Integrand size = 27, antiderivative size = 248 \[ \int (a+b x) (c+d x)^2 \sqrt {e+f x} (g+h x) \, dx=-\frac {2 (b e-a f) (d e-c f)^2 (f g-e h) (e+f x)^{3/2}}{3 f^5}+\frac {2 (d e-c f) (b d e (3 f g-4 e h)-b c f (f g-2 e h)-a f (2 d f g-3 d e h+c f h)) (e+f x)^{5/2}}{5 f^5}+\frac {2 \left (a d f (d f g-3 d e h+2 c f h)+b \left (c^2 f^2 h+2 c d f (f g-3 e h)-3 d^2 e (f g-2 e h)\right )\right ) (e+f x)^{7/2}}{7 f^5}+\frac {2 d (a d f h+b (d f g-4 d e h+2 c f h)) (e+f x)^{9/2}}{9 f^5}+\frac {2 b d^2 h (e+f x)^{11/2}}{11 f^5} \] Output:
-2/3*(-a*f+b*e)*(-c*f+d*e)^2*(-e*h+f*g)*(f*x+e)^(3/2)/f^5+2/5*(-c*f+d*e)*( b*d*e*(-4*e*h+3*f*g)-b*c*f*(-2*e*h+f*g)-a*f*(c*f*h-3*d*e*h+2*d*f*g))*(f*x+ e)^(5/2)/f^5+2/7*(a*d*f*(2*c*f*h-3*d*e*h+d*f*g)+b*(c^2*f^2*h+2*c*d*f*(-3*e *h+f*g)-3*d^2*e*(-2*e*h+f*g)))*(f*x+e)^(7/2)/f^5+2/9*d*(a*d*f*h+b*(2*c*f*h -4*d*e*h+d*f*g))*(f*x+e)^(9/2)/f^5+2/11*b*d^2*h*(f*x+e)^(11/2)/f^5
Time = 0.24 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.24 \[ \int (a+b x) (c+d x)^2 \sqrt {e+f x} (g+h x) \, dx=\frac {2 (e+f x)^{3/2} \left (11 a f \left (21 c^2 f^2 (5 f g-2 e h+3 f h x)+6 c d f \left (8 e^2 h+3 f^2 x (7 g+5 h x)-2 e f (7 g+6 h x)\right )+d^2 \left (-16 e^3 h+24 e^2 f (g+h x)-6 e f^2 x (6 g+5 h x)+5 f^3 x^2 (9 g+7 h x)\right )\right )+b \left (33 c^2 f^2 \left (8 e^2 h+3 f^2 x (7 g+5 h x)-2 e f (7 g+6 h x)\right )+22 c d f \left (-16 e^3 h+24 e^2 f (g+h x)-6 e f^2 x (6 g+5 h x)+5 f^3 x^2 (9 g+7 h x)\right )+d^2 \left (128 e^4 h+35 f^4 x^3 (11 g+9 h x)+24 e^2 f^2 x (11 g+10 h x)-16 e^3 f (11 g+12 h x)-10 e f^3 x^2 (33 g+28 h x)\right )\right )\right )}{3465 f^5} \] Input:
Integrate[(a + b*x)*(c + d*x)^2*Sqrt[e + f*x]*(g + h*x),x]
Output:
(2*(e + f*x)^(3/2)*(11*a*f*(21*c^2*f^2*(5*f*g - 2*e*h + 3*f*h*x) + 6*c*d*f *(8*e^2*h + 3*f^2*x*(7*g + 5*h*x) - 2*e*f*(7*g + 6*h*x)) + d^2*(-16*e^3*h + 24*e^2*f*(g + h*x) - 6*e*f^2*x*(6*g + 5*h*x) + 5*f^3*x^2*(9*g + 7*h*x))) + b*(33*c^2*f^2*(8*e^2*h + 3*f^2*x*(7*g + 5*h*x) - 2*e*f*(7*g + 6*h*x)) + 22*c*d*f*(-16*e^3*h + 24*e^2*f*(g + h*x) - 6*e*f^2*x*(6*g + 5*h*x) + 5*f^ 3*x^2*(9*g + 7*h*x)) + d^2*(128*e^4*h + 35*f^4*x^3*(11*g + 9*h*x) + 24*e^2 *f^2*x*(11*g + 10*h*x) - 16*e^3*f*(11*g + 12*h*x) - 10*e*f^3*x^2*(33*g + 2 8*h*x)))))/(3465*f^5)
Time = 0.49 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, 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.
