Integrand size = 30, antiderivative size = 254 \[ \int \frac {(c+d x) (e+f x) \left (A+B x+C x^2\right )}{\sqrt {a+b x}} \, dx=\frac {2 \left (A b^2-a (b B-a C)\right ) (b c-a d) (b e-a f) \sqrt {a+b x}}{b^5}-\frac {2 \left (4 a^3 C d f-b^3 (B c e+A d e+A c f)+2 a b^2 (c C e+B d e+B c f+A d f)-3 a^2 b (C d e+c C f+B d f)\right ) (a+b x)^{3/2}}{3 b^5}+\frac {2 \left (6 a^2 C d f+b^2 (c C e+B d e+B c f+A d f)-3 a b (C d e+c C f+B d f)\right ) (a+b x)^{5/2}}{5 b^5}-\frac {2 (4 a C d f-b (C d e+c C f+B d f)) (a+b x)^{7/2}}{7 b^5}+\frac {2 C d f (a+b x)^{9/2}}{9 b^5} \] Output:
2*(A*b^2-a*(B*b-C*a))*(-a*d+b*c)*(-a*f+b*e)*(b*x+a)^(1/2)/b^5-2/3*(4*a^3*C *d*f-b^3*(A*c*f+A*d*e+B*c*e)+2*a*b^2*(A*d*f+B*c*f+B*d*e+C*c*e)-3*a^2*b*(B* d*f+C*c*f+C*d*e))*(b*x+a)^(3/2)/b^5+2/5*(6*a^2*C*d*f+b^2*(A*d*f+B*c*f+B*d* e+C*c*e)-3*a*b*(B*d*f+C*c*f+C*d*e))*(b*x+a)^(5/2)/b^5-2/7*(4*a*C*d*f-b*(B* d*f+C*c*f+C*d*e))*(b*x+a)^(7/2)/b^5+2/9*C*d*f*(b*x+a)^(9/2)/b^5
Time = 0.24 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.09 \[ \int \frac {(c+d x) (e+f x) \left (A+B x+C x^2\right )}{\sqrt {a+b x}} \, dx=\frac {2 \sqrt {a+b x} \left (128 a^4 C d f+24 a^2 b^2 (C d x (3 e+2 f x)+B d (7 e+3 f x)+c (7 C e+7 B f+3 C f x))-16 a^3 b (9 B d f+C (9 d e+9 c f+4 d f x))+21 A b^2 \left (8 a^2 d f-2 a b (5 d e+5 c f+2 d f x)+b^2 (5 c (3 e+f x)+d x (5 e+3 f x))\right )+b^4 x (3 B (7 c (5 e+3 f x)+3 d x (7 e+5 f x))+C x (9 c (7 e+5 f x)+5 d x (9 e+7 f x)))-2 a b^3 (3 B (7 c (5 e+2 f x)+d x (14 e+9 f x))+C x (3 c (14 e+9 f x)+d x (27 e+20 f x)))\right )}{315 b^5} \] Input:
Integrate[((c + d*x)*(e + f*x)*(A + B*x + C*x^2))/Sqrt[a + b*x],x]
Output:
(2*Sqrt[a + b*x]*(128*a^4*C*d*f + 24*a^2*b^2*(C*d*x*(3*e + 2*f*x) + B*d*(7 *e + 3*f*x) + c*(7*C*e + 7*B*f + 3*C*f*x)) - 16*a^3*b*(9*B*d*f + C*(9*d*e + 9*c*f + 4*d*f*x)) + 21*A*b^2*(8*a^2*d*f - 2*a*b*(5*d*e + 5*c*f + 2*d*f*x ) + b^2*(5*c*(3*e + f*x) + d*x*(5*e + 3*f*x))) + b^4*x*(3*B*(7*c*(5*e + 3* f*x) + 3*d*x*(7*e + 5*f*x)) + C*x*(9*c*(7*e + 5*f*x) + 5*d*x*(9*e + 7*f*x) )) - 2*a*b^3*(3*B*(7*c*(5*e + 2*f*x) + d*x*(14*e + 9*f*x)) + C*x*(3*c*(14* e + 9*f*x) + d*x*(27*e + 20*f*x)))))/(315*b^5)
Time = 0.57 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2115, 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.
