Integrand size = 24, antiderivative size = 242 \[ \int (d+e x)^3 \sqrt {f+g x} \left (a+c x^2\right ) \, dx=-\frac {2 (e f-d g)^3 \left (c f^2+a g^2\right ) (f+g x)^{3/2}}{3 g^6}+\frac {2 (e f-d g)^2 \left (3 a e g^2+c f (5 e f-2 d g)\right ) (f+g x)^{5/2}}{5 g^6}-\frac {2 (e f-d g) \left (3 a e^2 g^2+c \left (10 e^2 f^2-8 d e f g+d^2 g^2\right )\right ) (f+g x)^{7/2}}{7 g^6}+\frac {2 e \left (a e^2 g^2+c \left (10 e^2 f^2-12 d e f g+3 d^2 g^2\right )\right ) (f+g x)^{9/2}}{9 g^6}-\frac {2 c e^2 (5 e f-3 d g) (f+g x)^{11/2}}{11 g^6}+\frac {2 c e^3 (f+g x)^{13/2}}{13 g^6} \] Output:
-2/3*(-d*g+e*f)^3*(a*g^2+c*f^2)*(g*x+f)^(3/2)/g^6+2/5*(-d*g+e*f)^2*(3*a*e* g^2+c*f*(-2*d*g+5*e*f))*(g*x+f)^(5/2)/g^6-2/7*(-d*g+e*f)*(3*a*e^2*g^2+c*(d ^2*g^2-8*d*e*f*g+10*e^2*f^2))*(g*x+f)^(7/2)/g^6+2/9*e*(a*e^2*g^2+c*(3*d^2* g^2-12*d*e*f*g+10*e^2*f^2))*(g*x+f)^(9/2)/g^6-2/11*c*e^2*(-3*d*g+5*e*f)*(g *x+f)^(11/2)/g^6+2/13*c*e^3*(g*x+f)^(13/2)/g^6
Time = 0.31 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.17 \[ \int (d+e x)^3 \sqrt {f+g x} \left (a+c x^2\right ) \, dx=\frac {2 (f+g x)^{3/2} \left (143 a g^2 \left (105 d^3 g^3+63 d^2 e g^2 (-2 f+3 g x)+9 d e^2 g \left (8 f^2-12 f g x+15 g^2 x^2\right )+e^3 \left (-16 f^3+24 f^2 g x-30 f g^2 x^2+35 g^3 x^3\right )\right )+c \left (429 d^3 g^3 \left (8 f^2-12 f g x+15 g^2 x^2\right )+429 d^2 e g^2 \left (-16 f^3+24 f^2 g x-30 f g^2 x^2+35 g^3 x^3\right )+39 d e^2 g \left (128 f^4-192 f^3 g x+240 f^2 g^2 x^2-280 f g^3 x^3+315 g^4 x^4\right )-5 e^3 \left (256 f^5-384 f^4 g x+480 f^3 g^2 x^2-560 f^2 g^3 x^3+630 f g^4 x^4-693 g^5 x^5\right )\right )\right )}{45045 g^6} \] Input:
Integrate[(d + e*x)^3*Sqrt[f + g*x]*(a + c*x^2),x]
Output:
(2*(f + g*x)^(3/2)*(143*a*g^2*(105*d^3*g^3 + 63*d^2*e*g^2*(-2*f + 3*g*x) + 9*d*e^2*g*(8*f^2 - 12*f*g*x + 15*g^2*x^2) + e^3*(-16*f^3 + 24*f^2*g*x - 3 0*f*g^2*x^2 + 35*g^3*x^3)) + c*(429*d^3*g^3*(8*f^2 - 12*f*g*x + 15*g^2*x^2 ) + 429*d^2*e*g^2*(-16*f^3 + 24*f^2*g*x - 30*f*g^2*x^2 + 35*g^3*x^3) + 39* d*e^2*g*(128*f^4 - 192*f^3*g*x + 240*f^2*g^2*x^2 - 280*f*g^3*x^3 + 315*g^4 *x^4) - 5*e^3*(256*f^5 - 384*f^4*g*x + 480*f^3*g^2*x^2 - 560*f^2*g^3*x^3 + 630*f*g^4*x^4 - 693*g^5*x^5))))/(45045*g^6)
Time = 0.42 (sec) , antiderivative size = 242, 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.
