Integrand size = 29, antiderivative size = 352 \[ \int \frac {(c+d x)^2 \sqrt {e+f x} (g+h x)}{(a+b x)^3} \, dx=\frac {2 d (b d g+2 b c h-3 a d h) \sqrt {e+f x}}{b^4}-\frac {(b c-a d)^2 (b g-a h) \sqrt {e+f x}}{2 b^4 (a+b x)^2}-\frac {(b c-a d) \left (13 a^2 d f h+b^2 (8 d e g+c f g+4 c e h)-a b (9 d f g+12 d e h+5 c f h)\right ) \sqrt {e+f x}}{4 b^4 (b e-a f) (a+b x)}+\frac {2 d^2 h (e+f x)^{3/2}}{3 b^3 f}+\frac {\left (35 a^3 d^2 f^2 h-15 a^2 b d f (d f g+4 d e h+2 c f h)-b^3 \left (8 d^2 e^2 g-c^2 f (f g-4 e h)+8 c d e (f g+2 e h)\right )+3 a b^2 \left (c^2 f^2 h+8 d^2 e (f g+e h)+2 c d f (f g+8 e h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{4 b^{9/2} (b e-a f)^{3/2}} \] Output:
2*d*(-3*a*d*h+2*b*c*h+b*d*g)*(f*x+e)^(1/2)/b^4-1/2*(-a*d+b*c)^2*(-a*h+b*g) *(f*x+e)^(1/2)/b^4/(b*x+a)^2-1/4*(-a*d+b*c)*(13*a^2*d*f*h+b^2*(4*c*e*h+c*f *g+8*d*e*g)-a*b*(5*c*f*h+12*d*e*h+9*d*f*g))*(f*x+e)^(1/2)/b^4/(-a*f+b*e)/( b*x+a)+2/3*d^2*h*(f*x+e)^(3/2)/b^3/f+1/4*(35*a^3*d^2*f^2*h-15*a^2*b*d*f*(2 *c*f*h+4*d*e*h+d*f*g)-b^3*(8*d^2*e^2*g-c^2*f*(-4*e*h+f*g)+8*c*d*e*(2*e*h+f *g))+3*a*b^2*(c^2*f^2*h+8*d^2*e*(e*h+f*g)+2*c*d*f*(8*e*h+f*g)))*arctanh(b^ (1/2)*(f*x+e)^(1/2)/(-a*f+b*e)^(1/2))/b^(9/2)/(-a*f+b*e)^(3/2)
Time = 1.26 (sec) , antiderivative size = 465, normalized size of antiderivative = 1.32 \[ \int \frac {(c+d x)^2 \sqrt {e+f x} (g+h x)}{(a+b x)^3} \, dx=\frac {\sqrt {e+f x} \left (105 a^4 d^2 f^2 h-5 a^3 b d f (18 c f h+d (9 f g+22 e h-35 f h x))+b^4 \left (-24 c d e f x (g-2 h x)-3 c^2 f (2 e g+f g x+4 e h x)+8 d^2 e x^2 (3 f g+e h+f h x)\right )+a b^3 \left (6 c d f (-2 e (g-12 h x)+f x (5 g-8 h x))+3 c^2 f (-2 e h+f (g+5 h x))-8 d^2 x \left (-2 e^2 h+f^2 x (3 g+h x)+e f (-9 g+8 h x)\right )\right )+a^2 b^2 \left (9 c^2 f^2 h+6 c d f (3 f g+14 e h-25 f h x)+d^2 \left (8 e^2 h+2 e f (21 g-94 h x)+f^2 x (-75 g+56 h x)\right )\right )\right )}{12 b^4 f (b e-a f) (a+b x)^2}-\frac {\left (-35 a^3 d^2 f^2 h+15 a^2 b d f (d f g+4 d e h+2 c f h)+b^3 \left (8 d^2 e^2 g+8 c d e (f g+2 e h)+c^2 f (-f g+4 e h)\right )-3 a b^2 \left (c^2 f^2 h+8 d^2 e (f g+e h)+2 c d f (f g+8 e h)\right )\right ) \arctan \left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {-b e+a f}}\right )}{4 b^{9/2} (-b e+a f)^{3/2}} \] Input:
Integrate[((c + d*x)^2*Sqrt[e + f*x]*(g + h*x))/(a + b*x)^3,x]
Output:
(Sqrt[e + f*x]*(105*a^4*d^2*f^2*h - 5*a^3*b*d*f*(18*c*f*h + d*(9*f*g + 22* e*h - 35*f*h*x)) + b^4*(-24*c*d*e*f*x*(g - 2*h*x) - 3*c^2*f*(2*e*g + f*g*x + 4*e*h*x) + 8*d^2*e*x^2*(3*f*g + e*h + f*h*x)) + a*b^3*(6*c*d*f*(-2*e*(g - 12*h*x) + f*x*(5*g - 8*h*x)) + 3*c^2*f*(-2*e*h + f*(g + 5*h*x)) - 8*d^2 *x*(-2*e^2*h + f^2*x*(3*g + h*x) + e*f*(-9*g + 8*h*x))) + a^2*b^2*(9*c^2*f ^2*h + 6*c*d*f*(3*f*g + 14*e*h - 25*f*h*x) + d^2*(8*e^2*h + 2*e*f*(21*g - 94*h*x) + f^2*x*(-75*g + 56*h*x)))))/(12*b^4*f*(b*e - a*f)*(a + b*x)^2) - ((-35*a^3*d^2*f^2*h + 15*a^2*b*d*f*(d*f*g + 4*d*e*h + 2*c*f*h) + b^3*(8*d^ 2*e^2*g + 8*c*d*e*(f*g + 2*e*h) + c^2*f*(-(f*g) + 4*e*h)) - 3*a*b^2*(c^2*f ^2*h + 8*d^2*e*(f*g + e*h) + 2*c*d*f*(f*g + 8*e*h)))*ArcTan[(Sqrt[b]*Sqrt[ e + f*x])/Sqrt[-(b*e) + a*f]])/(4*b^(9/2)*(-(b*e) + a*f)^(3/2))
Time = 0.58 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.24, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {166, 27, 163, 60, 73, 221}
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.