| \(\displaystyle \int \frac {(c+d x) (e+f x) \left (A+B x+C x^2\right )}{\sqrt {a+b x}} \, dx\) |
| \(\Big \downarrow \) 2115 |
| \(\displaystyle \int \left (\frac {(a+b x)^{3/2} \left (6 a^2 C d f-3 a b (B d f+c C f+C d e)+b^2 (A d f+B c f+B d e+c C e)\right )}{b^4}+\frac {\sqrt {a+b x} \left (-4 a^3 C d f+3 a^2 b (B d f+c C f+C d e)-2 a b^2 (A d f+B c f+B d e+c C e)+b^3 (A c f+A d e+B c e)\right )}{b^4}+\frac {(b c-a d) (b e-a f) \left (A b^2-a (b B-a C)\right )}{b^4 \sqrt {a+b x}}+\frac {(a+b x)^{5/2} (b (B d f+c C f+C d e)-4 a C d f)}{b^4}+\frac {C d f (a+b x)^{7/2}}{b^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 (a+b x)^{5/2} \left (6 a^2 C d f-3 a b (B d f+c C f+C d e)+b^2 (A d f+B c f+B d e+c C e)\right )}{5 b^5}-\frac {2 (a+b x)^{3/2} \left (4 a^3 C d f-3 a^2 b (B d f+c C f+C d e)+2 a b^2 (A d f+B c f+B d e+c C e)-b^3 (A c f+A d e+B c e)\right )}{3 b^5}+\frac {2 \sqrt {a+b x} (b c-a d) (b e-a f) \left (A b^2-a (b B-a C)\right )}{b^5}-\frac {2 (a+b x)^{7/2} (4 a C d f-b (B d f+c C f+C d e))}{7 b^5}+\frac {2 C d f (a+b x)^{9/2}}{9 b^5}\) |
Input:
Int[((c + d*x)*(e + f*x)*(A + B*x + C*x^2))/Sqrt[a + b*x],x]
Output:
(2*(A*b^2 - a*(b*B - a*C))*(b*c - a*d)*(b*e - a*f)*Sqrt[a + b*x])/b^5 - (2 *(4*a^3*C*d*f - b^3*(B*c*e + A*d*e + A*c*f) + 2*a*b^2*(c*C*e + B*d*e + B*c *f + A*d*f) - 3*a^2*b*(C*d*e + c*C*f + B*d*f))*(a + b*x)^(3/2))/(3*b^5) + (2*(6*a^2*C*d*f + b^2*(c*C*e + B*d*e + B*c*f + A*d*f) - 3*a*b*(C*d*e + c*C *f + B*d*f))*(a + b*x)^(5/2))/(5*b^5) - (2*(4*a*C*d*f - b*(C*d*e + c*C*f + B*d*f))*(a + b*x)^(7/2))/(7*b^5) + (2*C*d*f*(a + b*x)^(9/2))/(9*b^5)
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f _.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^ n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && IntegersQ[m, n]
Time = 1.54 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.00
| method | result | size |
| derivativedivides | \(\frac {\frac {2 C f d \left (b x +a \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (\left (-a d +b c \right ) f +\left (-a f +b e \right ) d \right ) C +d f \left (B b -2 C a \right )\right ) \left (b x +a \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (-a d +b c \right ) \left (-a f +b e \right ) C +\left (\left (-a d +b c \right ) f +\left (-a f +b e \right ) d \right ) \left (B b -2 C a \right )+d f \left (b^{2} A -a b B +a^{2} C \right )\right ) \left (b x +a \right )^{\frac {5}{2}}}{5}+\frac {2 \left (\left (-a d +b c \right ) \left (-a f +b e \right ) \left (B b -2 C a \right )+\left (\left (-a d +b c \right ) f +\left (-a f +b e \right ) d \right ) \left (b^{2} A -a b B +a^{2} C \right )\right ) \left (b x +a \right )^{\frac {3}{2}}}{3}+2 \left (-a d +b c \right ) \left (-a f +b e \right ) \left (b^{2} A -a b B +a^{2} C \right ) \sqrt {b x +a}}{b^{5}}\) | \(253\) |
