Integrand size = 29, antiderivative size = 427 \[ \int \frac {(a+b x)^3 \sqrt {e+f x} (g+h x)}{c+d x} \, dx=-\frac {2 (b c-a d)^3 (d g-c h) \sqrt {e+f x}}{d^5}+\frac {2 \left (a^3 d^3 f^3 h+3 a^2 b d^2 f^2 (d f g-d e h-c f h)+3 a b^2 d f \left (c^2 f^2 h-d^2 e (f g-e h)-c d f (f g-e h)\right )-b^3 \left (c^3 f^3 h-d^3 e^2 (f g-e h)-c d^2 e f (f g-e h)-c^2 d f^2 (f g-e h)\right )\right ) (e+f x)^{3/2}}{3 d^4 f^4}+\frac {2 b \left (3 a^2 d^2 f^2 h+3 a b d f (d f g-2 d e h-c f h)+b^2 \left (c^2 f^2 h-d^2 e (2 f g-3 e h)-c d f (f g-2 e h)\right )\right ) (e+f x)^{5/2}}{5 d^3 f^4}+\frac {2 b^2 (3 a d f h+b (d f g-3 d e h-c f h)) (e+f x)^{7/2}}{7 d^2 f^4}+\frac {2 b^3 h (e+f x)^{9/2}}{9 d f^4}+\frac {2 (b c-a d)^3 \sqrt {d e-c f} (d g-c h) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{11/2}} \] Output:
-2*(-a*d+b*c)^3*(-c*h+d*g)*(f*x+e)^(1/2)/d^5+2/3*(a^3*d^3*f^3*h+3*a^2*b*d^ 2*f^2*(-c*f*h-d*e*h+d*f*g)+3*a*b^2*d*f*(c^2*f^2*h-d^2*e*(-e*h+f*g)-c*d*f*( -e*h+f*g))-b^3*(c^3*f^3*h-d^3*e^2*(-e*h+f*g)-c*d^2*e*f*(-e*h+f*g)-c^2*d*f^ 2*(-e*h+f*g)))*(f*x+e)^(3/2)/d^4/f^4+2/5*b*(3*a^2*d^2*f^2*h+3*a*b*d*f*(-c* f*h-2*d*e*h+d*f*g)+b^2*(c^2*f^2*h-d^2*e*(-3*e*h+2*f*g)-c*d*f*(-2*e*h+f*g)) )*(f*x+e)^(5/2)/d^3/f^4+2/7*b^2*(3*a*d*f*h+b*(-c*f*h-3*d*e*h+d*f*g))*(f*x+ e)^(7/2)/d^2/f^4+2/9*b^3*h*(f*x+e)^(9/2)/d/f^4+2*(-a*d+b*c)^3*(-c*f+d*e)^( 1/2)*(-c*h+d*g)*arctanh(d^(1/2)*(f*x+e)^(1/2)/(-c*f+d*e)^(1/2))/d^(11/2)
Time = 0.77 (sec) , antiderivative size = 456, normalized size of antiderivative = 1.07 \[ \int \frac {(a+b x)^3 \sqrt {e+f x} (g+h x)}{c+d x} \, dx=\frac {2 \sqrt {e+f x} \left (105 a^3 d^3 f^3 (-3 c f h+d (3 f g+e h+f h x))+63 a^2 b d^2 f^2 \left (15 c^2 f^2 h-d^2 (e+f x) (-5 f g+2 e h-3 f h x)-5 c d f (3 f g+e h+f h x)\right )+9 a b^2 d f \left (-105 c^3 f^3 h+35 c^2 d f^2 (3 f g+e h+f h x)-7 c d^2 f (e+f x) (5 f g-2 e h+3 f h x)+d^3 (e+f x) \left (8 e^2 h+3 f^2 x (7 g+5 h x)-2 e f (7 g+6 h x)\right )\right )+b^3 \left (315 c^4 f^4 h-105 c^3 d f^3 (3 f g+e h+f h x)+21 c^2 d^2 f^2 (e+f x) (5 f g-2 e h+3 f h x)-3 c d^3 f (e+f x) \left (8 e^2 h+3 f^2 x (7 g+5 h x)-2 e f (7 g+6 h x)\right )-d^4 (e+f x) \left (16 e^3 h-24 e^2 f (g+h x)+6 e f^2 x (6 g+5 h x)-5 f^3 x^2 (9 g+7 h x)\right )\right )\right )}{315 d^5 f^4}-\frac {2 (-b c+a d)^3 \sqrt {-d e+c f} (d g-c h) \arctan \left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{d^{11/2}} \] Input:
Integrate[((a + b*x)^3*Sqrt[e + f*x]*(g + h*x))/(c + d*x),x]
Output:
(2*Sqrt[e + f*x]*(105*a^3*d^3*f^3*(-3*c*f*h + d*(3*f*g + e*h + f*h*x)) + 6 3*a^2*b*d^2*f^2*(15*c^2*f^2*h - d^2*(e + f*x)*(-5*f*g + 2*e*h - 3*f*h*x) - 5*c*d*f*(3*f*g + e*h + f*h*x)) + 9*a*b^2*d*f*(-105*c^3*f^3*h + 35*c^2*d*f ^2*(3*f*g + e*h + f*h*x) - 7*c*d^2*f*(e + f*x)*(5*f*g - 2*e*h + 3*f*h*x) + d^3*(e + f*x)*(8*e^2*h + 3*f^2*x*(7*g + 5*h*x) - 2*e*f*(7*g + 6*h*x))) + b^3*(315*c^4*f^4*h - 105*c^3*d*f^3*(3*f*g + e*h + f*h*x) + 21*c^2*d^2*f^2* (e + f*x)*(5*f*g - 2*e*h + 3*f*h*x) - 3*c*d^3*f*(e + f*x)*(8*e^2*h + 3*f^2 *x*(7*g + 5*h*x) - 2*e*f*(7*g + 6*h*x)) - d^4*(e + f*x)*(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)))))/(315* d^5*f^4) - (2*(-(b*c) + a*d)^3*Sqrt[-(d*e) + c*f]*(d*g - c*h)*ArcTan[(Sqrt [d]*Sqrt[e + f*x])/Sqrt[-(d*e) + c*f]])/d^(11/2)
