Integrand size = 22, antiderivative size = 126 \[ \int (c+d x)^2 (e+f x)^{3/2} (g+h x) \, dx=\frac {2 (d e-c f)^2 (f g-e h) (e+f x)^{5/2}}{5 f^4}-\frac {2 (d e-c f) (2 d f g-3 d e h+c f h) (e+f x)^{7/2}}{7 f^4}+\frac {2 d (d f g-3 d e h+2 c f h) (e+f x)^{9/2}}{9 f^4}+\frac {2 d^2 h (e+f x)^{11/2}}{11 f^4} \] Output:
2/5*(-c*f+d*e)^2*(-e*h+f*g)*(f*x+e)^(5/2)/f^4-2/7*(-c*f+d*e)*(c*f*h-3*d*e* h+2*d*f*g)*(f*x+e)^(7/2)/f^4+2/9*d*(2*c*f*h-3*d*e*h+d*f*g)*(f*x+e)^(9/2)/f ^4+2/11*d^2*h*(f*x+e)^(11/2)/f^4
Time = 0.10 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.06 \[ \int (c+d x)^2 (e+f x)^{3/2} (g+h x) \, dx=\frac {2 (e+f x)^{5/2} \left (99 c^2 f^2 (7 f g-2 e h+5 f h x)+22 c d f \left (8 e^2 h+5 f^2 x (9 g+7 h x)-2 e f (9 g+10 h x)\right )+d^2 \left (-48 e^3 h+35 f^3 x^2 (11 g+9 h x)+8 e^2 f (11 g+15 h x)-10 e f^2 x (22 g+21 h x)\right )\right )}{3465 f^4} \] Input:
Integrate[(c + d*x)^2*(e + f*x)^(3/2)*(g + h*x),x]
Output:
(2*(e + f*x)^(5/2)*(99*c^2*f^2*(7*f*g - 2*e*h + 5*f*h*x) + 22*c*d*f*(8*e^2 *h + 5*f^2*x*(9*g + 7*h*x) - 2*e*f*(9*g + 10*h*x)) + d^2*(-48*e^3*h + 35*f ^3*x^2*(11*g + 9*h*x) + 8*e^2*f*(11*g + 15*h*x) - 10*e*f^2*x*(22*g + 21*h* x))))/(3465*f^4)
Time = 0.27 (sec) , antiderivative size = 126, 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 = {86, 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.
| \(\displaystyle \int (c+d x)^2 (e+f x)^{3/2} (g+h x) \, dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (\frac {d (e+f x)^{7/2} (2 c f h-3 d e h+d f g)}{f^3}+\frac {(e+f x)^{5/2} (c f-d e) (c f h-3 d e h+2 d f g)}{f^3}+\frac {(e+f x)^{3/2} (c f-d e)^2 (f g-e h)}{f^3}+\frac {d^2 h (e+f x)^{9/2}}{f^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 d (e+f x)^{9/2} (2 c f h-3 d e h+d f g)}{9 f^4}-\frac {2 (e+f x)^{7/2} (d e-c f) (c f h-3 d e h+2 d f g)}{7 f^4}+\frac {2 (e+f x)^{5/2} (d e-c f)^2 (f g-e h)}{5 f^4}+\frac {2 d^2 h (e+f x)^{11/2}}{11 f^4}\) |
Input:
Int[(c + d*x)^2*(e + f*x)^(3/2)*(g + h*x),x]
Output:
(2*(d*e - c*f)^2*(f*g - e*h)*(e + f*x)^(5/2))/(5*f^4) - (2*(d*e - c*f)*(2* d*f*g - 3*d*e*h + c*f*h)*(e + f*x)^(7/2))/(7*f^4) + (2*d*(d*f*g - 3*d*e*h + 2*c*f*h)*(e + f*x)^(9/2))/(9*f^4) + (2*d^2*h*(e + f*x)^(11/2))/(11*f^4)
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Time = 0.64 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.94
| method | result | size |
