Integrand size = 22, antiderivative size = 161 \[ \int \frac {\left (c+d x+e x^2\right )^2}{\sqrt {a+b x}} \, dx=\frac {2 \left (b^2 c-a b d+a^2 e\right )^2 \sqrt {a+b x}}{b^5}+\frac {4 (b d-2 a e) \left (b^2 c-a b d+a^2 e\right ) (a+b x)^{3/2}}{3 b^5}-\frac {2 \left (6 a b d e-6 a^2 e^2-b^2 \left (d^2+2 c e\right )\right ) (a+b x)^{5/2}}{5 b^5}+\frac {4 e (b d-2 a e) (a+b x)^{7/2}}{7 b^5}+\frac {2 e^2 (a+b x)^{9/2}}{9 b^5} \] Output:
2*(a^2*e-a*b*d+b^2*c)^2*(b*x+a)^(1/2)/b^5+4/3*(-2*a*e+b*d)*(a^2*e-a*b*d+b^ 2*c)*(b*x+a)^(3/2)/b^5-2/5*(6*a*b*d*e-6*a^2*e^2-b^2*(2*c*e+d^2))*(b*x+a)^( 5/2)/b^5+4/7*e*(-2*a*e+b*d)*(b*x+a)^(7/2)/b^5+2/9*e^2*(b*x+a)^(9/2)/b^5
Time = 0.09 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.96 \[ \int \frac {\left (c+d x+e x^2\right )^2}{\sqrt {a+b x}} \, dx=\frac {2 \sqrt {a+b x} \left (128 a^4 e^2-32 a^3 b e (9 d+2 e x)+24 a^2 b^2 \left (7 d^2+6 d e x+2 e \left (7 c+e x^2\right )\right )-4 a b^3 \left (21 c (5 d+2 e x)+x \left (21 d^2+27 d e x+10 e^2 x^2\right )\right )+b^4 \left (315 c^2+42 c x (5 d+3 e x)+x^2 \left (63 d^2+90 d e x+35 e^2 x^2\right )\right )\right )}{315 b^5} \] Input:
Integrate[(c + d*x + e*x^2)^2/Sqrt[a + b*x],x]
Output:
(2*Sqrt[a + b*x]*(128*a^4*e^2 - 32*a^3*b*e*(9*d + 2*e*x) + 24*a^2*b^2*(7*d ^2 + 6*d*e*x + 2*e*(7*c + e*x^2)) - 4*a*b^3*(21*c*(5*d + 2*e*x) + x*(21*d^ 2 + 27*d*e*x + 10*e^2*x^2)) + b^4*(315*c^2 + 42*c*x*(5*d + 3*e*x) + x^2*(6 3*d^2 + 90*d*e*x + 35*e^2*x^2))))/(315*b^5)
Time = 0.34 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1140, 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.
\(\displaystyle \int \frac {\left (c+d x+e x^2\right )^2}{\sqrt {a+b x}} \, dx\) |
\(\Big \downarrow \) 1140 |
\(\displaystyle \int \left (\frac {(a+b x)^{3/2} \left (6 a^2 e^2-6 a b d e+b^2 \left (2 c e+d^2\right )\right )}{b^4}+\frac {2 \sqrt {a+b x} (b d-2 a e) \left (a^2 e-a b d+b^2 c\right )}{b^4}+\frac {\left (a^2 e-a b d+b^2 c\right )^2}{b^4 \sqrt {a+b x}}+\frac {2 e (a+b x)^{5/2} (b d-2 a e)}{b^4}+\frac {e^2 (a+b x)^{7/2}}{b^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 (a+b x)^{5/2} \left (-6 a^2 e^2+6 a b d e-\left (b^2 \left (2 c e+d^2\right )\right )\right )}{5 b^5}+\frac {4 (a+b x)^{3/2} (b d-2 a e) \left (a^2 e-a b d+b^2 c\right )}{3 b^5}+\frac {2 \sqrt {a+b x} \left (a^2 e-a b d+b^2 c\right )^2}{b^5}+\frac {4 e (a+b x)^{7/2} (b d-2 a e)}{7 b^5}+\frac {2 e^2 (a+b x)^{9/2}}{9 b^5}\) |
Input:
Int[(c + d*x + e*x^2)^2/Sqrt[a + b*x],x]
Output:
(2*(b^2*c - a*b*d + a^2*e)^2*Sqrt[a + b*x])/b^5 + (4*(b*d - 2*a*e)*(b^2*c - a*b*d + a^2*e)*(a + b*x)^(3/2))/(3*b^5) - (2*(6*a*b*d*e - 6*a^2*e^2 - b^ 2*(d^2 + 2*c*e))*(a + b*x)^(5/2))/(5*b^5) + (4*e*(b*d - 2*a*e)*(a + b*x)^( 7/2))/(7*b^5) + (2*e^2*(a + b*x)^(9/2))/(9*b^5)
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x _Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0]
Time = 0.79 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.84
method | result | size |
derivativedivides | \(\frac {\frac {2 e^{2} \left (b x +a \right )^{\frac {9}{2}}}{9}+\frac {4 \left (-2 a e +d b \right ) e \left (b x +a \right )^{\frac {7}{2}}}{7}+\frac {2 \left (2 \left (a^{2} e -a b d +b^{2} c \right ) e +\left (-2 a e +d b \right )^{2}\right ) \left (b x +a \right )^{\frac {5}{2}}}{5}+\frac {4 \left (a^{2} e -a b d +b^{2} c \right ) \left (-2 a e +d b \right ) \left (b x +a \right )^{\frac {3}{2}}}{3}+2 \left (a^{2} e -a b d +b^{2} c \right )^{2} \sqrt {b x +a}}{b^{5}}\) | \(135\) |
default | \(\frac {\frac {2 e^{2} \left (b x +a \right )^{\frac {9}{2}}}{9}+\frac {4 \left (-2 a e +d b \right ) e \left (b x +a \right )^{\frac {7}{2}}}{7}+\frac {2 \left (2 \left (a^{2} e -a b d +b^{2} c \right ) e +\left (-2 a e +d b \right )^{2}\right ) \left (b x +a \right )^{\frac {5}{2}}}{5}+\frac {4 \left (a^{2} e -a b d +b^{2} c \right ) \left (-2 a e +d b \right ) \left (b x +a \right )^{\frac {3}{2}}}{3}+2 \left (a^{2} e -a b d +b^{2} c \right )^{2} \sqrt {b x +a}}{b^{5}}\) | \(135\) |
pseudoelliptic | \(\frac {256 \left (\frac {7 \left (\frac {5 e^{2} x^{4}}{2}+9 \left (\frac {5 x d}{7}+c \right ) x^{2} e +\frac {9 d^{2} x^{2}}{2}+15 c d x +\frac {45 c^{2}}{2}\right ) b^{4}}{64}-\frac {105 a \left (\frac {2 e^{2} x^{3}}{21}+\frac {2 \left (\frac {9 x d}{14}+c \right ) x e}{5}+d \left (\frac {x d}{5}+c \right )\right ) b^{3}}{32}+\frac {21 a^{2} \left (\frac {e^{2} x^{2}}{7}+\left (\frac {3 x d}{7}+c \right ) e +\frac {d^{2}}{2}\right ) b^{2}}{8}-\frac {9 a^{3} \left (\frac {2 e x}{9}+d \right ) e b}{4}+a^{4} e^{2}\right ) \sqrt {b x +a}}{315 b^{5}}\) | \(141\) |
gosper | \(\frac {2 \sqrt {b x +a}\, \left (35 e^{2} x^{4} b^{4}-40 a \,b^{3} e^{2} x^{3}+90 b^{4} d e \,x^{3}+48 a^{2} b^{2} e^{2} x^{2}-108 a \,b^{3} d e \,x^{2}+126 b^{4} c e \,x^{2}+63 b^{4} d^{2} x^{2}-64 a^{3} b \,e^{2} x +144 a^{2} b^{2} d e x -168 a \,b^{3} c e x -84 a \,b^{3} d^{2} x +210 b^{4} c d x +128 a^{4} e^{2}-288 a^{3} b d e +336 a^{2} b^{2} c e +168 a^{2} b^{2} d^{2}-420 a \,b^{3} c d +315 b^{4} c^{2}\right )}{315 b^{5}}\) | \(194\) |
trager | \(\frac {2 \sqrt {b x +a}\, \left (35 e^{2} x^{4} b^{4}-40 a \,b^{3} e^{2} x^{3}+90 b^{4} d e \,x^{3}+48 a^{2} b^{2} e^{2} x^{2}-108 a \,b^{3} d e \,x^{2}+126 b^{4} c e \,x^{2}+63 b^{4} d^{2} x^{2}-64 a^{3} b \,e^{2} x +144 a^{2} b^{2} d e x -168 a \,b^{3} c e x -84 a \,b^{3} d^{2} x +210 b^{4} c d x +128 a^{4} e^{2}-288 a^{3} b d e +336 a^{2} b^{2} c e +168 a^{2} b^{2} d^{2}-420 a \,b^{3} c d +315 b^{4} c^{2}\right )}{315 b^{5}}\) | \(194\) |
risch | \(\frac {2 \sqrt {b x +a}\, \left (35 e^{2} x^{4} b^{4}-40 a \,b^{3} e^{2} x^{3}+90 b^{4} d e \,x^{3}+48 a^{2} b^{2} e^{2} x^{2}-108 a \,b^{3} d e \,x^{2}+126 b^{4} c e \,x^{2}+63 b^{4} d^{2} x^{2}-64 a^{3} b \,e^{2} x +144 a^{2} b^{2} d e x -168 a \,b^{3} c e x -84 a \,b^{3} d^{2} x +210 b^{4} c d x +128 a^{4} e^{2}-288 a^{3} b d e +336 a^{2} b^{2} c e +168 a^{2} b^{2} d^{2}-420 a \,b^{3} c d +315 b^{4} c^{2}\right )}{315 b^{5}}\) | \(194\) |
orering | \(\frac {2 \sqrt {b x +a}\, \left (35 e^{2} x^{4} b^{4}-40 a \,b^{3} e^{2} x^{3}+90 b^{4} d e \,x^{3}+48 a^{2} b^{2} e^{2} x^{2}-108 a \,b^{3} d e \,x^{2}+126 b^{4} c e \,x^{2}+63 b^{4} d^{2} x^{2}-64 a^{3} b \,e^{2} x +144 a^{2} b^{2} d e x -168 a \,b^{3} c e x -84 a \,b^{3} d^{2} x +210 b^{4} c d x +128 a^{4} e^{2}-288 a^{3} b d e +336 a^{2} b^{2} c e +168 a^{2} b^{2} d^{2}-420 a \,b^{3} c d +315 b^{4} c^{2}\right )}{315 b^{5}}\) | \(194\) |
Input:
int((e*x^2+d*x+c)^2/(b*x+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
2/b^5*(1/9*e^2*(b*x+a)^(9/2)+2/7*(-2*a*e+b*d)*e*(b*x+a)^(7/2)+1/5*(2*(a^2* e-a*b*d+b^2*c)*e+(-2*a*e+b*d)^2)*(b*x+a)^(5/2)+2/3*(a^2*e-a*b*d+b^2*c)*(-2 *a*e+b*d)*(b*x+a)^(3/2)+(a^2*e-a*b*d+b^2*c)^2*(b*x+a)^(1/2))
Time = 0.07 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.19 \[ \int \frac {\left (c+d x+e x^2\right )^2}{\sqrt {a+b x}} \, dx=\frac {2 \, {\left (35 \, b^{4} e^{2} x^{4} + 315 \, b^{4} c^{2} - 420 \, a b^{3} c d + 168 \, a^{2} b^{2} d^{2} + 128 \, a^{4} e^{2} + 10 \, {\left (9 \, b^{4} d e - 4 \, a b^{3} e^{2}\right )} x^{3} + 3 \, {\left (21 \, b^{4} d^{2} + 16 \, a^{2} b^{2} e^{2} + 6 \, {\left (7 \, b^{4} c - 6 \, a b^{3} d\right )} e\right )} x^{2} + 48 \, {\left (7 \, a^{2} b^{2} c - 6 \, a^{3} b d\right )} e + 2 \, {\left (105 \, b^{4} c d - 42 \, a b^{3} d^{2} - 32 \, a^{3} b e^{2} - 12 \, {\left (7 \, a b^{3} c - 6 \, a^{2} b^{2} d\right )} e\right )} x\right )} \sqrt {b x + a}}{315 \, b^{5}} \] Input:
integrate((e*x^2+d*x+c)^2/(b*x+a)^(1/2),x, algorithm="fricas")
Output:
2/315*(35*b^4*e^2*x^4 + 315*b^4*c^2 - 420*a*b^3*c*d + 168*a^2*b^2*d^2 + 12 8*a^4*e^2 + 10*(9*b^4*d*e - 4*a*b^3*e^2)*x^3 + 3*(21*b^4*d^2 + 16*a^2*b^2* e^2 + 6*(7*b^4*c - 6*a*b^3*d)*e)*x^2 + 48*(7*a^2*b^2*c - 6*a^3*b*d)*e + 2* (105*b^4*c*d - 42*a*b^3*d^2 - 32*a^3*b*e^2 - 12*(7*a*b^3*c - 6*a^2*b^2*d)* e)*x)*sqrt(b*x + a)/b^5
Time = 0.77 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.71 \[ \int \frac {\left (c+d x+e x^2\right )^2}{\sqrt {a+b x}} \, dx=\begin {cases} \frac {2 \left (\frac {e^{2} \left (a + b x\right )^{\frac {9}{2}}}{9 b^{4}} + \frac {\left (a + b x\right )^{\frac {7}{2}} \left (- 4 a e^{2} + 2 b d e\right )}{7 b^{4}} + \frac {\left (a + b x\right )^{\frac {5}{2}} \cdot \left (6 a^{2} e^{2} - 6 a b d e + 2 b^{2} c e + b^{2} d^{2}\right )}{5 b^{4}} + \frac {\left (a + b x\right )^{\frac {3}{2}} \left (- 4 a^{3} e^{2} + 6 a^{2} b d e - 4 a b^{2} c e - 2 a b^{2} d^{2} + 2 b^{3} c d\right )}{3 b^{4}} + \frac {\sqrt {a + b x} \left (a^{4} e^{2} - 2 a^{3} b d e + 2 a^{2} b^{2} c e + a^{2} b^{2} d^{2} - 2 a b^{3} c d + b^{4} c^{2}\right )}{b^{4}}\right )}{b} & \text {for}\: b \neq 0 \\\frac {c^{2} x + c d x^{2} + \frac {d e x^{4}}{2} + \frac {e^{2} x^{5}}{5} + \frac {x^{3} \cdot \left (2 c e + d^{2}\right )}{3}}{\sqrt {a}} & \text {otherwise} \end {cases} \] Input:
integrate((e*x**2+d*x+c)**2/(b*x+a)**(1/2),x)
Output:
Piecewise((2*(e**2*(a + b*x)**(9/2)/(9*b**4) + (a + b*x)**(7/2)*(-4*a*e**2 + 2*b*d*e)/(7*b**4) + (a + b*x)**(5/2)*(6*a**2*e**2 - 6*a*b*d*e + 2*b**2* c*e + b**2*d**2)/(5*b**4) + (a + b*x)**(3/2)*(-4*a**3*e**2 + 6*a**2*b*d*e - 4*a*b**2*c*e - 2*a*b**2*d**2 + 2*b**3*c*d)/(3*b**4) + sqrt(a + b*x)*(a** 4*e**2 - 2*a**3*b*d*e + 2*a**2*b**2*c*e + a**2*b**2*d**2 - 2*a*b**3*c*d + b**4*c**2)/b**4)/b, Ne(b, 0)), ((c**2*x + c*d*x**2 + d*e*x**4/2 + e**2*x** 5/5 + x**3*(2*c*e + d**2)/3)/sqrt(a), True))
Time = 0.04 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.47 \[ \int \frac {\left (c+d x+e x^2\right )^2}{\sqrt {a+b x}} \, dx=\frac {2 \, {\left (315 \, \sqrt {b x + a} c^{2} + 42 \, c {\left (\frac {5 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} d}{b} + \frac {{\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} e}{b^{2}}\right )} + \frac {21 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} d^{2}}{b^{2}} + \frac {18 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )} d e}{b^{3}} + \frac {{\left (35 \, {\left (b x + a\right )}^{\frac {9}{2}} - 180 \, {\left (b x + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b x + a} a^{4}\right )} e^{2}}{b^{4}}\right )}}{315 \, b} \] Input:
integrate((e*x^2+d*x+c)^2/(b*x+a)^(1/2),x, algorithm="maxima")
Output:
2/315*(315*sqrt(b*x + a)*c^2 + 42*c*(5*((b*x + a)^(3/2) - 3*sqrt(b*x + a)* a)*d/b + (3*(b*x + a)^(5/2) - 10*(b*x + a)^(3/2)*a + 15*sqrt(b*x + a)*a^2) *e/b^2) + 21*(3*(b*x + a)^(5/2) - 10*(b*x + a)^(3/2)*a + 15*sqrt(b*x + a)* a^2)*d^2/b^2 + 18*(5*(b*x + a)^(7/2) - 21*(b*x + a)^(5/2)*a + 35*(b*x + a) ^(3/2)*a^2 - 35*sqrt(b*x + a)*a^3)*d*e/b^3 + (35*(b*x + a)^(9/2) - 180*(b* x + a)^(7/2)*a + 378*(b*x + a)^(5/2)*a^2 - 420*(b*x + a)^(3/2)*a^3 + 315*s qrt(b*x + a)*a^4)*e^2/b^4)/b
Time = 0.12 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.47 \[ \int \frac {\left (c+d x+e x^2\right )^2}{\sqrt {a+b x}} \, dx=\frac {2 \, {\left (315 \, \sqrt {b x + a} c^{2} + \frac {210 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} c d}{b} + \frac {21 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} d^{2}}{b^{2}} + \frac {42 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} c e}{b^{2}} + \frac {18 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )} d e}{b^{3}} + \frac {{\left (35 \, {\left (b x + a\right )}^{\frac {9}{2}} - 180 \, {\left (b x + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b x + a} a^{4}\right )} e^{2}}{b^{4}}\right )}}{315 \, b} \] Input:
integrate((e*x^2+d*x+c)^2/(b*x+a)^(1/2),x, algorithm="giac")