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.
| \(\displaystyle \int \frac {(a+b x)^3 \sqrt {e+f x} (g+h x)}{c+d x} \, dx\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle \frac {2 \int \frac {3 (a+b x)^2 \sqrt {e+f x} (3 a d f g-2 b c e h-a c f h+(2 a d f h+b (3 d f g-2 d e h-3 c f h)) x)}{2 (c+d x)}dx}{9 d f}+\frac {2 h (a+b x)^3 (e+f x)^{3/2}}{9 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int -\frac {(a+b x)^2 \sqrt {e+f x} (2 b c e h-a f (3 d g-c h)-(2 a d f h+b (3 d f g-2 d e h-3 c f h)) x)}{c+d x}dx}{3 d f}+\frac {2 h (a+b x)^3 (e+f x)^{3/2}}{9 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 h (a+b x)^3 (e+f x)^{3/2}}{9 d f}-\frac {\int -\frac {(a+b x)^2 \sqrt {e+f x} (3 a d f g-2 b c e h-a c f h+(2 a d f h+b (3 d f g-2 d e h-3 c f h)) x)}{c+d x}dx}{3 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(a+b x)^2 \sqrt {e+f x} (2 b c e h-a f (3 d g-c h)-(2 a d f h+b (3 d f g-2 d e h-3 c f h)) x)}{c+d x}dx}{3 d f}+\frac {2 h (a+b x)^3 (e+f x)^{3/2}}{9 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 h (a+b x)^3 (e+f x)^{3/2}}{9 d f}-\frac {\int -\frac {(a+b x)^2 \sqrt {e+f x} (3 a d f g-2 b c e h-a c f h+(2 a d f h+b (3 d f g-2 d e h-3 c f h)) x)}{c+d x}dx}{3 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(a+b x)^2 \sqrt {e+f x} (2 b c e h-a f (3 d g-c h)-(2 a d f h+b (3 d f g-2 d e h-3 c f h)) x)}{c+d x}dx}{3 d f}+\frac {2 h (a+b x)^3 (e+f x)^{3/2}}{9 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 h (a+b x)^3 (e+f x)^{3/2}}{9 d f}-\frac {\int -\frac {(a+b x)^2 \sqrt {e+f x} (3 a d f g-2 b c e h-a c f h+(2 a d f h+b (3 d f g-2 d e h-3 c f h)) x)}{c+d x}dx}{3 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(a+b x)^2 \sqrt {e+f x} (2 b c e h-a f (3 d g-c h)-(2 a d f h+b (3 d f g-2 d e h-3 c f h)) x)}{c+d x}dx}{3 d f}+\frac {2 h (a+b x)^3 (e+f x)^{3/2}}{9 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 h (a+b x)^3 (e+f x)^{3/2}}{9 d f}-\frac {\int -\frac {(a+b x)^2 \sqrt {e+f x} (3 a d f g-2 b c e h-a c f h+(2 a d f h+b (3 d f g-2 d e h-3 c f h)) x)}{c+d x}dx}{3 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(a+b x)^2 \sqrt {e+f x} (2 b c e h-a f (3 d g-c h)-(2 a d f h+b (3 d f g-2 d e h-3 c f h)) x)}{c+d x}dx}{3 d f}+\frac {2 h (a+b x)^3 (e+f x)^{3/2}}{9 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 h (a+b x)^3 (e+f x)^{3/2}}{9 d f}-\frac {\int -\frac {(a+b x)^2 \sqrt {e+f x} (3 a d f g-2 b c e h-a c f h+(2 a d f h+b (3 d f g-2 d e h-3 c f h)) x)}{c+d x}dx}{3 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(a+b x)^2 \sqrt {e+f x} (2 b c e h-a f (3 d g-c h)-(2 a d f h+b (3 d f g-2 d e h-3 c f h)) x)}{c+d x}dx}{3 d f}+\frac {2 h (a+b x)^3 (e+f x)^{3/2}}{9 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 