| default | \(\frac {\frac {2 C f d \left (b x +a \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (-\left (a d -b c \right ) f -d \left (a f -b e \right )\right ) C +d f \left (B b -2 C a \right )\right ) \left (b x +a \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (a d -b c \right ) \left (a f -b e \right ) C +\left (-\left (a d -b c \right ) f -d \left (a f -b e \right )\right ) \left (B b -2 C a \right )+d f \left (b^{2} A -a b B +a^{2} C \right )\right ) \left (b x +a \right )^{\frac {5}{2}}}{5}+\frac {2 \left (\left (a d -b c \right ) \left (a f -b e \right ) \left (B b -2 C a \right )+\left (-\left (a d -b c \right ) f -d \left (a f -b e \right )\right ) \left (b^{2} A -a b B +a^{2} C \right )\right ) \left (b x +a \right )^{\frac {3}{2}}}{3}+2 \left (a d -b c \right ) \left (a f -b e \right ) \left (b^{2} A -a b B +a^{2} C \right ) \sqrt {b x +a}}{b^{5}}\) | \(259\) |
| pseudoelliptic | \(\frac {2 \left (\left (\left (\frac {\left (\frac {128}{21} a^{4}-\frac {40}{21} a \,x^{3} b^{3}+\frac {16}{7} a^{2} b^{2} x^{2}-\frac {64}{21} a^{3} b x +\frac {5}{3} b^{4} x^{4}\right ) d}{3}+\frac {5 \left (b^{3} x^{3}-\frac {6}{5} a \,b^{2} x^{2}+\frac {8}{5} a^{2} b x -\frac {16}{5} a^{3}\right ) c b}{7}\right ) C +b \left (\left (\frac {\left (5 b^{3} x^{3}-6 a \,b^{2} x^{2}+8 a^{2} b x -16 a^{3}\right ) B}{7}+A b \left (b^{2} x^{2}+\frac {8}{3} a^{2}-\frac {4}{3} a b x \right )\right ) d +\frac {5 \left (\frac {\left (3 b^{2} x^{2}-4 a b x +8 a^{2}\right ) B}{5}+A b \left (b x -2 a \right )\right ) c b}{3}\right )\right ) f +\frac {5 \left (3 \left (\frac {\left (b^{3} x^{3}-\frac {6}{5} a \,b^{2} x^{2}+\frac {8}{5} a^{2} b x -\frac {16}{5} a^{3}\right ) d}{7}+\frac {c b \left (b^{2} x^{2}+\frac {8}{3} a^{2}-\frac {4}{3} a b x \right )}{5}\right ) C +\left (\left (\frac {\left (3 b^{2} x^{2}-4 a b x +8 a^{2}\right ) B}{5}+A b \left (b x -2 a \right )\right ) d +3 \left (\frac {\left (b x -2 a \right ) B}{3}+A b \right ) c b \right ) b \right ) e b}{3}\right ) \sqrt {b x +a}}{5 b^{5}}\) | \(313\) |
| gosper | \(\frac {2 \sqrt {b x +a}\, \left (35 C d f \,x^{4} b^{4}+45 B \,b^{4} d f \,x^{3}-40 C a \,b^{3} d f \,x^{3}+45 C \,b^{4} c f \,x^{3}+45 C \,b^{4} d e \,x^{3}+63 A \,b^{4} d f \,x^{2}-54 B a \,b^{3} d f \,x^{2}+63 B \,b^{4} c f \,x^{2}+63 B \,b^{4} d e \,x^{2}+48 C \,a^{2} b^{2} d f \,x^{2}-54 C a \,b^{3} c f \,x^{2}-54 C a \,b^{3} d e \,x^{2}+63 C \,b^{4} c e \,x^{2}-84 A a \,b^{3} d f x +105 A \,b^{4} c f x +105 A \,b^{4} d e x +72 B \,a^{2} b^{2} d f x -84 B a \,b^{3} c f x -84 B a \,b^{3} d e x +105 B \,b^{4} c e x -64 C \,a^{3} b d f x +72 C \,a^{2} b^{2} c f x +72 C \,a^{2} b^{2} d e x -84 C a \,b^{3} c e x +168 A \,a^{2} b^{2} d f -210 A a \,b^{3} c f -210 A a \,b^{3} d e +315 A c e \,b^{4}-144 B \,a^{3} b d f +168 B \,a^{2} b^{2} c f +168 B \,a^{2} b^{2} d e -210 B a \,b^{3} c e +128 C \,a^{4} d f -144 C \,a^{3} b c f -144 C \,a^{3} b d e +168 C \,a^{2} b^{2} c e \right )}{315 b^{5}}\) | \(391\) |
