Integrand size = 22, antiderivative size = 166 \[ \int (d+e x)^{3/2} \left (a+b x+c x^2\right )^2 \, dx=\frac {2 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{5/2}}{5 e^5}-\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^{7/2}}{7 e^5}+\frac {2 \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^{9/2}}{9 e^5}-\frac {4 c (2 c d-b e) (d+e x)^{11/2}}{11 e^5}+\frac {2 c^2 (d+e x)^{13/2}}{13 e^5} \]
2/5*(a*e^2-b*d*e+c*d^2)^2*(e*x+d)^(5/2)/e^5-4/7*(-b*e+2*c*d)*(a*e^2-b*d*e+ c*d^2)*(e*x+d)^(7/2)/e^5+2/9*(6*c^2*d^2+b^2*e^2-2*c*e*(-a*e+3*b*d))*(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 = 0.12 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.05 \[ \int (d+e x)^{3/2} \left (a+b x+c x^2\right )^2 \, dx=\frac {2 (d+e x)^{5/2} \left (3 c^2 \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )+143 e^2 \left (63 a^2 e^2+18 a b e (-2 d+5 e x)+b^2 \left (8 d^2-20 d e x+35 e^2 x^2\right )\right )-26 c e \left (-11 a e \left (8 d^2-20 d e x+35 e^2 x^2\right )+3 b \left (16 d^3-40 d^2 e x+70 d e^2 x^2-105 e^3 x^3\right )\right )\right )}{45045 e^5} \]
(2*(d + e*x)^(5/2)*(3*c^2*(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) + 143*e^2*(63*a^2*e^2 + 18*a*b*e*(-2*d + 5*e*x) + b^2*(8*d^2 - 20*d*e*x + 35*e^2*x^2)) - 26*c*e*(-11*a*e*(8*d^2 - 20*d*e*x + 35*e^2*x^2) + 3*b*(16*d^3 - 40*d^2*e*x + 70*d*e^2*x^2 - 105*e^3*x^3))))/ (45045*e^5)
Time = 0.30 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, 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.
| \(\displaystyle \int (d+e x)^{3/2} \left (a+b x+c x^2\right )^2 \, dx\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle \int \left (\frac {(d+e x)^{7/2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^4}+\frac {2 (d+e x)^{5/2} (b e-2 c d) \left (a e^2-b d e+c d^2\right )}{e^4}+\frac {(d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )^2}{e^4}-\frac {2 c (d+e x)^{9/2} (2 c d-b e)}{e^4}+\frac {c^2 (d+e x)^{11/2}}{e^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 (d+e x)^{9/2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{9 e^5}-\frac {4 (d+e x)^{7/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{7 e^5}+\frac {2 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )^2}{5 e^5}-\frac {4 c (d+e x)^{11/2} (2 c d-b e)}{11 e^5}+\frac {2 c^2 (d+e x)^{13/2}}{13 e^5}\) |
(2*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(5/2))/(5*e^5) - (4*(2*c*d - b*e)*( c*d^2 - b*d*e + a*e^2)*(d + e*x)^(7/2))/(7*e^5) + (2*(6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e))*(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)
3.23.76.3.1 Defintions of rubi rules used
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 = 0.22 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.82
| method | result | size |
| derivativedivides | \(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {4 c \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {4 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{5}}\) | \(136\) |
| default | \(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {4 c \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {4 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{5}}\) | \(136\) |