| \(\displaystyle \int \left (a+c x^2\right ) (d+e x)^3 \sqrt {f+g x} \, dx\) |
| \(\Big \downarrow \) 652 |
| \(\displaystyle \int \left (\frac {e (f+g x)^{7/2} \left (a e^2 g^2+c \left (3 d^2 g^2-12 d e f g+10 e^2 f^2\right )\right )}{g^5}+\frac {(f+g x)^{5/2} (e f-d g) \left (-3 a e^2 g^2-c \left (d^2 g^2-8 d e f g+10 e^2 f^2\right )\right )}{g^5}+\frac {\sqrt {f+g x} \left (a g^2+c f^2\right ) (d g-e f)^3}{g^5}+\frac {(f+g x)^{3/2} (e f-d g)^2 \left (3 a e g^2+c f (5 e f-2 d g)\right )}{g^5}-\frac {c e^2 (f+g x)^{9/2} (5 e f-3 d g)}{g^5}+\frac {c e^3 (f+g x)^{11/2}}{g^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 e (f+g x)^{9/2} \left (a e^2 g^2+c \left (3 d^2 g^2-12 d e f g+10 e^2 f^2\right )\right )}{9 g^6}-\frac {2 (f+g x)^{7/2} (e f-d g) \left (3 a e^2 g^2+c \left (d^2 g^2-8 d e f g+10 e^2 f^2\right )\right )}{7 g^6}-\frac {2 (f+g x)^{3/2} \left (a g^2+c f^2\right ) (e f-d g)^3}{3 g^6}+\frac {2 (f+g x)^{5/2} (e f-d g)^2 \left (3 a e g^2+c f (5 e f-2 d g)\right )}{5 g^6}-\frac {2 c e^2 (f+g x)^{11/2} (5 e f-3 d g)}{11 g^6}+\frac {2 c e^3 (f+g x)^{13/2}}{13 g^6}\) |
Input:
Int[(d + e*x)^3*Sqrt[f + g*x]*(a + c*x^2),x]
Output:
(-2*(e*f - d*g)^3*(c*f^2 + a*g^2)*(f + g*x)^(3/2))/(3*g^6) + (2*(e*f - d*g )^2*(3*a*e*g^2 + c*f*(5*e*f - 2*d*g))*(f + g*x)^(5/2))/(5*g^6) - (2*(e*f - d*g)*(3*a*e^2*g^2 + c*(10*e^2*f^2 - 8*d*e*f*g + d^2*g^2))*(f + g*x)^(7/2) )/(7*g^6) + (2*e*(a*e^2*g^2 + c*(10*e^2*f^2 - 12*d*e*f*g + 3*d^2*g^2))*(f + g*x)^(9/2))/(9*g^6) - (2*c*e^2*(5*e*f - 3*d*g)*(f + g*x)^(11/2))/(11*g^6 ) + (2*c*e^3*(f + g*x)^(13/2))/(13*g^6)
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 = 1.40 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.01
| method | result | size |
| derivativedivides | \(\frac {\frac {2 e^{3} c \left (g x +f \right )^{\frac {13}{2}}}{13}+\frac {2 \left (3 \left (d g -e f \right ) e^{2} c -2 f \,e^{3} c \right ) \left (g x +f \right )^{\frac {11}{2}}}{11}+\frac {2 \left (3 \left (d g -e f \right )^{2} e c -6 \left (d g -e f \right ) e^{2} c f +e^{3} \left (a \,g^{2}+c \,f^{2}\right )\right ) \left (g x +f \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (d g -e f \right )^{3} c -6 \left (d g -e f \right )^{2} e c f +3 \left (d g -e f \right ) e^{2} \left (a \,g^{2}+c \,f^{2}\right )\right ) \left (g x +f \right )^{\frac {7}{2}}}{7}+\frac {2 \left (-2 \left (d g -e f \right )^{3} c f +3 \left (d g -e f \right )^{2} e \left (a \,g^{2}+c \,f^{2}\right )\right ) \left (g x +f \right )^{\frac {5}{2}}}{5}+\frac {2 \left (d g -e f \right )^{3} \left (a \,g^{2}+c \,f^{2}\right ) \left (g x +f \right )^{\frac {3}{2}}}{3}}{g^{6}}\) | \(244\) |
