Integrand size = 25, antiderivative size = 157 \[ \int \frac {(a+b x) (c+d x) (g+h x)}{\sqrt {e+f x}} \, dx=\frac {2 (b e-a f) (d e-c f) (f g-e h) \sqrt {e+f x}}{f^4}-\frac {2 (b d e (2 f g-3 e h)-b c f (f g-2 e h)-a f (d f g-2 d e h+c f h)) (e+f x)^{3/2}}{3 f^4}+\frac {2 (a d f h+b (d f g-3 d e h+c f h)) (e+f x)^{5/2}}{5 f^4}+\frac {2 b d h (e+f x)^{7/2}}{7 f^4} \] Output:
2*(-a*f+b*e)*(-c*f+d*e)*(-e*h+f*g)*(f*x+e)^(1/2)/f^4-2/3*(b*d*e*(-3*e*h+2* f*g)-b*c*f*(-2*e*h+f*g)-a*f*(c*f*h-2*d*e*h+d*f*g))*(f*x+e)^(3/2)/f^4+2/5*( a*d*f*h+b*(c*f*h-3*d*e*h+d*f*g))*(f*x+e)^(5/2)/f^4+2/7*b*d*h*(f*x+e)^(7/2) /f^4
Time = 0.11 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.06 \[ \int \frac {(a+b x) (c+d x) (g+h x)}{\sqrt {e+f x}} \, dx=\frac {2 \sqrt {e+f x} \left (7 a f \left (5 c f (3 f g-2 e h+f h x)+d \left (8 e^2 h-2 e f (5 g+2 h x)+f^2 x (5 g+3 h x)\right )\right )+b \left (7 c f \left (8 e^2 h-2 e f (5 g+2 h x)+f^2 x (5 g+3 h x)\right )+d \left (-48 e^3 h+8 e^2 f (7 g+3 h x)+3 f^3 x^2 (7 g+5 h x)-2 e f^2 x (14 g+9 h x)\right )\right )\right )}{105 f^4} \] Input:
Integrate[((a + b*x)*(c + d*x)*(g + h*x))/Sqrt[e + f*x],x]
Output:
(2*Sqrt[e + f*x]*(7*a*f*(5*c*f*(3*f*g - 2*e*h + f*h*x) + d*(8*e^2*h - 2*e* f*(5*g + 2*h*x) + f^2*x*(5*g + 3*h*x))) + b*(7*c*f*(8*e^2*h - 2*e*f*(5*g + 2*h*x) + f^2*x*(5*g + 3*h*x)) + d*(-48*e^3*h + 8*e^2*f*(7*g + 3*h*x) + 3* f^3*x^2*(7*g + 5*h*x) - 2*e*f^2*x*(14*g + 9*h*x)))))/(105*f^4)
Time = 0.33 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {159, 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.
| \(\displaystyle \int \frac {(a+b x) (c+d x) (g+h x)}{\sqrt {e+f x}} \, dx\) |
| \(\Big \downarrow \) 159 |
| \(\displaystyle \int \left (\frac {(e+f x)^{3/2} (a d f h+b (c f h-3 d e h+d f g))}{f^3}+\frac {\sqrt {e+f x} (a f (c f h-2 d e h+d f g)+b c f (f g-2 e h)-b d e (2 f g-3 e h))}{f^3}+\frac {(a f-b e) (c f-d e) (f g-e h)}{f^3 \sqrt {e+f x}}+\frac {b d h (e+f x)^{5/2}}{f^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 (e+f x)^{5/2} (a d f h+b (c f h-3 d e h+d f g))}{5 f^4}-\frac {2 (e+f x)^{3/2} (-a f (c f h-2 d e h+d f g)-b c f (f g-2 e h)+b d e (2 f g-3 e h))}{3 f^4}+\frac {2 \sqrt {e+f x} (b e-a f) (d e-c f) (f g-e h)}{f^4}+\frac {2 b d h (e+f x)^{7/2}}{7 f^4}\) |
Input:
Int[((a + b*x)*(c + d*x)*(g + h*x))/Sqrt[e + f*x],x]
Output:
(2*(b*e - a*f)*(d*e - c*f)*(f*g - e*h)*Sqrt[e + f*x])/f^4 - (2*(b*d*e*(2*f *g - 3*e*h) - b*c*f*(f*g - 2*e*h) - a*f*(d*f*g - 2*d*e*h + c*f*h))*(e + f* x)^(3/2))/(3*f^4) + (2*(a*d*f*h + b*(d*f*g - 3*d*e*h + c*f*h))*(e + f*x)^( 5/2))/(5*f^4) + (2*b*d*h*(e + f*x)^(7/2))/(7*f^4)