| \(\displaystyle \int (a+b x) (c+d x)^2 \sqrt {e+f x} (g+h x) \, dx\) |
| \(\Big \downarrow \) 159 |
| \(\displaystyle \int \left (\frac {(e+f x)^{5/2} \left (a d f (2 c f h-3 d e h+d f g)+b \left (c^2 f^2 h+2 c d f (f g-3 e h)-3 d^2 e (f g-2 e h)\right )\right )}{f^4}+\frac {d (e+f x)^{7/2} (a d f h+b (2 c f h-4 d e h+d f g))}{f^4}+\frac {(e+f x)^{3/2} (d e-c f) (-a f (c f h-3 d e h+2 d f g)-b c f (f g-2 e h)+b d e (3 f g-4 e h))}{f^4}+\frac {\sqrt {e+f x} (a f-b e) (c f-d e)^2 (f g-e h)}{f^4}+\frac {b d^2 h (e+f x)^{9/2}}{f^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 (e+f x)^{7/2} \left (a d f (2 c f h-3 d e h+d f g)+b \left (c^2 f^2 h+2 c d f (f g-3 e h)-3 d^2 e (f g-2 e h)\right )\right )}{7 f^5}+\frac {2 d (e+f x)^{9/2} (a d f h+b (2 c f h-4 d e h+d f g))}{9 f^5}+\frac {2 (e+f x)^{5/2} (d e-c f) (-a f (c f h-3 d e h+2 d f g)-b c f (f g-2 e h)+b d e (3 f g-4 e h))}{5 f^5}-\frac {2 (e+f x)^{3/2} (b e-a f) (d e-c f)^2 (f g-e h)}{3 f^5}+\frac {2 b d^2 h (e+f x)^{11/2}}{11 f^5}\) |
Input:
Int[(a + b*x)*(c + d*x)^2*Sqrt[e + f*x]*(g + h*x),x]
Output:
(-2*(b*e - a*f)*(d*e - c*f)^2*(f*g - e*h)*(e + f*x)^(3/2))/(3*f^5) + (2*(d *e - c*f)*(b*d*e*(3*f*g - 4*e*h) - b*c*f*(f*g - 2*e*h) - a*f*(2*d*f*g - 3* d*e*h + c*f*h))*(e + f*x)^(5/2))/(5*f^5) + (2*(a*d*f*(d*f*g - 3*d*e*h + 2* c*f*h) + b*(c^2*f^2*h + 2*c*d*f*(f*g - 3*e*h) - 3*d^2*e*(f*g - 2*e*h)))*(e + f*x)^(7/2))/(7*f^5) + (2*d*(a*d*f*h + b*(d*f*g - 4*d*e*h + 2*c*f*h))*(e + f*x)^(9/2))/(9*f^5) + (2*b*d^2*h*(e + f*x)^(11/2))/(11*f^5)
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 = 0.90 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.03
| method | result | size |
| derivativedivides | \(\frac {\frac {2 h b \,d^{2} \left (f x +e \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (\left (a f -b e \right ) d^{2}+2 b d \left (c f -d e \right )\right ) h +b \,d^{2} \left (-e h +f g \right )\right ) \left (f x +e \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (2 \left (a f -b e \right ) d \left (c f -d e \right )+b \left (c f -d e \right )^{2}\right ) h +\left (\left (a f -b e \right ) d^{2}+2 b d \left (c f -d e \right )\right ) \left (-e h +f g \right )\right ) \left (f x +e \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (a f -b e \right ) \left (c f -d e \right )^{2} h +\left (2 \left (a f -b e \right ) d \left (c f -d e \right )+b \left (c f -d e \right )^{2}\right ) \left (-e h +f g \right )\right ) \left (f x +e \right )^{\frac {5}{2}}}{5}+\frac {2 \left (a f -b e \right ) \left (c f -d e \right )^{2} \left (-e h +f g \right ) \left (f x +e \right )^{\frac {3}{2}}}{3}}{f^{5}}\) | \(255\) |
| default | \(\frac {\frac {2 h b \,d^{2} \left (f x +e \right )^{\frac {11}{2}}}{11}+\frac {2 \left (-\left (\left (-a f +b e \right ) d^{2}-2 b d \left (c f -d e \right )\right ) h -b \,d^{2} \left (e h -f g \right )\right ) \left (f x +e \right )^{\frac {9}{2}}}{9}+\frac {2 \left (-\left (2 \left (-a f +b e \right ) d \left (c f -d e \right )-b \left (c f -d e \right )^{2}\right ) h +\left (\left (-a f +b e \right ) d^{2}-2 b d \left (c f -d e \right )\right ) \left (e h -f g \right )\right ) \left (f x +e \right )^{\frac {7}{2}}}{7}+\frac {2 \left (-\left (-a f +b e \right ) \left (c f -d e \right )^{2} h +\left (2 \left (-a f +b e \right ) d \left (c f -d e \right )-b \left (c f -d e \right )^{2}\right ) \left (e h -f g \right )\right ) \left (f x +e \right )^{\frac {5}{2}}}{5}+\frac {2 \left (-a f +b e \right ) \left (c f -d e \right )^{2} \left (e h -f g \right ) \left (f x +e \right )^{\frac {3}{2}}}{3}}{f^{5}}\) | \(261\) |