h (a+b x)^3 (e+f x)^{3/2}}{9 d f}-\frac {\int -\frac {(a+b x)^2 \sqrt {e+f x} (3 a d f g-2 b c e h-a c f h+(2 a d f h+b (3 d f g-2 d e h-3 c f h)) x)}{c+d x}dx}{3 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(a+b x)^2 \sqrt {e+f x} (2 b c e h-a f (3 d g-c h)-(2 a d f h+b (3 d f g-2 d e h-3 c f h)) x)}{c+d x}dx}{3 d f}+\frac {2 h (a+b x)^3 (e+f x)^{3/2}}{9 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 h (a+b x)^3 (e+f x)^{3/2}}{9 d f}-\frac {\int -\frac {(a+b x)^2 \sqrt {e+f x} (3 a d f g-2 b c e h-a c f h+(2 a d f h+b (3 d f g-2 d e h-3 c f h)) x)}{c+d x}dx}{3 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(a+b x)^2 \sqrt {e+f x} (2 b c e h-a f (3 d g-c h)-(2 a d f h+b (3 d f g-2 d e h-3 c f h)) x)}{c+d x}dx}{3 d f}+\frac {2 h (a+b x)^3 (e+f x)^{3/2}}{9 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 h (a+b x)^3 (e+f x)^{3/2}}{9 d f}-\frac {\int -\frac {(a+b x)^2 \sqrt {e+f x} (3 a d f g-2 b c e h-a c f h+(2 a d f h+b (3 d f g-2 d e h-3 c f h)) x)}{c+d x}dx}{3 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(a+b x)^2 \sqrt {e+f x} (2 b c e h-a f (3 d g-c h)-(2 a d f h+b (3 d f g-2 d e h-3 c f h)) x)}{c+d x}dx}{3 d f}+\frac {2 h (a+b x)^3 (e+f x)^{3/2}}{9 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 h (a+b x)^3 (e+f x)^{3/2}}{9 d f}-\frac {\int -\frac {(a+b x)^2 \sqrt {e+f x} (3 a d f g-2 b c e h-a c f h+(2 a d f h+b (3 d f g-2 d e h-3 c f h)) x)}{c+d x}dx}{3 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(a+b x)^2 \sqrt {e+f x} (2 b c e h-a f (3 d g-c h)-(2 a d f h+b (3 d f g-2 d e h-3 c f h)) x)}{c+d x}dx}{3 d f}+\frac {2 h (a+b x)^3 (e+f x)^{3/2}}{9 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 h (a+b x)^3 (e+f x)^{3/2}}{9 d f}-\frac {\int -\frac {(a+b x)^2 \sqrt {e+f x} (3 a d f g-2 b c e h-a c f h+(2 a d f h+b (3 d f g-2 d e h-3 c f h)) x)}{c+d x}dx}{3 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(a+b x)^2 \sqrt {e+f x} (2 b c e h-a f (3 d g-c h)-(2 a d f h+b (3 d f g-2 d e h-3 c f h)) x)}{c+d x}dx}{3 d f}+\frac {2 h (a+b x)^3 (e+f x)^{3/2}}{9 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 h (a+b x)^3 (e+f x)^{3/2}}{9 d f}-\frac {\int -\frac {(a+b x)^2 \sqrt {e+f x} (3 a d f g-2 b c e h-a c f h+(2 a d f h+b (3 d f g-2 d e h-3 c f h)) x)}{c+d x}dx}{3 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(a+b x)^2 \sqrt {e+f x} (2 b c e h-a f (3 d g-c h)-(2 a d f h+b (3 d f g-2 d e h-3 c f h)) x)}{c+d x}dx}{3 d f}+\frac {2 h (a+b x)^3 (e+f x)^{3/2}}{9 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 h (a+b x)^3 (e+f x)^{3/2}}{9 d f}-\frac {\int -\frac {(a+b x)^2 \sqrt {e+f x} (3 a d f g-2 b c e h-a c f h+(2 a d f h+b (3 d f g-2 d e h-3 c f h)) x)}{c+d x}dx}{3 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(a+b x)^2 \sqrt {e+f x} (2 b c e h-a f (3 d g-c h)-(2 a d f h+b (3 d f g-2 d e h-3 c f h)) x)}{c+d x}dx}{3 d f}+\frac {2 h (a+b x)^3 (e+f x)^{3/2}}{9 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 h (a+b