| pseudoelliptic | \(-\frac {4 \left (\left (-\frac {35 x^{2} \left (\frac {9 h x}{11}+g \right ) d^{2}}{18}-5 x c \left (\frac {7 h x}{9}+g \right ) d -\frac {7 c^{2} \left (\frac {5 h x}{7}+g \right )}{2}\right ) f^{3}+e \left (\frac {10 x \left (\frac {21 h x}{22}+g \right ) d^{2}}{9}+2 c \left (\frac {10 h x}{9}+g \right ) d +h \,c^{2}\right ) f^{2}-\frac {8 e^{2} d \left (\left (\frac {15 h x}{22}+\frac {g}{2}\right ) d +c h \right ) f}{9}+\frac {8 d^{2} e^{3} h}{33}\right ) \left (f x +e \right )^{\frac {5}{2}}}{35 f^{4}}\) | \(118\) |
| derivativedivides | \(\frac {\frac {2 d^{2} h \left (f x +e \right )^{\frac {11}{2}}}{11}+\frac {2 \left (2 d \left (c f -d e \right ) h +d^{2} \left (-e h +f g \right )\right ) \left (f x +e \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (c f -d e \right )^{2} h +2 d \left (c f -d e \right ) \left (-e h +f g \right )\right ) \left (f x +e \right )^{\frac {7}{2}}}{7}+\frac {2 \left (c f -d e \right )^{2} \left (-e h +f g \right ) \left (f x +e \right )^{\frac {5}{2}}}{5}}{f^{4}}\) | \(122\) |
| default | \(\frac {\frac {2 d^{2} h \left (f x +e \right )^{\frac {11}{2}}}{11}-\frac {2 \left (-2 d \left (c f -d e \right ) h +d^{2} \left (e h -f g \right )\right ) \left (f x +e \right )^{\frac {9}{2}}}{9}-\frac {2 \left (-\left (c f -d e \right )^{2} h +2 d \left (c f -d e \right ) \left (e h -f g \right )\right ) \left (f x +e \right )^{\frac {7}{2}}}{7}-\frac {2 \left (c f -d e \right )^{2} \left (e h -f g \right ) \left (f x +e \right )^{\frac {5}{2}}}{5}}{f^{4}}\) | \(123\) |
| gosper | \(-\frac {2 \left (f x +e \right )^{\frac {5}{2}} \left (-315 d^{2} h \,x^{3} f^{3}-770 c d \,f^{3} h \,x^{2}+210 d^{2} e \,f^{2} h \,x^{2}-385 d^{2} f^{3} g \,x^{2}-495 c^{2} f^{3} h x +440 c d e \,f^{2} h x -990 c d \,f^{3} g x -120 d^{2} e^{2} f h x +220 d^{2} e \,f^{2} g x +198 c^{2} e \,f^{2} h -693 c^{2} g \,f^{3}-176 c d \,e^{2} f h +396 c d e \,f^{2} g +48 d^{2} e^{3} h -88 d^{2} e^{2} f g \right )}{3465 f^{4}}\) | \(169\) |
| orering | \(-\frac {2 \left (f x +e \right )^{\frac {5}{2}} \left (-315 d^{2} h \,x^{3} f^{3}-770 c d \,f^{3} h \,x^{2}+210 d^{2} e \,f^{2} h \,x^{2}-385 d^{2} f^{3} g \,x^{2}-495 c^{2} f^{3} h x +440 c d e \,f^{2} h x -990 c d \,f^{3} g x -120 d^{2} e^{2} f h x +220 d^{2} e \,f^{2} g x +198 c^{2} e \,f^{2} h -693 c^{2} g \,f^{3}-176 c d \,e^{2} f h +396 c d e \,f^{2} g +48 d^{2} e^{3} h -88 d^{2} e^{2} f g \right )}{3465 f^{4}}\) | \(169\) |