| \(\displaystyle \int \frac {(c+d x)^2 \sqrt {e+f x} (g+h x)}{(a+b x)^3} \, dx\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {\int \frac {(c+d x) \sqrt {e+f x} (4 b d e g-b c f g+4 b c e h-4 a d e h-3 a c f h+d (3 b f g+4 b e h-7 a f h) x)}{2 (a+b x)^2}dx}{2 b (b e-a f)}-\frac {(c+d x)^2 (e+f x)^{3/2} (b g-a h)}{2 b (a+b x)^2 (b e-a f)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(c+d x) \sqrt {e+f x} (4 b d e g-b c f g+4 b c e h-4 a d e h-3 a c f h+d (3 b f g+4 b e h-7 a f h) x)}{(a+b x)^2}dx}{4 b (b e-a f)}-\frac {(c+d x)^2 (e+f x)^{3/2} (b g-a h)}{2 b (a+b x)^2 (b e-a f)}\) |
| \(\Big \downarrow \) 163 |
| \(\displaystyle \frac {\frac {(e+f x)^{3/2} \left (35 a^3 d^2 f^2 h-a^2 b d f (30 c f h+46 d e h+15 d f g)+a b^2 \left (9 c^2 f^2 h+6 c d f (6 e h+f g)+2 d^2 e (4 e h+9 f g)\right )+2 b d^2 x (b e-a f) (-7 a f h+4 b e h+3 b f g)-3 b^3 c f (4 c e h-c f g+4 d e g)\right )}{3 b^2 f (a+b x) (b e-a f)}-\frac {\left (35 a^3 d^2 f^2 h-15 a^2 b d f (2 c f h+4 d e h+d f g)+3 a b^2 \left (c^2 f^2 h+2 c d f (8 e h+f g)+8 d^2 e (e h+f g)\right )-b^3 \left (c^2 (-f) (f g-4 e h)+8 c d e (2 e h+f g)+8 d^2 e^2 g\right )\right ) \int \frac {\sqrt {e+f x}}{a+b x}dx}{2 b^2 (b e-a f)}}{4 b (b e-a f)}-\frac {(c+d x)^2 (e+f x)^{3/2} (b g-a h)}{2 b (a+b x)^2 (b e-a f)}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\frac {(e+f x)^{3/2} \left (35 a^3 d^2 f^2 h-a^2 b d f (30 c f h+46 d e h+15 d f g)+a b^2 \left (9 c^2 f^2 h+6 c d f (6 e h+f g)+2 d^2 e (4 e h+9 f g)\right )+2 b d^2 x (b e-a f) (-7 a f h+4 b e h+3 b f g)-3 b^3 c f (4 c e h-c f g+4 d e g)\right )}{3 b^2 f (a+b x) (b e-a f)}-\frac {\left (35 a^3 d^2 f^2 h-15 a^2 b d f (2 c f h+4 d e h+d f g)+3 a b^2 \left (c^2 f^2 h+2 c d f (8 e h+f g)+8 d^2 e (e h+f g)\right )-b^3 \left (c^2 (-f) (f g-4 e h)+8 c d e (2 e h+f g)+8 d^2 e^2 g\right )\right ) \left (\frac {(b e-a f) \int \frac {1}{(a+b x) \sqrt {e+f x}}dx}{b}+\frac {2 \sqrt {e+f x}}{b}\right )}{2 b^2 (b e-a f)}}{4 b (b e-a f)}-\frac {(c+d x)^2 (e+f x)^{3/2} (b g-a h)}{2 b (a+b x)^2 (b e-a f)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {(e+f x)^{3/2} \left (35 a^3 d^2 f^2 h-a^2 b d f (30 c f h+46 d e h+15 d f g)+a b^2 \left (9 c^2 f^2 h+6 c d f (6 e h+f g)+2 d^2 e (4 e h+9 f g)\right )+2 b d^2 x (b e-a f) (-7 a f h+4 b e h+3 b f g)-3 b^3 c f (4 c e h-c f g+4 d e g)\right )}{3 b^2 f (a+b x) (b e-a f)}-\frac {\left (35 a^3 d^2 f^2 h-15 a^2 b d f (2 c f h+4 d e h+d f g)+3 a b^2 \left (c^2 f^2 h+2 c d f (8 e h+f g)+8 d^2 e (e h+f g)\right )-b^3 \left (c^2 (-f) (f g-4 e h)+8 c d e (2 e h+f g)+8 d^2 e^2 g\right )\right ) \left (\frac {2 (b e-a f) \int \frac {1}{a+\frac {b (e+f x)}{f}-\frac {b e}{f}}d\sqrt {e+f x}}{b f}+\frac {2 \sqrt {e+f x}}{b}\right )}{2 b^2 (b e-a f)}}{4 b (b e-a f)}-\frac {(c+d x)^2 (e+f x)^{3/2} (b g-a h)}{2 b (a+b x)^2 (b e-a f)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {(e+f x)^{3/2} \left (35 a^3 d^2 f^2 h-a^2 b d f (30 c f h+46 d e h+15 d f g)+a b^2 \left (9 c^2 f^2 h+6 c d f (6 e h+f g)+2 d^2 e (4 e h+9 f g)\right )+2 