| pseudoelliptic | \(-\frac {4 \left (f x +e \right )^{\frac {3}{2}} \left (\left (-\frac {15 x^{2} \left (\frac {7 b h \,x^{2}}{11}+\frac {7 \left (a h +b g \right ) x}{9}+g a \right ) d^{2}}{14}-3 x c \left (\frac {5 b h \,x^{2}}{9}+\frac {5 \left (a h +b g \right ) x}{7}+g a \right ) d -\frac {5 c^{2} \left (\frac {3 b h \,x^{2}}{7}+\frac {3 \left (a h +b g \right ) x}{5}+g a \right )}{2}\right ) f^{4}+e \left (\frac {6 x \left (\frac {70 b h \,x^{2}}{99}+\frac {5 \left (a h +b g \right ) x}{6}+g a \right ) d^{2}}{7}+2 c \left (\frac {5 b h \,x^{2}}{7}+\frac {6 \left (a h +b g \right ) x}{7}+g a \right ) d +c^{2} \left (\frac {6}{7} b h x +a h +b g \right )\right ) f^{3}-\frac {8 \left (\left (\frac {5 b h \,x^{2}}{11}+\frac {\left (a h +b g \right ) x}{2}+\frac {g a}{2}\right ) d^{2}+c \left (b h x +a h +b g \right ) d +\frac {h b \,c^{2}}{2}\right ) e^{2} f^{2}}{7}+\frac {8 d \left (\left (\frac {12}{11} b h x +a h +b g \right ) d +2 b c h \right ) e^{3} f}{21}-\frac {64 b \,d^{2} e^{4} h}{231}\right )}{15 f^{5}}\) | \(267\) |
| gosper | \(-\frac {2 \left (f x +e \right )^{\frac {3}{2}} \left (-315 h b \,d^{2} x^{4} f^{4}-385 a \,d^{2} f^{4} h \,x^{3}-770 b c d \,f^{4} h \,x^{3}+280 b \,d^{2} e \,f^{3} h \,x^{3}-385 b \,d^{2} f^{4} g \,x^{3}-990 a c d \,f^{4} h \,x^{2}+330 a \,d^{2} e \,f^{3} h \,x^{2}-495 a \,d^{2} f^{4} g \,x^{2}-495 b \,c^{2} f^{4} h \,x^{2}+660 b c d e \,f^{3} h \,x^{2}-990 b c d \,f^{4} g \,x^{2}-240 b \,d^{2} e^{2} f^{2} h \,x^{2}+330 b \,d^{2} e \,f^{3} g \,x^{2}-693 a \,c^{2} f^{4} h x +792 a c d e \,f^{3} h x -1386 a c d \,f^{4} g x -264 a \,d^{2} e^{2} f^{2} h x +396 a \,d^{2} e \,f^{3} g x +396 b \,c^{2} e \,f^{3} h x -693 b \,c^{2} f^{4} g x -528 b c d \,e^{2} f^{2} h x +792 b c d e \,f^{3} g x +192 b \,d^{2} e^{3} f h x -264 b \,d^{2} e^{2} f^{2} g x +462 a \,c^{2} e \,f^{3} h -1155 g a \,c^{2} f^{4}-528 a c d \,e^{2} f^{2} h +924 a c d e \,f^{3} g +176 a \,d^{2} e^{3} f h -264 a \,d^{2} e^{2} f^{2} g -264 b \,c^{2} e^{2} f^{2} h +462 b \,c^{2} e \,f^{3} g +352 b c d \,e^{3} f h -528 b c d \,e^{2} f^{2} g -128 b \,d^{2} e^{4} h +176 b \,d^{2} e^{3} f g \right )}{3465 f^{5}}\) | \(451\) |
| orering | \(-\frac {2 \left (f x +e \right )^{\frac {3}{2}} \left (-315 h b \,d^{2} x^{4} f^{4}-385 a \,d^{2} f^{4} h \,x^{3}-770 b c d \,f^{4} h \,x^{3}+280 b \,d^{2} e \,f^{3} h \,x^{3}-385 b \,d^{2} f^{4} g \,x^{3}-990 a c d \,f^{4} h \,x^{2}+330 a \,d^{2} e \,f^{3} h \,x^{2}-495 a \,d^{2} f^{4} g \,x^{2}-495 b \,c^{2} f^{4} h \,x^{2}+660 b c d e \,f^{3} h \,x^{2}-990 b c d \,f^{4} g \,x^{2}-240 b \,d^{2} e^{2} f^{2} h \,x^{2}+330 b \,d^{2} e \,f^{3} g \,x^{2}-693 a \,c^{2} f^{4} h x +792 a c d e \,f^{3} h x -1386 a c d \,f^{4} g x -264 a \,d^{2} e^{2} f^{2} h x +396 a \,d^{2} e \,f^{3} g x +396 b \,c^{2} e \,f^{3} h x -693 b \,c^{2} f^{4} g x -528 b c d \,e^{2} f^{2} h x +792 b c d e \,f^{3} g x +192 b \,d^{2} e^{3} f h x -264 b \,d^{2} e^{2} f^{2} g x +462 a \,c^{2} e \,f^{3} h -1155 g a \,c^{2} f^{4}-528 a c d \,e^{2} f^{2} h +924 a c d e \,f^{3} g +176 a \,d^{2} e^{3} f h -264 a \,d^{2} e^{2} f^{2} g -264 b \,c^{2} e^{2} f^{2} h +462 b \,c^{2} e \,f^{3} g +352 b c d \,e^{3} f h -528 b c d \,e^{2} f^{2} g -128 b \,d^{2} e^{4} h +176 b \,d^{2} e^{3} f g \right )}{3465 f^{5}}\) | \(451\) |