| trager | \(-\frac {2 \left (-315 d^{2} f^{5} h \,x^{5}-770 c d \,f^{5} h \,x^{4}-420 d^{2} e \,f^{4} h \,x^{4}-385 d^{2} f^{5} g \,x^{4}-495 c^{2} f^{5} h \,x^{3}-1100 c d e \,f^{4} h \,x^{3}-990 c d \,f^{5} g \,x^{3}-15 d^{2} e^{2} f^{3} h \,x^{3}-550 d^{2} e \,f^{4} g \,x^{3}-792 c^{2} e \,f^{4} h \,x^{2}-693 c^{2} f^{5} g \,x^{2}-66 c d \,e^{2} f^{3} h \,x^{2}-1584 c d e \,f^{4} g \,x^{2}+18 d^{2} e^{3} f^{2} h \,x^{2}-33 d^{2} e^{2} f^{3} g \,x^{2}-99 c^{2} e^{2} f^{3} h x -1386 c^{2} e \,f^{4} g x +88 c d \,e^{3} f^{2} h x -198 c d \,e^{2} f^{3} g x -24 d^{2} e^{4} f h x +44 d^{2} e^{3} f^{2} g x +198 c^{2} e^{3} f^{2} h -693 c^{2} e^{2} f^{3} g -176 c d \,e^{4} f h +396 c d \,e^{3} f^{2} g +48 d^{2} e^{5} h -88 d^{2} e^{4} f g \right ) \sqrt {f x +e}}{3465 f^{4}}\) | \(341\) |
| risch | \(-\frac {2 \left (-315 d^{2} f^{5} h \,x^{5}-770 c d \,f^{5} h \,x^{4}-420 d^{2} e \,f^{4} h \,x^{4}-385 d^{2} f^{5} g \,x^{4}-495 c^{2} f^{5} h \,x^{3}-1100 c d e \,f^{4} h \,x^{3}-990 c d \,f^{5} g \,x^{3}-15 d^{2} e^{2} f^{3} h \,x^{3}-550 d^{2} e \,f^{4} g \,x^{3}-792 c^{2} e \,f^{4} h \,x^{2}-693 c^{2} f^{5} g \,x^{2}-66 c d \,e^{2} f^{3} h \,x^{2}-1584 c d e \,f^{4} g \,x^{2}+18 d^{2} e^{3} f^{2} h \,x^{2}-33 d^{2} e^{2} f^{3} g \,x^{2}-99 c^{2} e^{2} f^{3} h x -1386 c^{2} e \,f^{4} g x +88 c d \,e^{3} f^{2} h x -198 c d \,e^{2} f^{3} g x -24 d^{2} e^{4} f h x +44 d^{2} e^{3} f^{2} g x +198 c^{2} e^{3} f^{2} h -693 c^{2} e^{2} f^{3} g -176 c d \,e^{4} f h +396 c d \,e^{3} f^{2} g +48 d^{2} e^{5} h -88 d^{2} e^{4} f g \right ) \sqrt {f x +e}}{3465 f^{4}}\) | \(341\) |
Input:
int((d*x+c)^2*(f*x+e)^(3/2)*(h*x+g),x,method=_RETURNVERBOSE)
Output:
-4/35*((-35/18*x^2*(9/11*h*x+g)*d^2-5*x*c*(7/9*h*x+g)*d-7/2*c^2*(5/7*h*x+g ))*f^3+e*(10/9*x*(21/22*h*x+g)*d^2+2*c*(10/9*h*x+g)*d+h*c^2)*f^2-8/9*e^2*d *((15/22*h*x+1/2*g)*d+c*h)*f+8/33*d^2*e^3*h)*(f*x+e)^(5/2)/f^4
Leaf count of result is larger than twice the leaf count of optimal. 323 vs. \(2 (110) = 220\).
Time = 0.07 (sec) , antiderivative size = 323, normalized size of antiderivative = 2.56 \[ \int (c+d x)^2 (e+f x)^{3/2} (g+h x) \, dx=\frac {2 \, {\left (315 \, d^{2} f^{5} h x^{5} + 35 \, {\left (11 \, d^{2} f^{5} g + 2 \, {\left (6 \, d^{2} e f^{4} + 11 \, c d f^{5}\right )} h\right )} x^{4} + 5 \, {\left (22 \, {\left (5 \, d^{2} e f^{4} + 9 \, c d f^{5}\right )} g + {\left (3 \, d^{2} e^{2} f^{3} + 220 \, c d e f^{4} + 99 \, c^{2} f^{5}\right )} h\right )} x^{3} + 3 \, {\left (11 \, {\left (d^{2} e^{2} f^{3} + 48 \, c d e f^{4} + 21 \, c^{2} f^{5}\right )} g - 2 \, {\left (3 \, d^{2} e^{3} f^{2} - 11 \, c d e^{2} f^{3} - 132 \, c^{2} e f^{4}\right )} h\right )} x^{2} + 11 \, {\left (8 \, d^{2} e^{4} f - 36 \, c d e^{3} f^{2} + 63 \, c^{2} e^{2} f^{3}\right )} g - 2 \, {\left (24 \, d^{2} e^{5} - 88 \, c d e^{4} f + 99 \, c^{2} e^{3} f^{2}\right )} h - {\left (22 \, {\left (2 \, d^{2} e^{3} f^{2} - 9 \, c d e^{2} f^{3} - 63 \, c^{2} e f^{4}\right )} g - {\left (24 \, d^{2} e^{4} f - 88 \, c d e^{3} f^{2} + 99 \, c^{2} e^{2} f^{3}\right )} h\right )} x\right )} \sqrt {f x + e}}{3465 \, f^{4}} \] Input:
integrate((d*x+c)^2*(f*x+e)^(3/2)*(h*x+g),x, algorithm="fricas")
Output:
2/3465*(315*d^2*f^5*h*x^5 + 35*(11*d^2*f^5*g + 2*(6*d^2*e*f^4 + 11*c*d*f^5 )*h)*x^4 + 5*(22*(5*d^2*e*f^4 + 9*c*d*f^5)*g + (3*d^2*e^2*f^3 + 220*c*d*e* f^4 + 99*c^2*f^5)*h)*x^3 + 3*(11*(d^2*e^2*f^3 + 48*c*d*e*f^4 + 21*c^2*f^5) *g - 2*(3*d^2*e^3*f^2 - 11*c*d*e^2*f^3 - 132*c^2*e*f^4)*h)*x^2 + 11*(8*d^2 *e^4*f - 36*c*d*e^3*f^2 + 63*c^2*e^2*f^3)*g - 2*(24*d^2*e^5 - 88*c*d*e^4*f + 99*c^2*e^3*f^2)*h - (22*(2*d^2*e^3*f^2 - 9*c*d*e^2*f^3 - 63*c^2*e*f^4)* g - (24*d^2*e^4*f - 88*c*d*e^3*f^2 + 99*c^2*e^2*f^3)*h)*x)*sqrt(f*x + e)/f ^4
Time = 1.08 (sec) , antiderivative size = 257, normalized size of antiderivative = 2.04 \[ \int (c+d x)^2 (e+f x)^{3/2} (g+h x) \, dx=\begin {cases} \frac {2 \left (\frac {d^{2} h \left (e + f x\right )^{\frac {11}{2}}}{11 f^{3}} + \frac {\left (e + f x\right )^{\frac {9}{2}} \cdot \left (2 c d f h - 3 d^{2} e h + d^{2} f g\right )}{9 f^{3}} + \frac {\left (e + f x\right )^{\frac {7}{2}} \left (c^{2} f^{2} h - 4 c d e f h + 2 c d f^{2} g + 3 d^{2} e^{2} h - 2 d^{2} e f g\right )}{7 f^{3}} + \frac {\left (e + f x\right )^{\frac {5}{2}} \left (- c^{2} e f^{2} h + c^{2} f^{3} g + 2 c d e^{2} f h - 2 c d e f^{2} g - d^{2} e^{3} h + d^{2} e^{2} f g\right )}{5 f^{3}}\right )}{f} & \text {for}\: f \neq 0 \\e^{\frac {3}{2}} \left (c^{2} g x + \frac {d^{2} h x^{4}}{4} + \frac {x^{3} \cdot \left (2 c d h + d^{2} g\right )}{3} + \frac {x^{2} \left (c^{2} h + 2 c d g\right )}{2}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((d*x+c)**2*(f*x+e)**(3/2)*(h*x+g),x)
Output:
Piecewise((2*(d**2*h*(e + f*x)**(11/2)/(11*f**3) + (e + f*x)**(9/2)*(2*c*d *f*h - 3*d**2*e*h + d**2*f*g)/(9*f**3) + (e + f*x)**(7/2)*(c**2*f**2*h - 4 *c*d*e*f*h + 2*c*d*f**2*g + 3*d**2*e**2*h - 2*d**2*e*f*g)/(7*f**3) + (e + f*x)**(5/2)*(-c**2*e*f**2*h + c**2*f**3*g + 2*c*d*e**2*f*h - 2*c*d*e*f**2* g - d**2*e**3*h + d**2*e**2*f*g)/(5*f**3))/f, Ne(f, 0)), (e**(3/2)*(c**2*g *x + d**2*h*x**4/4 + x**3*(2*c*d*h + d**2*g)/3 + x**2*(c**2*h + 2*c*d*g)/2 ), True))