b d^2 x (b e-a f) (-7 a f h+4 b e h+3 b f g)-3 b^3 c f (4 c e h-c f g+4 d e g)\right )}{3 b^2 f (a+b x) (b e-a f)}-\frac {\left (\frac {2 \sqrt {e+f x}}{b}-\frac {2 \sqrt {b e-a f} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{b^{3/2}}\right ) \left (35 a^3 d^2 f^2 h-15 a^2 b d f (2 c f h+4 d e h+d f g)+3 a b^2 \left (c^2 f^2 h+2 c d f (8 e h+f g)+8 d^2 e (e h+f g)\right )-b^3 \left (c^2 (-f) (f g-4 e h)+8 c d e (2 e h+f g)+8 d^2 e^2 g\right )\right )}{2 b^2 (b e-a f)}}{4 b (b e-a f)}-\frac {(c+d x)^2 (e+f x)^{3/2} (b g-a h)}{2 b (a+b x)^2 (b e-a f)}\) |
Input:
Int[((c + d*x)^2*Sqrt[e + f*x]*(g + h*x))/(a + b*x)^3,x]
Output:
-1/2*((b*g - a*h)*(c + d*x)^2*(e + f*x)^(3/2))/(b*(b*e - a*f)*(a + b*x)^2) + (((e + f*x)^(3/2)*(35*a^3*d^2*f^2*h - 3*b^3*c*f*(4*d*e*g - c*f*g + 4*c* e*h) - a^2*b*d*f*(15*d*f*g + 46*d*e*h + 30*c*f*h) + a*b^2*(9*c^2*f^2*h + 2 *d^2*e*(9*f*g + 4*e*h) + 6*c*d*f*(f*g + 6*e*h)) + 2*b*d^2*(b*e - a*f)*(3*b *f*g + 4*b*e*h - 7*a*f*h)*x))/(3*b^2*f*(b*e - a*f)*(a + b*x)) - ((35*a^3*d ^2*f^2*h - 15*a^2*b*d*f*(d*f*g + 4*d*e*h + 2*c*f*h) - b^3*(8*d^2*e^2*g - c ^2*f*(f*g - 4*e*h) + 8*c*d*e*(f*g + 2*e*h)) + 3*a*b^2*(c^2*f^2*h + 8*d^2*e *(f*g + e*h) + 2*c*d*f*(f*g + 8*e*h)))*((2*Sqrt[e + f*x])/b - (2*Sqrt[b*e - a*f]*ArcTanh[(Sqrt[b]*Sqrt[e + f*x])/Sqrt[b*e - a*f]])/b^(3/2)))/(2*b^2* (b*e - a*f)))/(4*b*(b*e - a*f))
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_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_) )*((g_.) + (h_.)*(x_)), x_] :> Simp[((a^2*d*f*h*(n + 2) + b^2*d*e*g*(m + n + 3) + a*b*(c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b*f*h*(b*c - a*d)* (m + 1)*x)/(b^2*d*(b*c - a*d)*(m + 1)*(m + n + 3)))*(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x] - Simp[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f *h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c* d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2* d*(b*c - a*d)*(m + 1)*(m + n + 3)) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && ((GeQ[m, -2] && LtQ[m, - 1]) || SumSimplerQ[m, 1]) && NeQ[m, -1] && NeQ[m + n + 3, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h )*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Time = 0.64 (sec) , antiderivative size = 492, normalized size of antiderivative = 1.40
| method | result | size |
| pseudoelliptic | \(-\frac {35 \left (-\left (\frac {\left (g \,f^{2} c^{2}-4 c e \left (c h +2 d g \right ) f -16 \left (c h +\frac {d g}{2}\right ) d \,e^{2}\right ) b^{3}}{35}+\frac {3 \left (\left (h \,c^{2}+2 c d g \right ) f^{2}+8 \left (2 c d h +d^{2} g \right ) e f +8 d^{2} e^{2} h \right ) a \,b^{2}}{35}-\frac {6 a^{2} \left (\left (c h +\frac {d g}{2}\right ) f +2 d e h \right ) d f b}{7}+a^{3} d^{2} f^{2} h \right ) \left (b x +a \right )^{2} f \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )+\left (\frac {\left (-c^{2} f^{2} g x -2 \left (4 \left (-\frac {1}{3} h \,x^{3}-g \,x^{2}\right ) d^{2}+4 c x \left (-2 h x +g \right ) d +c^{2} \left (2 h x +g \right )\right ) e f +\frac {8 d^{2} e^{2} h \,x^{2}}{3}\right ) b^{4}}{35}-\frac {2 a \left (\left (4 \left (\frac {1}{3} h \,x^{3}+g \,x^{2}\right ) d^{2}-5 x \left (-\frac {8 h x}{5}+g \right ) c d -\frac {c^{2} \left (5 h x +g \right )}{2}\right ) f^{2}+\left (-12 x \left (-\frac {8 h x}{9}+g \right ) d^{2}+2 c \left (-12 h x +g \right ) d +h \,c^{2}\right ) e f -\frac {8 d^{2} e^{2} h x}{3}\right ) b^{3}}{35}+\frac {3 a^{2} \left (\left (\frac {\left (\frac {56}{3} h \,x^{2}-25 g x \right ) d^{2}}{3}+2 c \left (-\frac {25 h x}{3}+g \right ) d +h \,c^{2}\right ) f^{2}+\frac {28 \left (\left (-\frac {47 h x}{21}+\frac {g}{2}\right ) d +c h \right ) d e f}{3}+\frac {8 d^{2} e^{2} h}{9}\right ) b^{2}}{35}-\frac {6 a^{3} d \left (\left (\frac {\left (-\frac {35 h x}{9}+g \right ) d}{2}+c h \right ) f +\frac {11 d e h}{9}\right ) f b}{7}+a^{4} d^{2} f^{2} h \right ) \sqrt {f x +e}\, \sqrt {\left (a f -b e \right ) b}\right )}{4 \sqrt {\left (a f -b e \right ) b}\, \left (b x +a \right )^{2} b^{4} \left (a f -b e \right ) f}\) | \(492\) |
| risch | \(-\frac {2 d \left (-h x d b f +9 a d f h -6 b c f h -b d e h -3 b d f g \right ) \sqrt {f x +e}}{3 f \,b^{4}}+\frac {\frac {-\frac {b f \left (13 a^{3} d^{2} f h -18 a^{2} b c d f h -12 a^{2} b \,d^{2} e h -9 a^{2} b \,d^{2} f g +5 a \,b^{2} c^{2} f h +16 a \,b^{2} c d e h +10 a \,b^{2} c d f g +8 a \,b^{2} d^{2} e g -4 b^{3} c^{2} e h -b^{3} c^{2} f g -8 b^{3} c d e g \right ) \left (f x +e \right )^{\frac {3}{2}}}{4 \left (a f -b e \right )}-\frac {f \left (11 a^{3} d^{2} f h -14 a^{2} b c d f h -12 a^{2} b \,d^{2} e h -7 a^{2} b \,d^{2} f g +3 a \,b^{2} c^{2} f h +16 a \,b^{2} c d e h +6 a \,b^{2} c d f g +8 a \,b^{2} d^{2} e g -4 b^{3} c^{2} e h +b^{3} c^{2} f g -8 b^{3} c d e g \right ) \sqrt {f x +e}}{4}}{\left (\left (f x +e \right ) b +a f -b e \right )^{2}}+\frac {\left (35 a^{3} d^{2} f^{2} h -30 a^{2} b c d \,f^{2} h -60 a^{2} b \,d^{2} e f h -15 a^{2} b \,d^{2} f^{2} g +3 a \,b^{2} c^{2} f^{2} h +48 a \,b^{2} c d e f h +6 a \,b^{2} c d \,f^{2} g +24 a \,b^{2} d^{2} e^{2} h +24 a \,b^{2} d^{2} e f g -4 b^{3} c^{2} e f h +b^{3} c^{2} f^{2} g -16 b^{3} c d \,e^{2} h -8 b^{3} c d e f g -8 b^{3} d^{2} e^{2} g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{4 \left (a f -b e \right ) \sqrt {\left (a f -b e \right ) b}}}{b^{4}}\) | \(545\) |
| derivativedivides | \(\frac {-\frac {2 d \left (-\frac {d h \left (f x +e \right )^{\frac {3}{2}} b}{3}+3 a d f h \sqrt {f x +e}-2 b c f h \sqrt {f x +e}-b d f g \sqrt {f x +e}\right )}{b^{4}}+\frac {2 f \left (\frac {-\frac {b f \left (13 a^{3} d^{2} f h -18 a^{2} b c d f h -12 a^{2} b \,d^{2} e h -9 a^{2} b \,d^{2} f g +5 a \,b^{2} c^{2} f h +16 a \,b^{2} c d e h +10 a \,b^{2} c d f g +8 a \,b^{2} d^{2} e g -4 b^{3} c^{2} e h -b^{3} c^{2} f g -8 b^{3} c d e g \right ) \left (f x +e \right )^{\frac {3}{2}}}{8 \left (a f -b e \right )}-\frac {f \left (11 a^{3} d^{2} f h -14 a^{2} b c d f h -12 a^{2} b \,d^{2} e h -7 a^{2} b \,d^{2} f g +3 a \,b^{2} c^{2} f h +16 a \,b^{2} c d e h +6 a \,b^{2} c d f g +8 a \,b^{2} d^{2} e