Integrand size = 26, antiderivative size = 340 \[ \int (d+e x)^2 (f+g x)^{3/2} \left (a+c x^2\right )^2 \, dx=\frac {2 (e f-d g)^2 \left (c f^2+a g^2\right )^2 (f+g x)^{5/2}}{5 g^7}-\frac {4 (e f-d g) \left (c f^2+a g^2\right ) \left (a e g^2+c f (3 e f-2 d g)\right ) (f+g x)^{7/2}}{7 g^7}+\frac {2 \left (a^2 e^2 g^4+2 a c g^2 \left (6 e^2 f^2-6 d e f g+d^2 g^2\right )+c^2 f^2 \left (15 e^2 f^2-20 d e f g+6 d^2 g^2\right )\right ) (f+g x)^{9/2}}{9 g^7}-\frac {8 c \left (a e g^2 (2 e f-d g)+c f \left (5 e^2 f^2-5 d e f g+d^2 g^2\right )\right ) (f+g x)^{11/2}}{11 g^7}+\frac {2 c \left (2 a e^2 g^2+c \left (15 e^2 f^2-10 d e f g+d^2 g^2\right )\right ) (f+g x)^{13/2}}{13 g^7}-\frac {4 c^2 e (3 e f-d g) (f+g x)^{15/2}}{15 g^7}+\frac {2 c^2 e^2 (f+g x)^{17/2}}{17 g^7} \] Output:
2/5*(-d*g+e*f)^2*(a*g^2+c*f^2)^2*(g*x+f)^(5/2)/g^7-4/7*(-d*g+e*f)*(a*g^2+c *f^2)*(a*e*g^2+c*f*(-2*d*g+3*e*f))*(g*x+f)^(7/2)/g^7+2/9*(a^2*e^2*g^4+2*a* c*g^2*(d^2*g^2-6*d*e*f*g+6*e^2*f^2)+c^2*f^2*(6*d^2*g^2-20*d*e*f*g+15*e^2*f ^2))*(g*x+f)^(9/2)/g^7-8/11*c*(a*e*g^2*(-d*g+2*e*f)+c*f*(d^2*g^2-5*d*e*f*g +5*e^2*f^2))*(g*x+f)^(11/2)/g^7+2/13*c*(2*a*e^2*g^2+c*(d^2*g^2-10*d*e*f*g+ 15*e^2*f^2))*(g*x+f)^(13/2)/g^7-4/15*c^2*e*(-d*g+3*e*f)*(g*x+f)^(15/2)/g^7 +2/17*c^2*e^2*(g*x+f)^(17/2)/g^7
Time = 0.39 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.07 \[ \int (d+e x)^2 (f+g x)^{3/2} \left (a+c x^2\right )^2 \, dx=\frac {2 (f+g x)^{5/2} \left (2431 a^2 g^4 \left (63 d^2 g^2+18 d e g (-2 f+5 g x)+e^2 \left (8 f^2-20 f g x+35 g^2 x^2\right )\right )+34 a c g^2 \left (143 d^2 g^2 \left (8 f^2-20 f g x+35 g^2 x^2\right )+78 d e g \left (-16 f^3+40 f^2 g x-70 f g^2 x^2+105 g^3 x^3\right )+3 e^2 \left (128 f^4-320 f^3 g x+560 f^2 g^2 x^2-840 f g^3 x^3+1155 g^4 x^4\right )\right )+c^2 \left (51 d^2 g^2 \left (128 f^4-320 f^3 g x+560 f^2 g^2 x^2-840 f g^3 x^3+1155 g^4 x^4\right )+34 d e g \left (-256 f^5+640 f^4 g x-1120 f^3 g^2 x^2+1680 f^2 g^3 x^3-2310 f g^4 x^4+3003 g^5 x^5\right )+3 e^2 \left (1024 f^6-2560 f^5 g x+4480 f^4 g^2 x^2-6720 f^3 g^3 x^3+9240 f^2 g^4 x^4-12012 f g^5 x^5+15015 g^6 x^6\right )\right )\right )}{765765 g^7} \] Input:
Integrate[(d + e*x)^2*(f + g*x)^(3/2)*(a + c*x^2)^2,x]
Output:
(2*(f + g*x)^(5/2)*(2431*a^2*g^4*(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)) + 34*a*c*g^2*(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)) + c^2*(51*d^2*g^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) + 34*d*e*g*(-256*f^5 + 640*f^4*g*x - 112 0*f^3*g^2*x^2 + 1680*f^2*g^3*x^3 - 2310*f*g^4*x^4 + 3003*g^5*x^5) + 3*e^2* (1024*f^6 - 2560*f^5*g*x + 4480*f^4*g^2*x^2 - 6720*f^3*g^3*x^3 + 9240*f^2* g^4*x^4 - 12012*f*g^5*x^5 + 15015*g^6*x^6))))/(765765*g^7)
Time = 0.54 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, 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.
| \(\displaystyle \int \left (a+c x^2\right )^2 (d+e x)^2 (f+g x)^{3/2} \, dx\) |
| \(\Big \downarrow \) 652 |
| \(\displaystyle \int \left (\frac {(f+g x)^{7/2} \left (a^2 e^2 g^4+2 a c g^2 \left (d^2 g^2-6 d e f g+6 e^2 f^2\right )+c^2 f^2 \left (6 d^2 g^2-20 d e f g+15 e^2 f^2\right )\right )}{g^6}+\frac {c (f+g x)^{11/2} \left (2 a e^2 g^2+c \left (d^2 g^2-10 d e f g+15 e^2 f^2\right )\right )}{g^6}+\frac {4 c (f+g x)^{9/2} \left (-a e g^2 (2 e f-d g)-c f \left (d^2 g^2-5 d e f g+5 e^2 f^2\right )\right )}{g^6}+\frac {2 (f+g x)^{5/2} \left (a g^2+c f^2\right ) (e f-d g) \left (-a e g^2-c f (3 e f-2 d g)\right )}{g^6}+\frac {(f+g x)^{3/2} \left (a g^2+c f^2\right )^2 (d g-e f)^2}{g^6}-\frac {2 c^2 e (f+g x)^{13/2} (3 e f-d g)}{g^6}+\frac {c^2 e^2 (f+g x)^{15/2}}{g^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 (f+g x)^{9/2} \left (a^2 e^2 g^4+2 a c g^2 \left (d^2 g^2-6 d e f g+6 e^2 f^2\right )+c^2 f^2 \left (6 d^2 g^2-20 d e f g+15 e^2 f^2\right )\right )}{9 g^7}+\frac {2 c (f+g x)^{13/2} \left (2 a e^2 g^2+c \left (d^2 g^2-10 d e f g+15 e^2 f^2\right )\right )}{13 g^7}-\frac {8 c (f+g x)^{11/2} \left (a e g^2 (2 e f-d g)+c f \left (d^2 g^2-5 d e f g+5 e^2 f^2\right )\right )}{11 g^7}-\frac {4 (f+g x)^{7/2} \left (a g^2+c f^2\right ) (e f-d g) \left (a e g^2+c f (3 e f-2 d g)\right )}{7 g^7}+\frac {2 (f+g x)^{5/2} \left (a g^2+c f^2\right )^2 (e f-d g)^2}{5 g^7}-\frac {4 c^2 e (f+g x)^{15/2} (3 e f-d g)}{15 g^7}+\frac {2 c^2 e^2 (f+g x)^{17/2}}{17 g^7}\) |
Input:
Int[(d + e*x)^2*(f + g*x)^(3/2)*(a + c*x^2)^2,x]
Output:
(2*(e*f - d*g)^2*(c*f^2 + a*g^2)^2*(f + g*x)^(5/2))/(5*g^7) - (4*(e*f - d* g)*(c*f^2 + a*g^2)*(a*e*g^2 + c*f*(3*e*f - 2*d*g))*(f + g*x)^(7/2))/(7*g^7 ) + (2*(a^2*e^2*g^4 + 2*a*c*g^2*(6*e^2*f^2 - 6*d*e*f*g + d^2*g^2) + c^2*f^ 2*(15*e^2*f^2 - 20*d*e*f*g + 6*d^2*g^2))*(f + g*x)^(9/2))/(9*g^7) - (8*c*( a*e*g^2*(2*e*f - d*g) + c*f*(5*e^2*f^2 - 5*d*e*f*g + d^2*g^2))*(f + g*x)^( 11/2))/(11*g^7) + (2*c*(2*a*e^2*g^2 + c*(15*e^2*f^2 - 10*d*e*f*g + d^2*g^2 ))*(f + g*x)^(13/2))/(13*g^7) - (4*c^2*e*(3*e*f - d*g)*(f + g*x)^(15/2))/( 15*g^7) + (2*c^2*e^2*(f + g*x)^(17/2))/(17*g^7)
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 = 1.45 (sec) , antiderivative size = 325, normalized size of antiderivative = 0.96
| method | result | size |
| pseudoelliptic | \(\frac {2 \left (g x +f \right )^{\frac {5}{2}} \left (\left (\left (\frac {5}{17} e^{2} x^{6}+\frac {5}{13} d^{2} x^{4}+\frac {2}{3} d e \,x^{5}\right ) c^{2}+\frac {10 \left (\frac {9}{13} e^{2} x^{2}+\frac {18}{11} d e x +d^{2}\right ) x^{2} a c}{9}+a^{2} \left (d^{2}+\frac {5}{9} e^{2} x^{2}+\frac {10}{7} d e x \right )\right ) g^{6}-\frac {4 f \left (\left (\frac {7}{17} e^{2} x^{5}+\frac {35}{39} d e \,x^{4}+\frac {70}{143} d^{2} x^{3}\right ) c^{2}+\frac {10 \left (\frac {126}{143} e^{2} x^{2}+\frac {21}{11} d e x +d^{2}\right ) x a c}{9}+a^{2} e \left (\frac {5 e x}{9}+d \right )\right ) g^{5}}{7}+\frac {8 f^{2} \left (\frac {210 \left (\frac {33}{34} e^{2} x^{2}+2 d e x +d^{2}\right ) x^{2} c^{2}}{143}+2 \left (\frac {210}{143} e^{2} x^{2}+\frac {30}{11} d e x +d^{2}\right ) a c +e^{2} a^{2}\right ) g^{4}}{63}-\frac {64 \left (\frac {5 x \left (\frac {21}{17} e^{2} x^{2}+\frac {7}{3} d e x +d^{2}\right ) c}{13}+e a \left (\frac {10 e x}{13}+d \right )\right ) f^{3} c \,g^{3}}{231}+\frac {256 f^{4} c \left (\left (\frac {35}{34} e^{2} x^{2}+\frac {5}{3} d e x +\frac {1}{2} d^{2}\right ) c +a \,e^{2}\right ) g^{2}}{3003}-\frac {512 e \,f^{5} \left (\frac {15 e x}{17}+d \right ) c^{2} g}{9009}+\frac {1024 c^{2} e^{2} f^{6}}{51051}\right )}{5 g^{7}}\) | \(325\) |
| derivativedivides | \(\frac {\frac {2 c^{2} e^{2} \left (g x +f \right )^{\frac {17}{2}}}{17}+\frac {2 \left (2 e \left (d g -e f \right ) c^{2}-4 f \,c^{2} e^{2}\right ) \left (g x +f \right )^{\frac {15}{2}}}{15}+\frac {2 \left (\left (d g -e f \right )^{2} c^{2}-8 e \left (d g -e f \right ) c^{2} f +e^{2} \left (2 \left (a \,g^{2}+c \,f^{2}\right ) c +4 c^{2} f^{2}\right )\right ) \left (g x +f \right )^{\frac {13}{2}}}{13}+\frac {2 \left (-4 \left (d g -e f \right )^{2} c^{2} f +2 e \left (d g -e f \right ) \left (2 \left (a \,g^{2}+c \,f^{2}\right ) c +4 c^{2} f^{2}\right )-4 e^{2} \left (a \,g^{2}+c \,f^{2}\right ) c f \right ) \left (g x +f \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (d g -e f \right )^{2} \left (2 \left (a \,g^{2}+c \,f^{2}\right ) c +4 c^{2} f^{2}\right )-8 e \left (d g -e f \right ) \left (a \,g^{2}+c \,f^{2}\right ) c f +e^{2} \left (a \,g^{2}+c \,f^{2}\right )^{2}\right ) \left (g x +f \right )^{\frac {9}{2}}}{9}+\frac {2 \left (-4 \left (d g -e f \right )^{2} \left (a \,g^{2}+c \,f^{2}\right ) c f +2 e \left (d g -e f \right ) \left (a \,g^{2}+c \,f^{2}\right )^{2}\right ) \left (g x +f \right )^{\frac {7}{2}}}{7}+\frac {2 \left (d g -e f \right )^{2} \left (a \,g^{2}+c \,f^{2}\right )^{2} \left (g x +f \right )^{\frac {5}{2}}}{5}}{g^{7}}\) | \(375\) |
| default | \(\frac {\frac {2 c^{2} e^{2} \left (g x +f \right )^{\frac {17}{2}}}{17}+\frac {2 \left (2 e \left (d g -e f \right ) c^{2}-4 f \,c^{2} e^{2}\right ) \left (g x +f \right )^{\frac {15}{2}}}{15}+\frac {2 \left (\left (d g -e f \right )^{2} c^{2}-8 e \left (d g -e f \right ) c^{2} f +e^{2} \left (2 \left (a \,g^{2}+c \,f^{2}\right ) c +4 c^{2} f^{2}\right )\right ) \left (g x +f \right )^{\frac {13}{2}}}{13}+\frac {2 \left (-4 \left (d g -e f \right )^{2} c^{2} f +2 e \left (d g -e f \right ) \left (2 \left (a \,g^{2}+c \,f^{2}\right ) c +4 c^{2} f^{2}\right )-4 e^{2} \left (a \,g^{2}+c \,f^{2}\right ) c f \right ) \left (g x +f \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (d g -e f \right )^{2} \left (2 \left (a \,g^{2}+c \,f^{2}\right ) c +4 c^{2} f^{2}\right )-8 e \left (d g -e f \right ) \left (a \,g^{2}+c \,f^{2}\right ) c f +e^{2} \left (a \,g^{2}+c \,f^{2}\right )^{2}\right ) \left (g x +f \right )^{\frac {9}{2}}}{9}+\frac {2 \left (-4 \left (d g -e f \right )^{2} \left (a \,g^{2}+c \,f^{2}\right ) c f +2 e \left (d g -e f \right ) \left (a \,g^{2}+c \,f^{2}\right )^{2}\right ) \left (g x +f \right )^{\frac {7}{2}}}{7}+\frac {2 \left (d g -e f \right )^{2} \left (a \,g^{2}+c \,f^{2}\right )^{2} \left (g x +f \right )^{\frac {5}{2}}}{5}}{g^{7}}\) | \(375\) |
| gosper | \(\frac {2 \left (g x +f \right )^{\frac {5}{2}} \left (45045 c^{2} e^{2} x^{6} g^{6}+102102 c^{2} d e \,g^{6} x^{5}-36036 c^{2} e^{2} f \,g^{5} x^{5}+117810 a c \,e^{2} g^{6} x^{4}+58905 c^{2} d^{2} g^{6} x^{4}-78540 c^{2} d e f \,g^{5} x^{4}+27720 c^{2} e^{2} f^{2} g^{4} x^{4}+278460 a c d e \,g^{6} x^{3}-85680 a c \,e^{2} f \,g^{5} x^{3}-42840 c^{2} d^{2} f \,g^{5} x^{3}+57120 c^{2} d e \,f^{2} g^{4} x^{3}-20160 c^{2} e^{2} f^{3} g^{3} x^{3}+85085 a^{2} e^{2} g^{6} x^{2}+170170 a c \,d^{2} g^{6} x^{2}-185640 a c d e f \,g^{5} x^{2}+57120 a c \,e^{2} f^{2} g^{4} x^{2}+28560 c^{2} d^{2} f^{2} g^{4} x^{2}-38080 c^{2} d e \,f^{3} g^{3} x^{2}+13440 c^{2} e^{2} f^{4} g^{2} x^{2}+218790 a^{2} d e \,g^{6} x -48620 a^{2} e^{2} f \,g^{5} x -97240 a c \,d^{2} f \,g^{5} x +106080 a c d e \,f^{2} g^{4} x -32640 a c \,e^{2} f^{3} g^{3} x -16320 c^{2} d^{2} f^{3} g^{3} x +21760 c^{2} d e \,f^{4} g^{2} x -7680 c^{2} e^{2} f^{5} g x +153153 a^{2} d^{2} g^{6}-87516 a^{2} d e f \,g^{5}+19448 a^{2} e^{2} f^{2} g^{4}+38896 a c \,d^{2} f^{2} g^{4}-42432 a c d e \,f^{3} g^{3}+13056 a c \,e^{2} f^{4} g^{2}+6528 c^{2} d^{2} f^{4} g^{2}-8704 c^{2} d e \,f^{5} g +3072 c^{2} e^{2} f^{6}\right )}{765765 g^{7}}\) | \(509\) |
| orering | \(\frac {2 \left (g x +f \right )^{\frac {5}{2}} \left (45045 c^{2} e^{2} x^{6} g^{6}+102102 c^{2} d e \,g^{6} x^{5}-36036 c^{2} e^{2} f \,g^{5} x^{5}+117810 a c \,e^{2} g^{6} x^{4}+58905 c^{2} d^{2} g^{6} x^{4}-78540 c^{2} d e f \,g^{5} x^{4}+27720 c^{2} e^{2} f^{2} g^{4} x^{4}+278460 a c d e \,g^{6} x^{3}-85680 a c \,e^{2} f \,g^{5} x^{3}-42840 c^{2} d^{2} f \,g^{5} x^{3}+57120 c^{2} d e \,f^{2} g^{4} x^{3}-20160 c^{2} e^{2} f^{3} g^{3} x^{3}+85085 a^{2} e^{2} g^{6} x^{2}+170170 a c \,d^{2} g^{6} x^{2}-185640 a c d e f \,g^{5} x^{2}+57120 a c \,e^{2} f^{2} g^{4} x^{2}+28560 c^{2} d^{2} f^{2} g^{4} x^{2}-38080 c^{2} d e \,f^{3} g^{3} x^{2}+13440 c^{2} e^{2} f^{4} g^{2} x^{2}+218790 a^{2} d e \,g^{6} x -48620 a^{2} e^{2} f \,g^{5} x -97240 a c \,d^{2} f \,g^{5} x +106080 a c d e \,f^{2} g^{4} x -32640 a c \,e^{2} f^{3} g^{3} x -16320 c^{2} d^{2} f^{3} g^{3} x +21760 c^{2} d e \,f^{4} g^{2} x -7680 c^{2} e^{2} f^{5} g x +153153 a^{2} d^{2} g^{6}-87516 a^{2} d e f \,g^{5}+19448 a^{2} e^{2} f^{2} g^{4}+38896 a c \,d^{2} f^{2} g^{4}-42432 a c d e \,f^{3} g^{3}+13056 a c \,e^{2} f^{4} g^{2}+6528 c^{2} d^{2} f^{4} g^{2}-8704 c^{2} d e \,f^{5} g +3072 c^{2} e^{2} f^{6}\right )}{765765 g^{7}}\) | \(509\) |
| trager | \(\frac {2 \left (45045 c^{2} e^{2} g^{8} x^{8}+102102 c^{2} d e \,g^{8} x^{7}+54054 c^{2} e^{2} f \,g^{7} x^{7}+117810 a c \,e^{2} g^{8} x^{6}+58905 c^{2} d^{2} g^{8} x^{6}+125664 c^{2} d e f \,g^{7} x^{6}+693 c^{2} e^{2} f^{2} g^{6} x^{6}+278460 a c d e \,g^{8} x^{5}+149940 a c \,e^{2} f \,g^{7} x^{5}+74970 c^{2} d^{2} f \,g^{7} x^{5}+2142 c^{2} d e \,f^{2} g^{6} x^{5}-756 c^{2} e^{2} f^{3} g^{5} x^{5}+85085 a^{2} e^{2} g^{8} x^{4}+170170 a c \,d^{2} g^{8} x^{4}+371280 a c d e f \,g^{7} x^{4}+3570 a c \,e^{2} f^{2} g^{6} x^{4}+1785 c^{2} d^{2} f^{2} g^{6} x^{4}-2380 c^{2} d e \,f^{3} g^{5} x^{4}+840 c^{2} e^{2} f^{4} g^{4} x^{4}+218790 a^{2} d e \,g^{8} x^{3}+121550 a^{2} e^{2} f \,g^{7} x^{3}+243100 a c \,d^{2} f \,g^{7} x^{3}+13260 a c d e \,f^{2} g^{6} x^{3}-4080 a c \,e^{2} f^{3} g^{5} x^{3}-2040 c^{2} d^{2} f^{3} g^{5} x^{3}+2720 c^{2} d e \,f^{4} g^{4} x^{3}-960 c^{2} e^{2} f^{5} g^{3} x^{3}+153153 a^{2} d^{2} g^{8} x^{2}+350064 a^{2} d e f \,g^{7} x^{2}+7293 a^{2} e^{2} f^{2} g^{6} x^{2}+14586 a c \,d^{2} f^{2} g^{6} x^{2}-15912 a c d e \,f^{3} g^{5} x^{2}+4896 a c \,e^{2} f^{4} g^{4} x^{2}+2448 c^{2} d^{2} f^{4} g^{4} x^{2}-3264 c^{2} d e \,f^{5} g^{3} x^{2}+1152 c^{2} e^{2} f^{6} g^{2} x^{2}+306306 a^{2} d^{2} f \,g^{7} x +43758 a^{2} d e \,f^{2} g^{6} x -9724 a^{2} e^{2} f^{3} g^{5} x -19448 a c \,d^{2} f^{3} g^{5} x +21216 a c d e \,f^{4} g^{4} x -6528 a c \,e^{2} f^{5} g^{3} x -3264 c^{2} d^{2} f^{5} g^{3} x +4352 c^{2} d e \,f^{6} g^{2} x -1536 c^{2} e^{2} f^{7} g x +153153 a^{2} d^{2} f^{2} g^{6}-87516 a^{2} d e \,f^{3} g^{5}+19448 a^{2} e^{2} f^{4} g^{4}+38896 a c \,d^{2} f^{4} g^{4}-42432 a c d e \,f^{5} g^{3}+13056 a c \,e^{2} f^{6} g^{2}+6528 c^{2} d^{2} f^{6} g^{2}-8704 c^{2} d e \,f^{7} g +3072 c^{2} e^{2} f^{8}\right ) \sqrt {g x +f}}{765765 g^{7}}\) | \(799\) |
| risch | \(\frac {2 \left (45045 c^{2} e^{2} g^{8} x^{8}+102102 c^{2} d e \,g^{8} x^{7}+54054 c^{2} e^{2} f \,g^{7} x^{7}+117810 a c \,e^{2} g^{8} x^{6}+58905 c^{2} d^{2} g^{8} x^{6}+125664 c^{2} d e f \,g^{7} x^{6}+693 c^{2} e^{2} f^{2} g^{6} x^{6}+278460 a c d e \,g^{8} x^{5}+149940 a c \,e^{2} f \,g^{7} x^{5}+74970 c^{2} d^{2} f \,g^{7} x^{5}+2142 c^{2} d e \,f^{2} g^{6} x^{5}-756 c^{2} e^{2} f^{3} g^{5} x^{5}+85085 a^{2} e^{2} g^{8} x^{4}+170170 a c \,d^{2} g^{8} x^{4}+371280 a c d e f \,g^{7} x^{4}+3570 a c \,e^{2} f^{2} g^{6} x^{4}+1785 c^{2} d^{2} f^{2} g^{6} x^{4}-2380 c^{2} d e \,f^{3} g^{5} x^{4}+840 c^{2} e^{2} f^{4} g^{4} x^{4}+218790 a^{2} d e \,g^{8} x^{3}+121550 a^{2} e^{2} f \,g^{7} x^{3}+243100 a c \,d^{2} f \,g^{7} x^{3}+13260 a c d e \,f^{2} g^{6} x^{3}-4080 a c \,e^{2} f^{3} g^{5} x^{3}-2040 c^{2} d^{2} f^{3} g^{5} x^{3}+2720 c^{2} d e \,f^{4} g^{4} x^{3}-960 c^{2} e^{2} f^{5} g^{3} x^{3}+153153 a^{2} d^{2} g^{8} x^{2}+350064 a^{2} d e f \,g^{7} x^{2}+7293 a^{2} e^{2} f^{2} g^{6} x^{2}+14586 a c \,d^{2} f^{2} g^{6} x^{2}-15912 a c d e \,f^{3} g^{5} x^{2}+4896 a c \,e^{2} f^{4} g^{4} x^{2}+2448 c^{2} d^{2} f^{4} g^{4} x^{2}-3264 c^{2} d e \,f^{5} g^{3} x^{2}+1152 c^{2} e^{2} f^{6} g^{2} x^{2}+306306 a^{2} d^{2} f \,g^{7} x +43758 a^{2} d e \,f^{2} g^{6} x -9724 a^{2} e^{2} f^{3} g^{5} x -19448 a c \,d^{2} f^{3} g^{5} x +21216 a c d e \,f^{4} g^{4} x -6528 a c \,e^{2} f^{5} g^{3} x -3264 c^{2} d^{2} f^{5} g^{3} x +4352 c^{2} d e \,f^{6} g^{2} x -1536 c^{2} e^{2} f^{7} g x +153153 a^{2} d^{2} f^{2} g^{6}-87516 a^{2} d e \,f^{3} g^{5}+19448 a^{2} e^{2} f^{4} g^{4}+38896 a c \,d^{2} f^{4} g^{4}-42432 a c d e \,f^{5} g^{3}+13056 a c \,e^{2} f^{6} g^{2}+6528 c^{2} d^{2} f^{6} g^{2}-8704 c^{2} d e \,f^{7} g +3072 c^{2} e^{2} f^{8}\right ) \sqrt {g x +f}}{765765 g^{7}}\) | \(799\) |
Input:
int((e*x+d)^2*(g*x+f)^(3/2)*(c*x^2+a)^2,x,method=_RETURNVERBOSE)
Output:
2/5*(g*x+f)^(5/2)*(((5/17*e^2*x^6+5/13*d^2*x^4+2/3*d*e*x^5)*c^2+10/9*(9/13 *e^2*x^2+18/11*d*e*x+d^2)*x^2*a*c+a^2*(d^2+5/9*e^2*x^2+10/7*d*e*x))*g^6-4/ 7*f*((7/17*e^2*x^5+35/39*d*e*x^4+70/143*d^2*x^3)*c^2+10/9*(126/143*e^2*x^2 +21/11*d*e*x+d^2)*x*a*c+a^2*e*(5/9*e*x+d))*g^5+8/63*f^2*(210/143*(33/34*e^ 2*x^2+2*d*e*x+d^2)*x^2*c^2+2*(210/143*e^2*x^2+30/11*d*e*x+d^2)*a*c+e^2*a^2 )*g^4-64/231*(5/13*x*(21/17*e^2*x^2+7/3*d*e*x+d^2)*c+e*a*(10/13*e*x+d))*f^ 3*c*g^3+256/3003*f^4*c*((35/34*e^2*x^2+5/3*d*e*x+1/2*d^2)*c+a*e^2)*g^2-512 /9009*e*f^5*(15/17*e*x+d)*c^2*g+1024/51051*c^2*e^2*f^6)/g^7
Leaf count of result is larger than twice the leaf count of optimal. 686 vs. \(2 (312) = 624\).
Time = 0.09 (sec) , antiderivative size = 686, normalized size of antiderivative = 2.02 \[ \int (d+e x)^2 (f+g x)^{3/2} \left (a+c x^2\right )^2 \, dx=\frac {2 \, {\left (45045 \, c^{2} e^{2} g^{8} x^{8} + 3072 \, c^{2} e^{2} f^{8} - 8704 \, c^{2} d e f^{7} g - 42432 \, a c d e f^{5} g^{3} - 87516 \, a^{2} d e f^{3} g^{5} + 153153 \, a^{2} d^{2} f^{2} g^{6} + 6528 \, {\left (c^{2} d^{2} + 2 \, a c e^{2}\right )} f^{6} g^{2} + 19448 \, {\left (2 \, a c d^{2} + a^{2} e^{2}\right )} f^{4} g^{4} + 6006 \, {\left (9 \, c^{2} e^{2} f g^{7} + 17 \, c^{2} d e g^{8}\right )} x^{7} + 231 \, {\left (3 \, c^{2} e^{2} f^{2} g^{6} + 544 \, c^{2} d e f g^{7} + 255 \, {\left (c^{2} d^{2} + 2 \, a c e^{2}\right )} g^{8}\right )} x^{6} - 126 \, {\left (6 \, c^{2} e^{2} f^{3} g^{5} - 17 \, c^{2} d e f^{2} g^{6} - 2210 \, a c d e g^{8} - 595 \, {\left (c^{2} d^{2} + 2 \, a c e^{2}\right )} f g^{7}\right )} x^{5} + 35 \, {\left (24 \, c^{2} e^{2} f^{4} g^{4} - 68 \, c^{2} d e f^{3} g^{5} + 10608 \, a c d e f g^{7} + 51 \, {\left (c^{2} d^{2} + 2 \, a c e^{2}\right )} f^{2} g^{6} + 2431 \, {\left (2 \, a c d^{2} + a^{2} e^{2}\right )} g^{8}\right )} x^{4} - 10 \, {\left (96 \, c^{2} e^{2} f^{5} g^{3} - 272 \, c^{2} d e f^{4} g^{4} - 1326 \, a c d e f^{2} g^{6} - 21879 \, a^{2} d e g^{8} + 204 \, {\left (c^{2} d^{2} + 2 \, a c e^{2}\right )} f^{3} g^{5} - 12155 \, {\left (2 \, a c d^{2} + a^{2} e^{2}\right )} f g^{7}\right )} x^{3} + 3 \, {\left (384 \, c^{2} e^{2} f^{6} g^{2} - 1088 \, c^{2} d e f^{5} g^{3} - 5304 \, a c d e f^{3} g^{5} + 116688 \, a^{2} d e f g^{7} + 51051 \, a^{2} d^{2} g^{8} + 816 \, {\left (c^{2} d^{2} + 2 \, a c e^{2}\right )} f^{4} g^{4} + 2431 \, {\left (2 \, a c d^{2} + a^{2} e^{2}\right )} f^{2} g^{6}\right )} x^{2} - 2 \, {\left (768 \, c^{2} e^{2} f^{7} g - 2176 \, c^{2} d e f^{6} g^{2} - 10608 \, a c d e f^{4} g^{4} - 21879 \, a^{2} d e f^{2} g^{6} - 153153 \, a^{2} d^{2} f g^{7} + 1632 \, {\left (c^{2} d^{2} + 2 \, a c e^{2}\right )} f^{5} g^{3} + 4862 \, {\left (2 \, a c d^{2} + a^{2} e^{2}\right )} f^{3} g^{5}\right )} x\right )} \sqrt {g x + f}}{765765 \, g^{7}} \] Input:
integrate((e*x+d)^2*(g*x+f)^(3/2)*(c*x^2+a)^2,x, algorithm="fricas")
Output:
2/765765*(45045*c^2*e^2*g^8*x^8 + 3072*c^2*e^2*f^8 - 8704*c^2*d*e*f^7*g - 42432*a*c*d*e*f^5*g^3 - 87516*a^2*d*e*f^3*g^5 + 153153*a^2*d^2*f^2*g^6 + 6 528*(c^2*d^2 + 2*a*c*e^2)*f^6*g^2 + 19448*(2*a*c*d^2 + a^2*e^2)*f^4*g^4 + 6006*(9*c^2*e^2*f*g^7 + 17*c^2*d*e*g^8)*x^7 + 231*(3*c^2*e^2*f^2*g^6 + 544 *c^2*d*e*f*g^7 + 255*(c^2*d^2 + 2*a*c*e^2)*g^8)*x^6 - 126*(6*c^2*e^2*f^3*g ^5 - 17*c^2*d*e*f^2*g^6 - 2210*a*c*d*e*g^8 - 595*(c^2*d^2 + 2*a*c*e^2)*f*g ^7)*x^5 + 35*(24*c^2*e^2*f^4*g^4 - 68*c^2*d*e*f^3*g^5 + 10608*a*c*d*e*f*g^ 7 + 51*(c^2*d^2 + 2*a*c*e^2)*f^2*g^6 + 2431*(2*a*c*d^2 + a^2*e^2)*g^8)*x^4 - 10*(96*c^2*e^2*f^5*g^3 - 272*c^2*d*e*f^4*g^4 - 1326*a*c*d*e*f^2*g^6 - 2 1879*a^2*d*e*g^8 + 204*(c^2*d^2 + 2*a*c*e^2)*f^3*g^5 - 12155*(2*a*c*d^2 + a^2*e^2)*f*g^7)*x^3 + 3*(384*c^2*e^2*f^6*g^2 - 1088*c^2*d*e*f^5*g^3 - 5304 *a*c*d*e*f^3*g^5 + 116688*a^2*d*e*f*g^7 + 51051*a^2*d^2*g^8 + 816*(c^2*d^2 + 2*a*c*e^2)*f^4*g^4 + 2431*(2*a*c*d^2 + a^2*e^2)*f^2*g^6)*x^2 - 2*(768*c ^2*e^2*f^7*g - 2176*c^2*d*e*f^6*g^2 - 10608*a*c*d*e*f^4*g^4 - 21879*a^2*d* e*f^2*g^6 - 153153*a^2*d^2*f*g^7 + 1632*(c^2*d^2 + 2*a*c*e^2)*f^5*g^3 + 48 62*(2*a*c*d^2 + a^2*e^2)*f^3*g^5)*x)*sqrt(g*x + f)/g^7
Time = 1.44 (sec) , antiderivative size = 666, normalized size of antiderivative = 1.96 \[ \int (d+e x)^2 (f+g x)^{3/2} \left (a+c x^2\right )^2 \, dx=\begin {cases} \frac {2 \left (\frac {c^{2} e^{2} \left (f + g x\right )^{\frac {17}{2}}}{17 g^{6}} + \frac {\left (f + g x\right )^{\frac {15}{2}} \cdot \left (2 c^{2} d e g - 6 c^{2} e^{2} f\right )}{15 g^{6}} + \frac {\left (f + g x\right )^{\frac {13}{2}} \cdot \left (2 a c e^{2} g^{2} + c^{2} d^{2} g^{2} - 10 c^{2} d e f g + 15 c^{2} e^{2} f^{2}\right )}{13 g^{6}} + \frac {\left (f + g x\right )^{\frac {11}{2}} \cdot \left (4 a c d e g^{3} - 8 a c e^{2} f g^{2} - 4 c^{2} d^{2} f g^{2} + 20 c^{2} d e f^{2} g - 20 c^{2} e^{2} f^{3}\right )}{11 g^{6}} + \frac {\left (f + g x\right )^{\frac {9}{2}} \left (a^{2} e^{2} g^{4} + 2 a c d^{2} g^{4} - 12 a c d e f g^{3} + 12 a c e^{2} f^{2} g^{2} + 6 c^{2} d^{2} f^{2} g^{2} - 20 c^{2} d e f^{3} g + 15 c^{2} e^{2} f^{4}\right )}{9 g^{6}} + \frac {\left (f + g x\right )^{\frac {7}{2}} \cdot \left (2 a^{2} d e g^{5} - 2 a^{2} e^{2} f g^{4} - 4 a c d^{2} f g^{4} + 12 a c d e f^{2} g^{3} - 8 a c e^{2} f^{3} g^{2} - 4 c^{2} d^{2} f^{3} g^{2} + 10 c^{2} d e f^{4} g - 6 c^{2} e^{2} f^{5}\right )}{7 g^{6}} + \frac {\left (f + g x\right )^{\frac {5}{2}} \left (a^{2} d^{2} g^{6} - 2 a^{2} d e f g^{5} + a^{2} e^{2} f^{2} g^{4} + 2 a c d^{2} f^{2} g^{4} - 4 a c d e f^{3} g^{3} + 2 a c e^{2} f^{4} g^{2} + c^{2} d^{2} f^{4} g^{2} - 2 c^{2} d e f^{5} g + c^{2} e^{2} f^{6}\right )}{5 g^{6}}\right )}{g} & \text {for}\: g \neq 0 \\f^{\frac {3}{2}} \left (a^{2} d^{2} x + a^{2} d e x^{2} + a c d e x^{4} + \frac {c^{2} d e x^{6}}{3} + \frac {c^{2} e^{2} x^{7}}{7} + \frac {x^{5} \cdot \left (2 a c e^{2} + c^{2} d^{2}\right )}{5} + \frac {x^{3} \left (a^{2} e^{2} + 2 a c d^{2}\right )}{3}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((e*x+d)**2*(g*x+f)**(3/2)*(c*x**2+a)**2,x)
Output:
Piecewise((2*(c**2*e**2*(f + g*x)**(17/2)/(17*g**6) + (f + g*x)**(15/2)*(2 *c**2*d*e*g - 6*c**2*e**2*f)/(15*g**6) + (f + g*x)**(13/2)*(2*a*c*e**2*g** 2 + c**2*d**2*g**2 - 10*c**2*d*e*f*g + 15*c**2*e**2*f**2)/(13*g**6) + (f + g*x)**(11/2)*(4*a*c*d*e*g**3 - 8*a*c*e**2*f*g**2 - 4*c**2*d**2*f*g**2 + 2 0*c**2*d*e*f**2*g - 20*c**2*e**2*f**3)/(11*g**6) + (f + g*x)**(9/2)*(a**2* e**2*g**4 + 2*a*c*d**2*g**4 - 12*a*c*d*e*f*g**3 + 12*a*c*e**2*f**2*g**2 + 6*c**2*d**2*f**2*g**2 - 20*c**2*d*e*f**3*g + 15*c**2*e**2*f**4)/(9*g**6) + (f + g*x)**(7/2)*(2*a**2*d*e*g**5 - 2*a**2*e**2*f*g**4 - 4*a*c*d**2*f*g** 4 + 12*a*c*d*e*f**2*g**3 - 8*a*c*e**2*f**3*g**2 - 4*c**2*d**2*f**3*g**2 + 10*c**2*d*e*f**4*g - 6*c**2*e**2*f**5)/(7*g**6) + (f + g*x)**(5/2)*(a**2*d **2*g**6 - 2*a**2*d*e*f*g**5 + a**2*e**2*f**2*g**4 + 2*a*c*d**2*f**2*g**4 - 4*a*c*d*e*f**3*g**3 + 2*a*c*e**2*f**4*g**2 + c**2*d**2*f**4*g**2 - 2*c** 2*d*e*f**5*g + c**2*e**2*f**6)/(5*g**6))/g, Ne(g, 0)), (f**(3/2)*(a**2*d** 2*x + a**2*d*e*x**2 + a*c*d*e*x**4 + c**2*d*e*x**6/3 + c**2*e**2*x**7/7 + x**5*(2*a*c*e**2 + c**2*d**2)/5 + x**3*(a**2*e**2 + 2*a*c*d**2)/3), True))
Time = 0.04 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.31 \[ \int (d+e x)^2 (f+g x)^{3/2} \left (a+c x^2\right )^2 \, dx=\frac {2 \, {\left (45045 \, {\left (g x + f\right )}^{\frac {17}{2}} c^{2} e^{2} - 102102 \, {\left (3 \, c^{2} e^{2} f - c^{2} d e g\right )} {\left (g x + f\right )}^{\frac {15}{2}} + 58905 \, {\left (15 \, c^{2} e^{2} f^{2} - 10 \, c^{2} d e f g + {\left (c^{2} d^{2} + 2 \, a c e^{2}\right )} g^{2}\right )} {\left (g x + f\right )}^{\frac {13}{2}} - 278460 \, {\left (5 \, c^{2} e^{2} f^{3} - 5 \, c^{2} d e f^{2} g - a c d e g^{3} + {\left (c^{2} d^{2} + 2 \, a c e^{2}\right )} f g^{2}\right )} {\left (g x + f\right )}^{\frac {11}{2}} + 85085 \, {\left (15 \, c^{2} e^{2} f^{4} - 20 \, c^{2} d e f^{3} g - 12 \, a c d e f g^{3} + 6 \, {\left (c^{2} d^{2} + 2 \, a c e^{2}\right )} f^{2} g^{2} + {\left (2 \, a c d^{2} + a^{2} e^{2}\right )} g^{4}\right )} {\left (g x + f\right )}^{\frac {9}{2}} - 218790 \, {\left (3 \, c^{2} e^{2} f^{5} - 5 \, c^{2} d e f^{4} g - 6 \, a c d e f^{2} g^{3} - a^{2} d e g^{5} + 2 \, {\left (c^{2} d^{2} + 2 \, a c e^{2}\right )} f^{3} g^{2} + {\left (2 \, a c d^{2} + a^{2} e^{2}\right )} f g^{4}\right )} {\left (g x + f\right )}^{\frac {7}{2}} + 153153 \, {\left (c^{2} e^{2} f^{6} - 2 \, c^{2} d e f^{5} g - 4 \, a c d e f^{3} g^{3} - 2 \, a^{2} d e f g^{5} + a^{2} d^{2} g^{6} + {\left (c^{2} d^{2} + 2 \, a c e^{2}\right )} f^{4} g^{2} + {\left (2 \, a c d^{2} + a^{2} e^{2}\right )} f^{2} g^{4}\right )} {\left (g x + f\right )}^{\frac {5}{2}}\right )}}{765765 \, g^{7}} \] Input:
integrate((e*x+d)^2*(g*x+f)^(3/2)*(c*x^2+a)^2,x, algorithm="maxima")
Output:
2/765765*(45045*(g*x + f)^(17/2)*c^2*e^2 - 102102*(3*c^2*e^2*f - c^2*d*e*g )*(g*x + f)^(15/2) + 58905*(15*c^2*e^2*f^2 - 10*c^2*d*e*f*g + (c^2*d^2 + 2 *a*c*e^2)*g^2)*(g*x + f)^(13/2) - 278460*(5*c^2*e^2*f^3 - 5*c^2*d*e*f^2*g - a*c*d*e*g^3 + (c^2*d^2 + 2*a*c*e^2)*f*g^2)*(g*x + f)^(11/2) + 85085*(15* c^2*e^2*f^4 - 20*c^2*d*e*f^3*g - 12*a*c*d*e*f*g^3 + 6*(c^2*d^2 + 2*a*c*e^2 )*f^2*g^2 + (2*a*c*d^2 + a^2*e^2)*g^4)*(g*x + f)^(9/2) - 218790*(3*c^2*e^2 *f^5 - 5*c^2*d*e*f^4*g - 6*a*c*d*e*f^2*g^3 - a^2*d*e*g^5 + 2*(c^2*d^2 + 2* a*c*e^2)*f^3*g^2 + (2*a*c*d^2 + a^2*e^2)*f*g^4)*(g*x + f)^(7/2) + 153153*( c^2*e^2*f^6 - 2*c^2*d*e*f^5*g - 4*a*c*d*e*f^3*g^3 - 2*a^2*d*e*f*g^5 + a^2* d^2*g^6 + (c^2*d^2 + 2*a*c*e^2)*f^4*g^2 + (2*a*c*d^2 + a^2*e^2)*f^2*g^4)*( g*x + f)^(5/2))/g^7
Leaf count of result is larger than twice the leaf count of optimal. 1819 vs. \(2 (312) = 624\).
Time = 0.14 (sec) , antiderivative size = 1819, normalized size of antiderivative = 5.35 \[ \int (d+e x)^2 (f+g x)^{3/2} \left (a+c x^2\right )^2 \, dx=\text {Too large to display} \] Input:
integrate((e*x+d)^2*(g*x+f)^(3/2)*(c*x^2+a)^2,x, algorithm="giac")
Output:
2/765765*(765765*sqrt(g*x + f)*a^2*d^2*f^2 + 510510*((g*x + f)^(3/2) - 3*s qrt(g*x + f)*f)*a^2*d^2*f + 510510*((g*x + f)^(3/2) - 3*sqrt(g*x + f)*f)*a ^2*d*e*f^2/g + 51051*(3*(g*x + f)^(5/2) - 10*(g*x + f)^(3/2)*f + 15*sqrt(g *x + f)*f^2)*a^2*d^2 + 102102*(3*(g*x + f)^(5/2) - 10*(g*x + f)^(3/2)*f + 15*sqrt(g*x + f)*f^2)*a*c*d^2*f^2/g^2 + 51051*(3*(g*x + f)^(5/2) - 10*(g*x + f)^(3/2)*f + 15*sqrt(g*x + f)*f^2)*a^2*e^2*f^2/g^2 + 204204*(3*(g*x + f )^(5/2) - 10*(g*x + f)^(3/2)*f + 15*sqrt(g*x + f)*f^2)*a^2*d*e*f/g + 87516 *(5*(g*x + f)^(7/2) - 21*(g*x + f)^(5/2)*f + 35*(g*x + f)^(3/2)*f^2 - 35*s qrt(g*x + f)*f^3)*a*c*d*e*f^2/g^3 + 87516*(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*c*d^2*f/g^2 + 43758*(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^2*e^2*f/g^2 + 43758*(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^2*d*e/g + 2431*(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^2*d^2*f^2/g^4 + 486 2*(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*c*e^2*f^2/g^4 + 19448*( 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*c*d*e*f/g^3 + 4862*(35*(g* x + f)^(9/2) - 180*(g*x + f)^(7/2)*f + 378*(g*x + f)^(5/2)*f^2 - 420*(g...
Time = 5.97 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.01 \[ \int (d+e x)^2 (f+g x)^{3/2} \left (a+c x^2\right )^2 \, dx=\frac {{\left (f+g\,x\right )}^{9/2}\,\left (2\,a^2\,e^2\,g^4+4\,a\,c\,d^2\,g^4-24\,a\,c\,d\,e\,f\,g^3+24\,a\,c\,e^2\,f^2\,g^2+12\,c^2\,d^2\,f^2\,g^2-40\,c^2\,d\,e\,f^3\,g+30\,c^2\,e^2\,f^4\right )}{9\,g^7}-\frac {{\left (f+g\,x\right )}^{11/2}\,\left (8\,c^2\,d^2\,f\,g^2-40\,c^2\,d\,e\,f^2\,g+40\,c^2\,e^2\,f^3-8\,a\,c\,d\,e\,g^3+16\,a\,c\,e^2\,f\,g^2\right )}{11\,g^7}+\frac {{\left (f+g\,x\right )}^{13/2}\,\left (2\,c^2\,d^2\,g^2-20\,c^2\,d\,e\,f\,g+30\,c^2\,e^2\,f^2+4\,a\,c\,e^2\,g^2\right )}{13\,g^7}+\frac {2\,c^2\,e^2\,{\left (f+g\,x\right )}^{17/2}}{17\,g^7}+\frac {2\,{\left (f+g\,x\right )}^{5/2}\,{\left (c\,f^2+a\,g^2\right )}^2\,{\left (d\,g-e\,f\right )}^2}{5\,g^7}+\frac {4\,c^2\,e\,{\left (f+g\,x\right )}^{15/2}\,\left (d\,g-3\,e\,f\right )}{15\,g^7}+\frac {4\,{\left (f+g\,x\right )}^{7/2}\,\left (c\,f^2+a\,g^2\right )\,\left (d\,g-e\,f\right )\,\left (3\,c\,e\,f^2-2\,c\,d\,f\,g+a\,e\,g^2\right )}{7\,g^7} \] Input:
int((f + g*x)^(3/2)*(a + c*x^2)^2*(d + e*x)^2,x)
Output:
((f + g*x)^(9/2)*(2*a^2*e^2*g^4 + 30*c^2*e^2*f^4 + 12*c^2*d^2*f^2*g^2 + 4* a*c*d^2*g^4 - 40*c^2*d*e*f^3*g + 24*a*c*e^2*f^2*g^2 - 24*a*c*d*e*f*g^3))/( 9*g^7) - ((f + g*x)^(11/2)*(40*c^2*e^2*f^3 + 8*c^2*d^2*f*g^2 + 16*a*c*e^2* f*g^2 - 40*c^2*d*e*f^2*g - 8*a*c*d*e*g^3))/(11*g^7) + ((f + g*x)^(13/2)*(2 *c^2*d^2*g^2 + 30*c^2*e^2*f^2 + 4*a*c*e^2*g^2 - 20*c^2*d*e*f*g))/(13*g^7) + (2*c^2*e^2*(f + g*x)^(17/2))/(17*g^7) + (2*(f + g*x)^(5/2)*(a*g^2 + c*f^ 2)^2*(d*g - e*f)^2)/(5*g^7) + (4*c^2*e*(f + g*x)^(15/2)*(d*g - 3*e*f))/(15 *g^7) + (4*(f + g*x)^(7/2)*(a*g^2 + c*f^2)*(d*g - e*f)*(a*e*g^2 + 3*c*e*f^ 2 - 2*c*d*f*g))/(7*g^7)
Time = 0.27 (sec) , antiderivative size = 797, normalized size of antiderivative = 2.34 \[ \int (d+e x)^2 (f+g x)^{3/2} \left (a+c x^2\right )^2 \, dx=\frac {2 \sqrt {g x +f}\, \left (45045 c^{2} e^{2} g^{8} x^{8}+102102 c^{2} d e \,g^{8} x^{7}+54054 c^{2} e^{2} f \,g^{7} x^{7}+117810 a c \,e^{2} g^{8} x^{6}+58905 c^{2} d^{2} g^{8} x^{6}+125664 c^{2} d e f \,g^{7} x^{6}+693 c^{2} e^{2} f^{2} g^{6} x^{6}+278460 a c d e \,g^{8} x^{5}+149940 a c \,e^{2} f \,g^{7} x^{5}+74970 c^{2} d^{2} f \,g^{7} x^{5}+2142 c^{2} d e \,f^{2} g^{6} x^{5}-756 c^{2} e^{2} f^{3} g^{5} x^{5}+85085 a^{2} e^{2} g^{8} x^{4}+170170 a c \,d^{2} g^{8} x^{4}+371280 a c d e f \,g^{7} x^{4}+3570 a c \,e^{2} f^{2} g^{6} x^{4}+1785 c^{2} d^{2} f^{2} g^{6} x^{4}-2380 c^{2} d e \,f^{3} g^{5} x^{4}+840 c^{2} e^{2} f^{4} g^{4} x^{4}+218790 a^{2} d e \,g^{8} x^{3}+121550 a^{2} e^{2} f \,g^{7} x^{3}+243100 a c \,d^{2} f \,g^{7} x^{3}+13260 a c d e \,f^{2} g^{6} x^{3}-4080 a c \,e^{2} f^{3} g^{5} x^{3}-2040 c^{2} d^{2} f^{3} g^{5} x^{3}+2720 c^{2} d e \,f^{4} g^{4} x^{3}-960 c^{2} e^{2} f^{5} g^{3} x^{3}+153153 a^{2} d^{2} g^{8} x^{2}+350064 a^{2} d e f \,g^{7} x^{2}+7293 a^{2} e^{2} f^{2} g^{6} x^{2}+14586 a c \,d^{2} f^{2} g^{6} x^{2}-15912 a c d e \,f^{3} g^{5} x^{2}+4896 a c \,e^{2} f^{4} g^{4} x^{2}+2448 c^{2} d^{2} f^{4} g^{4} x^{2}-3264 c^{2} d e \,f^{5} g^{3} x^{2}+1152 c^{2} e^{2} f^{6} g^{2} x^{2}+306306 a^{2} d^{2} f \,g^{7} x +43758 a^{2} d e \,f^{2} g^{6} x -9724 a^{2} e^{2} f^{3} g^{5} x -19448 a c \,d^{2} f^{3} g^{5} x +21216 a c d e \,f^{4} g^{4} x -6528 a c \,e^{2} f^{5} g^{3} x -3264 c^{2} d^{2} f^{5} g^{3} x +4352 c^{2} d e \,f^{6} g^{2} x -1536 c^{2} e^{2} f^{7} g x +153153 a^{2} d^{2} f^{2} g^{6}-87516 a^{2} d e \,f^{3} g^{5}+19448 a^{2} e^{2} f^{4} g^{4}+38896 a c \,d^{2} f^{4} g^{4}-42432 a c d e \,f^{5} g^{3}+13056 a c \,e^{2} f^{6} g^{2}+6528 c^{2} d^{2} f^{6} g^{2}-8704 c^{2} d e \,f^{7} g +3072 c^{2} e^{2} f^{8}\right )}{765765 g^{7}} \] Input:
int((e*x+d)^2*(g*x+f)^(3/2)*(c*x^2+a)^2,x)
Output:
(2*sqrt(f + g*x)*(153153*a**2*d**2*f**2*g**6 + 306306*a**2*d**2*f*g**7*x + 153153*a**2*d**2*g**8*x**2 - 87516*a**2*d*e*f**3*g**5 + 43758*a**2*d*e*f* *2*g**6*x + 350064*a**2*d*e*f*g**7*x**2 + 218790*a**2*d*e*g**8*x**3 + 1944 8*a**2*e**2*f**4*g**4 - 9724*a**2*e**2*f**3*g**5*x + 7293*a**2*e**2*f**2*g **6*x**2 + 121550*a**2*e**2*f*g**7*x**3 + 85085*a**2*e**2*g**8*x**4 + 3889 6*a*c*d**2*f**4*g**4 - 19448*a*c*d**2*f**3*g**5*x + 14586*a*c*d**2*f**2*g* *6*x**2 + 243100*a*c*d**2*f*g**7*x**3 + 170170*a*c*d**2*g**8*x**4 - 42432* a*c*d*e*f**5*g**3 + 21216*a*c*d*e*f**4*g**4*x - 15912*a*c*d*e*f**3*g**5*x* *2 + 13260*a*c*d*e*f**2*g**6*x**3 + 371280*a*c*d*e*f*g**7*x**4 + 278460*a* c*d*e*g**8*x**5 + 13056*a*c*e**2*f**6*g**2 - 6528*a*c*e**2*f**5*g**3*x + 4 896*a*c*e**2*f**4*g**4*x**2 - 4080*a*c*e**2*f**3*g**5*x**3 + 3570*a*c*e**2 *f**2*g**6*x**4 + 149940*a*c*e**2*f*g**7*x**5 + 117810*a*c*e**2*g**8*x**6 + 6528*c**2*d**2*f**6*g**2 - 3264*c**2*d**2*f**5*g**3*x + 2448*c**2*d**2*f **4*g**4*x**2 - 2040*c**2*d**2*f**3*g**5*x**3 + 1785*c**2*d**2*f**2*g**6*x **4 + 74970*c**2*d**2*f*g**7*x**5 + 58905*c**2*d**2*g**8*x**6 - 8704*c**2* d*e*f**7*g + 4352*c**2*d*e*f**6*g**2*x - 3264*c**2*d*e*f**5*g**3*x**2 + 27 20*c**2*d*e*f**4*g**4*x**3 - 2380*c**2*d*e*f**3*g**5*x**4 + 2142*c**2*d*e* f**2*g**6*x**5 + 125664*c**2*d*e*f*g**7*x**6 + 102102*c**2*d*e*g**8*x**7 + 3072*c**2*e**2*f**8 - 1536*c**2*e**2*f**7*g*x + 1152*c**2*e**2*f**6*g**2* x**2 - 960*c**2*e**2*f**5*g**3*x**3 + 840*c**2*e**2*f**4*g**4*x**4 - 75...