| trager | \(\frac {2 \sqrt {b x +a}\, \left (35 C d f \,x^{4} b^{4}+45 B \,b^{4} d f \,x^{3}-40 C a \,b^{3} d f \,x^{3}+45 C \,b^{4} c f \,x^{3}+45 C \,b^{4} d e \,x^{3}+63 A \,b^{4} d f \,x^{2}-54 B a \,b^{3} d f \,x^{2}+63 B \,b^{4} c f \,x^{2}+63 B \,b^{4} d e \,x^{2}+48 C \,a^{2} b^{2} d f \,x^{2}-54 C a \,b^{3} c f \,x^{2}-54 C a \,b^{3} d e \,x^{2}+63 C \,b^{4} c e \,x^{2}-84 A a \,b^{3} d f x +105 A \,b^{4} c f x +105 A \,b^{4} d e x +72 B \,a^{2} b^{2} d f x -84 B a \,b^{3} c f x -84 B a \,b^{3} d e x +105 B \,b^{4} c e x -64 C \,a^{3} b d f x +72 C \,a^{2} b^{2} c f x +72 C \,a^{2} b^{2} d e x -84 C a \,b^{3} c e x +168 A \,a^{2} b^{2} d f -210 A a \,b^{3} c f -210 A a \,b^{3} d e +315 A c e \,b^{4}-144 B \,a^{3} b d f +168 B \,a^{2} b^{2} c f +168 B \,a^{2} b^{2} d e -210 B a \,b^{3} c e +128 C \,a^{4} d f -144 C \,a^{3} b c f -144 C \,a^{3} b d e +168 C \,a^{2} b^{2} c e \right )}{315 b^{5}}\) | \(391\) |
| risch | \(\frac {2 \sqrt {b x +a}\, \left (35 C d f \,x^{4} b^{4}+45 B \,b^{4} d f \,x^{3}-40 C a \,b^{3} d f \,x^{3}+45 C \,b^{4} c f \,x^{3}+45 C \,b^{4} d e \,x^{3}+63 A \,b^{4} d f \,x^{2}-54 B a \,b^{3} d f \,x^{2}+63 B \,b^{4} c f \,x^{2}+63 B \,b^{4} d e \,x^{2}+48 C \,a^{2} b^{2} d f \,x^{2}-54 C a \,b^{3} c f \,x^{2}-54 C a \,b^{3} d e \,x^{2}+63 C \,b^{4} c e \,x^{2}-84 A a \,b^{3} d f x +105 A \,b^{4} c f x +105 A \,b^{4} d e x +72 B \,a^{2} b^{2} d f x -84 B a \,b^{3} c f x -84 B a \,b^{3} d e x +105 B \,b^{4} c e x -64 C \,a^{3} b d f x +72 C \,a^{2} b^{2} c f x +72 C \,a^{2} b^{2} d e x -84 C a \,b^{3} c e x +168 A \,a^{2} b^{2} d f -210 A a \,b^{3} c f -210 A a \,b^{3} d e +315 A c e \,b^{4}-144 B \,a^{3} b d f +168 B \,a^{2} b^{2} c f +168 B \,a^{2} b^{2} d e -210 B a \,b^{3} c e +128 C \,a^{4} d f -144 C \,a^{3} b c f -144 C \,a^{3} b d e +168 C \,a^{2} b^{2} c e \right )}{315 b^{5}}\) | \(391\) |
| orering | \(\frac {2 \sqrt {b x +a}\, \left (35 C d f \,x^{4} b^{4}+45 B \,b^{4} d f \,x^{3}-40 C a \,b^{3} d f \,x^{3}+45 C \,b^{4} c f \,x^{3}+45 C \,b^{4} d e \,x^{3}+63 A \,b^{4} d f \,x^{2}-54 B a \,b^{3} d f \,x^{2}+63 B \,b^{4} c f \,x^{2}+63 B \,b^{4} d e \,x^{2}+48 C \,a^{2} b^{2} d f \,x^{2}-54 C a \,b^{3} c f \,x^{2}-54 C a \,b^{3} d e \,x^{2}+63 C \,b^{4} c e \,x^{2}-84 A a \,b^{3} d f x +105 A \,b^{4} c f x +105 A \,b^{4} d e x +72 B \,a^{2} b^{2} d f x -84 B a \,b^{3} c f x -84 B a \,b^{3} d e x +105 B \,b^{4} c e x -64 C \,a^{3} b d f x +72 C \,a^{2} b^{2} c f x +72 C \,a^{2} b^{2} d e x -84 C a \,b^{3} c e x +168 A \,a^{2} b^{2} d f -210 A a \,b^{3} c f -210 A a \,b^{3} d e +315 A c e \,b^{4}-144 B \,a^{3} b d f +168 B \,a^{2} b^{2} c f +168 B \,a^{2} b^{2} d e -210 B a \,b^{3} c e +128 C \,a^{4} d f -144 C \,a^{3} b c f -144 C \,a^{3} b d e +168 C \,a^{2} b^{2} c e \right )}{315 b^{5}}\) | \(391\) |
Input:
int((d*x+c)*(f*x+e)*(C*x^2+B*x+A)/(b*x+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
2/b^5*(1/9*C*f*d*(b*x+a)^(9/2)+1/7*(((-a*d+b*c)*f+(-a*f+b*e)*d)*C+d*f*(B*b -2*C*a))*(b*x+a)^(7/2)+1/5*((-a*d+b*c)*(-a*f+b*e)*C+((-a*d+b*c)*f+(-a*f+b* e)*d)*(B*b-2*C*a)+d*f*(A*b^2-B*a*b+C*a^2))*(b*x+a)^(5/2)+1/3*((-a*d+b*c)*( -a*f+b*e)*(B*b-2*C*a)+((-a*d+b*c)*f+(-a*f+b*e)*d)*(A*b^2-B*a*b+C*a^2))*(b* x+a)^(3/2)+(-a*d+b*c)*(-a*f+b*e)*(A*b^2-B*a*b+C*a^2)*(b*x+a)^(1/2))
Time = 0.08 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.45 \[ \int \frac {(c+d x) (e+f x) \left (A+B x+C x^2\right )}{\sqrt {a+b x}} \, dx=\frac {2 \, {\left (35 \, C b^{4} d f x^{4} + 5 \, {\left (9 \, C b^{4} d e + {\left (9 \, C b^{4} c - {\left (8 \, C a b^{3} - 9 \, B b^{4}\right )} d\right )} f\right )} x^{3} + 3 \, {\left (3 \, {\left (7 \, C b^{4} c - {\left (6 \, C a b^{3} - 7 \, B b^{4}\right )} d\right )} e - {\left (3 \, {\left (6 \, C a b^{3} - 7 \, B b^{4}\right )} c - {\left (16 \, C a^{2} b^{2} - 18 \, B a b^{3} + 21 \, A b^{4}\right )} d\right )} f\right )} x^{2} + 3 \, {\left (7 \, {\left (8 \, C a^{2} b^{2} - 10 \, B a b^{3} + 15 \, A b^{4}\right )} c - 2 \, {\left (24 \, C a^{3} b - 28 \, B a^{2} b^{2} + 35 \, A a b^{3}\right )} d\right )} e - 2 \, {\left (3 \, {\left (24 \, C a^{3} b - 28 \, B a^{2} b^{2} + 35 \, A a b^{3}\right )} c - 4 \, {\left (16 \, C a^{4} - 18 \, B a^{3} b + 21 \, A a^{2} b^{2}\right )} d\right )} f - {\left (3 \, {\left (7 \, {\left (4 \, C a b^{3} - 5 \, B b^{4}\right )} c - {\left (24 \, C a^{2} b^{2} - 28 \, B a b^{3} + 35 \, A b^{4}\right )} d\right )} e - {\left (3 \, {\left (24 \, C a^{2} b^{2} - 28 \, B a b^{3} + 35 \, A b^{4}\right )} c - 4 \, {\left (16 \, C a^{3} b - 18 \, B a^{2} b^{2} + 21 \, A a b^{3}\right )} d\right )} f\right )} x\right )} \sqrt {b x + a}}{315 \, b^{5}} \] Input:
integrate((d*x+c)*(f*x+e)*(C*x^2+B*x+A)/(b*x+a)^(1/2),x, algorithm="fricas ")
Output:
2/315*(35*C*b^4*d*f*x^4 + 5*(9*C*b^4*d*e + (9*C*b^4*c - (8*C*a*b^3 - 9*B*b ^4)*d)*f)*x^3 + 3*(3*(7*C*b^4*c - (6*C*a*b^3 - 7*B*b^4)*d)*e - (3*(6*C*a*b ^3 - 7*B*b^4)*c - (16*C*a^2*b^2 - 18*B*a*b^3 + 21*A*b^4)*d)*f)*x^2 + 3*(7* (8*C*a^2*b^2 - 10*B*a*b^3 + 15*A*b^4)*c - 2*(24*C*a^3*b - 28*B*a^2*b^2 + 3 5*A*a*b^3)*d)*e - 2*(3*(24*C*a^3*b - 28*B*a^2*b^2 + 35*A*a*b^3)*c - 4*(16* C*a^4 - 18*B*a^3*b + 21*A*a^2*b^2)*d)*f - (3*(7*(4*C*a*b^3 - 5*B*b^4)*c - (24*C*a^2*b^2 - 28*B*a*b^3 + 35*A*b^4)*d)*e - (3*(24*C*a^2*b^2 - 28*B*a*b^ 3 + 35*A*b^4)*c - 4*(16*C*a^3*b - 18*B*a^2*b^2 + 21*A*a*b^3)*d)*f)*x)*sqrt (b*x + a)/b^5
Time = 1.47 (sec) , antiderivative size = 517, normalized size of antiderivative = 2.04 \[ \int \frac {(c+d x) (e+f x) \left (A+B x+C x^2\right )}{\sqrt {a+b x}} \, dx=\begin {cases} \frac {2 \left (\frac {C d f \left (a + b x\right )^{\frac {9}{2}}}{9 b^{4}} + \frac {\left (a + b x\right )^{\frac {7}{2}} \left (B b d f - 4 C a d f + C b c f + C b d e\right )}{7 b^{4}} + \frac {\left (a + b x\right )^{\frac {5}{2}} \left (A b^{2} d f - 3 B a b d f + B b^{2} c f + B b^{2} d e + 6 C a^{2} d f - 3 C a b c f - 3 C a b d e + C b^{2} c e\right )}{5 b^{4}} + \frac {\left (a + b x\right )^{\frac {3}{2}} \left (- 2 A a b^{2} d f + A b^{3} c f + A b^{3} d e + 3 B a^{2} b d f - 2 B a b^{2} c f - 2 B a b^{2} d e + B b^{3} c e - 4 C a^{3} d f + 3 C a^{2} b c f + 3 C a^{2} b d e - 2 C a b^{2} c e\right )}{3 b^{4}} + \frac {\sqrt {a + b x} \left (A a^{2} b^{2} d f - A a b^{3} c f - A a b^{3} d e + A b^{4} c e - B a^{3} b d f + B a^{2} b^{2} c f + B a^{2} b^{2} d e - B a b^{3} c e + C a^{4} d f - C a^{3} b c f - C a^{3} b d e + C a^{2} b^{2} c e\right )}{b^{4}}\right )}{b} & \text {for}\: b \neq 0 \\\frac {A c e x + \frac {C d f x^{5}}{5} + \frac {x^{4} \left (B d f + C c f + C d e\right )}{4} + \frac {x^{3} \left (A d f + B c f + B d e + C c e\right )}{3} + \frac {x^{2} \left (A c f + A d e + B c e\right )}{2}}{\sqrt {a}} & \text {otherwise} \end {cases} \] Input:
integrate((d*x+c)*(f*x+e)*(C*x**2+B*x+A)/(b*x+a)**(1/2),x)
Output:
Piecewise((2*(C*d*f*(a + b*x)**(9/2)/(9*b**4) + (a + b*x)**(7/2)*(B*b*d*f - 4*C*a*d*f + C*b*c*f + C*b*d*e)/(7*b**4) + (a + b*x)**(5/2)*(A*b**2*d*f - 3*B*a*b*d*f + B*b**2*c*f + B*b**2*d*e + 6*C*a**2*d*f - 3*C*a*b*c*f - 3*C* a*b*d*e + C*b**2*c*e)/(5*b**4) + (a + b*x)**(3/2)*(-2*A*a*b**2*d*f + A*b** 3*c*f + A*b**3*d*e + 3*B*a**2*b*d*f - 2*B*a*b**2*c*f - 2*B*a*b**2*d*e + B* b**3*c*e - 4*C*a**3*d*f + 3*C*a**2*b*c*f + 3*C*a**2*b*d*e - 2*C*a*b**2*c*e )/(3*b**4) + sqrt(a + b*x)*(A*a**2*b**2*d*f - A*a*b**3*c*f - A*a*b**3*d*e + A*b**4*c*e - B*a**3*b*d*f + B*a**2*b**2*c*f + B*a**2*b**2*d*e - B*a*b**3 *c*e + C*a**4*d*f - C*a**3*b*c*f - C*a**3*b*d*e + C*a**2*b**2*c*e)/b**4)/b , Ne(b, 0)), ((A*c*e*x + C*d*f*x**5/5 + x**4*(B*d*f + C*c*f + C*d*e)/4 + x **3*(A*d*f + B*c*f + B*d*e + C*c*e)/3 + x**2*(A*c*f + A*d*e + B*c*e)/2)/sq rt(a), True))
Time = 0.04 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.33 \[ \int \frac {(c+d x) (e+f x) \left (A+B x+C x^2\right )}{\sqrt {a+b x}} \, dx=\frac {2 \, {\left (35 \, {\left (b x + a\right )}^{\frac {9}{2}} C d f + 45 \, {\left (C b d e + {\left (C b c - {\left (4 \, C a - B b\right )} d\right )} f\right )} {\left (b x + a\right )}^{\frac {7}{2}} + 63 \, {\left ({\left (C b^{2} c - {\left (3 \, C a b - B b^{2}\right )} d\right )} e - {\left ({\left (3 \, C a b - B b^{2}\right )} c - {\left (6 \, C a^{2} - 3 \, B a b + A b^{2}\right )} d\right )} f\right )} {\left (b x + a\right )}^{\frac {5}{2}} - 105 \, {\left ({\left ({\left (2 \, C a b^{2} - B b^{3}\right )} c - {\left (3 \, C a^{2} b - 2 \, B a b^{2} + A b^{3}\right )} d\right )} e - {\left ({\left (3 \, C a^{2} b - 2 \, B a b^{2} + A b^{3}\right )} c - {\left (4 \, C a^{3} - 3 \, B a^{2} b + 2 \, A a b^{2}\right )} d\right )} f\right )} {\left (b x + a\right )}^{\frac {3}{2}} + 315 \, {\left ({\left ({\left (C a^{2} b^{2} - B a b^{3} + A b^{4}\right )} c - {\left (C a^{3} b - B a^{2} b^{2} + A a b^{3}\right )} d\right )} e - {\left ({\left (C a^{3} b - B a^{2} b^{2} + A a b^{3}\right )} c - {\left (C a^{4} - B a^{3} b + A a^{2} b^{2}\right )} d\right )} f\right )} \sqrt {b x + a}\right )}}{315 \, b^{5}} \] Input:
integrate((d*x+c)*(f*x+e)*(C*x^2+B*x+A)/(b*x+a)^(1/2),x, algorithm="maxima ")
Output:
2/315*(35*(b*x + a)^(9/2)*C*d*f + 45*(C*b*d*e + (C*b*c - (4*C*a - B*b)*d)* f)*(b*x + a)^(7/2) + 63*((C*b^2*c - (3*C*a*b - B*b^2)*d)*e - ((3*C*a*b - B *b^2)*c - (6*C*a^2 - 3*B*a*b + A*b^2)*d)*f)*(b*x + a)^(5/2) - 105*(((2*C*a *b^2 - B*b^3)*c - (3*C*a^2*b - 2*B*a*b^2 + A*b^3)*d)*e - ((3*C*a^2*b - 2*B *a*b^2 + A*b^3)*c - (4*C*a^3 - 3*B*a^2*b + 2*A*a*b^2)*d)*f)*(b*x + a)^(3/2 ) + 315*(((C*a^2*b^2 - B*a*b^3 + A*b^4)*c - (C*a^3*b - B*a^2*b^2 + A*a*b^3 )*d)*e - ((C*a^3*b - B*a^2*b^2 + A*a*b^3)*c - (C*a^4 - B*a^3*b + A*a^2*b^2 )*d)*f)*sqrt(b*x + a))/b^5
Leaf count of result is larger than twice the leaf count of optimal. 475 vs. \(2 (235) = 470\).
