^{6} c^{2} x^{6}+8190 b c \,e^{6} x^{5}+4410 c^{2} d \,e^{5} x^{5}+5005 b^{2} e^{6} x^{4}+10920 b c d \,e^{5} x^{4}+105 c^{2} d^{2} e^{4} x^{4}+7150 b^{2} d \,e^{5} x^{3}+390 b c \,d^{2} e^{4} x^{3}-120 c^{2} d^{3} e^{3} x^{3}+429 b^{2} d^{2} e^{4} x^{2}-468 b c \,d^{3} e^{3} x^{2}+144 c^{2} d^{4} e^{2} x^{2}-572 b^{2} d^{3} e^{3} x +624 b c \,d^{4} e^{2} x -192 c^{2} d^{5} e x +1144 b^{2} d^{4} e^{2}-1248 b c \,d^{5} e +384 d^{6} c^{2}\right ) \sqrt {e x +d}}{45045 e^{5}}\) | \(223\) |
| risch | \(\frac {2 \left (3465 e^{6} c^{2} x^{6}+8190 b c \,e^{6} x^{5}+4410 c^{2} d \,e^{5} x^{5}+5005 b^{2} e^{6} x^{4}+10920 b c d \,e^{5} x^{4}+105 c^{2} d^{2} e^{4} x^{4}+7150 b^{2} d \,e^{5} x^{3}+390 b c \,d^{2} e^{4} x^{3}-120 c^{2} d^{3} e^{3} x^{3}+429 b^{2} d^{2} e^{4} x^{2}-468 b c \,d^{3} e^{3} x^{2}+144 c^{2} d^{4} e^{2} x^{2}-572 b^{2} d^{3} e^{3} x +624 b c \,d^{4} e^{2} x -192 c^{2} d^{5} e x +1144 b^{2} d^{4} e^{2}-1248 b c \,d^{5} e +384 d^{6} c^{2}\right ) \sqrt {e x +d}}{45045 e^{5}}\) | \(223\) |
Input:
int((e*x+d)^(3/2)*(c*x^2+b*x)^2,x,method=_RETURNVERBOSE)
Output:
16/315*(e*x+d)^(5/2)*(35/8*x^2*(9/13*c^2*x^2+18/11*c*b*x+b^2)*e^4-5/2*(126 /143*c^2*x^2+21/11*c*b*x+b^2)*x*d*e^3+d^2*(210/143*c^2*x^2+30/11*c*b*x+b^2 )*e^2-12/11*(10/13*c*x+b)*c*d^3*e+48/143*c^2*d^4)/e^5
Time = 0.08 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.46 \[ \int (d+e x)^{3/2} \left (b x+c x^2\right )^2 \, dx=\frac {2 \, {\left (3465 \, c^{2} e^{6} x^{6} + 384 \, c^{2} d^{6} - 1248 \, b c d^{5} e + 1144 \, b^{2} d^{4} e^{2} + 630 \, {\left (7 \, c^{2} d e^{5} + 13 \, b c e^{6}\right )} x^{5} + 35 \, {\left (3 \, c^{2} d^{2} e^{4} + 312 \, b c d e^{5} + 143 \, b^{2} e^{6}\right )} x^{4} - 10 \, {\left (12 \, c^{2} d^{3} e^{3} - 39 \, b c d^{2} e^{4} - 715 \, b^{2} d e^{5}\right )} x^{3} + 3 \, {\left (48 \, c^{2} d^{4} e^{2} - 156 \, b c d^{3} e^{3} + 143 \, b^{2} d^{2} e^{4}\right )} x^{2} - 4 \, {\left (48 \, c^{2} d^{5} e - 156 \, b c d^{4} e^{2} + 143 \, b^{2} d^{3} e^{3}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{5}} \] Input:
integrate((e*x+d)^(3/2)*(c*x^2+b*x)^2,x, algorithm="fricas")
Output:
2/45045*(3465*c^2*e^6*x^6 + 384*c^2*d^6 - 1248*b*c*d^5*e + 1144*b^2*d^4*e^ 2 + 630*(7*c^2*d*e^5 + 13*b*c*e^6)*x^5 + 35*(3*c^2*d^2*e^4 + 312*b*c*d*e^5 + 143*b^2*e^6)*x^4 - 10*(12*c^2*d^3*e^3 - 39*b*c*d^2*e^4 - 715*b^2*d*e^5) *x^3 + 3*(48*c^2*d^4*e^2 - 156*b*c*d^3*e^3 + 143*b^2*d^2*e^4)*x^2 - 4*(48* c^2*d^5*e - 156*b*c*d^4*e^2 + 143*b^2*d^3*e^3)*x)*sqrt(e*x + d)/e^5
