Integrand size = 19, antiderivative size = 125 \[ \int \frac {\left (a+c x^2\right )^2}{\sqrt {f+g x}} \, dx=\frac {2 \left (c f^2+a g^2\right )^2 \sqrt {f+g x}}{g^5}-\frac {8 c f \left (c f^2+a g^2\right ) (f+g x)^{3/2}}{3 g^5}+\frac {4 c \left (3 c f^2+a g^2\right ) (f+g x)^{5/2}}{5 g^5}-\frac {8 c^2 f (f+g x)^{7/2}}{7 g^5}+\frac {2 c^2 (f+g x)^{9/2}}{9 g^5} \] Output:
2*(a*g^2+c*f^2)^2*(g*x+f)^(1/2)/g^5-8/3*c*f*(a*g^2+c*f^2)*(g*x+f)^(3/2)/g^ 5+4/5*c*(a*g^2+3*c*f^2)*(g*x+f)^(5/2)/g^5-8/7*c^2*f*(g*x+f)^(7/2)/g^5+2/9* c^2*(g*x+f)^(9/2)/g^5
Time = 0.07 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.77 \[ \int \frac {\left (a+c x^2\right )^2}{\sqrt {f+g x}} \, dx=\frac {2 \sqrt {f+g x} \left (315 a^2 g^4+42 a c g^2 \left (8 f^2-4 f g x+3 g^2 x^2\right )+c^2 \left (128 f^4-64 f^3 g x+48 f^2 g^2 x^2-40 f g^3 x^3+35 g^4 x^4\right )\right )}{315 g^5} \] Input:
Integrate[(a + c*x^2)^2/Sqrt[f + g*x],x]
Output:
(2*Sqrt[f + g*x]*(315*a^2*g^4 + 42*a*c*g^2*(8*f^2 - 4*f*g*x + 3*g^2*x^2) + c^2*(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))) /(315*g^5)
Time = 0.26 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {476, 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.
\(\displaystyle \int \frac {\left (a+c x^2\right )^2}{\sqrt {f+g x}} \, dx\) |
\(\Big \downarrow \) 476 |
\(\displaystyle \int \left (\frac {2 c (f+g x)^{3/2} \left (a g^2+3 c f^2\right )}{g^4}-\frac {4 c f \sqrt {f+g x} \left (a g^2+c f^2\right )}{g^4}+\frac {\left (a g^2+c f^2\right )^2}{g^4 \sqrt {f+g x}}+\frac {c^2 (f+g x)^{7/2}}{g^4}-\frac {4 c^2 f (f+g x)^{5/2}}{g^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {4 c (f+g x)^{5/2} \left (a g^2+3 c f^2\right )}{5 g^5}-\frac {8 c f (f+g x)^{3/2} \left (a g^2+c f^2\right )}{3 g^5}+\frac {2 \sqrt {f+g x} \left (a g^2+c f^2\right )^2}{g^5}+\frac {2 c^2 (f+g x)^{9/2}}{9 g^5}-\frac {8 c^2 f (f+g x)^{7/2}}{7 g^5}\) |
Input:
Int[(a + c*x^2)^2/Sqrt[f + g*x],x]
Output:
(2*(c*f^2 + a*g^2)^2*Sqrt[f + g*x])/g^5 - (8*c*f*(c*f^2 + a*g^2)*(f + g*x) ^(3/2))/(3*g^5) + (4*c*(3*c*f^2 + a*g^2)*(f + g*x)^(5/2))/(5*g^5) - (8*c^2 *f*(f + g*x)^(7/2))/(7*g^5) + (2*c^2*(f + g*x)^(9/2))/(9*g^5)
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ ExpandIntegrand[(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 0]
Time = 0.69 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.70
method | result | size |