| trager | \(-\frac {2 \left (-315 h b \,d^{2} f^{5} x^{5}-385 a \,d^{2} f^{5} h \,x^{4}-770 b c d \,f^{5} h \,x^{4}-35 b \,d^{2} e \,f^{4} h \,x^{4}-385 b \,d^{2} f^{5} g \,x^{4}-990 a c d \,f^{5} h \,x^{3}-55 a \,d^{2} e \,f^{4} h \,x^{3}-495 a \,d^{2} f^{5} g \,x^{3}-495 b \,c^{2} f^{5} h \,x^{3}-110 b c d e \,f^{4} h \,x^{3}-990 b c d \,f^{5} g \,x^{3}+40 b \,d^{2} e^{2} f^{3} h \,x^{3}-55 b \,d^{2} e \,f^{4} g \,x^{3}-693 a \,c^{2} f^{5} h \,x^{2}-198 a c d e \,f^{4} h \,x^{2}-1386 a c d \,f^{5} g \,x^{2}+66 a \,d^{2} e^{2} f^{3} h \,x^{2}-99 a \,d^{2} e \,f^{4} g \,x^{2}-99 b \,c^{2} e \,f^{4} h \,x^{2}-693 b \,c^{2} f^{5} g \,x^{2}+132 b c d \,e^{2} f^{3} h \,x^{2}-198 b c d e \,f^{4} g \,x^{2}-48 b \,d^{2} e^{3} f^{2} h \,x^{2}+66 b \,d^{2} e^{2} f^{3} g \,x^{2}-231 a \,c^{2} e \,f^{4} h x -1155 a \,c^{2} f^{5} g x +264 a c d \,e^{2} f^{3} h x -462 a c d e \,f^{4} g x -88 a \,d^{2} e^{3} f^{2} h x +132 a \,d^{2} e^{2} f^{3} g x +132 b \,c^{2} e^{2} f^{3} h x -231 b \,c^{2} e \,f^{4} g x -176 b c d \,e^{3} f^{2} h x +264 b c d \,e^{2} f^{3} g x +64 b \,d^{2} e^{4} f h x -88 b \,d^{2} e^{3} f^{2} g x +462 a \,c^{2} e^{2} f^{3} h -1155 a \,c^{2} e \,f^{4} g -528 a c d \,e^{3} f^{2} h +924 a c d \,e^{2} f^{3} g +176 a \,d^{2} e^{4} f h -264 a \,d^{2} e^{3} f^{2} g -264 b \,c^{2} e^{3} f^{2} h +462 b \,c^{2} e^{2} f^{3} g +352 b c d \,e^{4} f h -528 b c d \,e^{3} f^{2} g -128 b \,d^{2} e^{5} h +176 b \,d^{2} e^{4} f g \right ) \sqrt {f x +e}}{3465 f^{5}}\) | \(635\) |
| risch | \(-\frac {2 \left (-315 h b \,d^{2} f^{5} x^{5}-385 a \,d^{2} f^{5} h \,x^{4}-770 b c d \,f^{5} h \,x^{4}-35 b \,d^{2} e \,f^{4} h \,x^{4}-385 b \,d^{2} f^{5} g \,x^{4}-990 a c d \,f^{5} h \,x^{3}-55 a \,d^{2} e \,f^{4} h \,x^{3}-495 a \,d^{2} f^{5} g \,x^{3}-495 b \,c^{2} f^{5} h \,x^{3}-110 b c d e \,f^{4} h \,x^{3}-990 b c d \,f^{5} g \,x^{3}+40 b \,d^{2} e^{2} f^{3} h \,x^{3}-55 b \,d^{2} e \,f^{4} g \,x^{3}-693 a \,c^{2} f^{5} h \,x^{2}-198 a c d e \,f^{4} h \,x^{2}-1386 a c d \,f^{5} g \,x^{2}+66 a \,d^{2} e^{2} f^{3} h \,x^{2}-99 a \,d^{2} e \,f^{4} g \,x^{2}-99 b \,c^{2} e \,f^{4} h \,x^{2}-693 b \,c^{2} f^{5} g \,x^{2}+132 b c d \,e^{2} f^{3} h \,x^{2}-198 b c d e \,f^{4} g \,x^{2}-48 b \,d^{2} e^{3} f^{2} h \,x^{2}+66 b \,d^{2} e^{2} f^{3} g \,x^{2}-231 a \,c^{2} e \,f^{4} h x -1155 a \,c^{2} f^{5} g x +264 a c d \,e^{2} f^{3} h x -462 a c d e \,f^{4} g x -88 a \,d^{2} e^{3} f^{2} h x +132 a \,d^{2} e^{2} f^{3} g x +132 b \,c^{2} e^{2} f^{3} h x -231 b \,c^{2} e \,f^{4} g x -176 b c d \,e^{3} f^{2} h x +264 b c d \,e^{2} f^{3} g x +64 b \,d^{2} e^{4} f h x -88 b \,d^{2} e^{3} f^{2} g x +462 a \,c^{2} e^{2} f^{3} h -1155 a \,c^{2} e \,f^{4} g -528 a c d \,e^{3} f^{2} h +924 a c d \,e^{2} f^{3} g +176 a \,d^{2} e^{4} f h -264 a \,d^{2} e^{3} f^{2} g -264 b \,c^{2} e^{3} f^{2} h +462 b \,c^{2} e^{2} f^{3} g +352 b c d \,e^{4} f h -528 b c d \,e^{3} f^{2} g -128 b \,d^{2} e^{5} h +176 b \,d^{2} e^{4} f g \right ) \sqrt {f x +e}}{3465 f^{5}}\) | \(635\) |
Input:
int((b*x+a)*(d*x+c)^2*(f*x+e)^(1/2)*(h*x+g),x,method=_RETURNVERBOSE)
Output:
2/f^5*(1/11*h*b*d^2*(f*x+e)^(11/2)+1/9*(((a*f-b*e)*d^2+2*b*d*(c*f-d*e))*h+ b*d^2*(-e*h+f*g))*(f*x+e)^(9/2)+1/7*((2*(a*f-b*e)*d*(c*f-d*e)+b*(c*f-d*e)^ 2)*h+((a*f-b*e)*d^2+2*b*d*(c*f-d*e))*(-e*h+f*g))*(f*x+e)^(7/2)+1/5*((a*f-b *e)*(c*f-d*e)^2*h+(2*(a*f-b*e)*d*(c*f-d*e)+b*(c*f-d*e)^2)*(-e*h+f*g))*(f*x +e)^(5/2)+1/3*(a*f-b*e)*(c*f-d*e)^2*(-e*h+f*g)*(f*x+e)^(3/2))
Leaf count of result is larger than twice the leaf count of optimal. 519 vs. \(2 (228) = 456\).