g -4 b^{3} c^{2} e h +b^{3} c^{2} f g -8 b^{3} c d e g \right ) \sqrt {f x +e}}{8}}{\left (\left (f x +e \right ) b +a f -b e \right )^{2}}+\frac {\left (35 a^{3} d^{2} f^{2} h -30 a^{2} b c d \,f^{2} h -60 a^{2} b \,d^{2} e f h -15 a^{2} b \,d^{2} f^{2} g +3 a \,b^{2} c^{2} f^{2} h +48 a \,b^{2} c d e f h +6 a \,b^{2} c d \,f^{2} g +24 a \,b^{2} d^{2} e^{2} h +24 a \,b^{2} d^{2} e f g -4 b^{3} c^{2} e f h +b^{3} c^{2} f^{2} g -16 b^{3} c d \,e^{2} h -8 b^{3} c d e f g -8 b^{3} d^{2} e^{2} g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{8 \left (a f -b e \right ) \sqrt {\left (a f -b e \right ) b}}\right )}{b^{4}}}{f}\) | \(560\) |
| default | \(\frac {-\frac {2 d \left (-\frac {d h \left (f x +e \right )^{\frac {3}{2}} b}{3}+3 a d f h \sqrt {f x +e}-2 b c f h \sqrt {f x +e}-b d f g \sqrt {f x +e}\right )}{b^{4}}+\frac {2 f \left (\frac {-\frac {b f \left (13 a^{3} d^{2} f h -18 a^{2} b c d f h -12 a^{2} b \,d^{2} e h -9 a^{2} b \,d^{2} f g +5 a \,b^{2} c^{2} f h +16 a \,b^{2} c d e h +10 a \,b^{2} c d f g +8 a \,b^{2} d^{2} e g -4 b^{3} c^{2} e h -b^{3} c^{2} f g -8 b^{3} c d e g \right ) \left (f x +e \right )^{\frac {3}{2}}}{8 \left (a f -b e \right )}-\frac {f \left (11 a^{3} d^{2} f h -14 a^{2} b c d f h -12 a^{2} b \,d^{2} e h -7 a^{2} b \,d^{2} f g +3 a \,b^{2} c^{2} f h +16 a \,b^{2} c d e h +6 a \,b^{2} c d f g +8 a \,b^{2} d^{2} e g -4 b^{3} c^{2} e h +b^{3} c^{2} f g -8 b^{3} c d e g \right ) \sqrt {f x +e}}{8}}{\left (\left (f x +e \right ) b +a f -b e \right )^{2}}+\frac {\left (35 a^{3} d^{2} f^{2} h -30 a^{2} b c d \,f^{2} h -60 a^{2} b \,d^{2} e f h -15 a^{2} b \,d^{2} f^{2} g +3 a \,b^{2} c^{2} f^{2} h +48 a \,b^{2} c d e f h +6 a \,b^{2} c d \,f^{2} g +24 a \,b^{2} d^{2} e^{2} h +24 a \,b^{2} d^{2} e f g -4 b^{3} c^{2} e f h +b^{3} c^{2} f^{2} g -16 b^{3} c d \,e^{2} h -8 b^{3} c d e f g -8 b^{3} d^{2} e^{2} g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{8 \left (a f -b e \right ) \sqrt {\left (a f -b e \right ) b}}\right )}{b^{4}}}{f}\) | \(560\) |
Input:
int((d*x+c)^2*(f*x+e)^(1/2)*(h*x+g)/(b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
-35/4/((a*f-b*e)*b)^(1/2)*(-(1/35*(g*f^2*c^2-4*c*e*(c*h+2*d*g)*f-16*(c*h+1 /2*d*g)*d*e^2)*b^3+3/35*((c^2*h+2*c*d*g)*f^2+8*(2*c*d*h+d^2*g)*e*f+8*d^2*e ^2*h)*a*b^2-6/7*a^2*((c*h+1/2*d*g)*f+2*d*e*h)*d*f*b+a^3*d^2*f^2*h)*(b*x+a) ^2*f*arctan(b*(f*x+e)^(1/2)/((a*f-b*e)*b)^(1/2))+(1/35*(-c^2*f^2*g*x-2*(4* (-1/3*h*x^3-g*x^2)*d^2+4*c*x*(-2*h*x+g)*d+c^2*(2*h*x+g))*e*f+8/3*d^2*e^2*h *x^2)*b^4-2/35*a*((4*(1/3*h*x^3+g*x^2)*d^2-5*x*(-8/5*h*x+g)*c*d-1/2*c^2*(5 *h*x+g))*f^2+(-12*x*(-8/9*h*x+g)*d^2+2*c*(-12*h*x+g)*d+h*c^2)*e*f-8/3*d^2* e^2*h*x)*b^3+3/35*a^2*((1/3*(56/3*h*x^2-25*g*x)*d^2+2*c*(-25/3*h*x+g)*d+h* c^2)*f^2+28/3*((-47/21*h*x+1/2*g)*d+c*h)*d*e*f+8/9*d^2*e^2*h)*b^2-6/7*a^3* d*((1/2*(-35/9*h*x+g)*d+c*h)*f+11/9*d*e*h)*f*b+a^4*d^2*f^2*h)*(f*x+e)^(1/2 )*((a*f-b*e)*b)^(1/2))/(b*x+a)^2/b^4/(a*f-b*e)/f
Leaf count of result is larger than twice the leaf count of optimal. 1332 vs. \(2 (326) = 652\).