Time = 0.03 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.31 \[ \int (c+d x)^2 (e+f x)^{3/2} (g+h x) \, dx=\frac {2 \, {\left (315 \, {\left (f x + e\right )}^{\frac {11}{2}} d^{2} h + 385 \, {\left (d^{2} f g - {\left (3 \, d^{2} e - 2 \, c d f\right )} h\right )} {\left (f x + e\right )}^{\frac {9}{2}} - 495 \, {\left (2 \, {\left (d^{2} e f - c d f^{2}\right )} g - {\left (3 \, d^{2} e^{2} - 4 \, c d e f + c^{2} f^{2}\right )} h\right )} {\left (f x + e\right )}^{\frac {7}{2}} + 693 \, {\left ({\left (d^{2} e^{2} f - 2 \, c d e f^{2} + c^{2} f^{3}\right )} g - {\left (d^{2} e^{3} - 2 \, c d e^{2} f + c^{2} e f^{2}\right )} h\right )} {\left (f x + e\right )}^{\frac {5}{2}}\right )}}{3465 \, f^{4}} \] Input:
integrate((d*x+c)^2*(f*x+e)^(3/2)*(h*x+g),x, algorithm="maxima")
Output:
2/3465*(315*(f*x + e)^(11/2)*d^2*h + 385*(d^2*f*g - (3*d^2*e - 2*c*d*f)*h) *(f*x + e)^(9/2) - 495*(2*(d^2*e*f - c*d*f^2)*g - (3*d^2*e^2 - 4*c*d*e*f + c^2*f^2)*h)*(f*x + e)^(7/2) + 693*((d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3)*g - (d^2*e^3 - 2*c*d*e^2*f + c^2*e*f^2)*h)*(f*x + e)^(5/2))/f^4
Leaf count of result is larger than twice the leaf count of optimal. 854 vs. \(2 (110) = 220\).
Time = 0.13 (sec) , antiderivative size = 854, normalized size of antiderivative = 6.78 \[ \int (c+d x)^2 (e+f x)^{3/2} (g+h x) \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)^2*(f*x+e)^(3/2)*(h*x+g),x, algorithm="giac")
Output:
2/3465*(3465*sqrt(f*x + e)*c^2*e^2*g + 2310*((f*x + e)^(3/2) - 3*sqrt(f*x + e)*e)*c^2*e*g + 2310*((f*x + e)^(3/2) - 3*sqrt(f*x + e)*e)*c*d*e^2*g/f + 1155*((f*x + e)^(3/2) - 3*sqrt(f*x + e)*e)*c^2*e^2*h/f + 231*(3*(f*x + e) ^(5/2) - 10*(f*x + e)^(3/2)*e + 15*sqrt(f*x + e)*e^2)*c^2*g + 231*(3*(f*x + e)^(5/2) - 10*(f*x + e)^(3/2)*e + 15*sqrt(f*x + e)*e^2)*d^2*e^2*g/f^2 + 924*(3*(f*x + e)^(5/2) - 10*(f*x + e)^(3/2)*e + 15*sqrt(f*x + e)*e^2)*c*d* e*g/f + 462*(3*(f*x + e)^(5/2) - 10*(f*x + e)^(3/2)*e + 15*sqrt(f*x + e)*e ^2)*c*d*e^2*h/f^2 + 462*(3*(f*x + e)^(5/2) - 10*(f*x + e)^(3/2)*e + 15*sqr t(f*x + e)*e^2)*c^2*e*h/f + 198*(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)*d^2*e*g/f^2 + 198*(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)*c*d*g/f + 99*(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)*d^2*e^2*h/f^3 + 396*(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)*c*d*e*h/f^2 + 99*(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)*c^2*h/f + 11*(35*(f*x + e)^(9/2) - 18 0*(f*x + e)^(7/2)*e + 378*(f*x + e)^(5/2)*e^2 - 420*(f*x + e)^(3/2)*e^3 + 315*sqrt(f*x + e)*e^4)*d^2*g/f^2 + 22*(35*(f*x + e)^(9/2) - 180*(f*x + e)^ (7/2)*e + 378*(f*x + e)^(5/2)*e^2 - 420*(f*x + e)^(3/2)*e^3 + 315*sqrt(f*x + e)*e^4)*d^2*e*h/f^3 + 22*(35*(f*x + e)^(9/2) - 180*(f*x + e)^(7/2)*e...