Time = 0.08 (sec) , antiderivative size = 519, normalized size of antiderivative = 2.09 \[ \int (a+b x) (c+d x)^2 \sqrt {e+f x} (g+h x) \, dx=\frac {2 \, {\left (315 \, b d^{2} f^{5} h x^{5} + 35 \, {\left (11 \, b d^{2} f^{5} g + {\left (b d^{2} e f^{4} + 11 \, {\left (2 \, b c d + a d^{2}\right )} f^{5}\right )} h\right )} x^{4} + 5 \, {\left (11 \, {\left (b d^{2} e f^{4} + 9 \, {\left (2 \, b c d + a d^{2}\right )} f^{5}\right )} g - {\left (8 \, b d^{2} e^{2} f^{3} - 11 \, {\left (2 \, b c d + a d^{2}\right )} e f^{4} - 99 \, {\left (b c^{2} + 2 \, a c d\right )} f^{5}\right )} h\right )} x^{3} - 3 \, {\left (11 \, {\left (2 \, b d^{2} e^{2} f^{3} - 3 \, {\left (2 \, b c d + a d^{2}\right )} e f^{4} - 21 \, {\left (b c^{2} + 2 \, a c d\right )} f^{5}\right )} g - {\left (16 \, b d^{2} e^{3} f^{2} + 231 \, a c^{2} f^{5} - 22 \, {\left (2 \, b c d + a d^{2}\right )} e^{2} f^{3} + 33 \, {\left (b c^{2} + 2 \, a c d\right )} e f^{4}\right )} h\right )} x^{2} - 11 \, {\left (16 \, b d^{2} e^{4} f - 105 \, a c^{2} e f^{4} - 24 \, {\left (2 \, b c d + a d^{2}\right )} e^{3} f^{2} + 42 \, {\left (b c^{2} + 2 \, a c d\right )} e^{2} f^{3}\right )} g + 2 \, {\left (64 \, b d^{2} e^{5} - 231 \, a c^{2} e^{2} f^{3} - 88 \, {\left (2 \, b c d + a d^{2}\right )} e^{4} f + 132 \, {\left (b c^{2} + 2 \, a c d\right )} e^{3} f^{2}\right )} h + {\left (11 \, {\left (8 \, b d^{2} e^{3} f^{2} + 105 \, a c^{2} f^{5} - 12 \, {\left (2 \, b c d + a d^{2}\right )} e^{2} f^{3} + 21 \, {\left (b c^{2} + 2 \, a c d\right )} e f^{4}\right )} g - {\left (64 \, b d^{2} e^{4} f - 231 \, a c^{2} e f^{4} - 88 \, {\left (2 \, b c d + a d^{2}\right )} e^{3} f^{2} + 132 \, {\left (b c^{2} + 2 \, a c d\right )} e^{2} f^{3}\right )} h\right )} x\right )} \sqrt {f x + e}}{3465 \, f^{5}} \] Input:
integrate((b*x+a)*(d*x+c)^2*(f*x+e)^(1/2)*(h*x+g),x, algorithm="fricas")
Output:
2/3465*(315*b*d^2*f^5*h*x^5 + 35*(11*b*d^2*f^5*g + (b*d^2*e*f^4 + 11*(2*b* c*d + a*d^2)*f^5)*h)*x^4 + 5*(11*(b*d^2*e*f^4 + 9*(2*b*c*d + a*d^2)*f^5)*g - (8*b*d^2*e^2*f^3 - 11*(2*b*c*d + a*d^2)*e*f^4 - 99*(b*c^2 + 2*a*c*d)*f^ 5)*h)*x^3 - 3*(11*(2*b*d^2*e^2*f^3 - 3*(2*b*c*d + a*d^2)*e*f^4 - 21*(b*c^2 + 2*a*c*d)*f^5)*g - (16*b*d^2*e^3*f^2 + 231*a*c^2*f^5 - 22*(2*b*c*d + a*d ^2)*e^2*f^3 + 33*(b*c^2 + 2*a*c*d)*e*f^4)*h)*x^2 - 11*(16*b*d^2*e^4*f - 10 5*a*c^2*e*f^4 - 24*(2*b*c*d + a*d^2)*e^3*f^2 + 42*(b*c^2 + 2*a*c*d)*e^2*f^ 3)*g + 2*(64*b*d^2*e^5 - 231*a*c^2*e^2*f^3 - 88*(2*b*c*d + a*d^2)*e^4*f + 132*(b*c^2 + 2*a*c*d)*e^3*f^2)*h + (11*(8*b*d^2*e^3*f^2 + 105*a*c^2*f^5 - 12*(2*b*c*d + a*d^2)*e^2*f^3 + 21*(b*c^2 + 2*a*c*d)*e*f^4)*g - (64*b*d^2*e ^4*f - 231*a*c^2*e*f^4 - 88*(2*b*c*d + a*d^2)*e^3*f^2 + 132*(b*c^2 + 2*a*c *d)*e^2*f^3)*h)*x)*sqrt(f*x + e)/f^5
Leaf count of result is larger than twice the leaf count of optimal. 620 vs. \(2 (265) = 530\).
Time = 1.65 (sec) , antiderivative size = 620, normalized size of antiderivative = 2.50 \[ \int (a+b x) (c+d x)^2 \sqrt {e+f x} (g+h x) \, dx=\begin {cases} \frac {2 \left (\frac {b d^{2} h \left (e + f x\right )^{\frac {11}{2}}}{11 f^{4}} + \frac {\left (e + f x\right )^{\frac {9}{2}} \left (a d^{2} f h + 2 b c d f h - 4 b d^{2} e h + b d^{2} f g\right )}{9 f^{4}} + \frac {\left (e + f x\right )^{\frac {7}{2}} \cdot \left (2 a c d f^{2} h - 3 a d^{2} e f h + a d^{2} f^{2} g + b c^{2} f^{2} h - 6 b c d e f h + 2 b c d f^{2} g + 6 b d^{2} e^{2} h - 3 b d^{2} e f g\right )}{7 f^{4}} + \frac {\left (e + f x\right )^{\frac {5}{2}} \left (a c^{2} f^{3} h - 4 a c d e f^{2} h + 2 a c d f^{3} g + 3 a d^{2} e^{2} f h - 2 a d^{2} e f^{2} g - 2 b c^{2} e f^{2} h + b c^{2} f^{3} g + 6 b c d e^{2} f h - 4 b c d e f^{2} g - 4 b d^{2} e^{3} h + 3 b d^{2} e^{2} f g\right )}{5 f^{4}} + \frac {\left (e + f x\right )^{\frac {3}{2}} \left (- a c^{2} e f^{3} h + a c^{2} f^{4} g + 2 a c d e^{2} f^{2} h - 2 a c d