pseudoelliptic | \(\frac {2 \sqrt {g x +f}\, \left (\left (\frac {2}{5} a c \,x^{2}+a^{2}+\frac {1}{9} c^{2} x^{4}\right ) g^{4}-\frac {8 x \left (\frac {5 c \,x^{2}}{21}+a \right ) c f \,g^{3}}{15}+\frac {16 c \,f^{2} \left (\frac {c \,x^{2}}{7}+a \right ) g^{2}}{15}-\frac {64 c^{2} f^{3} g x}{315}+\frac {128 c^{2} f^{4}}{315}\right )}{g^{5}}\) | \(88\) |
gosper | \(\frac {2 \sqrt {g x +f}\, \left (35 c^{2} x^{4} g^{4}-40 c^{2} f \,x^{3} g^{3}+126 a c \,g^{4} x^{2}+48 c^{2} f^{2} g^{2} x^{2}-168 a c f \,g^{3} x -64 c^{2} f^{3} g x +315 a^{2} g^{4}+336 a c \,f^{2} g^{2}+128 c^{2} f^{4}\right )}{315 g^{5}}\) | \(106\) |
trager | \(\frac {2 \sqrt {g x +f}\, \left (35 c^{2} x^{4} g^{4}-40 c^{2} f \,x^{3} g^{3}+126 a c \,g^{4} x^{2}+48 c^{2} f^{2} g^{2} x^{2}-168 a c f \,g^{3} x -64 c^{2} f^{3} g x +315 a^{2} g^{4}+336 a c \,f^{2} g^{2}+128 c^{2} f^{4}\right )}{315 g^{5}}\) | \(106\) |
risch | \(\frac {2 \sqrt {g x +f}\, \left (35 c^{2} x^{4} g^{4}-40 c^{2} f \,x^{3} g^{3}+126 a c \,g^{4} x^{2}+48 c^{2} f^{2} g^{2} x^{2}-168 a c f \,g^{3} x -64 c^{2} f^{3} g x +315 a^{2} g^{4}+336 a c \,f^{2} g^{2}+128 c^{2} f^{4}\right )}{315 g^{5}}\) | \(106\) |
orering | \(\frac {2 \sqrt {g x +f}\, \left (35 c^{2} x^{4} g^{4}-40 c^{2} f \,x^{3} g^{3}+126 a c \,g^{4} x^{2}+48 c^{2} f^{2} g^{2} x^{2}-168 a c f \,g^{3} x -64 c^{2} f^{3} g x +315 a^{2} g^{4}+336 a c \,f^{2} g^{2}+128 c^{2} f^{4}\right )}{315 g^{5}}\) | \(106\) |
derivativedivides | \(\frac {\frac {2 c^{2} \left (g x +f \right )^{\frac {9}{2}}}{9}-\frac {8 c^{2} f \left (g x +f \right )^{\frac {7}{2}}}{7}+\frac {2 \left (2 \left (a \,g^{2}+c \,f^{2}\right ) c +4 c^{2} f^{2}\right ) \left (g x +f \right )^{\frac {5}{2}}}{5}-\frac {8 \left (a \,g^{2}+c \,f^{2}\right ) c f \left (g x +f \right )^{\frac {3}{2}}}{3}+2 \left (a \,g^{2}+c \,f^{2}\right )^{2} \sqrt {g x +f}}{g^{5}}\) | \(107\) |
default | \(\frac {\frac {2 c^{2} \left (g x +f \right )^{\frac {9}{2}}}{9}-\frac {8 c^{2} f \left (g x +f \right )^{\frac {7}{2}}}{7}+\frac {2 \left (2 \left (a \,g^{2}+c \,f^{2}\right ) c +4 c^{2} f^{2}\right ) \left (g x +f \right )^{\frac {5}{2}}}{5}-\frac {8 \left (a \,g^{2}+c \,f^{2}\right ) c f \left (g x +f \right )^{\frac {3}{2}}}{3}+2 \left (a \,g^{2}+c \,f^{2}\right )^{2} \sqrt {g x +f}}{g^{5}}\) | \(107\) |
Input:
int((c*x^2+a)^2/(g*x+f)^(1/2),x,method=_RETURNVERBOSE)
Output:
2*(g*x+f)^(1/2)*((2/5*a*c*x^2+a^2+1/9*c^2*x^4)*g^4-8/15*x*(5/21*c*x^2+a)*c *f*g^3+16/15*c*f^2*(1/7*c*x^2+a)*g^2-64/315*c^2*f^3*g*x+128/315*c^2*f^4)/g ^5