| pseudoelliptic | \(\frac {2 \left (\left (\frac {5 c^{2} x^{4}}{13}+\frac {10 \left (\frac {9 b x}{11}+a \right ) x^{2} c}{9}+\frac {5 b^{2} x^{2}}{9}+\frac {10 a b x}{7}+a^{2}\right ) e^{4}-\frac {4 d \left (\frac {70 c^{2} x^{3}}{143}+\frac {10 x \left (\frac {21 b x}{22}+a \right ) c}{9}+b \left (\frac {5 b x}{9}+a \right )\right ) e^{3}}{7}+\frac {16 \left (\frac {105 c^{2} x^{2}}{143}+\left (\frac {15 b x}{11}+a \right ) c +\frac {b^{2}}{2}\right ) d^{2} e^{2}}{63}-\frac {32 \left (\frac {10 c x}{13}+b \right ) c \,d^{3} e}{231}+\frac {128 c^{2} d^{4}}{3003}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5 e^{5}}\) | \(139\) |
| gosper | \(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (3465 c^{2} x^{4} e^{4}+8190 b c \,e^{4} x^{3}-2520 c^{2} d \,e^{3} x^{3}+10010 a c \,e^{4} x^{2}+5005 b^{2} e^{4} x^{2}-5460 b c d \,e^{3} x^{2}+1680 c^{2} d^{2} e^{2} x^{2}+12870 a b \,e^{4} x -5720 a c d \,e^{3} x -2860 b^{2} d \,e^{3} x +3120 b c \,d^{2} e^{2} x -960 c^{2} d^{3} e x +9009 a^{2} e^{4}-5148 a b d \,e^{3}+2288 a c \,d^{2} e^{2}+1144 b^{2} d^{2} e^{2}-1248 b c \,d^{3} e +384 c^{2} d^{4}\right )}{45045 e^{5}}\) | \(194\) |
| trager | \(\frac {2 \left (3465 c^{2} e^{6} x^{6}+8190 b c \,e^{6} x^{5}+4410 c^{2} d \,e^{5} x^{5}+10010 a c \,e^{6} x^{4}+5005 b^{2} e^{6} x^{4}+10920 b c d \,e^{5} x^{4}+105 c^{2} d^{2} e^{4} x^{4}+12870 a b \,e^{6} x^{3}+14300 a c d \,e^{5} x^{3}+7150 b^{2} d \,e^{5} x^{3}+390 b c \,d^{2} e^{4} x^{3}-120 c^{2} d^{3} e^{3} x^{3}+9009 a^{2} e^{6} x^{2}+20592 a b d \,e^{5} x^{2}+858 a c \,d^{2} e^{4} x^{2}+429 b^{2} d^{2} e^{4} x^{2}-468 b c \,d^{3} e^{3} x^{2}+144 c^{2} d^{4} e^{2} x^{2}+18018 a^{2} d \,e^{5} x +2574 a b \,d^{2} e^{4} x -1144 a c \,d^{3} e^{3} x -572 b^{2} d^{3} e^{3} x +624 b c \,d^{4} e^{2} x -192 c^{2} d^{5} e x +9009 a^{2} d^{2} e^{4}-5148 a b \,d^{3} e^{3}+2288 a c \,d^{4} e^{2}+1144 b^{2} d^{4} e^{2}-1248 b c \,d^{5} e +384 c^{2} d^{6}\right ) \sqrt {e x +d}}{45045 e^{5}}\) | \(352\) |
| risch | \(\frac {2 \left (3465 c^{2} e^{6} x^{6}+8190 b c \,e^{6} x^{5}+4410 c^{2} d \,e^{5} x^{5}+10010 a c \,e^{6} x^{4}+5005 b^{2} e^{6} x^{4}+10920 b c d \,e^{5} x^{4}+105 c^{2} d^{2} e^{4} x^{4}+12870 a b \,e^{6} x^{3}+14300 a c d \,e^{5} x^{3}+7150 b^{2} d \,e^{5} x^{3}+390 b c \,d^{2} e^{4} x^{3}-120 c^{2} d^{3} e^{3} x^{3}+9009 a^{2} e^{6} x^{2}+20592 a b d \,e^{5} x^{2}+858 a c \,d^{2} e^{4} x^{2}+429 b^{2} d^{2} e^{4} x^{2}-468 b c \,d^{3} e^{3} x^{2}+144 c^{2} d^{4} e^{2} x^{2}+18018 a^{2} d \,e^{5} x +2574 a b \,d^{2} e^{4} x -1144 a c \,d^{3} e^{3} x -572 b^{2} d^{3} e^{3} x +624 b c \,d^{4} e^{2} x -192 c^{2} d^{5} e x +9009 a^{2} d^{2} e^{4}-5148 a b \,d^{3} e^{3}+2288 a c \,d^{4} e^{2}+1144 b^{2} d^{4} e^{2}-1248 b c \,d^{5} e +384 c^{2} d^{6}\right ) \sqrt {e x +d}}{45045 e^{5}}\) | \(352\) |
2/e^5*(1/13*c^2*(e*x+d)^(13/2)+2/11*c*(b*e-2*c*d)*(e*x+d)^(11/2)+1/9*(2*(a *e^2-b*d*e+c*d^2)*c+(b*e-2*c*d)^2)*(e*x+d)^(9/2)+2/7*(a*e^2-b*d*e+c*d^2)*( b*e-2*c*d)*(e*x+d)^(7/2)+1/5*(a*e^2-b*d*e+c*d^2)^2*(e*x+d)^(5/2))
Leaf count of result is larger than twice the leaf count of optimal. 302 vs. \(2 (146) = 292\).