e f^{3} g - a d^{2} e^{3} f h + a d^{2} e^{2} f^{2} g + b c^{2} e^{2} f^{2} h - b c^{2} e f^{3} g - 2 b c d e^{3} f h + 2 b c d e^{2} f^{2} g + b d^{2} e^{4} h - b d^{2} e^{3} f g\right )}{3 f^{4}}\right )}{f} & \text {for}\: f \neq 0 \\\sqrt {e} \left (a c^{2} g x + \frac {b d^{2} h x^{5}}{5} + \frac {x^{4} \left (a d^{2} h + 2 b c d h + b d^{2} g\right )}{4} + \frac {x^{3} \cdot \left (2 a c d h + a d^{2} g + b c^{2} h + 2 b c d g\right )}{3} + \frac {x^{2} \left (a c^{2} h + 2 a c d g + b c^{2} g\right )}{2}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((b*x+a)*(d*x+c)**2*(f*x+e)**(1/2)*(h*x+g),x)
Output:
Piecewise((2*(b*d**2*h*(e + f*x)**(11/2)/(11*f**4) + (e + f*x)**(9/2)*(a*d **2*f*h + 2*b*c*d*f*h - 4*b*d**2*e*h + b*d**2*f*g)/(9*f**4) + (e + f*x)**( 7/2)*(2*a*c*d*f**2*h - 3*a*d**2*e*f*h + a*d**2*f**2*g + b*c**2*f**2*h - 6* b*c*d*e*f*h + 2*b*c*d*f**2*g + 6*b*d**2*e**2*h - 3*b*d**2*e*f*g)/(7*f**4) + (e + f*x)**(5/2)*(a*c**2*f**3*h - 4*a*c*d*e*f**2*h + 2*a*c*d*f**3*g + 3* a*d**2*e**2*f*h - 2*a*d**2*e*f**2*g - 2*b*c**2*e*f**2*h + b*c**2*f**3*g + 6*b*c*d*e**2*f*h - 4*b*c*d*e*f**2*g - 4*b*d**2*e**3*h + 3*b*d**2*e**2*f*g) /(5*f**4) + (e + f*x)**(3/2)*(-a*c**2*e*f**3*h + a*c**2*f**4*g + 2*a*c*d*e **2*f**2*h - 2*a*c*d*e*f**3*g - a*d**2*e**3*f*h + a*d**2*e**2*f**2*g + b*c **2*e**2*f**2*h - b*c**2*e*f**3*g - 2*b*c*d*e**3*f*h + 2*b*c*d*e**2*f**2*g + b*d**2*e**4*h - b*d**2*e**3*f*g)/(3*f**4))/f, Ne(f, 0)), (sqrt(e)*(a*c* *2*g*x + b*d**2*h*x**5/5 + x**4*(a*d**2*h + 2*b*c*d*h + b*d**2*g)/4 + x**3 *(2*a*c*d*h + a*d**2*g + b*c**2*h + 2*b*c*d*g)/3 + x**2*(a*c**2*h + 2*a*c* d*g + b*c**2*g)/2), True))
Time = 0.05 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.52 \[ \int (a+b x) (c+d x)^2 \sqrt {e+f x} (g+h x) \, dx=\frac {2 \, {\left (315 \, {\left (f x + e\right )}^{\frac {11}{2}} b d^{2} h + 385 \, {\left (b d^{2} f g - {\left (4 \, b d^{2} e - {\left (2 \, b c d + a d^{2}\right )} f\right )} h\right )} {\left (f x + e\right )}^{\frac {9}{2}} - 495 \, {\left ({\left (3 \, b d^{2} e f - {\left (2 \, b c d + a d^{2}\right )} f^{2}\right )} g - {\left (6 \, b d^{2} e^{2} - 3 \, {\left (2 \, b c d + a d^{2}\right )} e f + {\left (b c^{2} + 2 \, a c d\right )} f^{2}\right )} h\right )} {\left (f x + e\right )}^{\frac {7}{2}} + 693 \, {\left ({\left (3 \, b d^{2} e^{2} f - 2 \, {\left (2 \, b c d + a d^{2}\right )} e f^{2} + {\left (b c^{2} + 2 \, a c d\right )} f^{3}\right )} g - {\left (4 \, b d^{2} e^{3} - a c^{2} f^{3} - 3 \, {\left (2 \, b c d + a d^{2}\right )} e^{2} f + 2 \, {\left (b c^{2} + 2 \, a c d\right )} e f^{2}\right )} h\right )} {\left (f x + e\right )}^{\frac {5}{2}} - 1155 \, {\left ({\left (b d^{2} e^{3} f - a c^{2} f^{4} - {\left (2 \, b c d + a d^{2}\right )} e^{2} f^{2} + {\left (b c^{2} + 2 \, a c d\right )} e f^{3}\right )} g - {\left (b d^{2} e^{4} - a c^{2} e f^{3} - {\left (2 \, b c d + a d^{2}\right )} e^{3} f + {\left (b c^{2} + 2 \, a c d\right )} e^{2} f^{2}\right )} h\right )} {\left (f x + e\right )}^{\frac {3}{2}}\right )}}{3465 \, f^{5}} \] Input:
integrate((b*x+a)*(d*x+c)^2*(f*x+e)^(1/2)*(h*x+g),x, algorithm="maxima")
Output:
2/3465*(315*(f*x + e)^(11/2)*b*d^2*h + 385*(b*d^2*f*g - (4*b*d^2*e - (2*b* c*d + a*d^2)*f)*h)*(f*x + e)^(9/2) - 495*((3*b*d^2*e*f - (2*b*c*d + a*d^2) *f^2)*g - (6*b*d^2*e^2 - 3*(2*b*c*d + a*d^2)*e*f + (b*c^2 + 2*a*c*d)*f^2)* h)*(f*x + e)^(7/2) + 693*((3*b*d^2*e^2*f - 2*(2*b*c*d + a*d^2)*e*f^2 + (b* c^2 + 2*a*c*d)*f^3)*g - (4*b*d^2*e^3 - a*c^2*f^3 - 3*(2*b*c*d + a*d^2)*e^2 *f + 2*(b*c^2 + 2*a*c*d)*e*f^2)*h)*(f*x + e)^(5/2) - 1155*((b*d^2*e^3*f - a*c^2*f^4 - (2*b*c*d + a*d^2)*e^2*f^2 + (b*c^2 + 2*a*c*d)*e*f^3)*g - (b*d^ 2*e^4 - a*c^2*e*f^3 - (2*b*c*d + a*d^2)*e^3*f + (b*c^2 + 2*a*c*d)*e^2*f^2) *h)*(f*x + e)^(3/2))/f^5
Leaf count of result is larger than twice the leaf count of optimal. 1147 vs. \(2 (228) = 456\).
