Integrand size = 24, antiderivative size = 177 \[ \int (d+e x)^2 (f+g x)^{3/2} \left (a+c x^2\right ) \, dx=\frac {2 (e f-d g)^2 \left (c f^2+a g^2\right ) (f+g x)^{5/2}}{5 g^5}-\frac {4 (e f-d g) \left (a e g^2+c f (2 e f-d g)\right ) (f+g x)^{7/2}}{7 g^5}+\frac {2 \left (a e^2 g^2+c \left (6 e^2 f^2-6 d e f g+d^2 g^2\right )\right ) (f+g x)^{9/2}}{9 g^5}-\frac {4 c e (2 e f-d g) (f+g x)^{11/2}}{11 g^5}+\frac {2 c e^2 (f+g x)^{13/2}}{13 g^5} \] Output:
2/5*(-d*g+e*f)^2*(a*g^2+c*f^2)*(g*x+f)^(5/2)/g^5-4/7*(-d*g+e*f)*(a*e*g^2+c *f*(-d*g+2*e*f))*(g*x+f)^(7/2)/g^5+2/9*(a*e^2*g^2+c*(d^2*g^2-6*d*e*f*g+6*e ^2*f^2))*(g*x+f)^(9/2)/g^5-4/11*c*e*(-d*g+2*e*f)*(g*x+f)^(11/2)/g^5+2/13*c *e^2*(g*x+f)^(13/2)/g^5
Time = 0.20 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.01 \[ \int (d+e x)^2 (f+g x)^{3/2} \left (a+c x^2\right ) \, dx=\frac {2 (f+g x)^{5/2} \left (143 a g^2 \left (63 d^2 g^2+18 d e g (-2 f+5 g x)+e^2 \left (8 f^2-20 f g x+35 g^2 x^2\right )\right )+c \left (143 d^2 g^2 \left (8 f^2-20 f g x+35 g^2 x^2\right )+78 d e g \left (-16 f^3+40 f^2 g x-70 f g^2 x^2+105 g^3 x^3\right )+3 e^2 \left (128 f^4-320 f^3 g x+560 f^2 g^2 x^2-840 f g^3 x^3+1155 g^4 x^4\right )\right )\right )}{45045 g^5} \] Input:
Integrate[(d + e*x)^2*(f + g*x)^(3/2)*(a + c*x^2),x]
Output:
(2*(f + g*x)^(5/2)*(143*a*g^2*(63*d^2*g^2 + 18*d*e*g*(-2*f + 5*g*x) + e^2* (8*f^2 - 20*f*g*x + 35*g^2*x^2)) + c*(143*d^2*g^2*(8*f^2 - 20*f*g*x + 35*g ^2*x^2) + 78*d*e*g*(-16*f^3 + 40*f^2*g*x - 70*f*g^2*x^2 + 105*g^3*x^3) + 3 *e^2*(128*f^4 - 320*f^3*g*x + 560*f^2*g^2*x^2 - 840*f*g^3*x^3 + 1155*g^4*x ^4))))/(45045*g^5)
Time = 0.33 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {652, 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.
| \(\displaystyle \int \left (a+c x^2\right ) (d+e x)^2 (f+g x)^{3/2} \, dx\) |
| \(\Big \downarrow \) 652 |
| \(\displaystyle \int \left (\frac {(f+g x)^{7/2} \left (a e^2 g^2+c \left (d^2 g^2-6 d e f g+6 e^2 f^2\right )\right )}{g^4}+\frac {(f+g x)^{3/2} \left (a g^2+c f^2\right ) (d g-e f)^2}{g^4}+\frac {2 (f+g x)^{5/2} (e f-d g) \left (-a e g^2-c f (2 e f-d g)\right )}{g^4}-\frac {2 c e (f+g x)^{9/2} (2 e f-d g)}{g^4}+\frac {c e^2 (f+g x)^{11/2}}{g^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 (f+g x)^{9/2} \left (a e^2 g^2+c \left (d^2 g^2-6 d e f g+6 e^2 f^2\right )\right )}{9 g^5}+\frac {2 (f+g x)^{5/2} \left (a g^2+c f^2\right ) (e f-d g)^2}{5 g^5}-\frac {4 (f+g x)^{7/2} (e f-d g) \left (a e g^2+c f (2 e f-d g)\right )}{7 g^5}-\frac {4 c e (f+g x)^{11/2} (2 e f-d g)}{11 g^5}+\frac {2 c e^2 (f+g x)^{13/2}}{13 g^5}\) |
Input:
Int[(d + e*x)^2*(f + g*x)^(3/2)*(a + c*x^2),x]
Output:
(2*(e*f - d*g)^2*(c*f^2 + a*g^2)*(f + g*x)^(5/2))/(5*g^5) - (4*(e*f - d*g) *(a*e*g^2 + c*f*(2*e*f - d*g))*(f + g*x)^(7/2))/(7*g^5) + (2*(a*e^2*g^2 + c*(6*e^2*f^2 - 6*d*e*f*g + d^2*g^2))*(f + g*x)^(9/2))/(9*g^5) - (4*c*e*(2* e*f - d*g)*(f + g*x)^(11/2))/(11*g^5) + (2*c*e^2*(f + g*x)^(13/2))/(13*g^5 )
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (c_.)*(x_ )^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + c *x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && IGtQ[p, 0]
Time = 1.42 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.88
| method | result | size |
| pseudoelliptic | \(\frac {2 \left (g x +f \right )^{\frac {5}{2}} \left (\left (\frac {5 \left (\frac {9 c \,x^{2}}{13}+a \right ) x^{2} e^{2}}{9}+\frac {10 \left (\frac {7 c \,x^{2}}{11}+a \right ) x d e}{7}+d^{2} \left (\frac {5 c \,x^{2}}{9}+a \right )\right ) g^{4}-\frac {4 f \left (\left (\frac {70}{143} c \,x^{3}+\frac {5}{9} a x \right ) e^{2}+d \left (\frac {35 c \,x^{2}}{33}+a \right ) e +\frac {5 c \,d^{2} x}{9}\right ) g^{3}}{7}+\frac {8 f^{2} \left (\left (\frac {210 c \,x^{2}}{143}+a \right ) e^{2}+\frac {30 c d x e}{11}+c \,d^{2}\right ) g^{2}}{63}-\frac {32 \left (\frac {10 e x}{13}+d \right ) e \,f^{3} c g}{231}+\frac {128 c \,e^{2} f^{4}}{3003}\right )}{5 g^{5}}\) | \(155\) |
| derivativedivides | \(\frac {\frac {2 c \,e^{2} \left (g x +f \right )^{\frac {13}{2}}}{13}+\frac {2 \left (2 e \left (d g -e f \right ) c -2 f c \,e^{2}\right ) \left (g x +f \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (d g -e f \right )^{2} c -4 e \left (d g -e f \right ) c f +e^{2} \left (a \,g^{2}+c \,f^{2}\right )\right ) \left (g x +f \right )^{\frac {9}{2}}}{9}+\frac {2 \left (-2 \left (d g -e f \right )^{2} c f +2 e \left (d g -e f \right ) \left (a \,g^{2}+c \,f^{2}\right )\right ) \left (g x +f \right )^{\frac {7}{2}}}{7}+\frac {2 \left (d g -e f \right )^{2} \left (a \,g^{2}+c \,f^{2}\right ) \left (g x +f \right )^{\frac {5}{2}}}{5}}{g^{5}}\) | \(175\) |
| default | \(\frac {\frac {2 c \,e^{2} \left (g x +f \right )^{\frac {13}{2}}}{13}+\frac {2 \left (2 e \left (d g -e f \right ) c -2 f c \,e^{2}\right ) \left (g x +f \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (d g -e f \right )^{2} c -4 e \left (d g -e f \right ) c f +e^{2} \left (a \,g^{2}+c \,f^{2}\right )\right ) \left (g x +f \right )^{\frac {9}{2}}}{9}+\frac {2 \left (-2 \left (d g -e f \right )^{2} c f +2 e \left (d g -e f \right ) \left (a \,g^{2}+c \,f^{2}\right )\right ) \left (g x +f \right )^{\frac {7}{2}}}{7}+\frac {2 \left (d g -e f \right )^{2} \left (a \,g^{2}+c \,f^{2}\right ) \left (g x +f \right )^{\frac {5}{2}}}{5}}{g^{5}}\) | \(175\) |
| gosper | \(\frac {2 \left (g x +f \right )^{\frac {5}{2}} \left (3465 c \,e^{2} x^{4} g^{4}+8190 c d e \,g^{4} x^{3}-2520 c \,e^{2} f \,g^{3} x^{3}+5005 a \,e^{2} g^{4} x^{2}+5005 c \,d^{2} g^{4} x^{2}-5460 c d e f \,g^{3} x^{2}+1680 c \,e^{2} f^{2} g^{2} x^{2}+12870 a d e \,g^{4} x -2860 a \,e^{2} f \,g^{3} x -2860 c \,d^{2} f \,g^{3} x +3120 c d e \,f^{2} g^{2} x -960 c \,e^{2} f^{3} g x +9009 a \,d^{2} g^{4}-5148 a d e f \,g^{3}+1144 a \,e^{2} f^{2} g^{2}+1144 c \,d^{2} f^{2} g^{2}-1248 c d e \,f^{3} g +384 c \,e^{2} f^{4}\right )}{45045 g^{5}}\) | \(215\) |
| orering | \(\frac {2 \left (g x +f \right )^{\frac {5}{2}} \left (3465 c \,e^{2} x^{4} g^{4}+8190 c d e \,g^{4} x^{3}-2520 c \,e^{2} f \,g^{3} x^{3}+5005 a \,e^{2} g^{4} x^{2}+5005 c \,d^{2} g^{4} x^{2}-5460 c d e f \,g^{3} x^{2}+1680 c \,e^{2} f^{2} g^{2} x^{2}+12870 a d e \,g^{4} x -2860 a \,e^{2} f \,g^{3} x -2860 c \,d^{2} f \,g^{3} x +3120 c d e \,f^{2} g^{2} x -960 c \,e^{2} f^{3} g x +9009 a \,d^{2} g^{4}-5148 a d e f \,g^{3}+1144 a \,e^{2} f^{2} g^{2}+1144 c \,d^{2} f^{2} g^{2}-1248 c d e \,f^{3} g +384 c \,e^{2} f^{4}\right )}{45045 g^{5}}\) | \(215\) |
| trager | \(\frac {2 \left (3465 c \,e^{2} g^{6} x^{6}+8190 c d e \,g^{6} x^{5}+4410 c \,e^{2} f \,g^{5} x^{5}+5005 a \,e^{2} g^{6} x^{4}+5005 c \,d^{2} g^{6} x^{4}+10920 c d e f \,g^{5} x^{4}+105 c \,e^{2} f^{2} g^{4} x^{4}+12870 a d e \,g^{6} x^{3}+7150 a \,e^{2} f \,g^{5} x^{3}+7150 c \,d^{2} f \,g^{5} x^{3}+390 c d e \,f^{2} g^{4} x^{3}-120 c \,e^{2} f^{3} g^{3} x^{3}+9009 a \,d^{2} g^{6} x^{2}+20592 a d e f \,g^{5} x^{2}+429 a \,e^{2} f^{2} g^{4} x^{2}+429 c \,d^{2} f^{2} g^{4} x^{2}-468 c d e \,f^{3} g^{3} x^{2}+144 c \,e^{2} f^{4} g^{2} x^{2}+18018 a \,d^{2} f \,g^{5} x +2574 a d e \,f^{2} g^{4} x -572 a \,e^{2} f^{3} g^{3} x -572 c \,d^{2} f^{3} g^{3} x +624 c d e \,f^{4} g^{2} x -192 c \,e^{2} f^{5} g x +9009 a \,d^{2} f^{2} g^{4}-5148 a d e \,f^{3} g^{3}+1144 a \,e^{2} f^{4} g^{2}+1144 c \,d^{2} f^{4} g^{2}-1248 c d e \,f^{5} g +384 c \,e^{2} f^{6}\right ) \sqrt {g x +f}}{45045 g^{5}}\) | \(387\) |
| risch | \(\frac {2 \left (3465 c \,e^{2} g^{6} x^{6}+8190 c d e \,g^{6} x^{5}+4410 c \,e^{2} f \,g^{5} x^{5}+5005 a \,e^{2} g^{6} x^{4}+5005 c \,d^{2} g^{6} x^{4}+10920 c d e f \,g^{5} x^{4}+105 c \,e^{2} f^{2} g^{4} x^{4}+12870 a d e \,g^{6} x^{3}+7150 a \,e^{2} f \,g^{5} x^{3}+7150 c \,d^{2} f \,g^{5} x^{3}+390 c d e \,f^{2} g^{4} x^{3}-120 c \,e^{2} f^{3} g^{3} x^{3}+9009 a \,d^{2} g^{6} x^{2}+20592 a d e f \,g^{5} x^{2}+429 a \,e^{2} f^{2} g^{4} x^{2}+429 c \,d^{2} f^{2} g^{4} x^{2}-468 c d e \,f^{3} g^{3} x^{2}+144 c \,e^{2} f^{4} g^{2} x^{2}+18018 a \,d^{2} f \,g^{5} x +2574 a d e \,f^{2} g^{4} x -572 a \,e^{2} f^{3} g^{3} x -572 c \,d^{2} f^{3} g^{3} x +624 c d e \,f^{4} g^{2} x -192 c \,e^{2} f^{5} g x +9009 a \,d^{2} f^{2} g^{4}-5148 a d e \,f^{3} g^{3}+1144 a \,e^{2} f^{4} g^{2}+1144 c \,d^{2} f^{4} g^{2}-1248 c d e \,f^{5} g +384 c \,e^{2} f^{6}\right ) \sqrt {g x +f}}{45045 g^{5}}\) | \(387\) |
Input:
int((e*x+d)^2*(g*x+f)^(3/2)*(c*x^2+a),x,method=_RETURNVERBOSE)
Output:
2/5*(g*x+f)^(5/2)*((5/9*(9/13*c*x^2+a)*x^2*e^2+10/7*(7/11*c*x^2+a)*x*d*e+d ^2*(5/9*c*x^2+a))*g^4-4/7*f*((70/143*c*x^3+5/9*a*x)*e^2+d*(35/33*c*x^2+a)* e+5/9*c*d^2*x)*g^3+8/63*f^2*((210/143*c*x^2+a)*e^2+30/11*c*d*x*e+c*d^2)*g^ 2-32/231*(10/13*e*x+d)*e*f^3*c*g+128/3003*c*e^2*f^4)/g^5
Leaf count of result is larger than twice the leaf count of optimal. 337 vs. \(2 (157) = 314\).
Time = 0.08 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.90 \[ \int (d+e x)^2 (f+g x)^{3/2} \left (a+c x^2\right ) \, dx=\frac {2 \, {\left (3465 \, c e^{2} g^{6} x^{6} + 384 \, c e^{2} f^{6} - 1248 \, c d e f^{5} g - 5148 \, a d e f^{3} g^{3} + 9009 \, a d^{2} f^{2} g^{4} + 1144 \, {\left (c d^{2} + a e^{2}\right )} f^{4} g^{2} + 630 \, {\left (7 \, c e^{2} f g^{5} + 13 \, c d e g^{6}\right )} x^{5} + 35 \, {\left (3 \, c e^{2} f^{2} g^{4} + 312 \, c d e f g^{5} + 143 \, {\left (c d^{2} + a e^{2}\right )} g^{6}\right )} x^{4} - 10 \, {\left (12 \, c e^{2} f^{3} g^{3} - 39 \, c d e f^{2} g^{4} - 1287 \, a d e g^{6} - 715 \, {\left (c d^{2} + a e^{2}\right )} f g^{5}\right )} x^{3} + 3 \, {\left (48 \, c e^{2} f^{4} g^{2} - 156 \, c d e f^{3} g^{3} + 6864 \, a d e f g^{5} + 3003 \, a d^{2} g^{6} + 143 \, {\left (c d^{2} + a e^{2}\right )} f^{2} g^{4}\right )} x^{2} - 2 \, {\left (96 \, c e^{2} f^{5} g - 312 \, c d e f^{4} g^{2} - 1287 \, a d e f^{2} g^{4} - 9009 \, a d^{2} f g^{5} + 286 \, {\left (c d^{2} + a e^{2}\right )} f^{3} g^{3}\right )} x\right )} \sqrt {g x + f}}{45045 \, g^{5}} \] Input:
integrate((e*x+d)^2*(g*x+f)^(3/2)*(c*x^2+a),x, algorithm="fricas")
Output:
2/45045*(3465*c*e^2*g^6*x^6 + 384*c*e^2*f^6 - 1248*c*d*e*f^5*g - 5148*a*d* e*f^3*g^3 + 9009*a*d^2*f^2*g^4 + 1144*(c*d^2 + a*e^2)*f^4*g^2 + 630*(7*c*e ^2*f*g^5 + 13*c*d*e*g^6)*x^5 + 35*(3*c*e^2*f^2*g^4 + 312*c*d*e*f*g^5 + 143 *(c*d^2 + a*e^2)*g^6)*x^4 - 10*(12*c*e^2*f^3*g^3 - 39*c*d*e*f^2*g^4 - 1287 *a*d*e*g^6 - 715*(c*d^2 + a*e^2)*f*g^5)*x^3 + 3*(48*c*e^2*f^4*g^2 - 156*c* d*e*f^3*g^3 + 6864*a*d*e*f*g^5 + 3003*a*d^2*g^6 + 143*(c*d^2 + a*e^2)*f^2* g^4)*x^2 - 2*(96*c*e^2*f^5*g - 312*c*d*e*f^4*g^2 - 1287*a*d*e*f^2*g^4 - 90 09*a*d^2*f*g^5 + 286*(c*d^2 + a*e^2)*f^3*g^3)*x)*sqrt(g*x + f)/g^5
Time = 1.36 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.77 \[ \int (d+e x)^2 (f+g x)^{3/2} \left (a+c x^2\right ) \, dx=\begin {cases} \frac {2 \left (\frac {c e^{2} \left (f + g x\right )^{\frac {13}{2}}}{13 g^{4}} + \frac {\left (f + g x\right )^{\frac {11}{2}} \cdot \left (2 c d e g - 4 c e^{2} f\right )}{11 g^{4}} + \frac {\left (f + g x\right )^{\frac {9}{2}} \left (a e^{2} g^{2} + c d^{2} g^{2} - 6 c d e f g + 6 c e^{2} f^{2}\right )}{9 g^{4}} + \frac {\left (f + g x\right )^{\frac {7}{2}} \cdot \left (2 a d e g^{3} - 2 a e^{2} f g^{2} - 2 c d^{2} f g^{2} + 6 c d e f^{2} g - 4 c e^{2} f^{3}\right )}{7 g^{4}} + \frac {\left (f + g x\right )^{\frac {5}{2}} \left (a d^{2} g^{4} - 2 a d e f g^{3} + a e^{2} f^{2} g^{2} + c d^{2} f^{2} g^{2} - 2 c d e f^{3} g + c e^{2} f^{4}\right )}{5 g^{4}}\right )}{g} & \text {for}\: g \neq 0 \\f^{\frac {3}{2}} \left (a d^{2} x + a d e x^{2} + \frac {c d e x^{4}}{2} + \frac {c e^{2} x^{5}}{5} + \frac {x^{3} \left (a e^{2} + c d^{2}\right )}{3}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((e*x+d)**2*(g*x+f)**(3/2)*(c*x**2+a),x)
Output:
Piecewise((2*(c*e**2*(f + g*x)**(13/2)/(13*g**4) + (f + g*x)**(11/2)*(2*c* d*e*g - 4*c*e**2*f)/(11*g**4) + (f + g*x)**(9/2)*(a*e**2*g**2 + c*d**2*g** 2 - 6*c*d*e*f*g + 6*c*e**2*f**2)/(9*g**4) + (f + g*x)**(7/2)*(2*a*d*e*g**3 - 2*a*e**2*f*g**2 - 2*c*d**2*f*g**2 + 6*c*d*e*f**2*g - 4*c*e**2*f**3)/(7* g**4) + (f + g*x)**(5/2)*(a*d**2*g**4 - 2*a*d*e*f*g**3 + a*e**2*f**2*g**2 + c*d**2*f**2*g**2 - 2*c*d*e*f**3*g + c*e**2*f**4)/(5*g**4))/g, Ne(g, 0)), (f**(3/2)*(a*d**2*x + a*d*e*x**2 + c*d*e*x**4/2 + c*e**2*x**5/5 + x**3*(a *e**2 + c*d**2)/3), True))
Time = 0.05 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.11 \[ \int (d+e x)^2 (f+g x)^{3/2} \left (a+c x^2\right ) \, dx=\frac {2 \, {\left (3465 \, {\left (g x + f\right )}^{\frac {13}{2}} c e^{2} - 8190 \, {\left (2 \, c e^{2} f - c d e g\right )} {\left (g x + f\right )}^{\frac {11}{2}} + 5005 \, {\left (6 \, c e^{2} f^{2} - 6 \, c d e f g + {\left (c d^{2} + a e^{2}\right )} g^{2}\right )} {\left (g x + f\right )}^{\frac {9}{2}} - 12870 \, {\left (2 \, c e^{2} f^{3} - 3 \, c d e f^{2} g - a d e g^{3} + {\left (c d^{2} + a e^{2}\right )} f g^{2}\right )} {\left (g x + f\right )}^{\frac {7}{2}} + 9009 \, {\left (c e^{2} f^{4} - 2 \, c d e f^{3} g - 2 \, a d e f g^{3} + a d^{2} g^{4} + {\left (c d^{2} + a e^{2}\right )} f^{2} g^{2}\right )} {\left (g x + f\right )}^{\frac {5}{2}}\right )}}{45045 \, g^{5}} \] Input:
integrate((e*x+d)^2*(g*x+f)^(3/2)*(c*x^2+a),x, algorithm="maxima")
Output:
2/45045*(3465*(g*x + f)^(13/2)*c*e^2 - 8190*(2*c*e^2*f - c*d*e*g)*(g*x + f )^(11/2) + 5005*(6*c*e^2*f^2 - 6*c*d*e*f*g + (c*d^2 + a*e^2)*g^2)*(g*x + f )^(9/2) - 12870*(2*c*e^2*f^3 - 3*c*d*e*f^2*g - a*d*e*g^3 + (c*d^2 + a*e^2) *f*g^2)*(g*x + f)^(7/2) + 9009*(c*e^2*f^4 - 2*c*d*e*f^3*g - 2*a*d*e*f*g^3 + a*d^2*g^4 + (c*d^2 + a*e^2)*f^2*g^2)*(g*x + f)^(5/2))/g^5
Leaf count of result is larger than twice the leaf count of optimal. 964 vs. \(2 (157) = 314\).