Time = 0.48 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.82 \[ \int (d+e x)^{3/2} \left (a+b x+c x^2\right )^2 \, dx=\frac {2 \, {\left (3465 \, c^{2} e^{6} x^{6} + 384 \, c^{2} d^{6} - 1248 \, b c d^{5} e - 5148 \, a b d^{3} e^{3} + 9009 \, a^{2} d^{2} e^{4} + 1144 \, {\left (b^{2} + 2 \, a c\right )} d^{4} e^{2} + 630 \, {\left (7 \, c^{2} d e^{5} + 13 \, b c e^{6}\right )} x^{5} + 35 \, {\left (3 \, c^{2} d^{2} e^{4} + 312 \, b c d e^{5} + 143 \, {\left (b^{2} + 2 \, a c\right )} e^{6}\right )} x^{4} - 10 \, {\left (12 \, c^{2} d^{3} e^{3} - 39 \, b c d^{2} e^{4} - 1287 \, a b e^{6} - 715 \, {\left (b^{2} + 2 \, a c\right )} d e^{5}\right )} x^{3} + 3 \, {\left (48 \, c^{2} d^{4} e^{2} - 156 \, b c d^{3} e^{3} + 6864 \, a b d e^{5} + 3003 \, a^{2} e^{6} + 143 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e^{4}\right )} x^{2} - 2 \, {\left (96 \, c^{2} d^{5} e - 312 \, b c d^{4} e^{2} - 1287 \, a b d^{2} e^{4} - 9009 \, a^{2} d e^{5} + 286 \, {\left (b^{2} + 2 \, a c\right )} d^{3} e^{3}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{5}} \]
2/45045*(3465*c^2*e^6*x^6 + 384*c^2*d^6 - 1248*b*c*d^5*e - 5148*a*b*d^3*e^ 3 + 9009*a^2*d^2*e^4 + 1144*(b^2 + 2*a*c)*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 + 2*a*c)*e^6)* x^4 - 10*(12*c^2*d^3*e^3 - 39*b*c*d^2*e^4 - 1287*a*b*e^6 - 715*(b^2 + 2*a* c)*d*e^5)*x^3 + 3*(48*c^2*d^4*e^2 - 156*b*c*d^3*e^3 + 6864*a*b*d*e^5 + 300 3*a^2*e^6 + 143*(b^2 + 2*a*c)*d^2*e^4)*x^2 - 2*(96*c^2*d^5*e - 312*b*c*d^4 *e^2 - 1287*a*b*d^2*e^4 - 9009*a^2*d*e^5 + 286*(b^2 + 2*a*c)*d^3*e^3)*x)*s qrt(e*x + d)/e^5
Time = 0.97 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.67 \[ \int (d+e x)^{3/2} \left (a+b x+c x^2\right )^2 \, dx=\begin {cases} \frac {2 \left (\frac {c^{2} \left (d + e x\right )^{\frac {13}{2}}}{13 e^{4}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (2 b c e - 4 c^{2} d\right )}{11 e^{4}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (2 a c e^{2} + b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right )}{9 e^{4}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (2 a b e^{3} - 4 a c d e^{2} - 2 b^{2} d e^{2} + 6 b c d^{2} e - 4 c^{2} d^{3}\right )}{7 e^{4}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (a^{2} e^{4} - 2 a b d e^{3} + 2 a c d^{2} e^{2} + b^{2} d^{2} e^{2} - 2 b c d^{3} e + c^{2} d^{4}\right )}{5 e^{4}}\right )}{e} & \text {for}\: e \neq 0 \\d^{\frac {3}{2}} \left (a^{2} x + a b x^{2} + \frac {b c x^{4}}{2} + \frac {c^{2} x^{5}}{5} + \frac {x^{3} \cdot \left (2 a c + b^{2}\right )}{3}\right ) & \text {otherwise} \end {cases} \]
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)*(2*a*c*e**2 + b**2*e**2 - 6*b*c *d*e + 6*c**2*d**2)/(9*e**4) + (d + e*x)**(7/2)*(2*a*b*e**3 - 4*a*c*d*e**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) *(a**2*e**4 - 2*a*b*d*e**3 + 2*a*c*d**2*e**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)*(a**2*x + a*b*x**2 + b*c *x**4/2 + c**2*x**5/5 + x**3*(2*a*c + b**2)/3), True))