Time = 0.08 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+c x^2\right )^2}{\sqrt {f+g x}} \, dx=\frac {2 \, {\left (35 \, c^{2} g^{4} x^{4} - 40 \, c^{2} f g^{3} x^{3} + 128 \, c^{2} f^{4} + 336 \, a c f^{2} g^{2} + 315 \, a^{2} g^{4} + 6 \, {\left (8 \, c^{2} f^{2} g^{2} + 21 \, a c g^{4}\right )} x^{2} - 8 \, {\left (8 \, c^{2} f^{3} g + 21 \, a c f g^{3}\right )} x\right )} \sqrt {g x + f}}{315 \, g^{5}} \] Input:
integrate((c*x^2+a)^2/(g*x+f)^(1/2),x, algorithm="fricas")
Output:
2/315*(35*c^2*g^4*x^4 - 40*c^2*f*g^3*x^3 + 128*c^2*f^4 + 336*a*c*f^2*g^2 + 315*a^2*g^4 + 6*(8*c^2*f^2*g^2 + 21*a*c*g^4)*x^2 - 8*(8*c^2*f^3*g + 21*a* c*f*g^3)*x)*sqrt(g*x + f)/g^5
Time = 0.64 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.40 \[ \int \frac {\left (a+c x^2\right )^2}{\sqrt {f+g x}} \, dx=\begin {cases} \frac {2 \left (- \frac {4 c^{2} f \left (f + g x\right )^{\frac {7}{2}}}{7 g^{4}} + \frac {c^{2} \left (f + g x\right )^{\frac {9}{2}}}{9 g^{4}} + \frac {\left (f + g x\right )^{\frac {5}{2}} \cdot \left (2 a c g^{2} + 6 c^{2} f^{2}\right )}{5 g^{4}} + \frac {\left (f + g x\right )^{\frac {3}{2}} \left (- 4 a c f g^{2} - 4 c^{2} f^{3}\right )}{3 g^{4}} + \frac {\sqrt {f + g x} \left (a^{2} g^{4} + 2 a c f^{2} g^{2} + c^{2} f^{4}\right )}{g^{4}}\right )}{g} & \text {for}\: g \neq 0 \\\frac {a^{2} x + \frac {2 a c x^{3}}{3} + \frac {c^{2} x^{5}}{5}}{\sqrt {f}} & \text {otherwise} \end {cases} \] Input:
integrate((c*x**2+a)**2/(g*x+f)**(1/2),x)
Output:
Piecewise((2*(-4*c**2*f*(f + g*x)**(7/2)/(7*g**4) + c**2*(f + g*x)**(9/2)/ (9*g**4) + (f + g*x)**(5/2)*(2*a*c*g**2 + 6*c**2*f**2)/(5*g**4) + (f + g*x )**(3/2)*(-4*a*c*f*g**2 - 4*c**2*f**3)/(3*g**4) + sqrt(f + g*x)*(a**2*g**4 + 2*a*c*f**2*g**2 + c**2*f**4)/g**4)/g, Ne(g, 0)), ((a**2*x + 2*a*c*x**3/ 3 + c**2*x**5/5)/sqrt(f), True))
Time = 0.04 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+c x^2\right )^2}{\sqrt {f+g x}} \, dx=\frac {2 \, {\left (315 \, \sqrt {g x + f} a^{2} + \frac {42 \, {\left (3 \, {\left (g x + f\right )}^{\frac {5}{2}} - 10 \, {\left (g x + f\right )}^{\frac {3}{2}} f + 15 \, \sqrt {g x + f} f^{2}\right )} a c}{g^{2}} + \frac {{\left (35 \, {\left (g x + f\right )}^{\frac {9}{2}} - 180 \, {\left (g x + f\right )}^{\frac {7}{2}} f + 378 \, {\left (g x + f\right )}^{\frac {5}{2}} f^{2} - 420 \, {\left (g x + f\right )}^{\frac {3}{2}} f^{3} + 315 \, \sqrt {g x + f} f^{4}\right )} c^{2}}{g^{4}}\right )}}{315 \, g} \] Input:
integrate((c*x^2+a)^2/(g*x+f)^(1/2),x, algorithm="maxima")
Output:
2/315*(315*sqrt(g*x + f)*a^2 + 42*(3*(g*x + f)^(5/2) - 10*(g*x + f)^(3/2)* f + 15*sqrt(g*x + f)*f^2)*a*c/g^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)*c^2/g^4)/g
Time = 0.12 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+c x^2\right )^2}{\sqrt {f+g x}} \, dx=\frac {2 \, {\left (315 \, \sqrt {g x + f} a^{2} + \frac {42 \, {\left (3 \, {\left (g x + f\right )}^{\frac {5}{2}} - 10 \, {\left (g x + f\right )}^{\frac {3}{2}} f + 15 \, \sqrt {g x + f} f^{2}\right )} a c}{g^{2}} + \frac {{\left (35 \, {\left (g x + f\right )}^{\frac {9}{2}} - 180 \, {\left (g x + f\right )}^{\frac {7}{2}} f + 378 \, {\left (g x + f\right )}^{\frac {5}{2}} f^{2} - 420 \, {\left (g x + f\right )}^{\frac {3}{2}} f^{3} + 315 \, \sqrt {g x + f} f^{4}\right )} c^{2}}{g^{4}}\right )}}{315 \, g} \] Input:
integrate((c*x^2+a)^2/(g*x+f)^(1/2),x, algorithm="giac")
Output:
2/315*(315*sqrt(g*x + f)*a^2 + 42*(3*(g*x + f)^(5/2) - 10*(g*x + f)^(3/2)* f + 15*sqrt(g*x + f)*f^2)*a*c/g^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)*c^2/g^4)/g
Time = 0.02 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+c x^2\right )^2}{\sqrt {f+g x}} \, dx=\frac {2\,c^2\,{\left (f+g\,x\right )}^{9/2}}{9\,g^5}-\frac {{\left (f+g\,x\right )}^{3/2}\,\left (8\,c^2\,f^3+8\,a\,c\,f\,g^2\right )}{3\,g^5}+\frac {2\,\sqrt {f+g\,x}\,{\left (c\,f^2+a\,g^2\right )}^2}{g^5}+\frac {{\left (f+g\,x\right )}^{5/2}\,\left (12\,c^2\,f^2+4\,a\,c\,g^2\right )}{5\,g^5}-\frac {8\,c^2\,f\,{\left (f+g\,x\right )}^{7/2}}{7\,g^5} \] Input:
int((a + c*x^2)^2/(f + g*x)^(1/2),x)
Output:
(2*c^2*(f + g*x)^(9/2))/(9*g^5) - ((f + g*x)^(3/2)*(8*c^2*f^3 + 8*a*c*f*g^ 2))/(3*g^5) + (2*(f + g*x)^(1/2)*(a*g^2 + c*f^2)^2)/g^5 + ((f + g*x)^(5/2) *(12*c^2*f^2 + 4*a*c*g^2))/(5*g^5) - (8*c^2*f*(f + g*x)^(7/2))/(7*g^5)
Time = 0.22 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.83 \[ \int \frac {\left (a+c x^2\right )^2}{\sqrt {f+g x}} \, dx=\frac {2 \sqrt {g x +f}\, \left (35 c^{2} g^{4} x^{4}-40 c^{2} f \,g^{3} x^{3}+126 a c \,g^{4} x^{2}+48 c^{2} f^{2} g^{2} x^{2}-168 a c f \,g^{3} x -64 c^{2} f^{3} g x +315 a^{2} g^{4}+336 a c \,f^{2} g^{2}+128 c^{2} f^{4}\right )}{315 g^{5}} \] Input:
int((c*x^2+a)^2/(g*x+f)^(1/2),x)
Output:
(2*sqrt(f + g*x)*(315*a**2*g**4 + 336*a*c*f**2*g**2 - 168*a*c*f*g**3*x + 1 26*a*c*g**4*x**2 + 128*c**2*f**4 - 64*c**2*f**3*g*x + 48*c**2*f**2*g**2*x* *2 - 40*c**2*f*g**3*x**3 + 35*c**2*g**4*x**4))/(315*g**5)