x)^3 (e+f x)^{3/2}}{9 d f}-\frac {\int -\frac {(a+b x)^2 \sqrt {e+f x} (3 a d f g-2 b c e h-a c f h+(2 a d f h+b (3 d f g-2 d e h-3 c f h)) x)}{c+d x}dx}{3 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(a+b x)^2 \sqrt {e+f x} (2 b c e h-a f (3 d g-c h)-(2 a d f h+b (3 d f g-2 d e h-3 c f h)) x)}{c+d x}dx}{3 d f}+\frac {2 h (a+b x)^3 (e+f x)^{3/2}}{9 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 h (a+b x)^3 (e+f x)^{3/2}}{9 d f}-\frac {\int -\frac {(a+b x)^2 \sqrt {e+f x} (3 a d f g-2 b c e h-a c f h+(2 a d f h+b (3 d f g-2 d e h-3 c f h)) x)}{c+d x}dx}{3 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(a+b x)^2 \sqrt {e+f x} (2 b c e h-a f (3 d g-c h)-(2 a d f h+b (3 d f g-2 d e h-3 c f h)) x)}{c+d x}dx}{3 d f}+\frac {2 h (a+b x)^3 (e+f x)^{3/2}}{9 d f}\) |
Input:
Int[((a + b*x)^3*Sqrt[e + f*x]*(g + h*x))/(c + d*x),x]
Output:
$Aborted
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegerQ[m]
Time = 0.58 (sec) , antiderivative size = 513, normalized size of antiderivative = 1.20
| method | result | size |
| pseudoelliptic | \(-\frac {2 \left (-f^{4} \left (a d -b c \right )^{3} \left (c h -d g \right ) \left (c f -d e \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )+\left (\left (\left (-\frac {x^{3} \left (\frac {7 h x}{9}+g \right ) b^{3}}{7}-\frac {3 a \,x^{2} \left (\frac {5 h x}{7}+g \right ) b^{2}}{5}-a^{2} x \left (\frac {3 h x}{5}+g \right ) b -a^{3} \left (\frac {h x}{3}+g \right )\right ) d^{4}+c \left (\frac {x^{2} \left (\frac {5 h x}{7}+g \right ) b^{3}}{5}+a x \left (\frac {3 h x}{5}+g \right ) b^{2}+3 a^{2} \left (\frac {h x}{3}+g \right ) b +h \,a^{3}\right ) d^{3}-3 c^{2} \left (\frac {x \left (\frac {3 h x}{5}+g \right ) b^{2}}{9}+a \left (\frac {h x}{3}+g \right ) b +a^{2} h \right ) b \,d^{2}+3 \left (\frac {\left (\frac {h x}{3}+g \right ) b}{3}+a h \right ) c^{3} b^{2} d -c^{4} h \,b^{3}\right ) f^{4}-\frac {\left (\left (\frac {3 x^{2} \left (\frac {5 h x}{9}+g \right ) b^{3}}{35}+\frac {3 a \left (\frac {3 h x}{7}+g \right ) x \,b^{2}}{5}+3 a^{2} \left (\frac {h x}{5}+g \right ) b +h \,a^{3}\right ) d^{3}-3 c b \left (\frac {\left (\frac {3 h x}{7}+g \right ) x \,b^{2}}{15}+a \left (\frac {h x}{5}+g \right ) b +a^{2} h \right ) d^{2}+3 c^{2} b^{2} \left (\frac {\left (\frac {h x}{5}+g \right ) b}{3}+a h \right ) d -b^{3} c^{3} h \right ) d e \,f^{3}}{3}+\frac {2 d^{2} b \left (\left (\frac {2 x \left (\frac {h x}{2}+g \right ) b^{2}}{21}+a \left (\frac {2 h x}{7}+g \right ) b +a^{2} h \right ) d^{2}-c \left (\frac {\left (\frac {2 h x}{7}+g \right ) b}{3}+a h \right ) b d +\frac {b^{2} c^{2} h}{3}\right ) e^{2} f^{2}}{5}-\frac {8 d^{3} \left (\left (\frac {\left (\frac {h x}{3}+g \right ) b}{3}+a h \right ) d -\frac {b c h}{3}\right ) b^{2} e^{3} f}{35}+\frac {16 b^{3} d^{4} e^{4} h}{315}\right ) \sqrt {f x +e}\, \sqrt {\left (c f -d e \right ) d}\right )}{\sqrt {\left (c f -d e \right ) d}\, f^{4} d^{5}}\) | \(513\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1008\) |
| default | \(\text {Expression too large to display}\) | \(1008\) |
| risch | \(\text {Expression too large to display}\) | \(1077\) |
Input:
int((b*x+a)^3*(f*x+e)^(1/2)*(h*x+g)/(d*x+c),x,method=_RETURNVERBOSE)
Output:
-2/((c*f-d*e)*d)^(1/2)*(-f^4*(a*d-b*c)^3*(c*h-d*g)*(c*f-d*e)*arctan(d*(f*x +e)^(1/2)/((c*f-d*e)*d)^(1/2))+(((-1/7*x^3*(7/9*h*x+g)*b^3-3/5*a*x^2*(5/7* h*x+g)*b^2-a^2*x*(3/5*h*x+g)*b-a^3*(1/3*h*x+g))*d^4+c*(1/5*x^2*(5/7*h*x+g) *b^3+a*x*(3/5*h*x+g)*b^2+3*a^2*(1/3*h*x+g)*b+h*a^3)*d^3-3*c^2*(1/9*x*(3/5* h*x+g)*b^2+a*(1/3*h*x+g)*b+a^2*h)*b*d^2+3*(1/3*(1/3*h*x+g)*b+a*h)*c^3*b^2* d-c^4*h*b^3)*f^4-1/3*((3/35*x^2*(5/9*h*x+g)*b^3+3/5*a*(3/7*h*x+g)*x*b^2+3* a^2*(1/5*h*x+g)*b+h*a^3)*d^3-3*c*b*(1/15*(3/7*h*x+g)*x*b^2+a*(1/5*h*x+g)*b +a^2*h)*d^2+3*c^2*b^2*(1/3*(1/5*h*x+g)*b+a*h)*d-b^3*c^3*h)*d*e*f^3+2/5*d^2 *b*((2/21*x*(1/2*h*x+g)*b^2+a*(2/7*h*x+g)*b+a^2*h)*d^2-c*(1/3*(2/7*h*x+g)* b+a*h)*b*d+1/3*b^2*c^2*h)*e^2*f^2-8/35*d^3*((1/3*(1/3*h*x+g)*b+a*h)*d-1/3* b*c*h)*b^2*e^3*f+16/315*b^3*d^4*e^4*h)*(f*x+e)^(1/2)*((c*f-d*e)*d)^(1/2))/ f^4/d^5
Leaf count of result is larger than twice the leaf count of optimal. 843 vs. \(2 (399) = 798\).
Time = 0.10 (sec) , antiderivative size = 1697, normalized size of antiderivative = 3.97 \[ \int \frac {(a+b x)^3 \sqrt {e+f x} (g+h x)}{c+d x} \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^3*(f*x+e)^(1/2)*(h*x+g)/(d*x+c),x, algorithm="fricas")
Output:
[1/315*(315*((b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*f^4*g - (b^3*c^4 - 3*a*b^2*c^3*d + 3*a^2*b*c^2*d^2 - a^3*c*d^3)*f^4*h)*sqrt((d* e - c*f)/d)*log((d*f*x + 2*d*e - c*f + 2*sqrt(f*x + e)*d*sqrt((d*e - c*f)/ d))/(d*x + c)) + 2*(35*b^3*d^4*f^4*h*x^4 + 5*(9*b^3*d^4*f^4*g + (b^3*d^4*e *f^3 - 9*(b^3*c*d^3 - 3*a*b^2*d^4)*f^4)*h)*x^3 + 3*(3*(b^3*d^4*e*f^3 - 7*( b^3*c*d^3 - 3*a*b^2*d^4)*f^4)*g - (2*b^3*d^4*e^2*f^2 + 3*(b^3*c*d^3 - 3*a* b^2*d^4)*e*f^3 - 21*(b^3*c^2*d^2 - 3*a*b^2*c*d^3 + 3*a^2*b*d^4)*f^4)*h)*x^ 2 + 3*(8*b^3*d^4*e^3*f + 14*(b^3*c*d^3 - 3*a*b^2*d^4)*e^2*f^2 + 35*(b^3*c^ 2*d^2 - 3*a*b^2*c*d^3 + 3*a^2*b*d^4)*e*f^3 - 105*(b^3*c^3*d - 3*a*b^2*c^2* d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*f^4)*g - (16*b^3*d^4*e^4 + 24*(b^3*c*d^3 - 3*a*b^2*d^4)*e^3*f + 42*(b^3*c^2*d^2 - 3*a*b^2*c*d^3 + 3*a^2*b*d^4)*e^2*f^ 2 + 105*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*e*f^3 - 31 5*(b^3*c^4 - 3*a*b^2*c^3*d + 3*a^2*b*c^2*d^2 - a^3*c*d^3)*f^4)*h - (3*(4*b ^3*d^4*e^2*f^2 + 7*(b^3*c*d^3 - 3*a*b^2*d^4)*e*f^3 - 35*(b^3*c^2*d^2 - 3*a *b^2*c*d^3 + 3*a^2*b*d^4)*f^4)*g - (8*b^3*d^4*e^3*f + 12*(b^3*c*d^3 - 3*a* b^2*d^4)*e^2*f^2 + 21*(b^3*c^2*d^2 - 3*a*b^2*c*d^3 + 3*a^2*b*d^4)*e*f^3 - 105*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*f^4)*h)*x)*sqr t(f*x + e))/(d^5*f^4), 2/315*(315*((b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b* c*d^3 - a^3*d^4)*f^4*g - (b^3*c^4 - 3*a*b^2*c^3*d + 3*a^2*b*c^2*d^2 - a^3* c*d^3)*f^4*h)*sqrt(-(d*e - c*f)/d)*arctan(-sqrt(f*x + e)*d*sqrt(-(d*e -...
Leaf count of result is larger than twice the leaf count of optimal. 899 vs. \(2 (418) = 836\).
Time = 23.46 (sec) , antiderivative size = 899, normalized size of antiderivative = 2.11 \[ \int \frac {(a+b x)^3 \sqrt {e+f x} (g+h x)}{c+d x} \, dx =\text {Too large to display} \] Input:
integrate((b*x+a)**3*(f*x+e)**(1/2)*(h*x+g)/(d*x+c),x)
Output:
Piecewise((2*(b**3*h*(e + f*x)**(9/2)/(9*d*f**3) + (e + f*x)**(7/2)*(3*a*b **2*d*f*h - b**3*c*f*h - 3*b**3*d*e*h + b**3*d*f*g)/(7*d**2*f**3) + (e + f *x)**(5/2)*(3*a**2*b*d**2*f**2*h - 3*a*b**2*c*d*f**2*h - 6*a*b**2*d**2*e*f *h + 3*a*b**2*d**2*f**2*g + b**3*c**2*f**2*h + 2*b**3*c*d*e*f*h - b**3*c*d *f**2*g + 3*b**3*d**2*e**2*h - 2*b**3*d**2*e*f*g)/(5*d**3*f**3) + (e + f*x )**(3/2)*(a**3*d**3*f**3*h - 3*a**2*b*c*d**2*f**3*h - 3*a**2*b*d**3*e*f**2 *h + 3*a**2*b*d**3*f**3*g + 3*a*b**2*c**2*d*f**3*h + 3*a*b**2*c*d**2*e*f** 2*h - 3*a*b**2*c*d**2*f**3*g + 3*a*b**2*d**3*e**2*f*h - 3*a*b**2*d**3*e*f* *2*g - b**3*c**3*f**3*h - b**3*c**2*d*e*f**2*h + b**3*c**2*d*f**3*g - b**3 *c*d**2*e**2*f*h + b**3*c*d**2*e*f**2*g - b**3*d**3*e**3*h + b**3*d**3*e** 2*f*g)/(3*d**4*f**3) + sqrt(e + f*x)*(-a**3*c*d**3*f*h + a**3*d**4*f*g + 3 *a**2*b*c**2*d**2*f*h - 3*a**2*b*c*d**3*f*g - 3*a*b**2*c**3*d*f*h + 3*a*b* *2*c**2*d**2*f*g + b**3*c**4*f*h - b**3*c**3*d*f*g)/d**5 + f*(a*d - b*c)** 3*(c*f - d*e)*(c*h - d*g)*atan(sqrt(e + f*x)/sqrt((c*f - d*e)/d))/(d**6*sq rt((c*f - d*e)/d)))/f, Ne(f, 0)), (sqrt(e)*(b**3*h*x**4/(4*d) + x**3*(3*a* b**2*d*h - b**3*c*h + b**3*d*g)/(3*d**2) + x**2*(3*a**2*b*d**2*h - 3*a*b** 2*c*d*h + 3*a*b**2*d**2*g + b**3*c**2*h - b**3*c*d*g)/(2*d**3) + x*(a**3*d **3*h - 3*a**2*b*c*d**2*h + 3*a**2*b*d**3*g + 3*a*b**2*c**2*d*h - 3*a*b**2 *c*d**2*g - b**3*c**3*h + b**3*c**2*d*g)/d**4 - (a*d - b*c)**3*(c*h - d*g) *Piecewise((x/c, Eq(d, 0)), (log(c + d*x)/d, True))/d**4), True))
Exception generated. \[ \int \frac {(a+b x)^3 \sqrt {e+f x} (g+h x)}{c+d x} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((b*x+a)^3*(f*x+e)^(1/2)*(h*x+g)/(d*x+c),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(c*f-d*e>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 1050 vs. \(2 (399) = 798\).