Time = 0.13 (sec) , antiderivative size = 1147, normalized size of antiderivative = 4.62 \[ \int (a+b x) (c+d x)^2 \sqrt {e+f x} (g+h x) \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)*(d*x+c)^2*(f*x+e)^(1/2)*(h*x+g),x, algorithm="giac")
Output:
2/3465*(3465*sqrt(f*x + e)*a*c^2*e*g + 1155*((f*x + e)^(3/2) - 3*sqrt(f*x + e)*e)*a*c^2*g + 1155*((f*x + e)^(3/2) - 3*sqrt(f*x + e)*e)*b*c^2*e*g/f + 2310*((f*x + e)^(3/2) - 3*sqrt(f*x + e)*e)*a*c*d*e*g/f + 1155*((f*x + e)^ (3/2) - 3*sqrt(f*x + e)*e)*a*c^2*e*h/f + 462*(3*(f*x + e)^(5/2) - 10*(f*x + e)^(3/2)*e + 15*sqrt(f*x + e)*e^2)*b*c*d*e*g/f^2 + 231*(3*(f*x + e)^(5/2 ) - 10*(f*x + e)^(3/2)*e + 15*sqrt(f*x + e)*e^2)*a*d^2*e*g/f^2 + 231*(3*(f *x + e)^(5/2) - 10*(f*x + e)^(3/2)*e + 15*sqrt(f*x + e)*e^2)*b*c^2*g/f + 4 62*(3*(f*x + e)^(5/2) - 10*(f*x + e)^(3/2)*e + 15*sqrt(f*x + e)*e^2)*a*c*d *g/f + 231*(3*(f*x + e)^(5/2) - 10*(f*x + e)^(3/2)*e + 15*sqrt(f*x + e)*e^ 2)*b*c^2*e*h/f^2 + 462*(3*(f*x + e)^(5/2) - 10*(f*x + e)^(3/2)*e + 15*sqrt (f*x + e)*e^2)*a*c*d*e*h/f^2 + 231*(3*(f*x + e)^(5/2) - 10*(f*x + e)^(3/2) *e + 15*sqrt(f*x + e)*e^2)*a*c^2*h/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)*b*d^2*e*g/f^3 + 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)*b*c*d*g/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)*a*d^2*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*sq rt(f*x + e)*e^3)*b*c*d*e*h/f^3 + 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)*a*d^2*e*h/f^3 + 99*(5 *(f*x + e)^(7/2) - 21*(f*x + e)^(5/2)*e + 35*(f*x + e)^(3/2)*e^2 - 35*s...
Time = 0.07 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.04 \[ \int (a+b x) (c+d x)^2 \sqrt {e+f x} (g+h x) \, dx=\frac {{\left (e+f\,x\right )}^{9/2}\,\left (2\,a\,d^2\,f\,h-8\,b\,d^2\,e\,h+2\,b\,d^2\,f\,g+4\,b\,c\,d\,f\,h\right )}{9\,f^5}+\frac {{\left (e+f\,x\right )}^{7/2}\,\left (2\,a\,d^2\,f^2\,g+2\,b\,c^2\,f^2\,h+12\,b\,d^2\,e^2\,h+4\,a\,c\,d\,f^2\,h+4\,b\,c\,d\,f^2\,g-6\,a\,d^2\,e\,f\,h-6\,b\,d^2\,e\,f\,g-12\,b\,c\,d\,e\,f\,h\right )}{7\,f^5}+\frac {2\,{\left (e+f\,x\right )}^{5/2}\,\left (c\,f-d\,e\right )\,\left (a\,c\,f^2\,h+2\,a\,d\,f^2\,g+b\,c\,f^2\,g+4\,b\,d\,e^2\,h-3\,a\,d\,e\,f\,h-2\,b\,c\,e\,f\,h-3\,b\,d\,e\,f\,g\right )}{5\,f^5}+\frac {2\,b\,d^2\,h\,{\left (e+f\,x\right )}^{11/2}}{11\,f^5}-\frac {2\,{\left (e+f\,x\right )}^{3/2}\,\left (a\,f-b\,e\right )\,{\left (c\,f-d\,e\right )}^2\,\left (e\,h-f\,g\right )}{3\,f^5} \] Input:
int((e + f*x)^(1/2)*(g + h*x)*(a + b*x)*(c + d*x)^2,x)
Output:
((e + f*x)^(9/2)*(2*a*d^2*f*h - 8*b*d^2*e*h + 2*b*d^2*f*g + 4*b*c*d*f*h))/ (9*f^5) + ((e + f*x)^(7/2)*(2*a*d^2*f^2*g + 2*b*c^2*f^2*h + 12*b*d^2*e^2*h + 4*a*c*d*f^2*h + 4*b*c*d*f^2*g - 6*a*d^2*e*f*h - 6*b*d^2*e*f*g - 12*b*c* d*e*f*h))/(7*f^5) + (2*(e + f*x)^(5/2)*(c*f - d*e)*(a*c*f^2*h + 2*a*d*f^2* g + b*c*f^2*g + 4*b*d*e^2*h - 3*a*d*e*f*h - 2*b*c*e*f*h - 3*b*d*e*f*g))/(5 *f^5) + (2*b*d^2*h*(e + f*x)^(11/2))/(11*f^5) - (2*(e + f*x)^(3/2)*(a*f - b*e)*(c*f - d*e)^2*(e*h - f*g))/(3*f^5)