Time = 0.25 (sec) , antiderivative size = 2678, normalized size of antiderivative = 7.61 \[ \int \frac {(c+d x)^2 \sqrt {e+f x} (g+h x)}{(a+b x)^3} \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)^2*(f*x+e)^(1/2)*(h*x+g)/(b*x+a)^3,x, algorithm="fricas")
Output:
[1/24*(3*sqrt(b^2*e - a*b*f)*(((8*b^5*d^2*e^2*f + 8*(b^5*c*d - 3*a*b^4*d^2 )*e*f^2 - (b^5*c^2 + 6*a*b^4*c*d - 15*a^2*b^3*d^2)*f^3)*g + (8*(2*b^5*c*d - 3*a*b^4*d^2)*e^2*f + 4*(b^5*c^2 - 12*a*b^4*c*d + 15*a^2*b^3*d^2)*e*f^2 - (3*a*b^4*c^2 - 30*a^2*b^3*c*d + 35*a^3*b^2*d^2)*f^3)*h)*x^2 + (8*a^2*b^3* d^2*e^2*f + 8*(a^2*b^3*c*d - 3*a^3*b^2*d^2)*e*f^2 - (a^2*b^3*c^2 + 6*a^3*b ^2*c*d - 15*a^4*b*d^2)*f^3)*g + (8*(2*a^2*b^3*c*d - 3*a^3*b^2*d^2)*e^2*f + 4*(a^2*b^3*c^2 - 12*a^3*b^2*c*d + 15*a^4*b*d^2)*e*f^2 - (3*a^3*b^2*c^2 - 30*a^4*b*c*d + 35*a^5*d^2)*f^3)*h + 2*((8*a*b^4*d^2*e^2*f + 8*(a*b^4*c*d - 3*a^2*b^3*d^2)*e*f^2 - (a*b^4*c^2 + 6*a^2*b^3*c*d - 15*a^3*b^2*d^2)*f^3)* g + (8*(2*a*b^4*c*d - 3*a^2*b^3*d^2)*e^2*f + 4*(a*b^4*c^2 - 12*a^2*b^3*c*d + 15*a^3*b^2*d^2)*e*f^2 - (3*a^2*b^3*c^2 - 30*a^3*b^2*c*d + 35*a^4*b*d^2) *f^3)*h)*x)*log((b*f*x + 2*b*e - a*f - 2*sqrt(b^2*e - a*b*f)*sqrt(f*x + e) )/(b*x + a)) + 2*(8*(b^6*d^2*e^2*f - 2*a*b^5*d^2*e*f^2 + a^2*b^4*d^2*f^3)* h*x^3 + 8*(3*(b^6*d^2*e^2*f - 2*a*b^5*d^2*e*f^2 + a^2*b^4*d^2*f^3)*g + (b^ 6*d^2*e^3 + 3*(2*b^6*c*d - 3*a*b^5*d^2)*e^2*f - 3*(4*a*b^5*c*d - 5*a^2*b^4 *d^2)*e*f^2 + (6*a^2*b^4*c*d - 7*a^3*b^3*d^2)*f^3)*h)*x^2 - 3*(2*(b^6*c^2 + 2*a*b^5*c*d - 7*a^2*b^4*d^2)*e^2*f - (3*a*b^5*c^2 + 10*a^2*b^4*c*d - 29* a^3*b^3*d^2)*e*f^2 + (a^2*b^4*c^2 + 6*a^3*b^3*c*d - 15*a^4*b^2*d^2)*f^3)*g + (8*a^2*b^4*d^2*e^3 - 2*(3*a*b^5*c^2 - 42*a^2*b^4*c*d + 59*a^3*b^3*d^2)* e^2*f + (15*a^2*b^4*c^2 - 174*a^3*b^3*c*d + 215*a^4*b^2*d^2)*e*f^2 - 3*...
Timed out. \[ \int \frac {(c+d x)^2 \sqrt {e+f x} (g+h x)}{(a+b x)^3} \, dx=\text {Timed out} \] Input:
integrate((d*x+c)**2*(f*x+e)**(1/2)*(h*x+g)/(b*x+a)**3,x)
Output:
Timed out
Exception generated. \[ \int \frac {(c+d x)^2 \sqrt {e+f x} (g+h x)}{(a+b x)^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((d*x+c)^2*(f*x+e)^(1/2)*(h*x+g)/(b*x+a)^3,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(a*f-b*e>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 903 vs. \(2 (326) = 652\).
