Integrand size = 29, antiderivative size = 297 \[ \int \frac {(a+b x)^2 (e+f x)^{3/2} (g+h x)}{c+d x} \, dx=\frac {2 (b c-a d)^2 (d e-c f) (d g-c h) \sqrt {e+f x}}{d^5}+\frac {2 (b c-a d)^2 (d g-c h) (e+f x)^{3/2}}{3 d^4}+\frac {2 \left (d \left (2 a b d f^2 g+a^2 d f^2 h-b^2 e (d e+2 c f) h\right )-b (d e+c f) (2 a d f h+b (d f g-2 d e h-c f h))\right ) (e+f x)^{5/2}}{5 d^3 f^3}+\frac {2 b (2 a d f h+b (d f g-2 d e h-c f h)) (e+f x)^{7/2}}{7 d^2 f^3}+\frac {2 b^2 h (e+f x)^{9/2}}{9 d f^3}-\frac {2 (b c-a d)^2 (d e-c f)^{3/2} (d g-c h) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{11/2}} \] Output:
2*(-a*d+b*c)^2*(-c*f+d*e)*(-c*h+d*g)*(f*x+e)^(1/2)/d^5+2/3*(-a*d+b*c)^2*(- c*h+d*g)*(f*x+e)^(3/2)/d^4+2/5*(d*(2*a*b*d*f^2*g+a^2*d*f^2*h-b^2*e*(2*c*f+ d*e)*h)-b*(c*f+d*e)*(2*a*d*f*h+b*(-c*f*h-2*d*e*h+d*f*g)))*(f*x+e)^(5/2)/d^ 3/f^3+2/7*b*(2*a*d*f*h+b*(-c*f*h-2*d*e*h+d*f*g))*(f*x+e)^(7/2)/d^2/f^3+2/9 *b^2*h*(f*x+e)^(9/2)/d/f^3-2*(-a*d+b*c)^2*(-c*f+d*e)^(3/2)*(-c*h+d*g)*arct anh(d^(1/2)*(f*x+e)^(1/2)/(-c*f+d*e)^(1/2))/d^(11/2)
Time = 0.59 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.39 \[ \int \frac {(a+b x)^2 (e+f x)^{3/2} (g+h x)}{c+d x} \, dx=\frac {2 \sqrt {e+f x} \left (21 a^2 d^2 f^2 \left (15 c^2 f^2 h-5 c d f (3 f g+4 e h+f h x)+d^2 \left (3 e^2 h+f^2 x (5 g+3 h x)+e f (20 g+6 h x)\right )\right )+6 a b d f \left (-105 c^3 f^3 h-3 d^3 (e+f x)^2 (-7 f g+2 e h-5 f h x)+35 c^2 d f^2 (3 f g+4 e h+f h x)-7 c d^2 f \left (3 e^2 h+f^2 x (5 g+3 h x)+e f (20 g+6 h x)\right )\right )+b^2 \left (315 c^4 f^4 h-105 c^3 d f^3 (3 f g+4 e h+f h x)-9 c d^3 f (e+f x)^2 (7 f g-2 e h+5 f h x)+21 c^2 d^2 f^2 \left (3 e^2 h+f^2 x (5 g+3 h x)+e f (20 g+6 h x)\right )+d^4 (e+f x)^2 \left (8 e^2 h+5 f^2 x (9 g+7 h x)-2 e f (9 g+10 h x)\right )\right )\right )}{315 d^5 f^3}+\frac {2 (b c-a d)^2 (-d e+c f)^{3/2} (d g-c h) \arctan \left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{d^{11/2}} \] Input:
Integrate[((a + b*x)^2*(e + f*x)^(3/2)*(g + h*x))/(c + d*x),x]
Output:
(2*Sqrt[e + f*x]*(21*a^2*d^2*f^2*(15*c^2*f^2*h - 5*c*d*f*(3*f*g + 4*e*h + f*h*x) + d^2*(3*e^2*h + f^2*x*(5*g + 3*h*x) + e*f*(20*g + 6*h*x))) + 6*a*b *d*f*(-105*c^3*f^3*h - 3*d^3*(e + f*x)^2*(-7*f*g + 2*e*h - 5*f*h*x) + 35*c ^2*d*f^2*(3*f*g + 4*e*h + f*h*x) - 7*c*d^2*f*(3*e^2*h + f^2*x*(5*g + 3*h*x ) + e*f*(20*g + 6*h*x))) + b^2*(315*c^4*f^4*h - 105*c^3*d*f^3*(3*f*g + 4*e *h + f*h*x) - 9*c*d^3*f*(e + f*x)^2*(7*f*g - 2*e*h + 5*f*h*x) + 21*c^2*d^2 *f^2*(3*e^2*h + f^2*x*(5*g + 3*h*x) + e*f*(20*g + 6*h*x)) + d^4*(e + f*x)^ 2*(8*e^2*h + 5*f^2*x*(9*g + 7*h*x) - 2*e*f*(9*g + 10*h*x)))))/(315*d^5*f^3 ) + (2*(b*c - a*d)^2*(-(d*e) + c*f)^(3/2)*(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.
| \(\displaystyle \int \frac {(a+b x)^2 (e+f x)^{3/2} (g+h x)}{c+d x} \, dx\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle \frac {2 \int -\frac {(a+b x) (e+f x)^{3/2} (4 b c e h-a f (9 d g-5 c h)-(4 a d f h+b (9 d f g-4 d e h-9 c f h)) x)}{2 (c+d x)}dx}{9 d f}+\frac {2 h (a+b x)^2 (e+f x)^{5/2}}{9 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 h (a+b x)^2 (e+f x)^{5/2}}{9 d f}-\frac {\int -\frac {(a+b x) (e+f x)^{3/2} (9 a d f g-4 b c e h-5 a c f h+(4 a d f h+b (9 d f g-4 d e h-9 c f h)) x)}{c+d x}dx}{9 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(a+b x) (e+f x)^{3/2} (4 b c e h-a f (9 d g-5 c h)-(4 a d f h+b (9 d f g-4 d e h-9 c f h)) x)}{c+d x}dx}{9 d f}+\frac {2 h (a+b x)^2 (e+f x)^{5/2}}{9 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 h (a+b x)^2 (e+f x)^{5/2}}{9 d f}-\frac {\int -\frac {(a+b x) (e+f x)^{3/2} (9 a d f g-4 b c e h-5 a c f h+(4 a d f h+b (9 d f g-4 d e h-9 c f h)) x)}{c+d x}dx}{9 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(a+b x) (e+f x)^{3/2} (4 b c e h-a f (9 d g-5 c h)-(4 a d f h+b (9 d f g-4 d e h-9 c f h)) x)}{c+d x}dx}{9 d f}+\frac {2 h (a+b x)^2 (e+f x)^{5/2}}{9 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 h (a+b x)^2 (e+f x)^{5/2}}{9 d f}-\frac {\int -\frac {(a+b x) (e+f x)^{3/2} (9 a d f g-4 b c e h-5 a c f h+(4 a d f h+b (9 d f g-4 d e h-9 c f h)) x)}{c+d x}dx}{9 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(a+b x) (e+f x)^{3/2} (4 b c e h-a f (9 d g-5 c h)-(4 a d f h+b (9 d f g-4 d e h-9 c f h)) x)}{c+d x}dx}{9 d f}+\frac {2 h (a+b x)^2 (e+f x)^{5/2}}{9 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 h (a+b x)^2 (e+f x)^{5/2}}{9 d f}-\frac {\int -\frac {(a+b x) (e+f x)^{3/2} (9 a d f g-4 b c e h-5 a c f h+(4 a d f h+b (9 d f g-4 d e h-9 c f h)) x)}{c+d x}dx}{9 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(a+b x) (e+f x)^{3/2} (4 b c e h-a f (9 d g-5 c h)-(4 a d f h+b (9 d f g-4 d e h-9 c f h)) x)}{c+d x}dx}{9 d f}+\frac {2 h (a+b x)^2 (e+f x)^{5/2}}{9 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 h (a+b x)^2 (e+f x)^{5/2}}{9 d f}-\frac {\int -\frac {(a+b x) (e+f x)^{3/2} (9 a d f g-4 b c e h-5 a c f h+(4 a d f h+b (9 d f g-4 d e h-9 c f h)) x)}{c+d x}dx}{9 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(a+b x) (e+f x)^{3/2} (4 b c e h-a f (9 d g-5 c h)-(4 a d f h+b (9 d f g-4 d e h-9 c f h)) x)}{c+d x}dx}{9 d f}+\frac {2 h (a+b x)^2 (e+f x)^{5/2}}{9 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 h (a+b x)^2 (e+f x)^{5/2}}{9 d f}-\frac {\int -\frac {(a+b x) (e+f x)^{3/2} (9 a d f g-4 b c e h-5 a c f h+(4 a d f h+b (9 d f g-4 d e h-9 c f h)) x)}{c+d x}dx}{9 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(a+b x) (e+f x)^{3/2} (4 b c e h-a f (9 d g-5 c h)-(4 a d f h+b (9 d f g-4 d e h-9 c f h)) x)}{c+d x}dx}{9 d f}+\frac {2 h (a+b x)^2 (e+f x)^{5/2}}{9 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 h (a+b x)^2 (e+f x)^{5/2}}{9 d f}-\frac {\int -\frac {(a+b x) (e+f x)^{3/2} (9 a d f g-4 b c e h-5 a c f h+(4 a d f h+b (9 d f g-4 d e h-9 c f h)) x)}{c+d x}dx}{9 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(a+b x) (e+f x)^{3/2} (4 b c e h-a f (9 d g-5 c h)-(4 a d f h+b (9 d f g-4 d e h-9 c f h)) x)}{c+d x}dx}{9 d f}+\frac {2 h (a+b x)^2 (e+f x)^{5/2}}{9 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 h (a+b x)^2 (e+f x)^{5/2}}{9 d f}-\frac {\int -\frac {(a+b x) (e+f x)^{3/2} (9 a d f g-4 b c e h-5 a c f h+(4 a d f h+b (9 d f g-4 d e h-9 c f h)) x)}{c+d x}dx}{9 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(a+b x) (e+f x)^{3/2} (4 b c e h-a f (9 d g-5 c h)-(4 a d f h+b (9 d f g-4 d e h-9 c f h)) x)}{c+d x}dx}{9 d f}+\frac {2 h (a+b x)^2 (e+f x)^{5/2}}{9 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 h (a+b x)^2 (e+f x)^{5/2}}{9 d f}-\frac {\int -\frac {(a+b x) (e+f x)^{3/2} (9 a d f g-4 b c e h-5 a c f h+(4 a d f h+b (9 d f g-4 d e h-9 c f h)) x)}{c+d x}dx}{9 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(a+b x) (e+f x)^{3/2} (4 b c e h-a f (9 d g-5 c h)-(4 a d f h+b (9 d f g-4 d e h-9 c f h)) x)}{c+d x}dx}{9 d f}+\frac {2 h (a+b x)^2 (e+f x)^{5/2}}{9 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 h (a+b x)^2 (e+f x)^{5/2}}{9 d f}-\frac {\int -\frac {(a+b x) (e+f x)^{3/2} (9 a d f g-4 b c e h-5 a c f h+(4 a d f h+b (9 d f g-4 d e h-9 c f h)) x)}{c+d x}dx}{9 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(a+b x) (e+f x)^{3/2} (4 b c e h-a f (9 d g-5 c h)-(4 a d f h+b (9 d f g-4 d e h-9 c f h)) x)}{c+d x}dx}{9 d f}+\frac {2 h (a+b x)^2 (e+f x)^{5/2}}{9 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 h (a+b x)^2 (e+f x)^{5/2}}{9 d f}-\frac {\int -\frac {(a+b x) (e+f x)^{3/2} (9 a d f g-4 b c e h-5 a c f h+(4 a d f h+b (9 d f g-4 d e h-9 c f h)) x)}{c+d x}dx}{9 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(a+b x) (e+f x)^{3/2} (4 b c e h-a f (9 d g-5 c h)-(4 a d f h+b (9 d f g-4 d e h-9 c f h)) x)}{c+d x}dx}{9 d f}+\frac {2 h (a+b x)^2 (e+f x)^{5/2}}{9 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 h (a+b x)^2 (e+f x)^{5/2}}{9 d f}-\frac {\int -\frac {(a+b x) (e+f x)^{3/2} (9 a d f g-4 b c e h-5 a c f h+(4 a d f h+b (9 d f g-4 d e h-9 c f h)) x)}{c+d x}dx}{9 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(a+b x) (e+f x)^{3/2} (4 b c e h-a f (9 d g-5 c h)-(4 a d f h+b (9 d f g-4 d e h-9 c f h)) x)}{c+d x}dx}{9 d f}+\frac {2 h (a+b x)^2 (e+f x)^{5/2}}{9 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 h (a+b x)^2 (e+f x)^{5/2}}{9 d f}-\frac {\int -\frac {(a+b x) (e+f x)^{3/2} (9 a d f g-4 b c e h-5 a c f h+(4 a d f h+b (9 d f g-4 d e h-9 c f h)) x)}{c+d x}dx}{9 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(a+b x) (e+f x)^{3/2} (4 b c e h-a f (9 d g-5 c h)-(4 a d f h+b (9 d f g-4 d e h-9 c f h)) x)}{c+d x}dx}{9 d f}+\frac {2 h (a+b x)^2 (e+f x)^{5/2}}{9 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 h (a+b x)^2 (e+f x)^{5/2}}{9 d f}-\frac {\int -\frac {(a+b x) (e+f x)^{3/2} (9 a d f g-4 b c e h-5 a c f h+(4 a d f h+b (9 d f g-4 d e h-9 c f h)) x)}{c+d x}dx}{9 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(a+b x) (e+f x)^{3/2} (4 b c e h-a f (9 d g-5 c h)-(4 a d f h+b (9 d f g-4 d e h-9 c f h)) x)}{c+d x}dx}{9 d f}+\frac {2 h (a+b x)^2 (e+f x)^{5/2}}{9 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 h (a+b x)^2 (e+f x)^{5/2}}{9 d f}-\frac {\int -\frac {(a+b x) (e+f x)^{3/2} (9 a d f g-4 b c e h-5 a c f h+(4 a d f h+b (9 d f g-4 d e h-9 c f h)) x)}{c+d x}dx}{9 d f}\) |
Input:
Int[((a + b*x)^2*(e + f*x)^(3/2)*(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 = 0.53 (sec) , antiderivative size = 480, normalized size of antiderivative = 1.62
| method | result | size |
| pseudoelliptic | \(\frac {-2 f^{3} \left (c f -d e \right )^{2} \left (a d -b c \right )^{2} \left (c h -d g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )+2 \sqrt {\left (c f -d e \right ) d}\, \sqrt {f x +e}\, \left (\left (\frac {x \left (\frac {3 x^{2} \left (\frac {7 h x}{9}+g \right ) b^{2}}{7}+\frac {6 a x \left (\frac {5 h x}{7}+g \right ) b}{5}+a^{2} \left (\frac {3 h x}{5}+g \right )\right ) d^{4}}{3}-\left (\frac {x^{2} \left (\frac {5 h x}{7}+g \right ) b^{2}}{5}+\frac {2 a \left (\frac {3 h x}{5}+g \right ) x b}{3}+a^{2} \left (\frac {h x}{3}+g \right )\right ) c \,d^{3}+c^{2} \left (\frac {x \left (\frac {3 h x}{5}+g \right ) b^{2}}{3}+2 a \left (\frac {h x}{3}+g \right ) b +a^{2} h \right ) d^{2}-2 c^{3} \left (\left (\frac {h x}{6}+\frac {g}{2}\right ) b +a h \right ) b d +b^{2} c^{4} h \right ) f^{4}-\frac {4 \left (\left (\left (-\frac {5}{42} h \,x^{3}-\frac {6}{35} g \,x^{2}\right ) b^{2}-\frac {3 a x \left (\frac {4 h x}{7}+g \right ) b}{5}-a^{2} \left (\frac {3 h x}{10}+g \right )\right ) d^{3}+c \left (\frac {3 x \left (\frac {4 h x}{7}+g \right ) b^{2}}{10}+2 a \left (\frac {3 h x}{10}+g \right ) b +a^{2} h \right ) d^{2}-2 \left (\left (\frac {3 h x}{20}+\frac {g}{2}\right ) b +a h \right ) c^{2} b d +c^{3} h \,b^{2}\right ) d e \,f^{3}}{3}+\frac {d^{2} e^{2} \left (\left (\frac {x \left (\frac {h x}{3}+g \right ) b^{2}}{7}+2 a \left (\frac {h x}{7}+g \right ) b +a^{2} h \right ) d^{2}-2 \left (\left (\frac {h x}{14}+\frac {g}{2}\right ) b +a h \right ) c b d +b^{2} c^{2} h \right ) f^{2}}{5}-\frac {4 d^{3} \left (\left (\left (\frac {h x}{9}+\frac {g}{2}\right ) b +a h \right ) d -\frac {b c h}{2}\right ) b \,e^{3} f}{35}+\frac {8 b^{2} d^{4} e^{4} h}{315}\right )}{f^{3} d^{5} \sqrt {\left (c f -d e \right ) d}}\) | \(480\) |
| derivativedivides | \(\frac {\frac {2 \left (\frac {2 a b \,d^{4} f h \left (f x +e \right )^{\frac {7}{2}}}{7}+\frac {h \,b^{2} \left (f x +e \right )^{\frac {9}{2}} d^{4}}{9}-2 a b c \,d^{3} e \,f^{3} g \sqrt {f x +e}+2 a b \,c^{2} d^{2} e \,f^{3} h \sqrt {f x +e}-\frac {b^{2} c \,d^{3} f h \left (f x +e \right )^{\frac {7}{2}}}{7}+\frac {2 a b \,d^{4} f^{2} g \left (f x +e \right )^{\frac {5}{2}}}{5}+\frac {b^{2} c^{2} d^{2} f^{2} h \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {b^{2} c \,d^{3} f^{2} g \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {b^{2} d^{4} e f g \left (f x +e \right )^{\frac {5}{2}}}{5}+\frac {b^{2} c \,d^{3} e f h \left (f x +e \right )^{\frac {5}{2}}}{5}+2 a b \,c^{2} d^{2} f^{4} g \sqrt {f x +e}-b^{2} c^{3} d e \,f^{3} h \sqrt {f x +e}-\frac {2 a b c \,d^{3} f^{3} g \left (f x +e \right )^{\frac {3}{2}}}{3}-\frac {a^{2} c \,d^{3} f^{3} h \left (f x +e \right )^{\frac {3}{2}}}{3}-\frac {b^{2} c^{3} d \,f^{3} h \left (f x +e \right )^{\frac {3}{2}}}{3}+\frac {b^{2} c^{2} d^{2} f^{3} g \left (f x +e \right )^{\frac {3}{2}}}{3}+a^{2} c^{2} d^{2} f^{4} h \sqrt {f x +e}-a^{2} c \,d^{3} f^{4} g \sqrt {f x +e}+a^{2} d^{4} e \,f^{3} g \sqrt {f x +e}+b^{2} c^{2} d^{2} e \,f^{3} g \sqrt {f x +e}-\frac {2 a b \,d^{4} e f h \left (f x +e \right )^{\frac {5}{2}}}{5}+\frac {b^{2} d^{4} e^{2} h \left (f x +e \right )^{\frac {5}{2}}}{5}+\frac {a^{2} d^{4} f^{3} g \left (f x +e \right )^{\frac {3}{2}}}{3}+\frac {b^{2} d^{4} f g \left (f x +e \right )^{\frac {7}{2}}}{7}+\frac {a^{2} d^{4} f^{2} h \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {2 b^{2} d^{4} e h \left (f x +e \right )^{\frac {7}{2}}}{7}+b^{2} c^{4} f^{4} h \sqrt {f x +e}-b^{2} c^{3} d \,f^{4} g \sqrt {f x +e}-a^{2} c \,d^{3} e \,f^{3} h \sqrt {f x +e}-2 a b \,c^{3} d \,f^{4} h \sqrt {f x +e}+\frac {2 a b \,c^{2} d^{2} f^{3} h \left (f x +e \right )^{\frac {3}{2}}}{3}-\frac {2 a b c \,d^{3} f^{2} h \left (f x +e \right )^{\frac {5}{2}}}{5}\right )}{d^{5}}-\frac {2 f^{3} \left (a^{2} c^{3} d^{2} f^{2} h -2 a^{2} c^{2} d^{3} e f h -a^{2} c^{2} d^{3} f^{2} g +a^{2} c \,d^{4} e^{2} h +2 a^{2} c \,d^{4} e f g -g \,a^{2} e^{2} d^{5}-2 a b \,c^{4} d \,f^{2} h +4 a b \,c^{3} d^{2} e f h +2 a b \,c^{3} d^{2} f^{2} g -2 a b \,c^{2} d^{3} e^{2} h -4 a b \,c^{2} d^{3} e f g +2 a b c \,d^{4} e^{2} g +c^{5} b^{2} f^{2} h -2 b^{2} c^{4} d e f h -b^{2} c^{4} d \,f^{2} g +b^{2} c^{3} d^{2} e^{2} h +2 b^{2} c^{3} d^{2} e f g -b^{2} c^{2} d^{3} e^{2} g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{d^{5} \sqrt {\left (c f -d e \right ) d}}}{f^{3}}\) | \(911\) |
| default | \(\frac {\frac {2 \left (\frac {2 a b \,d^{4} f h \left (f x +e \right )^{\frac {7}{2}}}{7}+\frac {h \,b^{2} \left (f x +e \right )^{\frac {9}{2}} d^{4}}{9}-2 a b c \,d^{3} e \,f^{3} g \sqrt {f x +e}+2 a b \,c^{2} d^{2} e \,f^{3} h \sqrt {f x +e}-\frac {b^{2} c \,d^{3} f h \left (f x +e \right )^{\frac {7}{2}}}{7}+\frac {2 a b \,d^{4} f^{2} g \left (f x +e \right )^{\frac {5}{2}}}{5}+\frac {b^{2} c^{2} d^{2} f^{2} h \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {b^{2} c \,d^{3} f^{2} g \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {b^{2} d^{4} e f g \left (f x +e \right )^{\frac {5}{2}}}{5}+\frac {b^{2} c \,d^{3} e f h \left (f x +e \right )^{\frac {5}{2}}}{5}+2 a b \,c^{2} d^{2} f^{4} g \sqrt {f x +e}-b^{2} c^{3} d e \,f^{3} h \sqrt {f x +e}-\frac {2 a b c \,d^{3} f^{3} g \left (f x +e \right )^{\frac {3}{2}}}{3}-\frac {a^{2} c \,d^{3} f^{3} h \left (f x +e \right )^{\frac {3}{2}}}{3}-\frac {b^{2} c^{3} d \,f^{3} h \left (f x +e \right )^{\frac {3}{2}}}{3}+\frac {b^{2} c^{2} d^{2} f^{3} g \left (f x +e \right )^{\frac {3}{2}}}{3}+a^{2} c^{2} d^{2} f^{4} h \sqrt {f x +e}-a^{2} c \,d^{3} f^{4} g \sqrt {f x +e}+a^{2} d^{4} e \,f^{3} g \sqrt {f x +e}+b^{2} c^{2} d^{2} e \,f^{3} g \sqrt {f x +e}-\frac {2 a b \,d^{4} e f h \left (f x +e \right )^{\frac {5}{2}}}{5}+\frac {b^{2} d^{4} e^{2} h \left (f x +e \right )^{\frac {5}{2}}}{5}+\frac {a^{2} d^{4} f^{3} g \left (f x +e \right )^{\frac {3}{2}}}{3}+\frac {b^{2} d^{4} f g \left (f x +e \right )^{\frac {7}{2}}}{7}+\frac {a^{2} d^{4} f^{2} h \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {2 b^{2} d^{4} e h \left (f x +e \right )^{\frac {7}{2}}}{7}+b^{2} c^{4} f^{4} h \sqrt {f x +e}-b^{2} c^{3} d \,f^{4} g \sqrt {f x +e}-a^{2} c \,d^{3} e \,f^{3} h \sqrt {f x +e}-2 a b \,c^{3} d \,f^{4} h \sqrt {f x +e}+\frac {2 a b \,c^{2} d^{2} f^{3} h \left (f x +e \right )^{\frac {3}{2}}}{3}-\frac {2 a b c \,d^{3} f^{2} h \left (f x +e \right )^{\frac {5}{2}}}{5}\right )}{d^{5}}-\frac {2 f^{3} \left (a^{2} c^{3} d^{2} f^{2} h -2 a^{2} c^{2} d^{3} e f h -a^{2} c^{2} d^{3} f^{2} g +a^{2} c \,d^{4} e^{2} h +2 a^{2} c \,d^{4} e f g -g \,a^{2} e^{2} d^{5}-2 a b \,c^{4} d \,f^{2} h +4 a b \,c^{3} d^{2} e f h +2 a b \,c^{3} d^{2} f^{2} g -2 a b \,c^{2} d^{3} e^{2} h -4 a b \,c^{2} d^{3} e f g +2 a b c \,d^{4} e^{2} g +c^{5} b^{2} f^{2} h -2 b^{2} c^{4} d e f h -b^{2} c^{4} d \,f^{2} g +b^{2} c^{3} d^{2} e^{2} h +2 b^{2} c^{3} d^{2} e f g -b^{2} c^{2} d^{3} e^{2} g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{d^{5} \sqrt {\left (c f -d e \right ) d}}}{f^{3}}\) | \(911\) |
| risch | \(\text {Expression too large to display}\) | \(1027\) |
Input:
int((b*x+a)^2*(f*x+e)^(3/2)*(h*x+g)/(d*x+c),x,method=_RETURNVERBOSE)
Output:
2/((c*f-d*e)*d)^(1/2)*(-f^3*(c*f-d*e)^2*(a*d-b*c)^2*(c*h-d*g)*arctan(d*(f* x+e)^(1/2)/((c*f-d*e)*d)^(1/2))+((c*f-d*e)*d)^(1/2)*(f*x+e)^(1/2)*((1/3*x* (3/7*x^2*(7/9*h*x+g)*b^2+6/5*a*x*(5/7*h*x+g)*b+a^2*(3/5*h*x+g))*d^4-(1/5*x ^2*(5/7*h*x+g)*b^2+2/3*a*(3/5*h*x+g)*x*b+a^2*(1/3*h*x+g))*c*d^3+c^2*(1/3*x *(3/5*h*x+g)*b^2+2*a*(1/3*h*x+g)*b+a^2*h)*d^2-2*c^3*((1/6*h*x+1/2*g)*b+a*h )*b*d+b^2*c^4*h)*f^4-4/3*(((-5/42*h*x^3-6/35*g*x^2)*b^2-3/5*a*x*(4/7*h*x+g )*b-a^2*(3/10*h*x+g))*d^3+c*(3/10*x*(4/7*h*x+g)*b^2+2*a*(3/10*h*x+g)*b+a^2 *h)*d^2-2*((3/20*h*x+1/2*g)*b+a*h)*c^2*b*d+c^3*h*b^2)*d*e*f^3+1/5*d^2*e^2* ((1/7*x*(1/3*h*x+g)*b^2+2*a*(1/7*h*x+g)*b+a^2*h)*d^2-2*((1/14*h*x+1/2*g)*b +a*h)*c*b*d+b^2*c^2*h)*f^2-4/35*d^3*(((1/9*h*x+1/2*g)*b+a*h)*d-1/2*b*c*h)* b*e^3*f+8/315*b^2*d^4*e^4*h))/f^3/d^5
Leaf count of result is larger than twice the leaf count of optimal. 803 vs. \(2 (269) = 538\).
Time = 0.11 (sec) , antiderivative size = 1616, normalized size of antiderivative = 5.44 \[ \int \frac {(a+b x)^2 (e+f x)^{3/2} (g+h x)}{c+d x} \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^2*(f*x+e)^(3/2)*(h*x+g)/(d*x+c),x, algorithm="fricas")
Output:
[1/315*(315*(((b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*e*f^3 - (b^2*c^3*d - 2 *a*b*c^2*d^2 + a^2*c*d^3)*f^4)*g - ((b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3 )*e*f^3 - (b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2)*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^2*d^4*f^4*h*x^4 + 5*(9*b^2*d^4*f^4*g + (10*b^2*d^4*e*f^3 - 9*(b^2*c*d^3 - 2*a*b*d^4)*f^4)*h)*x^3 + 3*(3*(8*b^2*d^4*e*f^3 - 7*(b^2*c*d ^3 - 2*a*b*d^4)*f^4)*g + (b^2*d^4*e^2*f^2 - 24*(b^2*c*d^3 - 2*a*b*d^4)*e*f ^3 + 21*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*f^4)*h)*x^2 - 3*(6*b^2*d^4*e ^3*f + 21*(b^2*c*d^3 - 2*a*b*d^4)*e^2*f^2 - 140*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*e*f^3 + 105*(b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3)*f^4)*g + ( 8*b^2*d^4*e^4 + 18*(b^2*c*d^3 - 2*a*b*d^4)*e^3*f + 63*(b^2*c^2*d^2 - 2*a*b *c*d^3 + a^2*d^4)*e^2*f^2 - 420*(b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3)*e* f^3 + 315*(b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2)*f^4)*h + (3*(3*b^2*d^4*e^2 *f^2 - 42*(b^2*c*d^3 - 2*a*b*d^4)*e*f^3 + 35*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*f^4)*g - (4*b^2*d^4*e^3*f + 9*(b^2*c*d^3 - 2*a*b*d^4)*e^2*f^2 - 1 26*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*e*f^3 + 105*(b^2*c^3*d - 2*a*b*c^ 2*d^2 + a^2*c*d^3)*f^4)*h)*x)*sqrt(f*x + e))/(d^5*f^3), -2/315*(315*(((b^2 *c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*e*f^3 - (b^2*c^3*d - 2*a*b*c^2*d^2 + a^2 *c*d^3)*f^4)*g - ((b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3)*e*f^3 - (b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2)*f^4)*h)*sqrt(-(d*e - c*f)/d)*arctan(-sqrt...
Leaf count of result is larger than twice the leaf count of optimal. 677 vs. \(2 (289) = 578\).
Time = 11.67 (sec) , antiderivative size = 677, normalized size of antiderivative = 2.28 \[ \int \frac {(a+b x)^2 (e+f x)^{3/2} (g+h x)}{c+d x} \, dx=\begin {cases} \frac {2 \left (\frac {b^{2} h \left (e + f x\right )^{\frac {9}{2}}}{9 d f^{2}} + \frac {\left (e + f x\right )^{\frac {7}{2}} \cdot \left (2 a b d f h - b^{2} c f h - 2 b^{2} d e h + b^{2} d f g\right )}{7 d^{2} f^{2}} + \frac {\left (e + f x\right )^{\frac {5}{2}} \left (a^{2} d^{2} f^{2} h - 2 a b c d f^{2} h - 2 a b d^{2} e f h + 2 a b d^{2} f^{2} g + b^{2} c^{2} f^{2} h + b^{2} c d e f h - b^{2} c d f^{2} g + b^{2} d^{2} e^{2} h - b^{2} d^{2} e f g\right )}{5 d^{3} f^{2}} + \frac {\left (e + f x\right )^{\frac {3}{2}} \left (- a^{2} c d^{2} f h + a^{2} d^{3} f g + 2 a b c^{2} d f h - 2 a b c d^{2} f g - b^{2} c^{3} f h + b^{2} c^{2} d f g\right )}{3 d^{4}} + \frac {\sqrt {e + f x} \left (a^{2} c^{2} d^{2} f^{2} h - a^{2} c d^{3} e f h - a^{2} c d^{3} f^{2} g + a^{2} d^{4} e f g - 2 a b c^{3} d f^{2} h + 2 a b c^{2} d^{2} e f h + 2 a b c^{2} d^{2} f^{2} g - 2 a b c d^{3} e f g + b^{2} c^{4} f^{2} h - b^{2} c^{3} d e f h - b^{2} c^{3} d f^{2} g + b^{2} c^{2} d^{2} e f g\right )}{d^{5}} - \frac {f \left (a d - b c\right )^{2} \left (c f - d e\right )^{2} \left (c h - d g\right ) \operatorname {atan}{\left (\frac {\sqrt {e + f x}}{\sqrt {\frac {c f - d e}{d}}} \right )}}{d^{6} \sqrt {\frac {c f - d e}{d}}}\right )}{f} & \text {for}\: f \neq 0 \\e^{\frac {3}{2}} \left (\frac {b^{2} h x^{3}}{3 d} + \frac {x^{2} \cdot \left (2 a b d h - b^{2} c h + b^{2} d g\right )}{2 d^{2}} + \frac {x \left (a^{2} d^{2} h - 2 a b c d h + 2 a b d^{2} g + b^{2} c^{2} h - b^{2} c d g\right )}{d^{3}} - \frac {\left (a d - b c\right )^{2} \left (c h - d g\right ) \left (\begin {cases} \frac {x}{c} & \text {for}\: d = 0 \\\frac {\log {\left (c + d x \right )}}{d} & \text {otherwise} \end {cases}\right )}{d^{3}}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((b*x+a)**2*(f*x+e)**(3/2)*(h*x+g)/(d*x+c),x)
Output:
Piecewise((2*(b**2*h*(e + f*x)**(9/2)/(9*d*f**2) + (e + f*x)**(7/2)*(2*a*b *d*f*h - b**2*c*f*h - 2*b**2*d*e*h + b**2*d*f*g)/(7*d**2*f**2) + (e + f*x) **(5/2)*(a**2*d**2*f**2*h - 2*a*b*c*d*f**2*h - 2*a*b*d**2*e*f*h + 2*a*b*d* *2*f**2*g + b**2*c**2*f**2*h + b**2*c*d*e*f*h - b**2*c*d*f**2*g + b**2*d** 2*e**2*h - b**2*d**2*e*f*g)/(5*d**3*f**2) + (e + f*x)**(3/2)*(-a**2*c*d**2 *f*h + a**2*d**3*f*g + 2*a*b*c**2*d*f*h - 2*a*b*c*d**2*f*g - b**2*c**3*f*h + b**2*c**2*d*f*g)/(3*d**4) + sqrt(e + f*x)*(a**2*c**2*d**2*f**2*h - a**2 *c*d**3*e*f*h - a**2*c*d**3*f**2*g + a**2*d**4*e*f*g - 2*a*b*c**3*d*f**2*h + 2*a*b*c**2*d**2*e*f*h + 2*a*b*c**2*d**2*f**2*g - 2*a*b*c*d**3*e*f*g + b **2*c**4*f**2*h - b**2*c**3*d*e*f*h - b**2*c**3*d*f**2*g + b**2*c**2*d**2* e*f*g)/d**5 - f*(a*d - b*c)**2*(c*f - d*e)**2*(c*h - d*g)*atan(sqrt(e + f* x)/sqrt((c*f - d*e)/d))/(d**6*sqrt((c*f - d*e)/d)))/f, Ne(f, 0)), (e**(3/2 )*(b**2*h*x**3/(3*d) + x**2*(2*a*b*d*h - b**2*c*h + b**2*d*g)/(2*d**2) + x *(a**2*d**2*h - 2*a*b*c*d*h + 2*a*b*d**2*g + b**2*c**2*h - b**2*c*d*g)/d** 3 - (a*d - b*c)**2*(c*h - d*g)*Piecewise((x/c, Eq(d, 0)), (log(c + d*x)/d, True))/d**3), True))
Exception generated. \[ \int \frac {(a+b x)^2 (e+f x)^{3/2} (g+h x)}{c+d x} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((b*x+a)^2*(f*x+e)^(3/2)*(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. 944 vs. \(2 (269) = 538\).
Time = 0.15 (sec) , antiderivative size = 944, normalized size of antiderivative = 3.18 \[ \int \frac {(a+b x)^2 (e+f x)^{3/2} (g+h x)}{c+d x} \, dx =\text {Too large to display} \] Input:
integrate((b*x+a)^2*(f*x+e)^(3/2)*(h*x+g)/(d*x+c),x, algorithm="giac")
Output:
2*(b^2*c^2*d^3*e^2*g - 2*a*b*c*d^4*e^2*g + a^2*d^5*e^2*g - 2*b^2*c^3*d^2*e *f*g + 4*a*b*c^2*d^3*e*f*g - 2*a^2*c*d^4*e*f*g + b^2*c^4*d*f^2*g - 2*a*b*c ^3*d^2*f^2*g + a^2*c^2*d^3*f^2*g - b^2*c^3*d^2*e^2*h + 2*a*b*c^2*d^3*e^2*h - a^2*c*d^4*e^2*h + 2*b^2*c^4*d*e*f*h - 4*a*b*c^3*d^2*e*f*h + 2*a^2*c^2*d ^3*e*f*h - b^2*c^5*f^2*h + 2*a*b*c^4*d*f^2*h - a^2*c^3*d^2*f^2*h)*arctan(s qrt(f*x + e)*d/sqrt(-d^2*e + c*d*f))/(sqrt(-d^2*e + c*d*f)*d^5) + 2/315*(4 5*(f*x + e)^(7/2)*b^2*d^8*f^25*g - 63*(f*x + e)^(5/2)*b^2*d^8*e*f^25*g - 6 3*(f*x + e)^(5/2)*b^2*c*d^7*f^26*g + 126*(f*x + e)^(5/2)*a*b*d^8*f^26*g + 105*(f*x + e)^(3/2)*b^2*c^2*d^6*f^27*g - 210*(f*x + e)^(3/2)*a*b*c*d^7*f^2 7*g + 105*(f*x + e)^(3/2)*a^2*d^8*f^27*g + 315*sqrt(f*x + e)*b^2*c^2*d^6*e *f^27*g - 630*sqrt(f*x + e)*a*b*c*d^7*e*f^27*g + 315*sqrt(f*x + e)*a^2*d^8 *e*f^27*g - 315*sqrt(f*x + e)*b^2*c^3*d^5*f^28*g + 630*sqrt(f*x + e)*a*b*c ^2*d^6*f^28*g - 315*sqrt(f*x + e)*a^2*c*d^7*f^28*g + 35*(f*x + e)^(9/2)*b^ 2*d^8*f^24*h - 90*(f*x + e)^(7/2)*b^2*d^8*e*f^24*h + 63*(f*x + e)^(5/2)*b^ 2*d^8*e^2*f^24*h - 45*(f*x + e)^(7/2)*b^2*c*d^7*f^25*h + 90*(f*x + e)^(7/2 )*a*b*d^8*f^25*h + 63*(f*x + e)^(5/2)*b^2*c*d^7*e*f^25*h - 126*(f*x + e)^( 5/2)*a*b*d^8*e*f^25*h + 63*(f*x + e)^(5/2)*b^2*c^2*d^6*f^26*h - 126*(f*x + e)^(5/2)*a*b*c*d^7*f^26*h + 63*(f*x + e)^(5/2)*a^2*d^8*f^26*h - 105*(f*x + e)^(3/2)*b^2*c^3*d^5*f^27*h + 210*(f*x + e)^(3/2)*a*b*c^2*d^6*f^27*h - 1 05*(f*x + e)^(3/2)*a^2*c*d^7*f^27*h - 315*sqrt(f*x + e)*b^2*c^3*d^5*e*f...
Time = 2.63 (sec) , antiderivative size = 855, normalized size of antiderivative = 2.88 \[ \int \frac {(a+b x)^2 (e+f x)^{3/2} (g+h x)}{c+d x} \, dx =\text {Too large to display} \] Input:
int(((e + f*x)^(3/2)*(g + h*x)*(a + b*x)^2)/(c + d*x),x)
Output:
(e + f*x)^(7/2)*((2*b^2*f*g - 6*b^2*e*h + 4*a*b*f*h)/(7*d*f^3) - (2*b^2*h* (c*f^4 - d*e*f^3))/(7*d^2*f^6)) - (e + f*x)^(3/2)*((2*(a*f - b*e)^2*(e*h - f*g))/(3*d*f^3) - (((((2*b^2*f*g - 6*b^2*e*h + 4*a*b*f*h)/(d*f^3) - (2*b^ 2*h*(c*f^4 - d*e*f^3))/(d^2*f^6))*(c*f^4 - d*e*f^3))/(d*f^3) - (2*(a*f - b *e)*(a*f*h - 3*b*e*h + 2*b*f*g))/(d*f^3))*(c*f^4 - d*e*f^3))/(3*d*f^3)) - (e + f*x)^(5/2)*((((2*b^2*f*g - 6*b^2*e*h + 4*a*b*f*h)/(d*f^3) - (2*b^2*h* (c*f^4 - d*e*f^3))/(d^2*f^6))*(c*f^4 - d*e*f^3))/(5*d*f^3) - (2*(a*f - b*e )*(a*f*h - 3*b*e*h + 2*b*f*g))/(5*d*f^3)) + (2*b^2*h*(e + f*x)^(9/2))/(9*d *f^3) + (2*atan((d^(1/2)*(e + f*x)^(1/2)*(a*d - b*c)^2*(c*f - d*e)^(3/2)*( c*h - d*g))/(a^2*d^5*e^2*g - b^2*c^5*f^2*h - a^2*c*d^4*e^2*h + b^2*c^4*d*f ^2*g + a^2*c^2*d^3*f^2*g + b^2*c^2*d^3*e^2*g - a^2*c^3*d^2*f^2*h - b^2*c^3 *d^2*e^2*h - 2*a*b*c*d^4*e^2*g + 2*a*b*c^4*d*f^2*h - 2*a^2*c*d^4*e*f*g + 2 *b^2*c^4*d*e*f*h + 2*a*b*c^2*d^3*e^2*h - 2*a*b*c^3*d^2*f^2*g + 2*a^2*c^2*d ^3*e*f*h - 2*b^2*c^3*d^2*e*f*g + 4*a*b*c^2*d^3*e*f*g - 4*a*b*c^3*d^2*e*f*h ))*(a*d - b*c)^2*(c*f - d*e)^(3/2)*(c*h - d*g))/d^(11/2) + ((e + f*x)^(1/2 )*(c*f^4 - d*e*f^3)*((2*(a*f - b*e)^2*(e*h - f*g))/(d*f^3) - (((((2*b^2*f* g - 6*b^2*e*h + 4*a*b*f*h)/(d*f^3) - (2*b^2*h*(c*f^4 - d*e*f^3))/(d^2*f^6) )*(c*f^4 - d*e*f^3))/(d*f^3) - (2*(a*f - b*e)*(a*f*h - 3*b*e*h + 2*b*f*g)) /(d*f^3))*(c*f^4 - d*e*f^3))/(d*f^3)))/(d*f^3)
Time = 0.17 (sec) , antiderivative size = 1635, normalized size of antiderivative = 5.51 \[ \int \frac {(a+b x)^2 (e+f x)^{3/2} (g+h x)}{c+d x} \, dx =\text {Too large to display} \] Input:
int((b*x+a)^2*(f*x+e)^(3/2)*(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**2*c**2*d**2*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**2*c*d**3*e*f**3*h + 315*sqrt(d)* sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a**2*c*d **3*f**4*g - 315*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*s qrt(c*f - d*e)))*a**2*d**4*e*f**3*g + 630*sqrt(d)*sqrt(c*f - d*e)*atan((sq rt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a*b*c**3*d*f**4*h - 630*sqrt(d)* sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a*b*c**2 *d**2*e*f**3*h - 630*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt( d)*sqrt(c*f - d*e)))*a*b*c**2*d**2*f**4*g + 630*sqrt(d)*sqrt(c*f - d*e)*at an((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a*b*c*d**3*e*f**3*g - 315* sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))* b**2*c**4*f**4*h + 315*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqr t(d)*sqrt(c*f - d*e)))*b**2*c**3*d*e*f**3*h + 315*sqrt(d)*sqrt(c*f - d*e)* atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*b**2*c**3*d*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**2*c**2*d**2*e*f**3*g + 315*sqrt(e + f*x)*a**2*c**2*d**3*f**4*h - 420*s qrt(e + f*x)*a**2*c*d**4*e*f**3*h - 315*sqrt(e + f*x)*a**2*c*d**4*f**4*g - 105*sqrt(e + f*x)*a**2*c*d**4*f**4*h*x + 63*sqrt(e + f*x)*a**2*d**5*e**2* f**2*h + 420*sqrt(e + f*x)*a**2*d**5*e*f**3*g + 126*sqrt(e + f*x)*a**2*...