Time = 0.15 (sec) , antiderivative size = 1050, normalized size of antiderivative = 2.46 \[ \int \frac {(a+b x)^3 \sqrt {e+f x} (g+h x)}{c+d x} \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^3*(f*x+e)^(1/2)*(h*x+g)/(d*x+c),x, algorithm="giac")
Output:
-2*(b^3*c^3*d^2*e*g - 3*a*b^2*c^2*d^3*e*g + 3*a^2*b*c*d^4*e*g - a^3*d^5*e* g - b^3*c^4*d*f*g + 3*a*b^2*c^3*d^2*f*g - 3*a^2*b*c^2*d^3*f*g + a^3*c*d^4* f*g - b^3*c^4*d*e*h + 3*a*b^2*c^3*d^2*e*h - 3*a^2*b*c^2*d^3*e*h + a^3*c*d^ 4*e*h + b^3*c^5*f*h - 3*a*b^2*c^4*d*f*h + 3*a^2*b*c^3*d^2*f*h - a^3*c^2*d^ 3*f*h)*arctan(sqrt(f*x + e)*d/sqrt(-d^2*e + c*d*f))/(sqrt(-d^2*e + c*d*f)* d^5) + 2/315*(45*(f*x + e)^(7/2)*b^3*d^8*f^33*g - 126*(f*x + e)^(5/2)*b^3* d^8*e*f^33*g + 105*(f*x + e)^(3/2)*b^3*d^8*e^2*f^33*g - 63*(f*x + e)^(5/2) *b^3*c*d^7*f^34*g + 189*(f*x + e)^(5/2)*a*b^2*d^8*f^34*g + 105*(f*x + e)^( 3/2)*b^3*c*d^7*e*f^34*g - 315*(f*x + e)^(3/2)*a*b^2*d^8*e*f^34*g + 105*(f* x + e)^(3/2)*b^3*c^2*d^6*f^35*g - 315*(f*x + e)^(3/2)*a*b^2*c*d^7*f^35*g + 315*(f*x + e)^(3/2)*a^2*b*d^8*f^35*g - 315*sqrt(f*x + e)*b^3*c^3*d^5*f^36 *g + 945*sqrt(f*x + e)*a*b^2*c^2*d^6*f^36*g - 945*sqrt(f*x + e)*a^2*b*c*d^ 7*f^36*g + 315*sqrt(f*x + e)*a^3*d^8*f^36*g + 35*(f*x + e)^(9/2)*b^3*d^8*f ^32*h - 135*(f*x + e)^(7/2)*b^3*d^8*e*f^32*h + 189*(f*x + e)^(5/2)*b^3*d^8 *e^2*f^32*h - 105*(f*x + e)^(3/2)*b^3*d^8*e^3*f^32*h - 45*(f*x + e)^(7/2)* b^3*c*d^7*f^33*h + 135*(f*x + e)^(7/2)*a*b^2*d^8*f^33*h + 126*(f*x + e)^(5 /2)*b^3*c*d^7*e*f^33*h - 378*(f*x + e)^(5/2)*a*b^2*d^8*e*f^33*h - 105*(f*x + e)^(3/2)*b^3*c*d^7*e^2*f^33*h + 315*(f*x + e)^(3/2)*a*b^2*d^8*e^2*f^33* h + 63*(f*x + e)^(5/2)*b^3*c^2*d^6*f^34*h - 189*(f*x + e)^(5/2)*a*b^2*c*d^ 7*f^34*h + 189*(f*x + e)^(5/2)*a^2*b*d^8*f^34*h - 105*(f*x + e)^(3/2)*b...
Time = 2.10 (sec) , antiderivative size = 865, normalized size of antiderivative = 2.03 \[ \int \frac {(a+b x)^3 \sqrt {e+f x} (g+h x)}{c+d x} \, dx =\text {Too large to display} \] Input:
int(((e + f*x)^(1/2)*(g + h*x)*(a + b*x)^3)/(c + d*x),x)
Output:
(e + f*x)^(7/2)*((2*b^3*f*g - 8*b^3*e*h + 6*a*b^2*f*h)/(7*d*f^4) - (2*b^3* h*(c*f^5 - d*e*f^4))/(7*d^2*f^8)) - (e + f*x)^(1/2)*((2*(a*f - b*e)^3*(e*h - f*g))/(d*f^4) + ((c*f^5 - d*e*f^4)*(((c*f^5 - d*e*f^4)*((((2*b^3*f*g - 8*b^3*e*h + 6*a*b^2*f*h)/(d*f^4) - (2*b^3*h*(c*f^5 - d*e*f^4))/(d^2*f^8))* (c*f^5 - d*e*f^4))/(d*f^4) - (6*b*(a*f - b*e)*(a*f*h - 2*b*e*h + b*f*g))/( d*f^4)))/(d*f^4) + (2*(a*f - b*e)^2*(a*f*h - 4*b*e*h + 3*b*f*g))/(d*f^4))) /(d*f^4)) - (e + f*x)^(5/2)*((((2*b^3*f*g - 8*b^3*e*h + 6*a*b^2*f*h)/(d*f^ 4) - (2*b^3*h*(c*f^5 - d*e*f^4))/(d^2*f^8))*(c*f^5 - d*e*f^4))/(5*d*f^4) - (6*b*(a*f - b*e)*(a*f*h - 2*b*e*h + b*f*g))/(5*d*f^4)) + (e + f*x)^(3/2)* (((c*f^5 - d*e*f^4)*((((2*b^3*f*g - 8*b^3*e*h + 6*a*b^2*f*h)/(d*f^4) - (2* b^3*h*(c*f^5 - d*e*f^4))/(d^2*f^8))*(c*f^5 - d*e*f^4))/(d*f^4) - (6*b*(a*f - b*e)*(a*f*h - 2*b*e*h + b*f*g))/(d*f^4)))/(3*d*f^4) + (2*(a*f - b*e)^2* (a*f*h - 4*b*e*h + 3*b*f*g))/(3*d*f^4)) + (2*b^3*h*(e + f*x)^(9/2))/(9*d*f ^4) + (atan((d^(1/2)*(e + f*x)^(1/2)*(a*d - b*c)^3*(d*e - c*f)^(1/2)*(c*h - d*g)*1i)/(a^3*d^5*e*g - b^3*c^5*f*h - a^3*c*d^4*e*h - a^3*c*d^4*f*g + b^ 3*c^4*d*e*h + b^3*c^4*d*f*g - b^3*c^3*d^2*e*g + a^3*c^2*d^3*f*h - 3*a^2*b* c*d^4*e*g + 3*a*b^2*c^4*d*f*h + 3*a*b^2*c^2*d^3*e*g - 3*a*b^2*c^3*d^2*e*h - 3*a*b^2*c^3*d^2*f*g + 3*a^2*b*c^2*d^3*e*h + 3*a^2*b*c^2*d^3*f*g - 3*a^2* b*c^3*d^2*f*h))*(a*d - b*c)^3*(d*e - c*f)^(1/2)*(c*h - d*g)*2i)/d^(11/2)
Time = 0.18 (sec) , antiderivative size = 1557, normalized size of antiderivative = 3.65 \[ \int \frac {(a+b x)^3 \sqrt {e+f x} (g+h x)}{c+d x} \, dx =\text {Too large to display} \] Input:
int((b*x+a)^3*(f*x+e)^(1/2)*(h*x+g)/(d*x+c),x)
Output:
(2*(315*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a**3*c*d**3*f**4*h - 315*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f* x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a**3*d**4*f**4*g - 945*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a**2*b*c**2*d**2*f **4*h + 945*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c *f - d*e)))*a**2*b*c*d**3*f**4*g + 945*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt( e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a*b**2*c**3*d*f**4*h - 945*sqrt(d)* sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a*b**2*c **2*d**2*f**4*g - 315*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt (d)*sqrt(c*f - d*e)))*b**3*c**4*f**4*h + 315*sqrt(d)*sqrt(c*f - d*e)*atan( (sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*b**3*c**3*d*f**4*g - 315*sqrt (e + f*x)*a**3*c*d**4*f**4*h + 105*sqrt(e + f*x)*a**3*d**5*e*f**3*h + 315* sqrt(e + f*x)*a**3*d**5*f**4*g + 105*sqrt(e + f*x)*a**3*d**5*f**4*h*x + 94 5*sqrt(e + f*x)*a**2*b*c**2*d**3*f**4*h - 315*sqrt(e + f*x)*a**2*b*c*d**4* e*f**3*h - 945*sqrt(e + f*x)*a**2*b*c*d**4*f**4*g - 315*sqrt(e + f*x)*a**2 *b*c*d**4*f**4*h*x - 126*sqrt(e + f*x)*a**2*b*d**5*e**2*f**2*h + 315*sqrt( e + f*x)*a**2*b*d**5*e*f**3*g + 63*sqrt(e + f*x)*a**2*b*d**5*e*f**3*h*x + 315*sqrt(e + f*x)*a**2*b*d**5*f**4*g*x + 189*sqrt(e + f*x)*a**2*b*d**5*f** 4*h*x**2 - 945*sqrt(e + f*x)*a*b**2*c**3*d**2*f**4*h + 315*sqrt(e + f*x)*a *b**2*c**2*d**3*e*f**3*h + 945*sqrt(e + f*x)*a*b**2*c**2*d**3*f**4*g + ...