Time = 0.72 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.39 \[ \int (d+e x)^{3/2} \left (b x+c x^2\right )^2 \, dx=\begin {cases} \frac {2 \left (\frac {c^{2} \left (d + e x\right )^{\frac {13}{2}}}{13 e^{4}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (2 b c e - 4 c^{2} d\right )}{11 e^{4}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \left (b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right )}{9 e^{4}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (- 2 b^{2} d e^{2} + 6 b c d^{2} e - 4 c^{2} d^{3}\right )}{7 e^{4}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (b^{2} d^{2} e^{2} - 2 b c d^{3} e + c^{2} d^{4}\right )}{5 e^{4}}\right )}{e} & \text {for}\: e \neq 0 \\d^{\frac {3}{2}} \left (\frac {b^{2} x^{3}}{3} + \frac {b c x^{4}}{2} + \frac {c^{2} x^{5}}{5}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((e*x+d)**(3/2)*(c*x**2+b*x)**2,x)
Output:
Piecewise((2*(c**2*(d + e*x)**(13/2)/(13*e**4) + (d + e*x)**(11/2)*(2*b*c* e - 4*c**2*d)/(11*e**4) + (d + e*x)**(9/2)*(b**2*e**2 - 6*b*c*d*e + 6*c**2 *d**2)/(9*e**4) + (d + e*x)**(7/2)*(-2*b**2*d*e**2 + 6*b*c*d**2*e - 4*c**2 *d**3)/(7*e**4) + (d + e*x)**(5/2)*(b**2*d**2*e**2 - 2*b*c*d**3*e + c**2*d **4)/(5*e**4))/e, Ne(e, 0)), (d**(3/2)*(b**2*x**3/3 + b*c*x**4/2 + c**2*x* *5/5), True))
Time = 0.03 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.95 \[ \int (d+e x)^{3/2} \left (b x+c x^2\right )^2 \, dx=\frac {2 \, {\left (3465 \, {\left (e x + d\right )}^{\frac {13}{2}} c^{2} - 8190 \, {\left (2 \, c^{2} d - b c e\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 5005 \, {\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 12870 \, {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 9009 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {5}{2}}\right )}}{45045 \, e^{5}} \] Input:
integrate((e*x+d)^(3/2)*(c*x^2+b*x)^2,x, algorithm="maxima")
Output:
2/45045*(3465*(e*x + d)^(13/2)*c^2 - 8190*(2*c^2*d - b*c*e)*(e*x + d)^(11/ 2) + 5005*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2)*(e*x + d)^(9/2) - 12870*(2*c^2 *d^3 - 3*b*c*d^2*e + b^2*d*e^2)*(e*x + d)^(7/2) + 9009*(c^2*d^4 - 2*b*c*d^ 3*e + b^2*d^2*e^2)*(e*x + d)^(5/2))/e^5
Leaf count of result is larger than twice the leaf count of optimal. 591 vs. \(2 (127) = 254\).