Time = 0.13 (sec) , antiderivative size = 964, normalized size of antiderivative = 5.45 \[ \int (d+e x)^2 (f+g x)^{3/2} \left (a+c x^2\right ) \, dx=\text {Too large to display} \] Input:
integrate((e*x+d)^2*(g*x+f)^(3/2)*(c*x^2+a),x, algorithm="giac")
Output:
2/45045*(45045*sqrt(g*x + f)*a*d^2*f^2 + 30030*((g*x + f)^(3/2) - 3*sqrt(g *x + f)*f)*a*d^2*f + 30030*((g*x + f)^(3/2) - 3*sqrt(g*x + f)*f)*a*d*e*f^2 /g + 3003*(3*(g*x + f)^(5/2) - 10*(g*x + f)^(3/2)*f + 15*sqrt(g*x + f)*f^2 )*a*d^2 + 3003*(3*(g*x + f)^(5/2) - 10*(g*x + f)^(3/2)*f + 15*sqrt(g*x + f )*f^2)*c*d^2*f^2/g^2 + 3003*(3*(g*x + f)^(5/2) - 10*(g*x + f)^(3/2)*f + 15 *sqrt(g*x + f)*f^2)*a*e^2*f^2/g^2 + 12012*(3*(g*x + f)^(5/2) - 10*(g*x + f )^(3/2)*f + 15*sqrt(g*x + f)*f^2)*a*d*e*f/g + 2574*(5*(g*x + f)^(7/2) - 21 *(g*x + f)^(5/2)*f + 35*(g*x + f)^(3/2)*f^2 - 35*sqrt(g*x + f)*f^3)*c*d*e* f^2/g^3 + 2574*(5*(g*x + f)^(7/2) - 21*(g*x + f)^(5/2)*f + 35*(g*x + f)^(3 /2)*f^2 - 35*sqrt(g*x + f)*f^3)*c*d^2*f/g^2 + 2574*(5*(g*x + f)^(7/2) - 21 *(g*x + f)^(5/2)*f + 35*(g*x + f)^(3/2)*f^2 - 35*sqrt(g*x + f)*f^3)*a*e^2* f/g^2 + 2574*(5*(g*x + f)^(7/2) - 21*(g*x + f)^(5/2)*f + 35*(g*x + f)^(3/2 )*f^2 - 35*sqrt(g*x + f)*f^3)*a*d*e/g + 143*(35*(g*x + f)^(9/2) - 180*(g*x + f)^(7/2)*f + 378*(g*x + f)^(5/2)*f^2 - 420*(g*x + f)^(3/2)*f^3 + 315*sq rt(g*x + f)*f^4)*c*e^2*f^2/g^4 + 572*(35*(g*x + f)^(9/2) - 180*(g*x + f)^( 7/2)*f + 378*(g*x + f)^(5/2)*f^2 - 420*(g*x + f)^(3/2)*f^3 + 315*sqrt(g*x + f)*f^4)*c*d*e*f/g^3 + 143*(35*(g*x + f)^(9/2) - 180*(g*x + f)^(7/2)*f + 378*(g*x + f)^(5/2)*f^2 - 420*(g*x + f)^(3/2)*f^3 + 315*sqrt(g*x + f)*f^4) *c*d^2/g^2 + 143*(35*(g*x + f)^(9/2) - 180*(g*x + f)^(7/2)*f + 378*(g*x + f)^(5/2)*f^2 - 420*(g*x + f)^(3/2)*f^3 + 315*sqrt(g*x + f)*f^4)*a*e^2/g...
Time = 0.03 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.90 \[ \int (d+e x)^2 (f+g x)^{3/2} \left (a+c x^2\right ) \, dx=\frac {{\left (f+g\,x\right )}^{9/2}\,\left (2\,c\,d^2\,g^2-12\,c\,d\,e\,f\,g+12\,c\,e^2\,f^2+2\,a\,e^2\,g^2\right )}{9\,g^5}+\frac {2\,{\left (f+g\,x\right )}^{5/2}\,\left (c\,f^2+a\,g^2\right )\,{\left (d\,g-e\,f\right )}^2}{5\,g^5}+\frac {4\,{\left (f+g\,x\right )}^{7/2}\,\left (d\,g-e\,f\right )\,\left (2\,c\,e\,f^2-c\,d\,f\,g+a\,e\,g^2\right )}{7\,g^5}+\frac {2\,c\,e^2\,{\left (f+g\,x\right )}^{13/2}}{13\,g^5}+\frac {4\,c\,e\,{\left (f+g\,x\right )}^{11/2}\,\left (d\,g-2\,e\,f\right )}{11\,g^5} \] Input:
int((f + g*x)^(3/2)*(a + c*x^2)*(d + e*x)^2,x)
Output:
((f + g*x)^(9/2)*(2*a*e^2*g^2 + 2*c*d^2*g^2 + 12*c*e^2*f^2 - 12*c*d*e*f*g) )/(9*g^5) + (2*(f + g*x)^(5/2)*(a*g^2 + c*f^2)*(d*g - e*f)^2)/(5*g^5) + (4 *(f + g*x)^(7/2)*(d*g - e*f)*(a*e*g^2 + 2*c*e*f^2 - c*d*f*g))/(7*g^5) + (2 *c*e^2*(f + g*x)^(13/2))/(13*g^5) + (4*c*e*(f + g*x)^(11/2)*(d*g - 2*e*f)) /(11*g^5)
Time = 0.21 (sec) , antiderivative size = 385, normalized size of antiderivative = 2.18 \[ \int (d+e x)^2 (f+g x)^{3/2} \left (a+c x^2\right ) \, dx=\frac {2 \sqrt {g x +f}\, \left (3465 c \,e^{2} g^{6} x^{6}+8190 c d e \,g^{6} x^{5}+4410 c \,e^{2} f \,g^{5} x^{5}+5005 a \,e^{2} g^{6} x^{4}+5005 c \,d^{2} g^{6} x^{4}+10920 c d e f \,g^{5} x^{4}+105 c \,e^{2} f^{2} g^{4} x^{4}+12870 a d e \,g^{6} x^{3}+7150 a \,e^{2} f \,g^{5} x^{3}+7150 c \,d^{2} f \,g^{5} x^{3}+390 c d e \,f^{2} g^{4} x^{3}-120 c \,e^{2} f^{3} g^{3} x^{3}+9009 a \,d^{2} g^{6} x^{2}+20592 a d e f \,g^{5} x^{2}+429 a \,e^{2} f^{2} g^{4} x^{2}+429 c \,d^{2} f^{2} g^{4} x^{2}-468 c d e \,f^{3} g^{3} x^{2}+144 c \,e^{2} f^{4} g^{2} x^{2}+18018 a \,d^{2} f \,g^{5} x +2574 a d e \,f^{2} g^{4} x -572 a \,e^{2} f^{3} g^{3} x -572 c \,d^{2} f^{3} g^{3} x +624 c d e \,f^{4} g^{2} x -192 c \,e^{2} f^{5} g x +9009 a \,d^{2} f^{2} g^{4}-5148 a d e \,f^{3} g^{3}+1144 a \,e^{2} f^{4} g^{2}+1144 c \,d^{2} f^{4} g^{2}-1248 c d e \,f^{5} g +384 c \,e^{2} f^{6}\right )}{45045 g^{5}} \] Input:
int((e*x+d)^2*(g*x+f)^(3/2)*(c*x^2+a),x)
Output:
(2*sqrt(f + g*x)*(9009*a*d**2*f**2*g**4 + 18018*a*d**2*f*g**5*x + 9009*a*d **2*g**6*x**2 - 5148*a*d*e*f**3*g**3 + 2574*a*d*e*f**2*g**4*x + 20592*a*d* e*f*g**5*x**2 + 12870*a*d*e*g**6*x**3 + 1144*a*e**2*f**4*g**2 - 572*a*e**2 *f**3*g**3*x + 429*a*e**2*f**2*g**4*x**2 + 7150*a*e**2*f*g**5*x**3 + 5005* a*e**2*g**6*x**4 + 1144*c*d**2*f**4*g**2 - 572*c*d**2*f**3*g**3*x + 429*c* d**2*f**2*g**4*x**2 + 7150*c*d**2*f*g**5*x**3 + 5005*c*d**2*g**6*x**4 - 12 48*c*d*e*f**5*g + 624*c*d*e*f**4*g**2*x - 468*c*d*e*f**3*g**3*x**2 + 390*c *d*e*f**2*g**4*x**3 + 10920*c*d*e*f*g**5*x**4 + 8190*c*d*e*g**6*x**5 + 384 *c*e**2*f**6 - 192*c*e**2*f**5*g*x + 144*c*e**2*f**4*g**2*x**2 - 120*c*e** 2*f**3*g**3*x**3 + 105*c*e**2*f**2*g**4*x**4 + 4410*c*e**2*f*g**5*x**5 + 3 465*c*e**2*g**6*x**6))/(45045*g**5)