Time = 0.16 (sec) , antiderivative size = 903, normalized size of antiderivative = 2.57 \[ \int \frac {(c+d x)^2 \sqrt {e+f x} (g+h x)}{(a+b x)^3} \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)^2*(f*x+e)^(1/2)*(h*x+g)/(b*x+a)^3,x, algorithm="giac")
Output:
1/4*(8*b^3*d^2*e^2*g + 8*b^3*c*d*e*f*g - 24*a*b^2*d^2*e*f*g - b^3*c^2*f^2* g - 6*a*b^2*c*d*f^2*g + 15*a^2*b*d^2*f^2*g + 16*b^3*c*d*e^2*h - 24*a*b^2*d ^2*e^2*h + 4*b^3*c^2*e*f*h - 48*a*b^2*c*d*e*f*h + 60*a^2*b*d^2*e*f*h - 3*a *b^2*c^2*f^2*h + 30*a^2*b*c*d*f^2*h - 35*a^3*d^2*f^2*h)*arctan(sqrt(f*x + e)*b/sqrt(-b^2*e + a*b*f))/((b^5*e - a*b^4*f)*sqrt(-b^2*e + a*b*f)) - 1/4* (8*(f*x + e)^(3/2)*b^4*c*d*e*f*g - 8*(f*x + e)^(3/2)*a*b^3*d^2*e*f*g - 8*s qrt(f*x + e)*b^4*c*d*e^2*f*g + 8*sqrt(f*x + e)*a*b^3*d^2*e^2*f*g + (f*x + e)^(3/2)*b^4*c^2*f^2*g - 10*(f*x + e)^(3/2)*a*b^3*c*d*f^2*g + 9*(f*x + e)^ (3/2)*a^2*b^2*d^2*f^2*g + sqrt(f*x + e)*b^4*c^2*e*f^2*g + 14*sqrt(f*x + e) *a*b^3*c*d*e*f^2*g - 15*sqrt(f*x + e)*a^2*b^2*d^2*e*f^2*g - sqrt(f*x + e)* a*b^3*c^2*f^3*g - 6*sqrt(f*x + e)*a^2*b^2*c*d*f^3*g + 7*sqrt(f*x + e)*a^3* b*d^2*f^3*g + 4*(f*x + e)^(3/2)*b^4*c^2*e*f*h - 16*(f*x + e)^(3/2)*a*b^3*c *d*e*f*h + 12*(f*x + e)^(3/2)*a^2*b^2*d^2*e*f*h - 4*sqrt(f*x + e)*b^4*c^2* e^2*f*h + 16*sqrt(f*x + e)*a*b^3*c*d*e^2*f*h - 12*sqrt(f*x + e)*a^2*b^2*d^ 2*e^2*f*h - 5*(f*x + e)^(3/2)*a*b^3*c^2*f^2*h + 18*(f*x + e)^(3/2)*a^2*b^2 *c*d*f^2*h - 13*(f*x + e)^(3/2)*a^3*b*d^2*f^2*h + 7*sqrt(f*x + e)*a*b^3*c^ 2*e*f^2*h - 30*sqrt(f*x + e)*a^2*b^2*c*d*e*f^2*h + 23*sqrt(f*x + e)*a^3*b* d^2*e*f^2*h - 3*sqrt(f*x + e)*a^2*b^2*c^2*f^3*h + 14*sqrt(f*x + e)*a^3*b*c *d*f^3*h - 11*sqrt(f*x + e)*a^4*d^2*f^3*h)/((b^5*e - a*b^4*f)*((f*x + e)*b - b*e + a*f)^2) + 2/3*(3*sqrt(f*x + e)*b^6*d^2*f^3*g + (f*x + e)^(3/2)...
Time = 2.17 (sec) , antiderivative size = 639, normalized size of antiderivative = 1.82 \[ \int \frac {(c+d x)^2 \sqrt {e+f x} (g+h x)}{(a+b x)^3} \, dx=\sqrt {e+f\,x}\,\left (\frac {2\,d^2\,f\,g-6\,d^2\,e\,h+4\,c\,d\,f\,h}{b^3\,f}-\frac {6\,d^2\,h\,\left (a\,f-b\,e\right )}{b^4\,f}\right )-\frac {\sqrt {e+f\,x}\,\left (\frac {11\,h\,a^3\,d^2\,f^2}{4}-\frac {7\,h\,a^2\,b\,c\,d\,f^2}{2}-\frac {7\,g\,a^2\,b\,d^2\,f^2}{4}-3\,e\,h\,a^2\,b\,d^2\,f+\frac {3\,h\,a\,b^2\,c^2\,f^2}{4}+\frac {3\,g\,a\,b^2\,c\,d\,f^2}{2}+4\,e\,h\,a\,b^2\,c\,d\,f+2\,e\,g\,a\,b^2\,d^2\,f+\frac {g\,b^3\,c^2\,f^2}{4}-e\,h\,b^3\,c^2\,f-2\,e\,g\,b^3\,c\,d\,f\right )-\frac {{\left (e+f\,x\right )}^{3/2}\,\left (-13\,h\,a^3\,b\,d^2\,f^2+18\,h\,a^2\,b^2\,c\,d\,f^2+9\,g\,a^2\,b^2\,d^2\,f^2+12\,e\,h\,a^2\,b^2\,d^2\,f-5\,h\,a\,b^3\,c^2\,f^2-10\,g\,a\,b^3\,c\,d\,f^2-16\,e\,h\,a\,b^3\,c\,d\,f-8\,e\,g\,a\,b^3\,d^2\,f+g\,b^4\,c^2\,f^2+4\,e\,h\,b^4\,c^2\,f+8\,e\,g\,b^4\,c\,d\,f\right )}{4\,\left (a\,f-b\,e\right )}}{b^6\,{\left (e+f\,x\right )}^2-\left (e+f\,x\right )\,\left (2\,b^6\,e-2\,a\,b^5\,f\right )+b^6\,e^2+a^2\,b^4\,f^2-2\,a\,b^5\,e\,f}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {e+f\,x}}{\sqrt {a\,f-b\,e}}\right )\,\left (35\,h\,a^3\,d^2\,f^2-30\,h\,a^2\,b\,c\,d\,f^2-60\,h\,a^2\,b\,d^2\,e\,f-15\,g\,a^2\,b\,d^2\,f^2+3\,h\,a\,b^2\,c^2\,f^2+48\,h\,a\,b^2\,c\,d\,e\,f+6\,g\,a\,b^2\,c\,d\,f^2+24\,h\,a\,b^2\,d^2\,e^2+24\,g\,a\,b^2\,d^2\,e\,f-4\,h\,b^3\,c^2\,e\,f+g\,b^3\,c^2\,f^2-16\,h\,b^3\,c\,d\,e^2-8\,g\,b^3\,c\,d\,e\,f-8\,g\,b^3\,d^2\,e^2\right )}{4\,b^{9/2}\,{\left (a\,f-b\,e\right )}^{3/2}}+\frac {2\,d^2\,h\,{\left (e+f\,x\right )}^{3/2}}{3\,b^3\,f} \] Input:
int(((e + f*x)^(1/2)*(g + h*x)*(c + d*x)^2)/(a + b*x)^3,x)
Output:
(e + f*x)^(1/2)*((2*d^2*f*g - 6*d^2*e*h + 4*c*d*f*h)/(b^3*f) - (6*d^2*h*(a *f - b*e))/(b^4*f)) - ((e + f*x)^(1/2)*((b^3*c^2*f^2*g)/4 + (11*a^3*d^2*f^ 2*h)/4 - b^3*c^2*e*f*h + (3*a*b^2*c^2*f^2*h)/4 - (7*a^2*b*d^2*f^2*g)/4 - 2 *b^3*c*d*e*f*g + (3*a*b^2*c*d*f^2*g)/2 - (7*a^2*b*c*d*f^2*h)/2 + 2*a*b^2*d ^2*e*f*g - 3*a^2*b*d^2*e*f*h + 4*a*b^2*c*d*e*f*h) - ((e + f*x)^(3/2)*(b^4* c^2*f^2*g + 4*b^4*c^2*e*f*h - 5*a*b^3*c^2*f^2*h - 13*a^3*b*d^2*f^2*h + 9*a ^2*b^2*d^2*f^2*g + 8*b^4*c*d*e*f*g - 10*a*b^3*c*d*f^2*g - 8*a*b^3*d^2*e*f* g + 18*a^2*b^2*c*d*f^2*h + 12*a^2*b^2*d^2*e*f*h - 16*a*b^3*c*d*e*f*h))/(4* (a*f - b*e)))/(b^6*(e + f*x)^2 - (e + f*x)*(2*b^6*e - 2*a*b^5*f) + b^6*e^2 + a^2*b^4*f^2 - 2*a*b^5*e*f) + (atan((b^(1/2)*(e + f*x)^(1/2))/(a*f - b*e )^(1/2))*(b^3*c^2*f^2*g - 8*b^3*d^2*e^2*g + 35*a^3*d^2*f^2*h - 16*b^3*c*d* e^2*h - 4*b^3*c^2*e*f*h + 3*a*b^2*c^2*f^2*h + 24*a*b^2*d^2*e^2*h - 15*a^2* b*d^2*f^2*g - 8*b^3*c*d*e*f*g + 6*a*b^2*c*d*f^2*g - 30*a^2*b*c*d*f^2*h + 2 4*a*b^2*d^2*e*f*g - 60*a^2*b*d^2*e*f*h + 48*a*b^2*c*d*e*f*h))/(4*b^(9/2)*( a*f - b*e)^(3/2)) + (2*d^2*h*(e + f*x)^(3/2))/(3*b^3*f)
Time = 0.19 (sec) , antiderivative size = 3292, normalized size of antiderivative = 9.35 \[ \int \frac {(c+d x)^2 \sqrt {e+f x} (g+h x)}{(a+b x)^3} \, dx =\text {Too large to display} \] Input:
int((d*x+c)^2*(f*x+e)^(1/2)*(h*x+g)/(b*x+a)^3,x)
Output:
(105*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b* e)))*a**5*d**2*f**3*h - 90*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/ (sqrt(b)*sqrt(a*f - b*e)))*a**4*b*c*d*f**3*h - 180*sqrt(b)*sqrt(a*f - b*e) *atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**4*b*d**2*e*f**2*h - 45*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e) ))*a**4*b*d**2*f**3*g + 210*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b) /(sqrt(b)*sqrt(a*f - b*e)))*a**4*b*d**2*f**3*h*x + 9*sqrt(b)*sqrt(a*f - b* e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*b**2*c**2*f**3*h + 144*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*b**2*c*d*e*f**2*h + 18*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*b**2*c*d*f**3*g - 180*sqrt(b)*sqrt (a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*b**2*c* d*f**3*h*x + 72*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sq rt(a*f - b*e)))*a**3*b**2*d**2*e**2*f*h + 72*sqrt(b)*sqrt(a*f - b*e)*atan( (sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*b**2*d**2*e*f**2*g - 360 *sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e))) *a**3*b**2*d**2*e*f**2*h*x - 90*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x )*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*b**2*d**2*f**3*g*x + 105*sqrt(b)*sqrt (a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*b**2*d* *2*f**3*h*x**2 - 12*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqr...