Integrand size = 24, antiderivative size = 212 \[ \int \frac {(d+e x) \left (a+c x^2\right )^2}{\sqrt {f+g x}} \, dx=-\frac {2 (e f-d g) \left (c f^2+a g^2\right )^2 \sqrt {f+g x}}{g^6}+\frac {2 \left (c f^2+a g^2\right ) \left (a e g^2+c f (5 e f-4 d g)\right ) (f+g x)^{3/2}}{3 g^6}-\frac {4 c \left (c f^2 (5 e f-3 d g)+a g^2 (3 e f-d g)\right ) (f+g x)^{5/2}}{5 g^6}+\frac {4 c \left (a e g^2+c f (5 e f-2 d g)\right ) (f+g x)^{7/2}}{7 g^6}-\frac {2 c^2 (5 e f-d g) (f+g x)^{9/2}}{9 g^6}+\frac {2 c^2 e (f+g x)^{11/2}}{11 g^6} \] Output:
-2*(-d*g+e*f)*(a*g^2+c*f^2)^2*(g*x+f)^(1/2)/g^6+2/3*(a*g^2+c*f^2)*(a*e*g^2 +c*f*(-4*d*g+5*e*f))*(g*x+f)^(3/2)/g^6-4/5*c*(c*f^2*(-3*d*g+5*e*f)+a*g^2*( -d*g+3*e*f))*(g*x+f)^(5/2)/g^6+4/7*c*(a*e*g^2+c*f*(-2*d*g+5*e*f))*(g*x+f)^ (7/2)/g^6-2/9*c^2*(-d*g+5*e*f)*(g*x+f)^(9/2)/g^6+2/11*c^2*e*(g*x+f)^(11/2) /g^6
Time = 0.20 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.98 \[ \int \frac {(d+e x) \left (a+c x^2\right )^2}{\sqrt {f+g x}} \, dx=\frac {2 \sqrt {f+g x} \left (1155 a^2 g^4 (-2 e f+3 d g+e g x)-66 a c g^2 \left (-7 d g \left (8 f^2-4 f g x+3 g^2 x^2\right )+3 e \left (16 f^3-8 f^2 g x+6 f g^2 x^2-5 g^3 x^3\right )\right )+c^2 \left (11 d g \left (128 f^4-64 f^3 g x+48 f^2 g^2 x^2-40 f g^3 x^3+35 g^4 x^4\right )-5 e \left (256 f^5-128 f^4 g x+96 f^3 g^2 x^2-80 f^2 g^3 x^3+70 f g^4 x^4-63 g^5 x^5\right )\right )\right )}{3465 g^6} \] Input:
Integrate[((d + e*x)*(a + c*x^2)^2)/Sqrt[f + g*x],x]
Output:
(2*Sqrt[f + g*x]*(1155*a^2*g^4*(-2*e*f + 3*d*g + e*g*x) - 66*a*c*g^2*(-7*d *g*(8*f^2 - 4*f*g*x + 3*g^2*x^2) + 3*e*(16*f^3 - 8*f^2*g*x + 6*f*g^2*x^2 - 5*g^3*x^3)) + c^2*(11*d*g*(128*f^4 - 64*f^3*g*x + 48*f^2*g^2*x^2 - 40*f*g ^3*x^3 + 35*g^4*x^4) - 5*e*(256*f^5 - 128*f^4*g*x + 96*f^3*g^2*x^2 - 80*f^ 2*g^3*x^3 + 70*f*g^4*x^4 - 63*g^5*x^5))))/(3465*g^6)
Time = 0.40 (sec) , antiderivative size = 212, 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.
| \(\displaystyle \int \frac {\left (a+c x^2\right )^2 (d+e x)}{\sqrt {f+g x}} \, dx\) |
| \(\Big \downarrow \) 652 |
| \(\displaystyle \int \left (\frac {2 c (f+g x)^{3/2} \left (-a g^2 (3 e f-d g)-c f^2 (5 e f-3 d g)\right )}{g^5}+\frac {\sqrt {f+g x} \left (a g^2+c f^2\right ) \left (a e g^2+c f (5 e f-4 d g)\right )}{g^5}+\frac {\left (a g^2+c f^2\right )^2 (d g-e f)}{g^5 \sqrt {f+g x}}+\frac {2 c (f+g x)^{5/2} \left (a e g^2+c f (5 e f-2 d g)\right )}{g^5}+\frac {c^2 (f+g x)^{7/2} (d g-5 e f)}{g^5}+\frac {c^2 e (f+g x)^{9/2}}{g^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {4 c (f+g x)^{5/2} \left (a g^2 (3 e f-d g)+c f^2 (5 e f-3 d g)\right )}{5 g^6}+\frac {2 (f+g x)^{3/2} \left (a g^2+c f^2\right ) \left (a e g^2+c f (5 e f-4 d g)\right )}{3 g^6}-\frac {2 \sqrt {f+g x} \left (a g^2+c f^2\right )^2 (e f-d g)}{g^6}+\frac {4 c (f+g x)^{7/2} \left (a e g^2+c f (5 e f-2 d g)\right )}{7 g^6}-\frac {2 c^2 (f+g x)^{9/2} (5 e f-d g)}{9 g^6}+\frac {2 c^2 e (f+g x)^{11/2}}{11 g^6}\) |
Input:
Int[((d + e*x)*(a + c*x^2)^2)/Sqrt[f + g*x],x]
Output:
(-2*(e*f - d*g)*(c*f^2 + a*g^2)^2*Sqrt[f + g*x])/g^6 + (2*(c*f^2 + a*g^2)* (a*e*g^2 + c*f*(5*e*f - 4*d*g))*(f + g*x)^(3/2))/(3*g^6) - (4*c*(c*f^2*(5* e*f - 3*d*g) + a*g^2*(3*e*f - d*g))*(f + g*x)^(5/2))/(5*g^6) + (4*c*(a*e*g ^2 + c*f*(5*e*f - 2*d*g))*(f + g*x)^(7/2))/(7*g^6) - (2*c^2*(5*e*f - d*g)* (f + g*x)^(9/2))/(9*g^6) + (2*c^2*e*(f + g*x)^(11/2))/(11*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 = 1.41 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.83
| method | result | size |
| pseudoelliptic | \(\frac {2 \left (\left (\frac {x^{4} \left (\frac {9 e x}{11}+d \right ) c^{2}}{9}+\frac {2 x^{2} \left (\frac {5 e x}{7}+d \right ) a c}{5}+a^{2} \left (\frac {e x}{3}+d \right )\right ) g^{5}-\frac {2 f \left (\left (\frac {5}{33} x^{4} e +\frac {4}{21} d \,x^{3}\right ) c^{2}+\frac {4 \left (\frac {9 e x}{14}+d \right ) x a c}{5}+a^{2} e \right ) g^{4}}{3}+\frac {16 f^{2} \left (\frac {x^{2} \left (\frac {25 e x}{33}+d \right ) c}{7}+a \left (\frac {3 e x}{7}+d \right )\right ) c \,g^{3}}{15}-\frac {32 f^{3} c \left (\left (\frac {5}{33} e \,x^{2}+\frac {2}{9} d x \right ) c +a e \right ) g^{2}}{35}+\frac {128 \left (\frac {5 e x}{11}+d \right ) f^{4} c^{2} g}{315}-\frac {256 c^{2} e \,f^{5}}{693}\right ) \sqrt {g x +f}}{g^{6}}\) | \(176\) |
| derivativedivides | \(\frac {\frac {2 e \,c^{2} \left (g x +f \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (d g -e f \right ) c^{2}-4 f e \,c^{2}\right ) \left (g x +f \right )^{\frac {9}{2}}}{9}+\frac {2 \left (-4 \left (d g -e f \right ) c^{2} f +e \left (2 \left (a \,g^{2}+c \,f^{2}\right ) c +4 c^{2} f^{2}\right )\right ) \left (g x +f \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (d g -e f \right ) \left (2 \left (a \,g^{2}+c \,f^{2}\right ) c +4 c^{2} f^{2}\right )-4 e \left (a \,g^{2}+c \,f^{2}\right ) c f \right ) \left (g x +f \right )^{\frac {5}{2}}}{5}+\frac {2 \left (-4 \left (d g -e f \right ) \left (a \,g^{2}+c \,f^{2}\right ) c f +e \left (a \,g^{2}+c \,f^{2}\right )^{2}\right ) \left (g x +f \right )^{\frac {3}{2}}}{3}+2 \left (d g -e f \right ) \left (a \,g^{2}+c \,f^{2}\right )^{2} \sqrt {g x +f}}{g^{6}}\) | \(233\) |
| default | \(\frac {\frac {2 e \,c^{2} \left (g x +f \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (d g -e f \right ) c^{2}-4 f e \,c^{2}\right ) \left (g x +f \right )^{\frac {9}{2}}}{9}+\frac {2 \left (-4 \left (d g -e f \right ) c^{2} f +e \left (2 \left (a \,g^{2}+c \,f^{2}\right ) c +4 c^{2} f^{2}\right )\right ) \left (g x +f \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (d g -e f \right ) \left (2 \left (a \,g^{2}+c \,f^{2}\right ) c +4 c^{2} f^{2}\right )-4 e \left (a \,g^{2}+c \,f^{2}\right ) c f \right ) \left (g x +f \right )^{\frac {5}{2}}}{5}+\frac {2 \left (-4 \left (d g -e f \right ) \left (a \,g^{2}+c \,f^{2}\right ) c f +e \left (a \,g^{2}+c \,f^{2}\right )^{2}\right ) \left (g x +f \right )^{\frac {3}{2}}}{3}+2 \left (d g -e f \right ) \left (a \,g^{2}+c \,f^{2}\right )^{2} \sqrt {g x +f}}{g^{6}}\) | \(233\) |
| gosper | \(\frac {2 \sqrt {g x +f}\, \left (315 c^{2} e \,x^{5} g^{5}+385 c^{2} d \,g^{5} x^{4}-350 c^{2} e f \,g^{4} x^{4}+990 a c e \,g^{5} x^{3}-440 c^{2} d f \,g^{4} x^{3}+400 c^{2} e \,f^{2} g^{3} x^{3}+1386 a c d \,g^{5} x^{2}-1188 a c e f \,g^{4} x^{2}+528 c^{2} d \,f^{2} g^{3} x^{2}-480 c^{2} e \,f^{3} g^{2} x^{2}+1155 a^{2} e \,g^{5} x -1848 a c d f \,g^{4} x +1584 a c e \,f^{2} g^{3} x -704 c^{2} d \,f^{3} g^{2} x +640 c^{2} e \,f^{4} g x +3465 a^{2} d \,g^{5}-2310 a^{2} e f \,g^{4}+3696 a c d \,f^{2} g^{3}-3168 a c e \,f^{3} g^{2}+1408 c^{2} d \,f^{4} g -1280 c^{2} e \,f^{5}\right )}{3465 g^{6}}\) | \(259\) |
| trager | \(\frac {2 \sqrt {g x +f}\, \left (315 c^{2} e \,x^{5} g^{5}+385 c^{2} d \,g^{5} x^{4}-350 c^{2} e f \,g^{4} x^{4}+990 a c e \,g^{5} x^{3}-440 c^{2} d f \,g^{4} x^{3}+400 c^{2} e \,f^{2} g^{3} x^{3}+1386 a c d \,g^{5} x^{2}-1188 a c e f \,g^{4} x^{2}+528 c^{2} d \,f^{2} g^{3} x^{2}-480 c^{2} e \,f^{3} g^{2} x^{2}+1155 a^{2} e \,g^{5} x -1848 a c d f \,g^{4} x +1584 a c e \,f^{2} g^{3} x -704 c^{2} d \,f^{3} g^{2} x +640 c^{2} e \,f^{4} g x +3465 a^{2} d \,g^{5}-2310 a^{2} e f \,g^{4}+3696 a c d \,f^{2} g^{3}-3168 a c e \,f^{3} g^{2}+1408 c^{2} d \,f^{4} g -1280 c^{2} e \,f^{5}\right )}{3465 g^{6}}\) | \(259\) |
| risch | \(\frac {2 \sqrt {g x +f}\, \left (315 c^{2} e \,x^{5} g^{5}+385 c^{2} d \,g^{5} x^{4}-350 c^{2} e f \,g^{4} x^{4}+990 a c e \,g^{5} x^{3}-440 c^{2} d f \,g^{4} x^{3}+400 c^{2} e \,f^{2} g^{3} x^{3}+1386 a c d \,g^{5} x^{2}-1188 a c e f \,g^{4} x^{2}+528 c^{2} d \,f^{2} g^{3} x^{2}-480 c^{2} e \,f^{3} g^{2} x^{2}+1155 a^{2} e \,g^{5} x -1848 a c d f \,g^{4} x +1584 a c e \,f^{2} g^{3} x -704 c^{2} d \,f^{3} g^{2} x +640 c^{2} e \,f^{4} g x +3465 a^{2} d \,g^{5}-2310 a^{2} e f \,g^{4}+3696 a c d \,f^{2} g^{3}-3168 a c e \,f^{3} g^{2}+1408 c^{2} d \,f^{4} g -1280 c^{2} e \,f^{5}\right )}{3465 g^{6}}\) | \(259\) |
| orering | \(\frac {2 \sqrt {g x +f}\, \left (315 c^{2} e \,x^{5} g^{5}+385 c^{2} d \,g^{5} x^{4}-350 c^{2} e f \,g^{4} x^{4}+990 a c e \,g^{5} x^{3}-440 c^{2} d f \,g^{4} x^{3}+400 c^{2} e \,f^{2} g^{3} x^{3}+1386 a c d \,g^{5} x^{2}-1188 a c e f \,g^{4} x^{2}+528 c^{2} d \,f^{2} g^{3} x^{2}-480 c^{2} e \,f^{3} g^{2} x^{2}+1155 a^{2} e \,g^{5} x -1848 a c d f \,g^{4} x +1584 a c e \,f^{2} g^{3} x -704 c^{2} d \,f^{3} g^{2} x +640 c^{2} e \,f^{4} g x +3465 a^{2} d \,g^{5}-2310 a^{2} e f \,g^{4}+3696 a c d \,f^{2} g^{3}-3168 a c e \,f^{3} g^{2}+1408 c^{2} d \,f^{4} g -1280 c^{2} e \,f^{5}\right )}{3465 g^{6}}\) | \(259\) |
Input:
int((e*x+d)*(c*x^2+a)^2/(g*x+f)^(1/2),x,method=_RETURNVERBOSE)
Output:
2*((1/9*x^4*(9/11*e*x+d)*c^2+2/5*x^2*(5/7*e*x+d)*a*c+a^2*(1/3*e*x+d))*g^5- 2/3*f*((5/33*x^4*e+4/21*d*x^3)*c^2+4/5*(9/14*e*x+d)*x*a*c+a^2*e)*g^4+16/15 *f^2*(1/7*x^2*(25/33*e*x+d)*c+a*(3/7*e*x+d))*c*g^3-32/35*f^3*c*((5/33*e*x^ 2+2/9*d*x)*c+a*e)*g^2+128/315*(5/11*e*x+d)*f^4*c^2*g-256/693*c^2*e*f^5)*(g *x+f)^(1/2)/g^6
Time = 0.09 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.17 \[ \int \frac {(d+e x) \left (a+c x^2\right )^2}{\sqrt {f+g x}} \, dx=\frac {2 \, {\left (315 \, c^{2} e g^{5} x^{5} - 1280 \, c^{2} e f^{5} + 1408 \, c^{2} d f^{4} g - 3168 \, a c e f^{3} g^{2} + 3696 \, a c d f^{2} g^{3} - 2310 \, a^{2} e f g^{4} + 3465 \, a^{2} d g^{5} - 35 \, {\left (10 \, c^{2} e f g^{4} - 11 \, c^{2} d g^{5}\right )} x^{4} + 10 \, {\left (40 \, c^{2} e f^{2} g^{3} - 44 \, c^{2} d f g^{4} + 99 \, a c e g^{5}\right )} x^{3} - 6 \, {\left (80 \, c^{2} e f^{3} g^{2} - 88 \, c^{2} d f^{2} g^{3} + 198 \, a c e f g^{4} - 231 \, a c d g^{5}\right )} x^{2} + {\left (640 \, c^{2} e f^{4} g - 704 \, c^{2} d f^{3} g^{2} + 1584 \, a c e f^{2} g^{3} - 1848 \, a c d f g^{4} + 1155 \, a^{2} e g^{5}\right )} x\right )} \sqrt {g x + f}}{3465 \, g^{6}} \] Input:
integrate((e*x+d)*(c*x^2+a)^2/(g*x+f)^(1/2),x, algorithm="fricas")
Output:
2/3465*(315*c^2*e*g^5*x^5 - 1280*c^2*e*f^5 + 1408*c^2*d*f^4*g - 3168*a*c*e *f^3*g^2 + 3696*a*c*d*f^2*g^3 - 2310*a^2*e*f*g^4 + 3465*a^2*d*g^5 - 35*(10 *c^2*e*f*g^4 - 11*c^2*d*g^5)*x^4 + 10*(40*c^2*e*f^2*g^3 - 44*c^2*d*f*g^4 + 99*a*c*e*g^5)*x^3 - 6*(80*c^2*e*f^3*g^2 - 88*c^2*d*f^2*g^3 + 198*a*c*e*f* g^4 - 231*a*c*d*g^5)*x^2 + (640*c^2*e*f^4*g - 704*c^2*d*f^3*g^2 + 1584*a*c *e*f^2*g^3 - 1848*a*c*d*f*g^4 + 1155*a^2*e*g^5)*x)*sqrt(g*x + f)/g^6
Time = 0.99 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.75 \[ \int \frac {(d+e x) \left (a+c x^2\right )^2}{\sqrt {f+g x}} \, dx=\begin {cases} \frac {2 \left (\frac {c^{2} e \left (f + g x\right )^{\frac {11}{2}}}{11 g^{5}} + \frac {\left (f + g x\right )^{\frac {9}{2}} \left (c^{2} d g - 5 c^{2} e f\right )}{9 g^{5}} + \frac {\left (f + g x\right )^{\frac {7}{2}} \cdot \left (2 a c e g^{2} - 4 c^{2} d f g + 10 c^{2} e f^{2}\right )}{7 g^{5}} + \frac {\left (f + g x\right )^{\frac {5}{2}} \cdot \left (2 a c d g^{3} - 6 a c e f g^{2} + 6 c^{2} d f^{2} g - 10 c^{2} e f^{3}\right )}{5 g^{5}} + \frac {\left (f + g x\right )^{\frac {3}{2}} \left (a^{2} e g^{4} - 4 a c d f g^{3} + 6 a c e f^{2} g^{2} - 4 c^{2} d f^{3} g + 5 c^{2} e f^{4}\right )}{3 g^{5}} + \frac {\sqrt {f + g x} \left (a^{2} d g^{5} - a^{2} e f g^{4} + 2 a c d f^{2} g^{3} - 2 a c e f^{3} g^{2} + c^{2} d f^{4} g - c^{2} e f^{5}\right )}{g^{5}}\right )}{g} & \text {for}\: g \neq 0 \\\frac {a^{2} d x + \frac {a^{2} e x^{2}}{2} + \frac {2 a c d x^{3}}{3} + \frac {a c e x^{4}}{2} + \frac {c^{2} d x^{5}}{5} + \frac {c^{2} e x^{6}}{6}}{\sqrt {f}} & \text {otherwise} \end {cases} \] Input:
integrate((e*x+d)*(c*x**2+a)**2/(g*x+f)**(1/2),x)
Output:
Piecewise((2*(c**2*e*(f + g*x)**(11/2)/(11*g**5) + (f + g*x)**(9/2)*(c**2* d*g - 5*c**2*e*f)/(9*g**5) + (f + g*x)**(7/2)*(2*a*c*e*g**2 - 4*c**2*d*f*g + 10*c**2*e*f**2)/(7*g**5) + (f + g*x)**(5/2)*(2*a*c*d*g**3 - 6*a*c*e*f*g **2 + 6*c**2*d*f**2*g - 10*c**2*e*f**3)/(5*g**5) + (f + g*x)**(3/2)*(a**2* e*g**4 - 4*a*c*d*f*g**3 + 6*a*c*e*f**2*g**2 - 4*c**2*d*f**3*g + 5*c**2*e*f **4)/(3*g**5) + sqrt(f + g*x)*(a**2*d*g**5 - a**2*e*f*g**4 + 2*a*c*d*f**2* g**3 - 2*a*c*e*f**3*g**2 + c**2*d*f**4*g - c**2*e*f**5)/g**5)/g, Ne(g, 0)) , ((a**2*d*x + a**2*e*x**2/2 + 2*a*c*d*x**3/3 + a*c*e*x**4/2 + c**2*d*x**5 /5 + c**2*e*x**6/6)/sqrt(f), True))
Time = 0.04 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.17 \[ \int \frac {(d+e x) \left (a+c x^2\right )^2}{\sqrt {f+g x}} \, dx=\frac {2 \, {\left (315 \, {\left (g x + f\right )}^{\frac {11}{2}} c^{2} e - 385 \, {\left (5 \, c^{2} e f - c^{2} d g\right )} {\left (g x + f\right )}^{\frac {9}{2}} + 990 \, {\left (5 \, c^{2} e f^{2} - 2 \, c^{2} d f g + a c e g^{2}\right )} {\left (g x + f\right )}^{\frac {7}{2}} - 1386 \, {\left (5 \, c^{2} e f^{3} - 3 \, c^{2} d f^{2} g + 3 \, a c e f g^{2} - a c d g^{3}\right )} {\left (g x + f\right )}^{\frac {5}{2}} + 1155 \, {\left (5 \, c^{2} e f^{4} - 4 \, c^{2} d f^{3} g + 6 \, a c e f^{2} g^{2} - 4 \, a c d f g^{3} + a^{2} e g^{4}\right )} {\left (g x + f\right )}^{\frac {3}{2}} - 3465 \, {\left (c^{2} e f^{5} - c^{2} d f^{4} g + 2 \, a c e f^{3} g^{2} - 2 \, a c d f^{2} g^{3} + a^{2} e f g^{4} - a^{2} d g^{5}\right )} \sqrt {g x + f}\right )}}{3465 \, g^{6}} \] Input:
integrate((e*x+d)*(c*x^2+a)^2/(g*x+f)^(1/2),x, algorithm="maxima")
Output:
2/3465*(315*(g*x + f)^(11/2)*c^2*e - 385*(5*c^2*e*f - c^2*d*g)*(g*x + f)^( 9/2) + 990*(5*c^2*e*f^2 - 2*c^2*d*f*g + a*c*e*g^2)*(g*x + f)^(7/2) - 1386* (5*c^2*e*f^3 - 3*c^2*d*f^2*g + 3*a*c*e*f*g^2 - a*c*d*g^3)*(g*x + f)^(5/2) + 1155*(5*c^2*e*f^4 - 4*c^2*d*f^3*g + 6*a*c*e*f^2*g^2 - 4*a*c*d*f*g^3 + a^ 2*e*g^4)*(g*x + f)^(3/2) - 3465*(c^2*e*f^5 - c^2*d*f^4*g + 2*a*c*e*f^3*g^2 - 2*a*c*d*f^2*g^3 + a^2*e*f*g^4 - a^2*d*g^5)*sqrt(g*x + f))/g^6
Time = 0.11 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.32 \[ \int \frac {(d+e x) \left (a+c x^2\right )^2}{\sqrt {f+g x}} \, dx=\frac {2 \, {\left (3465 \, \sqrt {g x + f} a^{2} d + \frac {1155 \, {\left ({\left (g x + f\right )}^{\frac {3}{2}} - 3 \, \sqrt {g x + f} f\right )} a^{2} e}{g} + \frac {462 \, {\left (3 \, {\left (g x + f\right )}^{\frac {5}{2}} - 10 \, {\left (g x + f\right )}^{\frac {3}{2}} f + 15 \, \sqrt {g x + f} f^{2}\right )} a c d}{g^{2}} + \frac {198 \, {\left (5 \, {\left (g x + f\right )}^{\frac {7}{2}} - 21 \, {\left (g x + f\right )}^{\frac {5}{2}} f + 35 \, {\left (g x + f\right )}^{\frac {3}{2}} f^{2} - 35 \, \sqrt {g x + f} f^{3}\right )} a c e}{g^{3}} + \frac {11 \, {\left (35 \, {\left (g x + f\right )}^{\frac {9}{2}} - 180 \, {\left (g x + f\right )}^{\frac {7}{2}} f + 378 \, {\left (g x + f\right )}^{\frac {5}{2}} f^{2} - 420 \, {\left (g x + f\right )}^{\frac {3}{2}} f^{3} + 315 \, \sqrt {g x + f} f^{4}\right )} c^{2} d}{g^{4}} + \frac {5 \, {\left (63 \, {\left (g x + f\right )}^{\frac {11}{2}} - 385 \, {\left (g x + f\right )}^{\frac {9}{2}} f + 990 \, {\left (g x + f\right )}^{\frac {7}{2}} f^{2} - 1386 \, {\left (g x + f\right )}^{\frac {5}{2}} f^{3} + 1155 \, {\left (g x + f\right )}^{\frac {3}{2}} f^{4} - 693 \, \sqrt {g x + f} f^{5}\right )} c^{2} e}{g^{5}}\right )}}{3465 \, g} \] Input:
integrate((e*x+d)*(c*x^2+a)^2/(g*x+f)^(1/2),x, algorithm="giac")
Output:
2/3465*(3465*sqrt(g*x + f)*a^2*d + 1155*((g*x + f)^(3/2) - 3*sqrt(g*x + f) *f)*a^2*e/g + 462*(3*(g*x + f)^(5/2) - 10*(g*x + f)^(3/2)*f + 15*sqrt(g*x + f)*f^2)*a*c*d/g^2 + 198*(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*e/g^3 + 11*(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/g^4 + 5*(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 + 115 5*(g*x + f)^(3/2)*f^4 - 693*sqrt(g*x + f)*f^5)*c^2*e/g^5)/g