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ ))*((g_.) + (h_.)*(x_)), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n *(e + f*x)*(g + h*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && (IGtQ [m, 0] || IntegersQ[m, n])
Time = 0.83 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.89
| method | result | size |
| pseudoelliptic | \(-\frac {4 \left (\frac {\left (-x \left (\frac {3 x \left (\frac {5 h x}{7}+g \right ) d}{5}+c \left (\frac {3 h x}{5}+g \right )\right ) b -3 a \left (\frac {x \left (\frac {3 h x}{5}+g \right ) d}{3}+c \left (\frac {h x}{3}+g \right )\right )\right ) f^{3}}{2}+\left (\left (\frac {2 x \left (\frac {9 h x}{14}+g \right ) d}{5}+c \left (\frac {2 h x}{5}+g \right )\right ) b +a \left (\left (\frac {2 h x}{5}+g \right ) d +c h \right )\right ) e \,f^{2}-\frac {4 \left (\left (\left (\frac {3 h x}{7}+g \right ) d +c h \right ) b +a d h \right ) e^{2} f}{5}+\frac {24 b d \,e^{3} h}{35}\right ) \sqrt {f x +e}}{3 f^{4}}\) | \(139\) |
| derivativedivides | \(\frac {\frac {2 d b h \left (f x +e \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (\left (a f -b e \right ) d +b \left (c f -d e \right )\right ) h +d b \left (-e h +f g \right )\right ) \left (f x +e \right )^{\frac {5}{2}}}{5}+\frac {2 \left (\left (a f -b e \right ) \left (c f -d e \right ) h +\left (\left (a f -b e \right ) d +b \left (c f -d e \right )\right ) \left (-e h +f g \right )\right ) \left (f x +e \right )^{\frac {3}{2}}}{3}+2 \left (a f -b e \right ) \left (c f -d e \right ) \left (-e h +f g \right ) \sqrt {f x +e}}{f^{4}}\) | \(153\) |
| default | \(\frac {\frac {2 d b h \left (f x +e \right )^{\frac {7}{2}}}{7}+\frac {2 \left (-\left (\left (-a f +b e \right ) d -b \left (c f -d e \right )\right ) h -d b \left (e h -f g \right )\right ) \left (f x +e \right )^{\frac {5}{2}}}{5}+\frac {2 \left (-\left (-a f +b e \right ) \left (c f -d e \right ) h +\left (\left (-a f +b e \right ) d -b \left (c f -d e \right )\right ) \left (e h -f g \right )\right ) \left (f x +e \right )^{\frac {3}{2}}}{3}+2 \left (-a f +b e \right ) \left (c f -d e \right ) \left (e h -f g \right ) \sqrt {f x +e}}{f^{4}}\) | \(158\) |