Time = 0.20 (sec) , antiderivative size = 633, normalized size of antiderivative = 2.55 \[ \int (a+b x) (c+d x)^2 \sqrt {e+f x} (g+h x) \, dx=\frac {2 \sqrt {f x +e}\, \left (315 b \,d^{2} f^{5} h \,x^{5}+385 a \,d^{2} f^{5} h \,x^{4}+770 b c d \,f^{5} h \,x^{4}+35 b \,d^{2} e \,f^{4} h \,x^{4}+385 b \,d^{2} f^{5} g \,x^{4}+990 a c d \,f^{5} h \,x^{3}+55 a \,d^{2} e \,f^{4} h \,x^{3}+495 a \,d^{2} f^{5} g \,x^{3}+495 b \,c^{2} f^{5} h \,x^{3}+110 b c d e \,f^{4} h \,x^{3}+990 b c d \,f^{5} g \,x^{3}-40 b \,d^{2} e^{2} f^{3} h \,x^{3}+55 b \,d^{2} e \,f^{4} g \,x^{3}+693 a \,c^{2} f^{5} h \,x^{2}+198 a c d e \,f^{4} h \,x^{2}+1386 a c d \,f^{5} g \,x^{2}-66 a \,d^{2} e^{2} f^{3} h \,x^{2}+99 a \,d^{2} e \,f^{4} g \,x^{2}+99 b \,c^{2} e \,f^{4} h \,x^{2}+693 b \,c^{2} f^{5} g \,x^{2}-132 b c d \,e^{2} f^{3} h \,x^{2}+198 b c d e \,f^{4} g \,x^{2}+48 b \,d^{2} e^{3} f^{2} h \,x^{2}-66 b \,d^{2} e^{2} f^{3} g \,x^{2}+231 a \,c^{2} e \,f^{4} h x +1155 a \,c^{2} f^{5} g x -264 a c d \,e^{2} f^{3} h x +462 a c d e \,f^{4} g x +88 a \,d^{2} e^{3} f^{2} h x -132 a \,d^{2} e^{2} f^{3} g x -132 b \,c^{2} e^{2} f^{3} h x +231 b \,c^{2} e \,f^{4} g x +176 b c d \,e^{3} f^{2} h x -264 b c d \,e^{2} f^{3} g x -64 b \,d^{2} e^{4} f h x +88 b \,d^{2} e^{3} f^{2} g x -462 a \,c^{2} e^{2} f^{3} h +1155 a \,c^{2} e \,f^{4} g +528 a c d \,e^{3} f^{2} h -924 a c d \,e^{2} f^{3} g -176 a \,d^{2} e^{4} f h +264 a \,d^{2} e^{3} f^{2} g +264 b \,c^{2} e^{3} f^{2} h -462 b \,c^{2} e^{2} f^{3} g -352 b c d \,e^{4} f h +528 b c d \,e^{3} f^{2} g +128 b \,d^{2} e^{5} h -176 b \,d^{2} e^{4} f g \right )}{3465 f^{5}} \] Input:
int((b*x+a)*(d*x+c)^2*(f*x+e)^(1/2)*(h*x+g),x)
Output:
(2*sqrt(e + f*x)*( - 462*a*c**2*e**2*f**3*h + 1155*a*c**2*e*f**4*g + 231*a *c**2*e*f**4*h*x + 1155*a*c**2*f**5*g*x + 693*a*c**2*f**5*h*x**2 + 528*a*c *d*e**3*f**2*h - 924*a*c*d*e**2*f**3*g - 264*a*c*d*e**2*f**3*h*x + 462*a*c *d*e*f**4*g*x + 198*a*c*d*e*f**4*h*x**2 + 1386*a*c*d*f**5*g*x**2 + 990*a*c *d*f**5*h*x**3 - 176*a*d**2*e**4*f*h + 264*a*d**2*e**3*f**2*g + 88*a*d**2* e**3*f**2*h*x - 132*a*d**2*e**2*f**3*g*x - 66*a*d**2*e**2*f**3*h*x**2 + 99 *a*d**2*e*f**4*g*x**2 + 55*a*d**2*e*f**4*h*x**3 + 495*a*d**2*f**5*g*x**3 + 385*a*d**2*f**5*h*x**4 + 264*b*c**2*e**3*f**2*h - 462*b*c**2*e**2*f**3*g - 132*b*c**2*e**2*f**3*h*x + 231*b*c**2*e*f**4*g*x + 99*b*c**2*e*f**4*h*x* *2 + 693*b*c**2*f**5*g*x**2 + 495*b*c**2*f**5*h*x**3 - 352*b*c*d*e**4*f*h + 528*b*c*d*e**3*f**2*g + 176*b*c*d*e**3*f**2*h*x - 264*b*c*d*e**2*f**3*g* x - 132*b*c*d*e**2*f**3*h*x**2 + 198*b*c*d*e*f**4*g*x**2 + 110*b*c*d*e*f** 4*h*x**3 + 990*b*c*d*f**5*g*x**3 + 770*b*c*d*f**5*h*x**4 + 128*b*d**2*e**5 *h - 176*b*d**2*e**4*f*g - 64*b*d**2*e**4*f*h*x + 88*b*d**2*e**3*f**2*g*x + 48*b*d**2*e**3*f**2*h*x**2 - 66*b*d**2*e**2*f**3*g*x**2 - 40*b*d**2*e**2 *f**3*h*x**3 + 55*b*d**2*e*f**4*g*x**3 + 35*b*d**2*e*f**4*h*x**4 + 385*b*d **2*f**5*g*x**4 + 315*b*d**2*f**5*h*x**5))/(3465*f**5)