Time = 5.71 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.93 \[ \int \frac {(d+e x) \left (a+c x^2\right )^2}{\sqrt {f+g x}} \, dx=\frac {{\left (f+g\,x\right )}^{7/2}\,\left (20\,e\,c^2\,f^2-8\,d\,c^2\,f\,g+4\,a\,e\,c\,g^2\right )}{7\,g^6}+\frac {2\,\sqrt {f+g\,x}\,{\left (c\,f^2+a\,g^2\right )}^2\,\left (d\,g-e\,f\right )}{g^6}+\frac {2\,{\left (f+g\,x\right )}^{3/2}\,\left (c\,f^2+a\,g^2\right )\,\left (5\,c\,e\,f^2-4\,c\,d\,f\,g+a\,e\,g^2\right )}{3\,g^6}+\frac {2\,c^2\,e\,{\left (f+g\,x\right )}^{11/2}}{11\,g^6}+\frac {4\,c\,{\left (f+g\,x\right )}^{5/2}\,\left (-5\,c\,e\,f^3+3\,c\,d\,f^2\,g-3\,a\,e\,f\,g^2+a\,d\,g^3\right )}{5\,g^6}+\frac {2\,c^2\,{\left (f+g\,x\right )}^{9/2}\,\left (d\,g-5\,e\,f\right )}{9\,g^6} \] Input:
int(((a + c*x^2)^2*(d + e*x))/(f + g*x)^(1/2),x)
Output:
((f + g*x)^(7/2)*(20*c^2*e*f^2 + 4*a*c*e*g^2 - 8*c^2*d*f*g))/(7*g^6) + (2* (f + g*x)^(1/2)*(a*g^2 + c*f^2)^2*(d*g - e*f))/g^6 + (2*(f + g*x)^(3/2)*(a *g^2 + c*f^2)*(a*e*g^2 + 5*c*e*f^2 - 4*c*d*f*g))/(3*g^6) + (2*c^2*e*(f + g *x)^(11/2))/(11*g^6) + (4*c*(f + g*x)^(5/2)*(a*d*g^3 - 5*c*e*f^3 - 3*a*e*f *g^2 + 3*c*d*f^2*g))/(5*g^6) + (2*c^2*(f + g*x)^(9/2)*(d*g - 5*e*f))/(9*g^ 6)
Time = 0.20 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.21 \[ \int \frac {(d+e x) \left (a+c x^2\right )^2}{\sqrt {f+g x}} \, dx=\frac {2 \sqrt {g x +f}\, \left (315 c^{2} e \,g^{5} x^{5}+385 c^{2} d \,g^{5} x^{4}-350 c^{2} e f \,g^{4} x^{4}+990 a c e \,g^{5} x^{3}-440 c^{2} d f \,g^{4} x^{3}+400 c^{2} e \,f^{2} g^{3} x^{3}+1386 a c d \,g^{5} x^{2}-1188 a c e f \,g^{4} x^{2}+528 c^{2} d \,f^{2} g^{3} x^{2}-480 c^{2} e \,f^{3} g^{2} x^{2}+1155 a^{2} e \,g^{5} x -1848 a c d f \,g^{4} x +1584 a c e \,f^{2} g^{3} x -704 c^{2} d \,f^{3} g^{2} x +640 c^{2} e \,f^{4} g x +3465 a^{2} d \,g^{5}-2310 a^{2} e f \,g^{4}+3696 a c d \,f^{2} g^{3}-3168 a c e \,f^{3} g^{2}+1408 c^{2} d \,f^{4} g -1280 c^{2} e \,f^{5}\right )}{3465 g^{6}} \] Input:
int((e*x+d)*(c*x^2+a)^2/(g*x+f)^(1/2),x)
Output:
(2*sqrt(f + g*x)*(3465*a**2*d*g**5 - 2310*a**2*e*f*g**4 + 1155*a**2*e*g**5 *x + 3696*a*c*d*f**2*g**3 - 1848*a*c*d*f*g**4*x + 1386*a*c*d*g**5*x**2 - 3 168*a*c*e*f**3*g**2 + 1584*a*c*e*f**2*g**3*x - 1188*a*c*e*f*g**4*x**2 + 99 0*a*c*e*g**5*x**3 + 1408*c**2*d*f**4*g - 704*c**2*d*f**3*g**2*x + 528*c**2 *d*f**2*g**3*x**2 - 440*c**2*d*f*g**4*x**3 + 385*c**2*d*g**5*x**4 - 1280*c **2*e*f**5 + 640*c**2*e*f**4*g*x - 480*c**2*e*f**3*g**2*x**2 + 400*c**2*e* f**2*g**3*x**3 - 350*c**2*e*f*g**4*x**4 + 315*c**2*e*g**5*x**5))/(3465*g** 6)