Time = 0.19 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.06 \[ \int (d+e x)^{3/2} \left (a+b x+c x^2\right )^2 \, dx=\frac {2 \, {\left (3465 \, {\left (e x + d\right )}^{\frac {13}{2}} c^{2} - 8190 \, {\left (2 \, c^{2} d - b c e\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 5005 \, {\left (6 \, c^{2} d^{2} - 6 \, b c d e + {\left (b^{2} + 2 \, a c\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 12870 \, {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} + {\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 9009 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + a^{2} e^{4} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {5}{2}}\right )}}{45045 \, e^{5}} \]
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 + 2*a*c)*e^2)*(e*x + d)^(9/2) - 12 870*(2*c^2*d^3 - 3*b*c*d^2*e - a*b*e^3 + (b^2 + 2*a*c)*d*e^2)*(e*x + d)^(7 /2) + 9009*(c^2*d^4 - 2*b*c*d^3*e - 2*a*b*d*e^3 + a^2*e^4 + (b^2 + 2*a*c)* d^2*e^2)*(e*x + d)^(5/2))/e^5
Leaf count of result is larger than twice the leaf count of optimal. 943 vs. \(2 (146) = 292\).
Time = 0.28 (sec) , antiderivative size = 943, normalized size of antiderivative = 5.68 \[ \int (d+e x)^{3/2} \left (a+b x+c x^2\right )^2 \, dx=\text {Too large to display} \]
2/45045*(45045*sqrt(e*x + d)*a^2*d^2 + 30030*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^2*d + 30030*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a*b*d^2/e + 3 003*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^2 + 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 + 6006*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a*c*d^2/e^2 + 12012*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a*b*d/e + 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*sqr t(e*x + d)*d^3)*b^2*d/e^2 + 5148*(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)*a*c*d/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)*a*b/e + 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 + 14 3*(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^2/e^2 + 286*(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)*a*c/e^2 + 130*(63*(e*x + d)^(11/2) ...
Time = 9.78 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.89 \[ \int (d+e x)^{3/2} \left (a+b x+c x^2\right )^2 \, dx=\frac {2\,c^2\,{\left (d+e\,x\right )}^{13/2}}{13\,e^5}+\frac {{\left (d+e\,x\right )}^{9/2}\,\left (2\,b^2\,e^2-12\,b\,c\,d\,e+12\,c^2\,d^2+4\,a\,c\,e^2\right )}{9\,e^5}+\frac {2\,{\left (d+e\,x\right )}^{5/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2}{5\,e^5}-\frac {\left (8\,c^2\,d-4\,b\,c\,e\right )\,{\left (d+e\,x\right )}^{11/2}}{11\,e^5}+\frac {4\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{7/2}\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{7\,e^5} \]