| default | \(\frac {\frac {2 e^{3} c \left (g x +f \right )^{\frac {13}{2}}}{13}+\frac {2 \left (3 \left (d g -e f \right ) e^{2} c -2 f \,e^{3} c \right ) \left (g x +f \right )^{\frac {11}{2}}}{11}+\frac {2 \left (3 \left (d g -e f \right )^{2} e c -6 \left (d g -e f \right ) e^{2} c f +e^{3} \left (a \,g^{2}+c \,f^{2}\right )\right ) \left (g x +f \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (d g -e f \right )^{3} c -6 \left (d g -e f \right )^{2} e c f +3 \left (d g -e f \right ) e^{2} \left (a \,g^{2}+c \,f^{2}\right )\right ) \left (g x +f \right )^{\frac {7}{2}}}{7}+\frac {2 \left (-2 \left (d g -e f \right )^{3} c f +3 \left (d g -e f \right )^{2} e \left (a \,g^{2}+c \,f^{2}\right )\right ) \left (g x +f \right )^{\frac {5}{2}}}{5}+\frac {2 \left (d g -e f \right )^{3} \left (a \,g^{2}+c \,f^{2}\right ) \left (g x +f \right )^{\frac {3}{2}}}{3}}{g^{6}}\) | \(244\) |
| pseudoelliptic | \(\frac {2 \left (\left (\frac {\left (\frac {9 c \,x^{2}}{13}+a \right ) x^{3} e^{3}}{3}+\frac {9 \left (\frac {7 c \,x^{2}}{11}+a \right ) x^{2} d \,e^{2}}{7}+\frac {9 \left (\frac {5 c \,x^{2}}{9}+a \right ) x \,d^{2} e}{5}+d^{3} \left (a +\frac {3 c \,x^{2}}{7}\right )\right ) g^{5}-\frac {6 f \left (\left (\frac {25}{143} c \,x^{4}+\frac {5}{21} a \,x^{2}\right ) e^{3}+\frac {6 x d \left (\frac {70 c \,x^{2}}{99}+a \right ) e^{2}}{7}+d^{2} \left (\frac {5 c \,x^{2}}{7}+a \right ) e +\frac {2 d^{3} c x}{7}\right ) g^{4}}{5}+\frac {24 f^{2} \left (\frac {x \left (\frac {350 c \,x^{2}}{429}+a \right ) e^{3}}{3}+d \left (\frac {10 c \,x^{2}}{11}+a \right ) e^{2}+c \,d^{2} e x +\frac {c \,d^{3}}{3}\right ) g^{3}}{35}-\frac {16 e \,f^{3} \left (\left (\frac {150 c \,x^{2}}{143}+a \right ) e^{2}+\frac {36 c d x e}{11}+3 c \,d^{2}\right ) g^{2}}{105}+\frac {128 e^{2} f^{4} \left (\frac {5 e x}{13}+d \right ) c g}{385}-\frac {256 c \,e^{3} f^{5}}{3003}\right ) \left (g x +f \right )^{\frac {3}{2}}}{3 g^{6}}\) | \(246\) |
| gosper | \(\frac {2 \left (g x +f \right )^{\frac {3}{2}} \left (3465 e^{3} c \,x^{5} g^{5}+12285 c d \,e^{2} g^{5} x^{4}-3150 c \,e^{3} f \,g^{4} x^{4}+5005 a \,e^{3} g^{5} x^{3}+15015 c \,d^{2} e \,g^{5} x^{3}-10920 c d \,e^{2} f \,g^{4} x^{3}+2800 c \,e^{3} f^{2} g^{3} x^{3}+19305 a d \,e^{2} g^{5} x^{2}-4290 a \,e^{3} f \,g^{4} x^{2}+6435 c \,d^{3} g^{5} x^{2}-12870 c \,d^{2} e f \,g^{4} x^{2}+9360 c d \,e^{2} f^{2} g^{3} x^{2}-2400 c \,e^{3} f^{3} g^{2} x^{2}+27027 a \,d^{2} e \,g^{5} x -15444 a d \,e^{2} f \,g^{4} x +3432 a \,e^{3} f^{2} g^{3} x -5148 c \,d^{3} f \,g^{4} x +10296 c \,d^{2} e \,f^{2} g^{3} x -7488 c d \,e^{2} f^{3} g^{2} x +1920 c \,e^{3} f^{4} g x +15015 a \,d^{3} g^{5}-18018 a \,d^{2} e f \,g^{4}+10296 a d \,e^{2} f^{2} g^{3}-2288 a \,e^{3} f^{3} g^{2}+3432 c \,d^{3} f^{2} g^{3}-6864 c \,d^{2} e \,f^{3} g^{2}+4992 c d \,e^{2} f^{4} g -1280 c \,e^{3} f^{5}\right )}{45045 g^{6}}\) | \(365\) |
| orering | \(\frac {2 \left (g x +f \right )^{\frac {3}{2}} \left (3465 e^{3} c \,x^{5} g^{5}+12285 c d \,e^{2} g^{5} x^{4}-3150 c \,e^{3} f \,g^{4} x^{4}+5005 a \,e^{3} g^{5} x^{3}+15015 c \,d^{2} e \,g^{5} x^{3}-10920 c d \,e^{2} f \,g^{4} x^{3}+2800 c \,e^{3} f^{2} g^{3} x^{3}+19305 a d \,e^{2} g^{5} x^{2}-4290 a \,e^{3} f \,g^{4} x^{2}+6435 c \,d^{3} g^{5} x^{2}-12870 c \,d^{2} e f \,g^{4} x^{2}+9360 c d \,e^{2} f^{2} g^{3} x^{2}-2400 c \,e^{3} f^{3} g^{2} x^{2}+27027 a \,d^{2} e \,g^{5} x -15444 a d \,e^{2} f \,g^{4} x +3432 a \,e^{3} f^{2} g^{3} x -5148 c \,d^{3} f \,g^{4} x +10296 c \,d^{2} e \,f^{2} g^{3} x -7488 c d \,e^{2} f^{3} g^{2} x +1920 c \,e^{3} f^{4} g x +15015 a \,d^{3} g^{5}-18018 a \,d^{2} e f \,g^{4}+10296 a d \,e^{2} f^{2} g^{3}-2288 a \,e^{3} f^{3} g^{2}+3432 c \,d^{3} f^{2} g^{3}-6864 c \,d^{2} e \,f^{3} g^{2}+4992 c d \,e^{2} f^{4} g -1280 c \,e^{3} f^{5}\right )}{45045 g^{6}}\) | \(365\) |
| trager | \(\frac {2 \left (3465 e^{3} c \,g^{6} x^{6}+12285 c d \,e^{2} g^{6} x^{5}+315 c \,e^{3} f \,g^{5} x^{5}+5005 a \,e^{3} g^{6} x^{4}+15015 c \,d^{2} e \,g^{6} x^{4}+1365 c d \,e^{2} f \,g^{5} x^{4}-350 c \,e^{3} f^{2} g^{4} x^{4}+19305 a d \,e^{2} g^{6} x^{3}+715 a \,e^{3} f \,g^{5} x^{3}+6435 c \,d^{3} g^{6} x^{3}+2145 c \,d^{2} e f \,g^{5} x^{3}-1560 c d \,e^{2} f^{2} g^{4} x^{3}+400 c \,e^{3} f^{3} g^{3} x^{3}+27027 a \,d^{2} e \,g^{6} x^{2}+3861 a d \,e^{2} f \,g^{5} x^{2}-858 a \,e^{3} f^{2} g^{4} x^{2}+1287 c \,d^{3} f \,g^{5} x^{2}-2574 c \,d^{2} e \,f^{2} g^{4} x^{2}+1872 c d \,e^{2} f^{3} g^{3} x^{2}-480 c \,e^{3} f^{4} g^{2} x^{2}+15015 a \,d^{3} g^{6} x +9009 a \,d^{2} e f \,g^{5} x -5148 a d \,e^{2} f^{2} g^{4} x +1144 a \,e^{3} f^{3} g^{3} x -1716 c \,d^{3} f^{2} g^{4} x +3432 c \,d^{2} e \,f^{3} g^{3} x -2496 c d \,e^{2} f^{4} g^{2} x +640 c \,e^{3} f^{5} g x +15015 a \,d^{3} f \,g^{5}-18018 a \,d^{2} e \,f^{2} g^{4}+10296 a d \,e^{2} f^{3} g^{3}-2288 a \,e^{3} f^{4} g^{2}+3432 c \,d^{3} f^{3} g^{3}-6864 c \,d^{2} e \,f^{4} g^{2}+4992 c d \,e^{2} f^{5} g -1280 c \,e^{3} f^{6}\right ) \sqrt {g x +f}}{45045 g^{6}}\) | \(485\) |
| risch | \(\frac {2 \left (3465 e^{3} c \,g^{6} x^{6}+12285 c d \,e^{2} g^{6} x^{5}+315 c \,e^{3} f \,g^{5} x^{5}+5005 a \,e^{3} g^{6} x^{4}+15015 c \,d^{2} e \,g^{6} x^{4}+1365 c d \,e^{2} f \,g^{5} x^{4}-350 c \,e^{3} f^{2} g^{4} x^{4}+19305 a d \,e^{2} g^{6} x^{3}+715 a \,e^{3} f \,g^{5} x^{3}+6435 c \,d^{3} g^{6} x^{3}+2145 c \,d^{2} e f \,g^{5} x^{3}-1560 c d \,e^{2} f^{2} g^{4} x^{3}+400 c \,e^{3} f^{3} g^{3} x^{3}+27027 a \,d^{2} e \,g^{6} x^{2}+3861 a d \,e^{2} f \,g^{5} x^{2}-858 a \,e^{3} f^{2} g^{4} x^{2}+1287 c \,d^{3} f \,g^{5} x^{2}-2574 c \,d^{2} e \,f^{2} g^{4} x^{2}+1872 c d \,e^{2} f^{3} g^{3} x^{2}-480 c \,e^{3} f^{4} g^{2} x^{2}+15015 a \,d^{3} g^{6} x +9009 a \,d^{2} e f \,g^{5} x -5148 a d \,e^{2} f^{2} g^{4} x +1144 a \,e^{3} f^{3} g^{3} x -1716 c \,d^{3} f^{2} g^{4} x +3432 c \,d^{2} e \,f^{3} g^{3} x -2496 c d \,e^{2} f^{4} g^{2} x +640 c \,e^{3} f^{5} g x +15015 a \,d^{3} f \,g^{5}-18018 a \,d^{2} e \,f^{2} g^{4}+10296 a d \,e^{2} f^{3} g^{3}-2288 a \,e^{3} f^{4} g^{2}+3432 c \,d^{3} f^{3} g^{3}-6864 c \,d^{2} e \,f^{4} g^{2}+4992 c d \,e^{2} f^{5} g -1280 c \,e^{3} f^{6}\right ) \sqrt {g x +f}}{45045 g^{6}}\) | \(485\) |
Input:
int((e*x+d)^3*(g*x+f)^(1/2)*(c*x^2+a),x,method=_RETURNVERBOSE)
Output:
2/g^6*(1/13*e^3*c*(g*x+f)^(13/2)+1/11*(3*(d*g-e*f)*e^2*c-2*f*e^3*c)*(g*x+f )^(11/2)+1/9*(3*(d*g-e*f)^2*e*c-6*(d*g-e*f)*e^2*c*f+e^3*(a*g^2+c*f^2))*(g* x+f)^(9/2)+1/7*((d*g-e*f)^3*c-6*(d*g-e*f)^2*e*c*f+3*(d*g-e*f)*e^2*(a*g^2+c *f^2))*(g*x+f)^(7/2)+1/5*(-2*(d*g-e*f)^3*c*f+3*(d*g-e*f)^2*e*(a*g^2+c*f^2) )*(g*x+f)^(5/2)+1/3*(d*g-e*f)^3*(a*g^2+c*f^2)*(g*x+f)^(3/2))
Time = 0.08 (sec) , antiderivative size = 419, normalized size of antiderivative = 1.73 \[ \int (d+e x)^3 \sqrt {f+g x} \left (a+c x^2\right ) \, dx=\frac {2 \, {\left (3465 \, c e^{3} g^{6} x^{6} - 1280 \, c e^{3} f^{6} + 4992 \, c d e^{2} f^{5} g - 18018 \, a d^{2} e f^{2} g^{4} + 15015 \, a d^{3} f g^{5} - 2288 \, {\left (3 \, c d^{2} e + a e^{3}\right )} f^{4} g^{2} + 3432 \, {\left (c d^{3} + 3 \, a d e^{2}\right )} f^{3} g^{3} + 315 \, {\left (c e^{3} f g^{5} + 39 \, c d e^{2} g^{6}\right )} x^{5} - 35 \, {\left (10 \, c e^{3} f^{2} g^{4} - 39 \, c d e^{2} f g^{5} - 143 \, {\left (3 \, c d^{2} e + a e^{3}\right )} g^{6}\right )} x^{4} + 5 \, {\left (80 \, c e^{3} f^{3} g^{3} - 312 \, c d e^{2} f^{2} g^{4} + 143 \, {\left (3 \, c d^{2} e + a e^{3}\right )} f g^{5} + 1287 \, {\left (c d^{3} + 3 \, a d e^{2}\right )} g^{6}\right )} x^{3} - 3 \, {\left (160 \, c e^{3} f^{4} g^{2} - 624 \, c d e^{2} f^{3} g^{3} - 9009 \, a d^{2} e g^{6} + 286 \, {\left (3 \, c d^{2} e + a e^{3}\right )} f^{2} g^{4} - 429 \, {\left (c d^{3} + 3 \, a d e^{2}\right )} f g^{5}\right )} x^{2} + {\left (640 \, c e^{3} f^{5} g - 2496 \, c d e^{2} f^{4} g^{2} + 9009 \, a d^{2} e f g^{5} + 15015 \, a d^{3} g^{6} + 1144 \, {\left (3 \, c d^{2} e + a e^{3}\right )} f^{3} g^{3} - 1716 \, {\left (c d^{3} + 3 \, a d e^{2}\right )} f^{2} g^{4}\right )} x\right )} \sqrt {g x + f}}{45045 \, g^{6}} \] Input:
integrate((e*x+d)^3*(g*x+f)^(1/2)*(c*x^2+a),x, algorithm="fricas")
Output:
2/45045*(3465*c*e^3*g^6*x^6 - 1280*c*e^3*f^6 + 4992*c*d*e^2*f^5*g - 18018* a*d^2*e*f^2*g^4 + 15015*a*d^3*f*g^5 - 2288*(3*c*d^2*e + a*e^3)*f^4*g^2 + 3 432*(c*d^3 + 3*a*d*e^2)*f^3*g^3 + 315*(c*e^3*f*g^5 + 39*c*d*e^2*g^6)*x^5 - 35*(10*c*e^3*f^2*g^4 - 39*c*d*e^2*f*g^5 - 143*(3*c*d^2*e + a*e^3)*g^6)*x^ 4 + 5*(80*c*e^3*f^3*g^3 - 312*c*d*e^2*f^2*g^4 + 143*(3*c*d^2*e + a*e^3)*f* g^5 + 1287*(c*d^3 + 3*a*d*e^2)*g^6)*x^3 - 3*(160*c*e^3*f^4*g^2 - 624*c*d*e ^2*f^3*g^3 - 9009*a*d^2*e*g^6 + 286*(3*c*d^2*e + a*e^3)*f^2*g^4 - 429*(c*d ^3 + 3*a*d*e^2)*f*g^5)*x^2 + (640*c*e^3*f^5*g - 2496*c*d*e^2*f^4*g^2 + 900 9*a*d^2*e*f*g^5 + 15015*a*d^3*g^6 + 1144*(3*c*d^2*e + a*e^3)*f^3*g^3 - 171 6*(c*d^3 + 3*a*d*e^2)*f^2*g^4)*x)*sqrt(g*x + f)/g^6
Leaf count of result is larger than twice the leaf count of optimal. 503 vs. \(2 (243) = 486\).
Time = 1.37 (sec) , antiderivative size = 503, normalized size of antiderivative = 2.08 \[ \int (d+e x)^3 \sqrt {f+g x} \left (a+c x^2\right ) \, dx=\begin {cases} \frac {2 \left (\frac {c e^{3} \left (f + g x\right )^{\frac {13}{2}}}{13 g^{5}} + \frac {\left (f + g x\right )^{\frac {11}{2}} \cdot \left (3 c d e^{2} g - 5 c e^{3} f\right )}{11 g^{5}} + \frac {\left (f + g x\right )^{\frac {9}{2}} \left (a e^{3} g^{2} + 3 c d^{2} e g^{2} - 12 c d e^{2} f g + 10 c e^{3} f^{2}\right )}{9 g^{5}} + \frac {\left (f + g x\right )^{\frac {7}{2}} \cdot \left (3 a d e^{2} g^{3} - 3 a e^{3} f g^{2} + c d^{3} g^{3} - 9 c d^{2} e f g^{2} + 18 c d e^{2} f^{2} g - 10 c e^{3} f^{3}\right )}{7 g^{5}} + \frac {\left (f + g x\right )^{\frac {5}{2}} \cdot \left (3 a d^{2} e g^{4} - 6 a d e^{2} f g^{3} + 3 a e^{3} f^{2} g^{2} - 2 c d^{3} f g^{3} + 9 c d^{2} e f^{2} g^{2} - 12 c d e^{2} f^{3} g + 5 c e^{3} f^{4}\right )}{5 g^{5}} + \frac {\left (f + g x\right )^{\frac {3}{2}} \left (a d^{3} g^{5} - 3 a d^{2} e f g^{4} + 3 a d e^{2} f^{2} g^{3} - a e^{3} f^{3} g^{2} + c d^{3} f^{2} g^{3} - 3 c d^{2} e f^{3} g^{2} + 3 c d e^{2} f^{4} g - c e^{3} f^{5}\right )}{3 g^{5}}\right )}{g} & \text {for}\: g \neq 0 \\\sqrt {f} \left (a d^{3} x + \frac {3 a d^{2} e x^{2}}{2} + \frac {3 c d e^{2} x^{5}}{5} + \frac {c e^{3} x^{6}}{6} + \frac {x^{4} \left (a e^{3} + 3 c d^{2} e\right )}{4} + \frac {x^{3} \cdot \left (3 a d e^{2} + c d^{3}\right )}{3}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((e*x+d)**3*(g*x+f)**(1/2)*(c*x**2+a),x)
Output:
Piecewise((2*(c*e**3*(f + g*x)**(13/2)/(13*g**5) + (f + g*x)**(11/2)*(3*c* d*e**2*g - 5*c*e**3*f)/(11*g**5) + (f + g*x)**(9/2)*(a*e**3*g**2 + 3*c*d** 2*e*g**2 - 12*c*d*e**2*f*g + 10*c*e**3*f**2)/(9*g**5) + (f + g*x)**(7/2)*( 3*a*d*e**2*g**3 - 3*a*e**3*f*g**2 + c*d**3*g**3 - 9*c*d**2*e*f*g**2 + 18*c *d*e**2*f**2*g - 10*c*e**3*f**3)/(7*g**5) + (f + g*x)**(5/2)*(3*a*d**2*e*g **4 - 6*a*d*e**2*f*g**3 + 3*a*e**3*f**2*g**2 - 2*c*d**3*f*g**3 + 9*c*d**2* e*f**2*g**2 - 12*c*d*e**2*f**3*g + 5*c*e**3*f**4)/(5*g**5) + (f + g*x)**(3 /2)*(a*d**3*g**5 - 3*a*d**2*e*f*g**4 + 3*a*d*e**2*f**2*g**3 - a*e**3*f**3* g**2 + c*d**3*f**2*g**3 - 3*c*d**2*e*f**3*g**2 + 3*c*d*e**2*f**4*g - c*e** 3*f**5)/(3*g**5))/g, Ne(g, 0)), (sqrt(f)*(a*d**3*x + 3*a*d**2*e*x**2/2 + 3 *c*d*e**2*x**5/5 + c*e**3*x**6/6 + x**4*(a*e**3 + 3*c*d**2*e)/4 + x**3*(3* a*d*e**2 + c*d**3)/3), True))
Time = 0.04 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.35 \[ \int (d+e x)^3 \sqrt {f+g x} \left (a+c x^2\right ) \, dx=\frac {2 \, {\left (3465 \, {\left (g x + f\right )}^{\frac {13}{2}} c e^{3} - 4095 \, {\left (5 \, c e^{3} f - 3 \, c d e^{2} g\right )} {\left (g x + f\right )}^{\frac {11}{2}} + 5005 \, {\left (10 \, c e^{3} f^{2} - 12 \, c d e^{2} f g + {\left (3 \, c d^{2} e + a e^{3}\right )} g^{2}\right )} {\left (g x + f\right )}^{\frac {9}{2}} - 6435 \, {\left (10 \, c e^{3} f^{3} - 18 \, c d e^{2} f^{2} g + 3 \, {\left (3 \, c d^{2} e + a e^{3}\right )} f g^{2} - {\left (c d^{3} + 3 \, a d e^{2}\right )} g^{3}\right )} {\left (g x + f\right )}^{\frac {7}{2}} + 9009 \, {\left (5 \, c e^{3} f^{4} - 12 \, c d e^{2} f^{3} g + 3 \, a d^{2} e g^{4} + 3 \, {\left (3 \, c d^{2} e + a e^{3}\right )} f^{2} g^{2} - 2 \, {\left (c d^{3} + 3 \, a d e^{2}\right )} f g^{3}\right )} {\left (g x + f\right )}^{\frac {5}{2}} - 15015 \, {\left (c e^{3} f^{5} - 3 \, c d e^{2} f^{4} g + 3 \, a d^{2} e f g^{4} - a d^{3} g^{5} + {\left (3 \, c d^{2} e + a e^{3}\right )} f^{3} g^{2} - {\left (c d^{3} + 3 \, a d e^{2}\right )} f^{2} g^{3}\right )} {\left (g x + f\right )}^{\frac {3}{2}}\right )}}{45045 \, g^{6}} \] Input:
integrate((e*x+d)^3*(g*x+f)^(1/2)*(c*x^2+a),x, algorithm="maxima")
Output:
2/45045*(3465*(g*x + f)^(13/2)*c*e^3 - 4095*(5*c*e^3*f - 3*c*d*e^2*g)*(g*x + f)^(11/2) + 5005*(10*c*e^3*f^2 - 12*c*d*e^2*f*g + (3*c*d^2*e + a*e^3)*g ^2)*(g*x + f)^(9/2) - 6435*(10*c*e^3*f^3 - 18*c*d*e^2*f^2*g + 3*(3*c*d^2*e + a*e^3)*f*g^2 - (c*d^3 + 3*a*d*e^2)*g^3)*(g*x + f)^(7/2) + 9009*(5*c*e^3 *f^4 - 12*c*d*e^2*f^3*g + 3*a*d^2*e*g^4 + 3*(3*c*d^2*e + a*e^3)*f^2*g^2 - 2*(c*d^3 + 3*a*d*e^2)*f*g^3)*(g*x + f)^(5/2) - 15015*(c*e^3*f^5 - 3*c*d*e^ 2*f^4*g + 3*a*d^2*e*f*g^4 - a*d^3*g^5 + (3*c*d^2*e + a*e^3)*f^3*g^2 - (c*d ^3 + 3*a*d*e^2)*f^2*g^3)*(g*x + f)^(3/2))/g^6
Leaf count of result is larger than twice the leaf count of optimal. 859 vs. \(2 (218) = 436\).
Time = 0.12 (sec) , antiderivative size = 859, normalized size of antiderivative = 3.55 \[ \int (d+e x)^3 \sqrt {f+g x} \left (a+c x^2\right ) \, dx=\text {Too large to display} \] Input:
integrate((e*x+d)^3*(g*x+f)^(1/2)*(c*x^2+a),x, algorithm="giac")
Output:
2/45045*(45045*sqrt(g*x + f)*a*d^3*f + 15015*((g*x + f)^(3/2) - 3*sqrt(g*x + f)*f)*a*d^3 + 45045*((g*x + f)^(3/2) - 3*sqrt(g*x + f)*f)*a*d^2*e*f/g + 3003*(3*(g*x + f)^(5/2) - 10*(g*x + f)^(3/2)*f + 15*sqrt(g*x + f)*f^2)*c* d^3*f/g^2 + 9009*(3*(g*x + f)^(5/2) - 10*(g*x + f)^(3/2)*f + 15*sqrt(g*x + f)*f^2)*a*d*e^2*f/g^2 + 9009*(3*(g*x + f)^(5/2) - 10*(g*x + f)^(3/2)*f + 15*sqrt(g*x + f)*f^2)*a*d^2*e/g + 3861*(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*e*f/g^3 + 12 87*(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^3*f/g^3 + 1287*(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^3/g^2 + 3861*( 5*(g*x + f)^(7/2) - 21*(g*x + f)^(5/2)*f + 35*(g*x + f)^(3/2)*f^2 - 35*sqr t(g*x + f)*f^3)*a*d*e^2/g^2 + 429*(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^2*f/g^4 + 429*(35*(g*x + f)^(9/2) - 180*(g*x + f)^(7/2)*f + 3 78*(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*e/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)*a*e^3/g^3 + 65*(63*(g*x + f)^(11/2) - 385*(g*x + f)^(9/2)*f + 990*(g*x + f)^(7/2)*f ^2 - 1386*(g*x + f)^(5/2)*f^3 + 1155*(g*x + f)^(3/2)*f^4 - 693*sqrt(g*x + f)*f^5)*c*e^3*f/g^5 + 195*(63*(g*x + f)^(11/2) - 385*(g*x + f)^(9/2)*f ...
Time = 0.07 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.92 \[ \int (d+e x)^3 \sqrt {f+g x} \left (a+c x^2\right ) \, dx=\frac {{\left (f+g\,x\right )}^{9/2}\,\left (6\,c\,d^2\,e\,g^2-24\,c\,d\,e^2\,f\,g+20\,c\,e^3\,f^2+2\,a\,e^3\,g^2\right )}{9\,g^6}+\frac {2\,{\left (f+g\,x\right )}^{3/2}\,\left (c\,f^2+a\,g^2\right )\,{\left (d\,g-e\,f\right )}^3}{3\,g^6}+\frac {2\,c\,e^3\,{\left (f+g\,x\right )}^{13/2}}{13\,g^6}+\frac {2\,{\left (f+g\,x\right )}^{5/2}\,{\left (d\,g-e\,f\right )}^2\,\left (5\,c\,e\,f^2-2\,c\,d\,f\,g+3\,a\,e\,g^2\right )}{5\,g^6}+\frac {2\,{\left (f+g\,x\right )}^{7/2}\,\left (d\,g-e\,f\right )\,\left (c\,d^2\,g^2-8\,c\,d\,e\,f\,g+10\,c\,e^2\,f^2+3\,a\,e^2\,g^2\right )}{7\,g^6}+\frac {2\,c\,e^2\,{\left (f+g\,x\right )}^{11/2}\,\left (3\,d\,g-5\,e\,f\right )}{11\,g^6} \] Input:
int((f + g*x)^(1/2)*(a + c*x^2)*(d + e*x)^3,x)
Output:
((f + g*x)^(9/2)*(2*a*e^3*g^2 + 20*c*e^3*f^2 + 6*c*d^2*e*g^2 - 24*c*d*e^2* f*g))/(9*g^6) + (2*(f + g*x)^(3/2)*(a*g^2 + c*f^2)*(d*g - e*f)^3)/(3*g^6) + (2*c*e^3*(f + g*x)^(13/2))/(13*g^6) + (2*(f + g*x)^(5/2)*(d*g - e*f)^2*( 3*a*e*g^2 + 5*c*e*f^2 - 2*c*d*f*g))/(5*g^6) + (2*(f + g*x)^(7/2)*(d*g - e* f)*(3*a*e^2*g^2 + c*d^2*g^2 + 10*c*e^2*f^2 - 8*c*d*e*f*g))/(7*g^6) + (2*c* e^2*(f + g*x)^(11/2)*(3*d*g - 5*e*f))/(11*g^6)
Time = 0.22 (sec) , antiderivative size = 483, normalized size of antiderivative = 2.00 \[ \int (d+e x)^3 \sqrt {f+g x} \left (a+c x^2\right ) \, dx=\frac {2 \sqrt {g x +f}\, \left (3465 c \,e^{3} g^{6} x^{6}+12285 c d \,e^{2} g^{6} x^{5}+315 c \,e^{3} f \,g^{5} x^{5}+5005 a \,e^{3} g^{6} x^{4}+15015 c \,d^{2} e \,g^{6} x^{4}+1365 c d \,e^{2} f \,g^{5} x^{4}-350 c \,e^{3} f^{2} g^{4} x^{4}+19305 a d \,e^{2} g^{6} x^{3}+715 a \,e^{3} f \,g^{5} x^{3}+6435 c \,d^{3} g^{6} x^{3}+2145 c \,d^{2} e f \,g^{5} x^{3}-1560 c d \,e^{2} f^{2} g^{4} x^{3}+400 c \,e^{3} f^{3} g^{3} x^{3}+27027 a \,d^{2} e \,g^{6} x^{2}+3861 a d \,e^{2} f \,g^{5} x^{2}-858 a \,e^{3} f^{2} g^{4} x^{2}+1287 c \,d^{3} f \,g^{5} x^{2}-2574 c \,d^{2} e \,f^{2} g^{4} x^{2}+1872 c d \,e^{2} f^{3} g^{3} x^{2}-480 c \,e^{3} f^{4} g^{2} x^{2}+15015 a \,d^{3} g^{6} x +9009 a \,d^{2} e f \,g^{5} x -5148 a d \,e^{2} f^{2} g^{4} x +1144 a \,e^{3} f^{3} g^{3} x -1716 c \,d^{3} f^{2} g^{4} x +3432 c \,d^{2} e \,f^{3} g^{3} x -2496 c d \,e^{2} f^{4} g^{2} x +640 c \,e^{3} f^{5} g x +15015 a \,d^{3} f \,g^{5}-18018 a \,d^{2} e \,f^{2} g^{4}+10296 a d \,e^{2} f^{3} g^{3}-2288 a \,e^{3} f^{4} g^{2}+3432 c \,d^{3} f^{3} g^{3}-6864 c \,d^{2} e \,f^{4} g^{2}+4992 c d \,e^{2} f^{5} g -1280 c \,e^{3} f^{6}\right )}{45045 g^{6}} \] Input:
int((e*x+d)^3*(g*x+f)^(1/2)*(c*x^2+a),x)
Output:
(2*sqrt(f + g*x)*(15015*a*d**3*f*g**5 + 15015*a*d**3*g**6*x - 18018*a*d**2 *e*f**2*g**4 + 9009*a*d**2*e*f*g**5*x + 27027*a*d**2*e*g**6*x**2 + 10296*a *d*e**2*f**3*g**3 - 5148*a*d*e**2*f**2*g**4*x + 3861*a*d*e**2*f*g**5*x**2 + 19305*a*d*e**2*g**6*x**3 - 2288*a*e**3*f**4*g**2 + 1144*a*e**3*f**3*g**3 *x - 858*a*e**3*f**2*g**4*x**2 + 715*a*e**3*f*g**5*x**3 + 5005*a*e**3*g**6 *x**4 + 3432*c*d**3*f**3*g**3 - 1716*c*d**3*f**2*g**4*x + 1287*c*d**3*f*g* *5*x**2 + 6435*c*d**3*g**6*x**3 - 6864*c*d**2*e*f**4*g**2 + 3432*c*d**2*e* f**3*g**3*x - 2574*c*d**2*e*f**2*g**4*x**2 + 2145*c*d**2*e*f*g**5*x**3 + 1 5015*c*d**2*e*g**6*x**4 + 4992*c*d*e**2*f**5*g - 2496*c*d*e**2*f**4*g**2*x + 1872*c*d*e**2*f**3*g**3*x**2 - 1560*c*d*e**2*f**2*g**4*x**3 + 1365*c*d* e**2*f*g**5*x**4 + 12285*c*d*e**2*g**6*x**5 - 1280*c*e**3*f**6 + 640*c*e** 3*f**5*g*x - 480*c*e**3*f**4*g**2*x**2 + 400*c*e**3*f**3*g**3*x**3 - 350*c *e**3*f**2*g**4*x**4 + 315*c*e**3*f*g**5*x**5 + 3465*c*e**3*g**6*x**6))/(4 5045*g**6)