Time = 0.12 (sec) , antiderivative size = 591, normalized size of antiderivative = 4.02 \[ \int (d+e x)^{3/2} \left (b x+c x^2\right )^2 \, dx =\text {Too large to display} \] Input:
integrate((e*x+d)^(3/2)*(c*x^2+b*x)^2,x, algorithm="giac")
Output:
2/45045*(3003*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d) *d^2)*b^2*d^2/e^2 + 2574*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e *x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*b*c*d^2/e^3 + 2574*(5*(e*x + d)^ (7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d ^3)*b^2*d/e^2 + 143*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*c^2*d^2 /e^4 + 572*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/ 2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*b*c*d/e^3 + 143* (35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 42 0*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*b^2/e^2 + 130*(63*(e*x + d) ^(11/2) - 385*(e*x + d)^(9/2)*d + 990*(e*x + d)^(7/2)*d^2 - 1386*(e*x + d) ^(5/2)*d^3 + 1155*(e*x + d)^(3/2)*d^4 - 693*sqrt(e*x + d)*d^5)*c^2*d/e^4 + 130*(63*(e*x + d)^(11/2) - 385*(e*x + d)^(9/2)*d + 990*(e*x + d)^(7/2)*d^ 2 - 1386*(e*x + d)^(5/2)*d^3 + 1155*(e*x + d)^(3/2)*d^4 - 693*sqrt(e*x + d )*d^5)*b*c/e^3 + 15*(231*(e*x + d)^(13/2) - 1638*(e*x + d)^(11/2)*d + 5005 *(e*x + d)^(9/2)*d^2 - 8580*(e*x + d)^(7/2)*d^3 + 9009*(e*x + d)^(5/2)*d^4 - 6006*(e*x + d)^(3/2)*d^5 + 3003*sqrt(e*x + d)*d^6)*c^2/e^4)/e
Time = 0.02 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.94 \[ \int (d+e x)^{3/2} \left (b x+c x^2\right )^2 \, dx=\frac {2\,c^2\,{\left (d+e\,x\right )}^{13/2}}{13\,e^5}-\frac {{\left (d+e\,x\right )}^{7/2}\,\left (4\,b^2\,d\,e^2-12\,b\,c\,d^2\,e+8\,c^2\,d^3\right )}{7\,e^5}+\frac {{\left (d+e\,x\right )}^{9/2}\,\left (2\,b^2\,e^2-12\,b\,c\,d\,e+12\,c^2\,d^2\right )}{9\,e^5}-\frac {\left (8\,c^2\,d-4\,b\,c\,e\right )\,{\left (d+e\,x\right )}^{11/2}}{11\,e^5}+\frac {2\,d^2\,{\left (b\,e-c\,d\right )}^2\,{\left (d+e\,x\right )}^{5/2}}{5\,e^5} \] Input:
int((b*x + c*x^2)^2*(d + e*x)^(3/2),x)
Output:
(2*c^2*(d + e*x)^(13/2))/(13*e^5) - ((d + e*x)^(7/2)*(8*c^2*d^3 + 4*b^2*d* e^2 - 12*b*c*d^2*e))/(7*e^5) + ((d + e*x)^(9/2)*(2*b^2*e^2 + 12*c^2*d^2 - 12*b*c*d*e))/(9*e^5) - ((8*c^2*d - 4*b*c*e)*(d + e*x)^(11/2))/(11*e^5) + ( 2*d^2*(b*e - c*d)^2*(d + e*x)^(5/2))/(5*e^5)
Time = 0.25 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.50 \[ \int (d+e x)^{3/2} \left (b x+c x^2\right )^2 \, dx=\frac {2 \sqrt {e x +d}\, \left (3465 c^{2} e^{6} x^{6}+8190 b c \,e^{6} x^{5}+4410 c^{2} d \,e^{5} x^{5}+5005 b^{2} e^{6} x^{4}+10920 b c d \,e^{5} x^{4}+105 c^{2} d^{2} e^{4} x^{4}+7150 b^{2} d \,e^{5} x^{3}+390 b c \,d^{2} e^{4} x^{3}-120 c^{2} d^{3} e^{3} x^{3}+429 b^{2} d^{2} e^{4} x^{2}-468 b c \,d^{3} e^{3} x^{2}+144 c^{2} d^{4} e^{2} x^{2}-572 b^{2} d^{3} e^{3} x +624 b c \,d^{4} e^{2} x -192 c^{2} d^{5} e x +1144 b^{2} d^{4} e^{2}-1248 b c \,d^{5} e +384 c^{2} d^{6}\right )}{45045 e^{5}} \] Input:
int((e*x+d)^(3/2)*(c*x^2+b*x)^2,x)
Output:
(2*sqrt(d + e*x)*(1144*b**2*d**4*e**2 - 572*b**2*d**3*e**3*x + 429*b**2*d* *2*e**4*x**2 + 7150*b**2*d*e**5*x**3 + 5005*b**2*e**6*x**4 - 1248*b*c*d**5 *e + 624*b*c*d**4*e**2*x - 468*b*c*d**3*e**3*x**2 + 390*b*c*d**2*e**4*x**3 + 10920*b*c*d*e**5*x**4 + 8190*b*c*e**6*x**5 + 384*c**2*d**6 - 192*c**2*d **5*e*x + 144*c**2*d**4*e**2*x**2 - 120*c**2*d**3*e**3*x**3 + 105*c**2*d** 2*e**4*x**4 + 4410*c**2*d*e**5*x**5 + 3465*c**2*e**6*x**6))/(45045*e**5)