Time = 0.13 (sec) , antiderivative size = 475, normalized size of antiderivative = 1.87 \[ \int \frac {(c+d x) (e+f x) \left (A+B x+C x^2\right )}{\sqrt {a+b x}} \, dx=\frac {2 \, {\left (315 \, \sqrt {b x + a} A c e + \frac {105 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} B c e}{b} + \frac {105 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} A d e}{b} + \frac {105 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} A c f}{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 )} C c e}{b^{2}} + \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 )} B d e}{b^{2}} + \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 )} B c f}{b^{2}} + \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 )} A d f}{b^{2}} + \frac {9 \, {\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 )} C d e}{b^{3}} + \frac {9 \, {\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 )} C c f}{b^{3}} + \frac {9 \, {\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 )} B d f}{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 )} C d f}{b^{4}}\right )}}{315 \, b} \] Input:
integrate((d*x+c)*(f*x+e)*(C*x^2+B*x+A)/(b*x+a)^(1/2),x, algorithm="giac")
Output:
2/315*(315*sqrt(b*x + a)*A*c*e + 105*((b*x + a)^(3/2) - 3*sqrt(b*x + a)*a) *B*c*e/b + 105*((b*x + a)^(3/2) - 3*sqrt(b*x + a)*a)*A*d*e/b + 105*((b*x + a)^(3/2) - 3*sqrt(b*x + a)*a)*A*c*f/b + 21*(3*(b*x + a)^(5/2) - 10*(b*x + a)^(3/2)*a + 15*sqrt(b*x + a)*a^2)*C*c*e/b^2 + 21*(3*(b*x + a)^(5/2) - 10 *(b*x + a)^(3/2)*a + 15*sqrt(b*x + a)*a^2)*B*d*e/b^2 + 21*(3*(b*x + a)^(5/ 2) - 10*(b*x + a)^(3/2)*a + 15*sqrt(b*x + a)*a^2)*B*c*f/b^2 + 21*(3*(b*x + a)^(5/2) - 10*(b*x + a)^(3/2)*a + 15*sqrt(b*x + a)*a^2)*A*d*f/b^2 + 9*(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)*C*d*e/b^3 + 9*(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)*C*c*f/b^3 + 9*(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 )*B*d*f/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)*C*d*f/b^4)/b
Time = 0.08 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.10 \[ \int \frac {(c+d x) (e+f x) \left (A+B x+C x^2\right )}{\sqrt {a+b x}} \, dx=\frac {{\left (a+b\,x\right )}^{7/2}\,\left (2\,B\,b\,d\,f-8\,C\,a\,d\,f+2\,C\,b\,c\,f+2\,C\,b\,d\,e\right )}{7\,b^5}+\frac {{\left (a+b\,x\right )}^{5/2}\,\left (2\,A\,b^2\,d\,f+2\,B\,b^2\,c\,f+2\,B\,b^2\,d\,e+2\,C\,b^2\,c\,e+12\,C\,a^2\,d\,f-6\,B\,a\,b\,d\,f-6\,C\,a\,b\,c\,f-6\,C\,a\,b\,d\,e\right )}{5\,b^5}+\frac {{\left (a+b\,x\right )}^{3/2}\,\left (2\,A\,b^3\,c\,f+2\,A\,b^3\,d\,e+2\,B\,b^3\,c\,e-8\,C\,a^3\,d\,f-4\,A\,a\,b^2\,d\,f-4\,B\,a\,b^2\,c\,f-4\,B\,a\,b^2\,d\,e-4\,C\,a\,b^2\,c\,e+6\,B\,a^2\,b\,d\,f+6\,C\,a^2\,b\,c\,f+6\,C\,a^2\,b\,d\,e\right )}{3\,b^5}+\frac {2\,\left (a\,d-b\,c\right )\,\left (a\,f-b\,e\right )\,\sqrt {a+b\,x}\,\left (C\,a^2-B\,a\,b+A\,b^2\right )}{b^5}+\frac {2\,C\,d\,f\,{\left (a+b\,x\right )}^{9/2}}{9\,b^5} \] Input:
int(((e + f*x)*(c + d*x)*(A + B*x + C*x^2))/(a + b*x)^(1/2),x)
Output:
((a + b*x)^(7/2)*(2*B*b*d*f - 8*C*a*d*f + 2*C*b*c*f + 2*C*b*d*e))/(7*b^5) + ((a + b*x)^(5/2)*(2*A*b^2*d*f + 2*B*b^2*c*f + 2*B*b^2*d*e + 2*C*b^2*c*e + 12*C*a^2*d*f - 6*B*a*b*d*f - 6*C*a*b*c*f - 6*C*a*b*d*e))/(5*b^5) + ((a + b*x)^(3/2)*(2*A*b^3*c*f + 2*A*b^3*d*e + 2*B*b^3*c*e - 8*C*a^3*d*f - 4*A*a *b^2*d*f - 4*B*a*b^2*c*f - 4*B*a*b^2*d*e - 4*C*a*b^2*c*e + 6*B*a^2*b*d*f + 6*C*a^2*b*c*f + 6*C*a^2*b*d*e))/(3*b^5) + (2*(a*d - b*c)*(a*f - b*e)*(a + b*x)^(1/2)*(A*b^2 + C*a^2 - B*a*b))/b^5 + (2*C*d*f*(a + b*x)^(9/2))/(9*b^ 5)
Time = 0.15 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.22 \[ \int \frac {(c+d x) (e+f x) \left (A+B x+C x^2\right )}{\sqrt {a+b x}} \, dx=\frac {2 \sqrt {b x +a}\, \left (35 b^{4} c d f \,x^{4}-40 a \,b^{3} c d f \,x^{3}+45 b^{5} d f \,x^{3}+45 b^{4} c^{2} f \,x^{3}+45 b^{4} c d e \,x^{3}+48 a^{2} b^{2} c d f \,x^{2}+9 a \,b^{4} d f \,x^{2}-54 a \,b^{3} c^{2} f \,x^{2}-54 a \,b^{3} c d e \,x^{2}+63 b^{5} c f \,x^{2}+63 b^{5} d e \,x^{2}+63 b^{4} c^{2} e \,x^{2}-64 a^{3} b c d f x -12 a^{2} b^{3} d f x +72 a^{2} b^{2} c^{2} f x +72 a^{2} b^{2} c d e x +21 a \,b^{4} c f x +21 a \,b^{4} d e x -84 a \,b^{3} c^{2} e x +105 b^{5} c e x +128 a^{4} c d f +24 a^{3} b^{2} d f -144 a^{3} b \,c^{2} f -144 a^{3} b c d e -42 a^{2} b^{3} c f -42 a^{2} b^{3} d e +168 a^{2} b^{2} c^{2} e +105 a \,b^{4} c e \right )}{315 b^{5}} \] Input:
int((d*x+c)*(f*x+e)*(C*x^2+B*x+A)/(b*x+a)^(1/2),x)
Output:
(2*sqrt(a + b*x)*(128*a**4*c*d*f + 24*a**3*b**2*d*f - 144*a**3*b*c**2*f - 144*a**3*b*c*d*e - 64*a**3*b*c*d*f*x - 42*a**2*b**3*c*f - 42*a**2*b**3*d*e - 12*a**2*b**3*d*f*x + 168*a**2*b**2*c**2*e + 72*a**2*b**2*c**2*f*x + 72* a**2*b**2*c*d*e*x + 48*a**2*b**2*c*d*f*x**2 + 105*a*b**4*c*e + 21*a*b**4*c *f*x + 21*a*b**4*d*e*x + 9*a*b**4*d*f*x**2 - 84*a*b**3*c**2*e*x - 54*a*b** 3*c**2*f*x**2 - 54*a*b**3*c*d*e*x**2 - 40*a*b**3*c*d*f*x**3 + 105*b**5*c*e *x + 63*b**5*c*f*x**2 + 63*b**5*d*e*x**2 + 45*b**5*d*f*x**3 + 63*b**4*c**2 *e*x**2 + 45*b**4*c**2*f*x**3 + 45*b**4*c*d*e*x**3 + 35*b**4*c*d*f*x**4))/ (315*b**5)