Output:
2/315*(315*sqrt(b*x + a)*c^2 + 210*((b*x + a)^(3/2) - 3*sqrt(b*x + a)*a)*c *d/b + 21*(3*(b*x + a)^(5/2) - 10*(b*x + a)^(3/2)*a + 15*sqrt(b*x + a)*a^2 )*d^2/b^2 + 42*(3*(b*x + a)^(5/2) - 10*(b*x + a)^(3/2)*a + 15*sqrt(b*x + a )*a^2)*c*e/b^2 + 18*(5*(b*x + a)^(7/2) - 21*(b*x + a)^(5/2)*a + 35*(b*x + a)^(3/2)*a^2 - 35*sqrt(b*x + a)*a^3)*d*e/b^3 + (35*(b*x + a)^(9/2) - 180*( b*x + a)^(7/2)*a + 378*(b*x + a)^(5/2)*a^2 - 420*(b*x + a)^(3/2)*a^3 + 315 *sqrt(b*x + a)*a^4)*e^2/b^4)/b
Time = 0.04 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.93 \[ \int \frac {\left (c+d x+e x^2\right )^2}{\sqrt {a+b x}} \, dx=\frac {2\,e^2\,{\left (a+b\,x\right )}^{9/2}}{9\,b^5}+\frac {{\left (a+b\,x\right )}^{5/2}\,\left (12\,a^2\,e^2-12\,a\,b\,d\,e+2\,b^2\,d^2+4\,c\,b^2\,e\right )}{5\,b^5}+\frac {2\,\sqrt {a+b\,x}\,{\left (e\,a^2-d\,a\,b+c\,b^2\right )}^2}{b^5}-\frac {\left (8\,a\,e^2-4\,b\,d\,e\right )\,{\left (a+b\,x\right )}^{7/2}}{7\,b^5}-\frac {4\,\left (2\,a\,e-b\,d\right )\,{\left (a+b\,x\right )}^{3/2}\,\left (e\,a^2-d\,a\,b+c\,b^2\right )}{3\,b^5} \] Input:
int((c + d*x + e*x^2)^2/(a + b*x)^(1/2),x)
Output:
(2*e^2*(a + b*x)^(9/2))/(9*b^5) + ((a + b*x)^(5/2)*(12*a^2*e^2 + 2*b^2*d^2 + 4*b^2*c*e - 12*a*b*d*e))/(5*b^5) + (2*(a + b*x)^(1/2)*(b^2*c + a^2*e - a*b*d)^2)/b^5 - ((8*a*e^2 - 4*b*d*e)*(a + b*x)^(7/2))/(7*b^5) - (4*(2*a*e - b*d)*(a + b*x)^(3/2)*(b^2*c + a^2*e - a*b*d))/(3*b^5)
Time = 0.14 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.19 \[ \int \frac {\left (c+d x+e x^2\right )^2}{\sqrt {a+b x}} \, dx=\frac {2 \sqrt {b x +a}\, \left (35 b^{4} e^{2} x^{4}-40 a \,b^{3} e^{2} x^{3}+90 b^{4} d e \,x^{3}+48 a^{2} b^{2} e^{2} x^{2}-108 a \,b^{3} d e \,x^{2}+126 b^{4} c e \,x^{2}+63 b^{4} d^{2} x^{2}-64 a^{3} b \,e^{2} x +144 a^{2} b^{2} d e x -168 a \,b^{3} c e x -84 a \,b^{3} d^{2} x +210 b^{4} c d x +128 a^{4} e^{2}-288 a^{3} b d e +336 a^{2} b^{2} c e +168 a^{2} b^{2} d^{2}-420 a \,b^{3} c d +315 b^{4} c^{2}\right )}{315 b^{5}} \] Input:
int((e*x^2+d*x+c)^2/(b*x+a)^(1/2),x)
Output:
(2*sqrt(a + b*x)*(128*a**4*e**2 - 288*a**3*b*d*e - 64*a**3*b*e**2*x + 336* a**2*b**2*c*e + 168*a**2*b**2*d**2 + 144*a**2*b**2*d*e*x + 48*a**2*b**2*e* *2*x**2 - 420*a*b**3*c*d - 168*a*b**3*c*e*x - 84*a*b**3*d**2*x - 108*a*b** 3*d*e*x**2 - 40*a*b**3*e**2*x**3 + 315*b**4*c**2 + 210*b**4*c*d*x + 126*b* *4*c*e*x**2 + 63*b**4*d**2*x**2 + 90*b**4*d*e*x**3 + 35*b**4*e**2*x**4))/( 315*b**5)