| gosper | \(-\frac {2 \sqrt {f x +e}\, \left (-15 b d h \,x^{3} f^{3}-21 a d \,f^{3} h \,x^{2}-21 b c \,f^{3} h \,x^{2}+18 b d e \,f^{2} h \,x^{2}-21 b d \,f^{3} g \,x^{2}-35 a c \,f^{3} h x +28 a d e \,f^{2} h x -35 a d \,f^{3} g x +28 b c e \,f^{2} h x -35 b c \,f^{3} g x -24 b d \,e^{2} f h x +28 b d e \,f^{2} g x +70 a c e \,f^{2} h -105 c a g \,f^{3}-56 a d \,e^{2} f h +70 a d e \,f^{2} g -56 b c \,e^{2} f h +70 b c e \,f^{2} g +48 b d \,e^{3} h -56 b d \,e^{2} f g \right )}{105 f^{4}}\) | \(207\) |
| trager | \(-\frac {2 \sqrt {f x +e}\, \left (-15 b d h \,x^{3} f^{3}-21 a d \,f^{3} h \,x^{2}-21 b c \,f^{3} h \,x^{2}+18 b d e \,f^{2} h \,x^{2}-21 b d \,f^{3} g \,x^{2}-35 a c \,f^{3} h x +28 a d e \,f^{2} h x -35 a d \,f^{3} g x +28 b c e \,f^{2} h x -35 b c \,f^{3} g x -24 b d \,e^{2} f h x +28 b d e \,f^{2} g x +70 a c e \,f^{2} h -105 c a g \,f^{3}-56 a d \,e^{2} f h +70 a d e \,f^{2} g -56 b c \,e^{2} f h +70 b c e \,f^{2} g +48 b d \,e^{3} h -56 b d \,e^{2} f g \right )}{105 f^{4}}\) | \(207\) |
| risch | \(-\frac {2 \sqrt {f x +e}\, \left (-15 b d h \,x^{3} f^{3}-21 a d \,f^{3} h \,x^{2}-21 b c \,f^{3} h \,x^{2}+18 b d e \,f^{2} h \,x^{2}-21 b d \,f^{3} g \,x^{2}-35 a c \,f^{3} h x +28 a d e \,f^{2} h x -35 a d \,f^{3} g x +28 b c e \,f^{2} h x -35 b c \,f^{3} g x -24 b d \,e^{2} f h x +28 b d e \,f^{2} g x +70 a c e \,f^{2} h -105 c a g \,f^{3}-56 a d \,e^{2} f h +70 a d e \,f^{2} g -56 b c \,e^{2} f h +70 b c e \,f^{2} g +48 b d \,e^{3} h -56 b d \,e^{2} f g \right )}{105 f^{4}}\) | \(207\) |
| orering | \(-\frac {2 \sqrt {f x +e}\, \left (-15 b d h \,x^{3} f^{3}-21 a d \,f^{3} h \,x^{2}-21 b c \,f^{3} h \,x^{2}+18 b d e \,f^{2} h \,x^{2}-21 b d \,f^{3} g \,x^{2}-35 a c \,f^{3} h x +28 a d e \,f^{2} h x -35 a d \,f^{3} g x +28 b c e \,f^{2} h x -35 b c \,f^{3} g x -24 b d \,e^{2} f h x +28 b d e \,f^{2} g x +70 a c e \,f^{2} h -105 c a g \,f^{3}-56 a d \,e^{2} f h +70 a d e \,f^{2} g -56 b c \,e^{2} f h +70 b c e \,f^{2} g +48 b d \,e^{3} h -56 b d \,e^{2} f g \right )}{105 f^{4}}\) | \(207\) |
Input:
int((b*x+a)*(d*x+c)*(h*x+g)/(f*x+e)^(1/2),x,method=_RETURNVERBOSE)
Output:
-4/3*(1/2*(-x*(3/5*x*(5/7*h*x+g)*d+c*(3/5*h*x+g))*b-3*a*(1/3*x*(3/5*h*x+g) *d+c*(1/3*h*x+g)))*f^3+((2/5*x*(9/14*h*x+g)*d+c*(2/5*h*x+g))*b+a*((2/5*h*x +g)*d+c*h))*e*f^2-4/5*(((3/7*h*x+g)*d+c*h)*b+a*d*h)*e^2*f+24/35*b*d*e^3*h) *(f*x+e)^(1/2)/f^4