Time = 0.06 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.91 \[ \int (c+d x)^2 (e+f x)^{3/2} (g+h x) \, dx=\frac {{\left (e+f\,x\right )}^{9/2}\,\left (2\,d^2\,f\,g-6\,d^2\,e\,h+4\,c\,d\,f\,h\right )}{9\,f^4}+\frac {2\,{\left (e+f\,x\right )}^{7/2}\,\left (c\,f-d\,e\right )\,\left (c\,f\,h-3\,d\,e\,h+2\,d\,f\,g\right )}{7\,f^4}-\frac {2\,{\left (e+f\,x\right )}^{5/2}\,{\left (c\,f-d\,e\right )}^2\,\left (e\,h-f\,g\right )}{5\,f^4}+\frac {2\,d^2\,h\,{\left (e+f\,x\right )}^{11/2}}{11\,f^4} \] Input:
int((e + f*x)^(3/2)*(g + h*x)*(c + d*x)^2,x)
Output:
((e + f*x)^(9/2)*(2*d^2*f*g - 6*d^2*e*h + 4*c*d*f*h))/(9*f^4) + (2*(e + f* x)^(7/2)*(c*f - d*e)*(c*f*h - 3*d*e*h + 2*d*f*g))/(7*f^4) - (2*(e + f*x)^( 5/2)*(c*f - d*e)^2*(e*h - f*g))/(5*f^4) + (2*d^2*h*(e + f*x)^(11/2))/(11*f ^4)
Time = 0.16 (sec) , antiderivative size = 339, normalized size of antiderivative = 2.69 \[ \int (c+d x)^2 (e+f x)^{3/2} (g+h x) \, dx=\frac {2 \sqrt {f x +e}\, \left (315 d^{2} f^{5} h \,x^{5}+770 c d \,f^{5} h \,x^{4}+420 d^{2} e \,f^{4} h \,x^{4}+385 d^{2} f^{5} g \,x^{4}+495 c^{2} f^{5} h \,x^{3}+1100 c d e \,f^{4} h \,x^{3}+990 c d \,f^{5} g \,x^{3}+15 d^{2} e^{2} f^{3} h \,x^{3}+550 d^{2} e \,f^{4} g \,x^{3}+792 c^{2} e \,f^{4} h \,x^{2}+693 c^{2} f^{5} g \,x^{2}+66 c d \,e^{2} f^{3} h \,x^{2}+1584 c d e \,f^{4} g \,x^{2}-18 d^{2} e^{3} f^{2} h \,x^{2}+33 d^{2} e^{2} f^{3} g \,x^{2}+99 c^{2} e^{2} f^{3} h x +1386 c^{2} e \,f^{4} g x -88 c d \,e^{3} f^{2} h x +198 c d \,e^{2} f^{3} g x +24 d^{2} e^{4} f h x -44 d^{2} e^{3} f^{2} g x -198 c^{2} e^{3} f^{2} h +693 c^{2} e^{2} f^{3} g +176 c d \,e^{4} f h -396 c d \,e^{3} f^{2} g -48 d^{2} e^{5} h +88 d^{2} e^{4} f g \right )}{3465 f^{4}} \] Input:
int((d*x+c)^2*(f*x+e)^(3/2)*(h*x+g),x)
Output:
(2*sqrt(e + f*x)*( - 198*c**2*e**3*f**2*h + 693*c**2*e**2*f**3*g + 99*c**2 *e**2*f**3*h*x + 1386*c**2*e*f**4*g*x + 792*c**2*e*f**4*h*x**2 + 693*c**2* f**5*g*x**2 + 495*c**2*f**5*h*x**3 + 176*c*d*e**4*f*h - 396*c*d*e**3*f**2* g - 88*c*d*e**3*f**2*h*x + 198*c*d*e**2*f**3*g*x + 66*c*d*e**2*f**3*h*x**2 + 1584*c*d*e*f**4*g*x**2 + 1100*c*d*e*f**4*h*x**3 + 990*c*d*f**5*g*x**3 + 770*c*d*f**5*h*x**4 - 48*d**2*e**5*h + 88*d**2*e**4*f*g + 24*d**2*e**4*f* h*x - 44*d**2*e**3*f**2*g*x - 18*d**2*e**3*f**2*h*x**2 + 33*d**2*e**2*f**3 *g*x**2 + 15*d**2*e**2*f**3*h*x**3 + 550*d**2*e*f**4*g*x**3 + 420*d**2*e*f **4*h*x**4 + 385*d**2*f**5*g*x**4 + 315*d**2*f**5*h*x**5))/(3465*f**4)