Time = 0.08 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.18 \[ \int \frac {(a+b x) (c+d x) (g+h x)}{\sqrt {e+f x}} \, dx=\frac {2 \, {\left (15 \, b d f^{3} h x^{3} + 3 \, {\left (7 \, b d f^{3} g - {\left (6 \, b d e f^{2} - 7 \, {\left (b c + a d\right )} f^{3}\right )} h\right )} x^{2} + 7 \, {\left (8 \, b d e^{2} f + 15 \, a c f^{3} - 10 \, {\left (b c + a d\right )} e f^{2}\right )} g - 2 \, {\left (24 \, b d e^{3} + 35 \, a c e f^{2} - 28 \, {\left (b c + a d\right )} e^{2} f\right )} h - {\left (7 \, {\left (4 \, b d e f^{2} - 5 \, {\left (b c + a d\right )} f^{3}\right )} g - {\left (24 \, b d e^{2} f + 35 \, a c f^{3} - 28 \, {\left (b c + a d\right )} e f^{2}\right )} h\right )} x\right )} \sqrt {f x + e}}{105 \, f^{4}} \] Input:
integrate((b*x+a)*(d*x+c)*(h*x+g)/(f*x+e)^(1/2),x, algorithm="fricas")
Output:
2/105*(15*b*d*f^3*h*x^3 + 3*(7*b*d*f^3*g - (6*b*d*e*f^2 - 7*(b*c + a*d)*f^ 3)*h)*x^2 + 7*(8*b*d*e^2*f + 15*a*c*f^3 - 10*(b*c + a*d)*e*f^2)*g - 2*(24* b*d*e^3 + 35*a*c*e*f^2 - 28*(b*c + a*d)*e^2*f)*h - (7*(4*b*d*e*f^2 - 5*(b* c + a*d)*f^3)*g - (24*b*d*e^2*f + 35*a*c*f^3 - 28*(b*c + a*d)*e*f^2)*h)*x) *sqrt(f*x + e)/f^4
Time = 1.12 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.92 \[ \int \frac {(a+b x) (c+d x) (g+h x)}{\sqrt {e+f x}} \, dx=\begin {cases} \frac {2 \left (\frac {b d h \left (e + f x\right )^{\frac {7}{2}}}{7 f^{3}} + \frac {\left (e + f x\right )^{\frac {5}{2}} \left (a d f h + b c f h - 3 b d e h + b d f g\right )}{5 f^{3}} + \frac {\left (e + f x\right )^{\frac {3}{2}} \left (a c f^{2} h - 2 a d e f h + a d f^{2} g - 2 b c e f h + b c f^{2} g + 3 b d e^{2} h - 2 b d e f g\right )}{3 f^{3}} + \frac {\sqrt {e + f x} \left (- a c e f^{2} h + a c f^{3} g + a d e^{2} f h - a d e f^{2} g + b c e^{2} f h - b c e f^{2} g - b d e^{3} h + b d e^{2} f g\right )}{f^{3}}\right )}{f} & \text {for}\: f \neq 0 \\\frac {a c g x + \frac {b d h x^{4}}{4} + \frac {x^{3} \left (a d h + b c h + b d g\right )}{3} + \frac {x^{2} \left (a c h + a d g + b c g\right )}{2}}{\sqrt {e}} & \text {otherwise} \end {cases} \] Input:
integrate((b*x+a)*(d*x+c)*(h*x+g)/(f*x+e)**(1/2),x)
Output:
Piecewise((2*(b*d*h*(e + f*x)**(7/2)/(7*f**3) + (e + f*x)**(5/2)*(a*d*f*h + b*c*f*h - 3*b*d*e*h + b*d*f*g)/(5*f**3) + (e + f*x)**(3/2)*(a*c*f**2*h - 2*a*d*e*f*h + a*d*f**2*g - 2*b*c*e*f*h + b*c*f**2*g + 3*b*d*e**2*h - 2*b* d*e*f*g)/(3*f**3) + sqrt(e + f*x)*(-a*c*e*f**2*h + a*c*f**3*g + a*d*e**2*f *h - a*d*e*f**2*g + b*c*e**2*f*h - b*c*e*f**2*g - b*d*e**3*h + b*d*e**2*f* g)/f**3)/f, Ne(f, 0)), ((a*c*g*x + b*d*h*x**4/4 + x**3*(a*d*h + b*c*h + b* d*g)/3 + x**2*(a*c*h + a*d*g + b*c*g)/2)/sqrt(e), True))
Time = 0.04 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.15 \[ \int \frac {(a+b x) (c+d x) (g+h x)}{\sqrt {e+f x}} \, dx=\frac {2 \, {\left (15 \, {\left (f x + e\right )}^{\frac {7}{2}} b d h + 21 \, {\left (b d f g - {\left (3 \, b d e - {\left (b c + a d\right )} f\right )} h\right )} {\left (f x + e\right )}^{\frac {5}{2}} - 35 \, {\left ({\left (2 \, b d e f - {\left (b c + a d\right )} f^{2}\right )} g - {\left (3 \, b d e^{2} + a c f^{2} - 2 \, {\left (b c + a d\right )} e f\right )} h\right )} {\left (f x + e\right )}^{\frac {3}{2}} + 105 \, {\left ({\left (b d e^{2} f + a c f^{3} - {\left (b c + a d\right )} e f^{2}\right )} g - {\left (b d e^{3} + a c e f^{2} - {\left (b c + a d\right )} e^{2} f\right )} h\right )} \sqrt {f x + e}\right )}}{105 \, f^{4}} \] Input:
integrate((b*x+a)*(d*x+c)*(h*x+g)/(f*x+e)^(1/2),x, algorithm="maxima")
Output:
2/105*(15*(f*x + e)^(7/2)*b*d*h + 21*(b*d*f*g - (3*b*d*e - (b*c + a*d)*f)* h)*(f*x + e)^(5/2) - 35*((2*b*d*e*f - (b*c + a*d)*f^2)*g - (3*b*d*e^2 + a* c*f^2 - 2*(b*c + a*d)*e*f)*h)*(f*x + e)^(3/2) + 105*((b*d*e^2*f + a*c*f^3 - (b*c + a*d)*e*f^2)*g - (b*d*e^3 + a*c*e*f^2 - (b*c + a*d)*e^2*f)*h)*sqrt (f*x + e))/f^4
Time = 0.12 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.71 \[ \int \frac {(a+b x) (c+d x) (g+h x)}{\sqrt {e+f x}} \, dx=\frac {2 \, {\left (105 \, \sqrt {f x + e} a c g + \frac {35 \, {\left ({\left (f x + e\right )}^{\frac {3}{2}} - 3 \, \sqrt {f x + e} e\right )} b c g}{f} + \frac {35 \, {\left ({\left (f x + e\right )}^{\frac {3}{2}} - 3 \, \sqrt {f x + e} e\right )} a d g}{f} + \frac {35 \, {\left ({\left (f x + e\right )}^{\frac {3}{2}} - 3 \, \sqrt {f x + e} e\right )} a c h}{f} + \frac {7 \, {\left (3 \, {\left (f x + e\right )}^{\frac {5}{2}} - 10 \, {\left (f x + e\right )}^{\frac {3}{2}} e + 15 \, \sqrt {f x + e} e^{2}\right )} b d g}{f^{2}} + \frac {7 \, {\left (3 \, {\left (f x + e\right )}^{\frac {5}{2}} - 10 \, {\left (f x + e\right )}^{\frac {3}{2}} e + 15 \, \sqrt {f x + e} e^{2}\right )} b c h}{f^{2}} + \frac {7 \, {\left (3 \, {\left (f x + e\right )}^{\frac {5}{2}} - 10 \, {\left (f x + e\right )}^{\frac {3}{2}} e + 15 \, \sqrt {f x + e} e^{2}\right )} a d h}{f^{2}} + \frac {3 \, {\left (5 \, {\left (f x + e\right )}^{\frac {7}{2}} - 21 \, {\left (f x + e\right )}^{\frac {5}{2}} e + 35 \, {\left (f x + e\right )}^{\frac {3}{2}} e^{2} - 35 \, \sqrt {f x + e} e^{3}\right )} b d h}{f^{3}}\right )}}{105 \, f} \] Input:
integrate((b*x+a)*(d*x+c)*(h*x+g)/(f*x+e)^(1/2),x, algorithm="giac")
Output:
2/105*(105*sqrt(f*x + e)*a*c*g + 35*((f*x + e)^(3/2) - 3*sqrt(f*x + e)*e)* b*c*g/f + 35*((f*x + e)^(3/2) - 3*sqrt(f*x + e)*e)*a*d*g/f + 35*((f*x + e) ^(3/2) - 3*sqrt(f*x + e)*e)*a*c*h/f + 7*(3*(f*x + e)^(5/2) - 10*(f*x + e)^ (3/2)*e + 15*sqrt(f*x + e)*e^2)*b*d*g/f^2 + 7*(3*(f*x + e)^(5/2) - 10*(f*x + e)^(3/2)*e + 15*sqrt(f*x + e)*e^2)*b*c*h/f^2 + 7*(3*(f*x + e)^(5/2) - 1 0*(f*x + e)^(3/2)*e + 15*sqrt(f*x + e)*e^2)*a*d*h/f^2 + 3*(5*(f*x + e)^(7/ 2) - 21*(f*x + e)^(5/2)*e + 35*(f*x + e)^(3/2)*e^2 - 35*sqrt(f*x + e)*e^3) *b*d*h/f^3)/f
Time = 2.46 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.99 \[ \int \frac {(a+b x) (c+d x) (g+h x)}{\sqrt {e+f x}} \, dx=\frac {{\left (e+f\,x\right )}^{3/2}\,\left (2\,a\,c\,f^2\,h+2\,a\,d\,f^2\,g+2\,b\,c\,f^2\,g+6\,b\,d\,e^2\,h-4\,a\,d\,e\,f\,h-4\,b\,c\,e\,f\,h-4\,b\,d\,e\,f\,g\right )}{3\,f^4}+\frac {{\left (e+f\,x\right )}^{5/2}\,\left (2\,a\,d\,f\,h+2\,b\,c\,f\,h-6\,b\,d\,e\,h+2\,b\,d\,f\,g\right )}{5\,f^4}-\frac {2\,\sqrt {e+f\,x}\,\left (a\,f-b\,e\right )\,\left (c\,f-d\,e\right )\,\left (e\,h-f\,g\right )}{f^4}+\frac {2\,b\,d\,h\,{\left (e+f\,x\right )}^{7/2}}{7\,f^4} \] Input:
int(((g + h*x)*(a + b*x)*(c + d*x))/(e + f*x)^(1/2),x)
Output:
((e + f*x)^(3/2)*(2*a*c*f^2*h + 2*a*d*f^2*g + 2*b*c*f^2*g + 6*b*d*e^2*h - 4*a*d*e*f*h - 4*b*c*e*f*h - 4*b*d*e*f*g))/(3*f^4) + ((e + f*x)^(5/2)*(2*a* d*f*h + 2*b*c*f*h - 6*b*d*e*h + 2*b*d*f*g))/(5*f^4) - (2*(e + f*x)^(1/2)*( a*f - b*e)*(c*f - d*e)*(e*h - f*g))/f^4 + (2*b*d*h*(e + f*x)^(7/2))/(7*f^4 )
Time = 0.17 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.31 \[ \int \frac {(a+b x) (c+d x) (g+h x)}{\sqrt {e+f x}} \, dx=\frac {2 \sqrt {f x +e}\, \left (15 b d \,f^{3} h \,x^{3}+21 a d \,f^{3} h \,x^{2}+21 b c \,f^{3} h \,x^{2}-18 b d e \,f^{2} h \,x^{2}+21 b d \,f^{3} g \,x^{2}+35 a c \,f^{3} h x -28 a d e \,f^{2} h x +35 a d \,f^{3} g x -28 b c e \,f^{2} h x +35 b c \,f^{3} g x +24 b d \,e^{2} f h x -28 b d e \,f^{2} g x -70 a c e \,f^{2} h +105 a c \,f^{3} g +56 a d \,e^{2} f h -70 a d e \,f^{2} g +56 b c \,e^{2} f h -70 b c e \,f^{2} g -48 b d \,e^{3} h +56 b d \,e^{2} f g \right )}{105 f^{4}} \] Input:
int((b*x+a)*(d*x+c)*(h*x+g)/(f*x+e)^(1/2),x)
Output:
(2*sqrt(e + f*x)*( - 70*a*c*e*f**2*h + 105*a*c*f**3*g + 35*a*c*f**3*h*x + 56*a*d*e**2*f*h - 70*a*d*e*f**2*g - 28*a*d*e*f**2*h*x + 35*a*d*f**3*g*x + 21*a*d*f**3*h*x**2 + 56*b*c*e**2*f*h - 70*b*c*e*f**2*g - 28*b*c*e*f**2*h*x + 35*b*c*f**3*g*x + 21*b*c*f**3*h*x**2 - 48*b*d*e**3*h + 56*b*d*e**2*f*g + 24*b*d*e**2*f*h*x - 28*b*d*e*f**2*g*x - 18*b*d*e*f**2*h*x**2 + 21*b*d*f* *3*g*x**2 + 15*b*d*f**3*h*x**3))/(105*f**4)