Integrand size = 29, antiderivative size = 353 \[ \int \frac {(a+b x)^2 \sqrt {e+f x} (g+h x)}{(c+d x)^3} \, dx=\frac {2 b (b d g-3 b c h+2 a d h) \sqrt {e+f x}}{d^4}-\frac {(b c-a d)^2 (d g-c h) \sqrt {e+f x}}{2 d^4 (c+d x)^2}+\frac {(b c-a d) \left (a d (d f g+4 d e h-5 c f h)+b \left (8 d^2 e g+13 c^2 f h-3 c d (3 f g+4 e h)\right )\right ) \sqrt {e+f x}}{4 d^4 (d e-c f) (c+d x)}+\frac {2 b^2 h (e+f x)^{3/2}}{3 d^3 f}+\frac {\left (a^2 d^2 f (d f g-4 d e h+3 c f h)-b^2 \left (8 d^3 e^2 g-35 c^3 f^2 h-24 c d^2 e (f g+e h)+15 c^2 d f (f g+4 e h)\right )-2 a b d \left (15 c^2 f^2 h+4 d^2 e (f g+2 e h)-3 c d f (f g+8 e h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{4 d^{9/2} (d e-c f)^{3/2}} \] Output:
2*b*(2*a*d*h-3*b*c*h+b*d*g)*(f*x+e)^(1/2)/d^4-1/2*(-a*d+b*c)^2*(-c*h+d*g)* (f*x+e)^(1/2)/d^4/(d*x+c)^2+1/4*(-a*d+b*c)*(a*d*(-5*c*f*h+4*d*e*h+d*f*g)+b *(8*d^2*e*g+13*c^2*f*h-3*c*d*(4*e*h+3*f*g)))*(f*x+e)^(1/2)/d^4/(-c*f+d*e)/ (d*x+c)+2/3*b^2*h*(f*x+e)^(3/2)/d^3/f+1/4*(a^2*d^2*f*(3*c*f*h-4*d*e*h+d*f* g)-b^2*(8*d^3*e^2*g-35*c^3*f^2*h-24*c*d^2*e*(e*h+f*g)+15*c^2*d*f*(4*e*h+f* g))-2*a*b*d*(15*c^2*f^2*h+4*d^2*e*(2*e*h+f*g)-3*c*d*f*(8*e*h+f*g)))*arctan h(d^(1/2)*(f*x+e)^(1/2)/(-c*f+d*e)^(1/2))/d^(9/2)/(-c*f+d*e)^(3/2)
Time = 1.27 (sec) , antiderivative size = 462, normalized size of antiderivative = 1.31 \[ \int \frac {(a+b x)^2 \sqrt {e+f x} (g+h x)}{(c+d x)^3} \, dx=\frac {\sqrt {e+f x} \left (3 a^2 d^2 f \left (3 c^2 f h-d^2 (2 e g+f g x+4 e h x)+c d (f g-2 e h+5 f h x)\right )-6 a b d f \left (15 c^3 f h+4 d^3 e x (g-2 h x)+c^2 d (-3 f g-14 e h+25 f h x)+c d^2 (2 e (g-12 h x)+f x (-5 g+8 h x))\right )+b^2 \left (105 c^4 f^2 h-5 c^3 d f (9 f g+22 e h-35 f h x)+8 d^4 e x^2 (3 f g+e h+f h x)-8 c d^3 x \left (-2 e^2 h+f^2 x (3 g+h x)+e f (-9 g+8 h x)\right )+c^2 d^2 \left (8 e^2 h+2 e f (21 g-94 h x)+f^2 x (-75 g+56 h x)\right )\right )\right )}{12 d^4 f (d e-c f) (c+d x)^2}-\frac {\left (-a^2 d^2 f (d f g-4 d e h+3 c f h)+b^2 \left (8 d^3 e^2 g-35 c^3 f^2 h-24 c d^2 e (f g+e h)+15 c^2 d f (f g+4 e h)\right )+2 a b d \left (15 c^2 f^2 h+4 d^2 e (f g+2 e h)-3 c d f (f g+8 e h)\right )\right ) \arctan \left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{4 d^{9/2} (-d e+c f)^{3/2}} \] Input:
Integrate[((a + b*x)^2*Sqrt[e + f*x]*(g + h*x))/(c + d*x)^3,x]
Output:
(Sqrt[e + f*x]*(3*a^2*d^2*f*(3*c^2*f*h - d^2*(2*e*g + f*g*x + 4*e*h*x) + c *d*(f*g - 2*e*h + 5*f*h*x)) - 6*a*b*d*f*(15*c^3*f*h + 4*d^3*e*x*(g - 2*h*x ) + c^2*d*(-3*f*g - 14*e*h + 25*f*h*x) + c*d^2*(2*e*(g - 12*h*x) + f*x*(-5 *g + 8*h*x))) + b^2*(105*c^4*f^2*h - 5*c^3*d*f*(9*f*g + 22*e*h - 35*f*h*x) + 8*d^4*e*x^2*(3*f*g + e*h + f*h*x) - 8*c*d^3*x*(-2*e^2*h + f^2*x*(3*g + h*x) + e*f*(-9*g + 8*h*x)) + c^2*d^2*(8*e^2*h + 2*e*f*(21*g - 94*h*x) + f^ 2*x*(-75*g + 56*h*x)))))/(12*d^4*f*(d*e - c*f)*(c + d*x)^2) - ((-(a^2*d^2* f*(d*f*g - 4*d*e*h + 3*c*f*h)) + b^2*(8*d^3*e^2*g - 35*c^3*f^2*h - 24*c*d^ 2*e*(f*g + e*h) + 15*c^2*d*f*(f*g + 4*e*h)) + 2*a*b*d*(15*c^2*f^2*h + 4*d^ 2*e*(f*g + 2*e*h) - 3*c*d*f*(f*g + 8*e*h)))*ArcTan[(Sqrt[d]*Sqrt[e + f*x]) /Sqrt[-(d*e) + c*f]])/(4*d^(9/2)*(-(d*e) + c*f)^(3/2))
Time = 0.64 (sec) , antiderivative size = 434, normalized size of antiderivative = 1.23, 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.
| \(\displaystyle \int \frac {(a+b x)^2 \sqrt {e+f x} (g+h x)}{(c+d x)^3} \, dx\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {\int \frac {(a+b x) \sqrt {e+f x} (4 b e (d g-c h)-a (d f g-4 d e h+3 c f h)+b (3 d f g+4 d e h-7 c f h) x)}{2 (c+d x)^2}dx}{2 d (d e-c f)}-\frac {(a+b x)^2 (e+f x)^{3/2} (d g-c h)}{2 d (c+d x)^2 (d e-c f)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(a+b x) \sqrt {e+f x} (4 b e (d g-c h)-a (d f g-4 d e h+3 c f h)+b (3 d f g+4 d e h-7 c f h) x)}{(c+d x)^2}dx}{4 d (d e-c f)}-\frac {(a+b x)^2 (e+f x)^{3/2} (d g-c h)}{2 d (c+d x)^2 (d e-c f)}\) |
| \(\Big \downarrow \) 163 |
| \(\displaystyle \frac {\frac {(e+f x)^{3/2} \left (3 a^2 d^2 f (3 c f h-4 d e h+d f g)-6 a b d f \left (5 c^2 f h-c d (6 e h+f g)+2 d^2 e g\right )+b^2 c \left (35 c^2 f^2 h-c d f (46 e h+15 f g)+2 d^2 e (4 e h+9 f g)\right )+2 b^2 d x (d e-c f) (-7 c f h+4 d e h+3 d f g)\right )}{3 d^2 f (c+d x) (d e-c f)}-\frac {\left (a^2 d^2 f (3 c f h-4 d e h+d f g)-2 a b d \left (15 c^2 f^2 h-3 c d f (8 e h+f g)+4 d^2 e (2 e h+f g)\right )-\left (b^2 \left (-35 c^3 f^2 h+15 c^2 d f (4 e h+f g)-24 c d^2 e (e h+f g)+8 d^3 e^2 g\right )\right )\right ) \int \frac {\sqrt {e+f x}}{c+d x}dx}{2 d^2 (d e-c f)}}{4 d (d e-c f)}-\frac {(a+b x)^2 (e+f x)^{3/2} (d g-c h)}{2 d (c+d x)^2 (d e-c f)}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\frac {(e+f x)^{3/2} \left (3 a^2 d^2 f (3 c f h-4 d e h+d f g)-6 a b d f \left (5 c^2 f h-c d (6 e h+f g)+2 d^2 e g\right )+b^2 c \left (35 c^2 f^2 h-c d f (46 e h+15 f g)+2 d^2 e (4 e h+9 f g)\right )+2 b^2 d x (d e-c f) (-7 c f h+4 d e h+3 d f g)\right )}{3 d^2 f (c+d x) (d e-c f)}-\frac {\left (a^2 d^2 f (3 c f h-4 d e h+d f g)-2 a b d \left (15 c^2 f^2 h-3 c d f (8 e h+f g)+4 d^2 e (2 e h+f g)\right )-\left (b^2 \left (-35 c^3 f^2 h+15 c^2 d f (4 e h+f g)-24 c d^2 e (e h+f g)+8 d^3 e^2 g\right )\right )\right ) \left (\frac {(d e-c f) \int \frac {1}{(c+d x) \sqrt {e+f x}}dx}{d}+\frac {2 \sqrt {e+f x}}{d}\right )}{2 d^2 (d e-c f)}}{4 d (d e-c f)}-\frac {(a+b x)^2 (e+f x)^{3/2} (d g-c h)}{2 d (c+d x)^2 (d e-c f)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {(e+f x)^{3/2} \left (3 a^2 d^2 f (3 c f h-4 d e h+d f g)-6 a b d f \left (5 c^2 f h-c d (6 e h+f g)+2 d^2 e g\right )+b^2 c \left (35 c^2 f^2 h-c d f (46 e h+15 f g)+2 d^2 e (4 e h+9 f g)\right )+2 b^2 d x (d e-c f) (-7 c f h+4 d e h+3 d f g)\right )}{3 d^2 f (c+d x) (d e-c f)}-\frac {\left (a^2 d^2 f (3 c f h-4 d e h+d f g)-2 a b d \left (15 c^2 f^2 h-3 c d f (8 e h+f g)+4 d^2 e (2 e h+f g)\right )-\left (b^2 \left (-35 c^3 f^2 h+15 c^2 d f (4 e h+f g)-24 c d^2 e (e h+f g)+8 d^3 e^2 g\right )\right )\right ) \left (\frac {2 (d e-c f) \int \frac {1}{c+\frac {d (e+f x)}{f}-\frac {d e}{f}}d\sqrt {e+f x}}{d f}+\frac {2 \sqrt {e+f x}}{d}\right )}{2 d^2 (d e-c f)}}{4 d (d e-c f)}-\frac {(a+b x)^2 (e+f x)^{3/2} (d g-c h)}{2 d (c+d x)^2 (d e-c f)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {(e+f x)^{3/2} \left (3 a^2 d^2 f (3 c f h-4 d e h+d f g)-6 a b d f \left (5 c^2 f h-c d (6 e h+f g)+2 d^2 e g\right )+b^2 c \left (35 c^2 f^2 h-c d f (46 e h+15 f g)+2 d^2 e (4 e h+9 f g)\right )+2 b^2 d x (d e-c f) (-7 c f h+4 d e h+3 d f g)\right )}{3 d^2 f (c+d x) (d e-c f)}-\frac {\left (\frac {2 \sqrt {e+f x}}{d}-\frac {2 \sqrt {d e-c f} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{3/2}}\right ) \left (a^2 d^2 f (3 c f h-4 d e h+d f g)-2 a b d \left (15 c^2 f^2 h-3 c d f (8 e h+f g)+4 d^2 e (2 e h+f g)\right )-\left (b^2 \left (-35 c^3 f^2 h+15 c^2 d f (4 e h+f g)-24 c d^2 e (e h+f g)+8 d^3 e^2 g\right )\right )\right )}{2 d^2 (d e-c f)}}{4 d (d e-c f)}-\frac {(a+b x)^2 (e+f x)^{3/2} (d g-c h)}{2 d (c+d x)^2 (d e-c f)}\) |
Input:
Int[((a + b*x)^2*Sqrt[e + f*x]*(g + h*x))/(c + d*x)^3,x]
Output:
-1/2*((d*g - c*h)*(a + b*x)^2*(e + f*x)^(3/2))/(d*(d*e - c*f)*(c + d*x)^2) + (((e + f*x)^(3/2)*(3*a^2*d^2*f*(d*f*g - 4*d*e*h + 3*c*f*h) - 6*a*b*d*f* (2*d^2*e*g + 5*c^2*f*h - c*d*(f*g + 6*e*h)) + b^2*c*(35*c^2*f^2*h + 2*d^2* e*(9*f*g + 4*e*h) - c*d*f*(15*f*g + 46*e*h)) + 2*b^2*d*(d*e - c*f)*(3*d*f* g + 4*d*e*h - 7*c*f*h)*x))/(3*d^2*f*(d*e - c*f)*(c + d*x)) - ((a^2*d^2*f*( d*f*g - 4*d*e*h + 3*c*f*h) - b^2*(8*d^3*e^2*g - 35*c^3*f^2*h - 24*c*d^2*e* (f*g + e*h) + 15*c^2*d*f*(f*g + 4*e*h)) - 2*a*b*d*(15*c^2*f^2*h + 4*d^2*e* (f*g + 2*e*h) - 3*c*d*f*(f*g + 8*e*h)))*((2*Sqrt[e + f*x])/d - (2*Sqrt[d*e - c*f]*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e - c*f]])/d^(3/2)))/(2*d^2 *(d*e - c*f)))/(4*d*(d*e - c*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 = 0.70 (sec) , antiderivative size = 492, normalized size of antiderivative = 1.39
| method | result | size |
| pseudoelliptic | \(-\frac {3 \left (-\left (x d +c \right )^{2} f \left (\frac {\left (a^{2} f^{2} g -4 a e \left (a h +2 b g \right ) f -16 b \,e^{2} \left (a h +\frac {b g}{2}\right )\right ) d^{3}}{3}+c \left (\left (a^{2} h +2 g a b \right ) f^{2}+8 \left (2 b h a +b^{2} g \right ) e f +8 b^{2} e^{2} h \right ) d^{2}-10 c^{2} \left (\left (a h +\frac {b g}{2}\right ) f +2 e h b \right ) b f d +\frac {35 b^{2} c^{3} f^{2} h}{3}\right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )+\sqrt {\left (c f -d e \right ) d}\, \sqrt {f x +e}\, \left (\frac {\left (-a^{2} f^{2} g x -2 \left (4 \left (-\frac {1}{3} h \,x^{3}-g \,x^{2}\right ) b^{2}+4 a x \left (-2 h x +g \right ) b +a^{2} \left (2 h x +g \right )\right ) e f +\frac {8 b^{2} e^{2} h \,x^{2}}{3}\right ) d^{4}}{3}-\frac {2 c \left (\left (4 \left (\frac {1}{3} h \,x^{3}+g \,x^{2}\right ) b^{2}-5 a x \left (-\frac {8 h x}{5}+g \right ) b -\frac {a^{2} \left (5 h x +g \right )}{2}\right ) f^{2}+\left (-12 x \left (-\frac {8 h x}{9}+g \right ) b^{2}+2 a \left (-12 h x +g \right ) b +a^{2} h \right ) e f -\frac {8 b^{2} e^{2} h x}{3}\right ) d^{3}}{3}+\left (\left (\frac {\left (\frac {56}{3} h \,x^{2}-25 g x \right ) b^{2}}{3}+2 a \left (-\frac {25 h x}{3}+g \right ) b +a^{2} h \right ) f^{2}+\frac {28 \left (\left (-\frac {47 h x}{21}+\frac {g}{2}\right ) b +a h \right ) b e f}{3}+\frac {8 b^{2} e^{2} h}{9}\right ) c^{2} d^{2}-10 c^{3} \left (\left (\frac {\left (-\frac {35 h x}{9}+g \right ) b}{2}+a h \right ) f +\frac {11 e h b}{9}\right ) b f d +\frac {35 b^{2} c^{4} f^{2} h}{3}\right )\right )}{4 \sqrt {\left (c f -d e \right ) d}\, \left (x d +c \right )^{2} d^{4} \left (c f -d e \right ) f}\) | \(492\) |
| risch | \(\frac {2 b \left (h x d b f +6 a d f h -9 b c f h +b d e h +3 b d f g \right ) \sqrt {f x +e}}{3 f \,d^{4}}+\frac {\frac {-\frac {d f \left (5 a^{2} c \,d^{2} f h -4 a^{2} d^{3} e h -a^{2} d^{3} f g -18 a b \,c^{2} d f h +16 a b c \,d^{2} e h +10 a b c \,d^{2} f g -8 a b \,d^{3} e g +13 b^{2} c^{3} f h -12 b^{2} c^{2} d e h -9 b^{2} c^{2} d f g +8 b^{2} c \,d^{2} e g \right ) \left (f x +e \right )^{\frac {3}{2}}}{4 \left (c f -d e \right )}-\frac {f \left (3 a^{2} c \,d^{2} f h -4 a^{2} d^{3} e h +a^{2} d^{3} f g -14 a b \,c^{2} d f h +16 a b c \,d^{2} e h +6 a b c \,d^{2} f g -8 a b \,d^{3} e g +11 b^{2} c^{3} f h -12 b^{2} c^{2} d e h -7 b^{2} c^{2} d f g +8 b^{2} c \,d^{2} e g \right ) \sqrt {f x +e}}{4}}{\left (\left (f x +e \right ) d +c f -d e \right )^{2}}+\frac {\left (3 a^{2} c \,d^{2} f^{2} h -4 a^{2} d^{3} e f h +a^{2} d^{3} f^{2} g -30 a b \,c^{2} d \,f^{2} h +48 a b c \,d^{2} e f h +6 a b c \,d^{2} f^{2} g -16 e^{2} h b a \,d^{3}-8 a b \,d^{3} e f g +35 b^{2} c^{3} f^{2} h -60 b^{2} c^{2} d e f h -15 b^{2} c^{2} d \,f^{2} g +24 b^{2} c \,d^{2} e^{2} h +24 b^{2} c \,d^{2} e f g -8 b^{2} d^{3} e^{2} g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{4 \left (c f -d e \right ) \sqrt {\left (c f -d e \right ) d}}}{d^{4}}\) | \(543\) |
| derivativedivides | \(\frac {\frac {2 b \left (\frac {d h \left (f x +e \right )^{\frac {3}{2}} b}{3}+2 a d f h \sqrt {f x +e}-3 b c f h \sqrt {f x +e}+b d f g \sqrt {f x +e}\right )}{d^{4}}-\frac {2 f \left (\frac {\frac {d f \left (5 a^{2} c \,d^{2} f h -4 a^{2} d^{3} e h -a^{2} d^{3} f g -18 a b \,c^{2} d f h +16 a b c \,d^{2} e h +10 a b c \,d^{2} f g -8 a b \,d^{3} e g +13 b^{2} c^{3} f h -12 b^{2} c^{2} d e h -9 b^{2} c^{2} d f g +8 b^{2} c \,d^{2} e g \right ) \left (f x +e \right )^{\frac {3}{2}}}{8 c f -8 d e}+\frac {f \left (3 a^{2} c \,d^{2} f h -4 a^{2} d^{3} e h +a^{2} d^{3} f g -14 a b \,c^{2} d f h +16 a b c \,d^{2} e h +6 a b c \,d^{2} f g -8 a b \,d^{3} e g +11 b^{2} c^{3} f h -12 b^{2} c^{2} d e h -7 b^{2} c^{2} d f g +8 b^{2} c \,d^{2} e g \right ) \sqrt {f x +e}}{8}}{\left (\left (f x +e \right ) d +c f -d e \right )^{2}}-\frac {\left (3 a^{2} c \,d^{2} f^{2} h -4 a^{2} d^{3} e f h +a^{2} d^{3} f^{2} g -30 a b \,c^{2} d \,f^{2} h +48 a b c \,d^{2} e f h +6 a b c \,d^{2} f^{2} g -16 e^{2} h b a \,d^{3}-8 a b \,d^{3} e f g +35 b^{2} c^{3} f^{2} h -60 b^{2} c^{2} d e f h -15 b^{2} c^{2} d \,f^{2} g +24 b^{2} c \,d^{2} e^{2} h +24 b^{2} c \,d^{2} e f g -8 b^{2} d^{3} e^{2} g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{8 \left (c f -d e \right ) \sqrt {\left (c f -d e \right ) d}}\right )}{d^{4}}}{f}\) | \(559\) |
| default | \(\frac {\frac {2 b \left (\frac {d h \left (f x +e \right )^{\frac {3}{2}} b}{3}+2 a d f h \sqrt {f x +e}-3 b c f h \sqrt {f x +e}+b d f g \sqrt {f x +e}\right )}{d^{4}}-\frac {2 f \left (\frac {\frac {d f \left (5 a^{2} c \,d^{2} f h -4 a^{2} d^{3} e h -a^{2} d^{3} f g -18 a b \,c^{2} d f h +16 a b c \,d^{2} e h +10 a b c \,d^{2} f g -8 a b \,d^{3} e g +13 b^{2} c^{3} f h -12 b^{2} c^{2} d e h -9 b^{2} c^{2} d f g +8 b^{2} c \,d^{2} e g \right ) \left (f x +e \right )^{\frac {3}{2}}}{8 c f -8 d e}+\frac {f \left (3 a^{2} c \,d^{2} f h -4 a^{2} d^{3} e h +a^{2} d^{3} f g -14 a b \,c^{2} d f h +16 a b c \,d^{2} e h +6 a b c \,d^{2} f g -8 a b \,d^{3} e g +11 b^{2} c^{3} f h -12 b^{2} c^{2} d e h -7 b^{2} c^{2} d f g +8 b^{2} c \,d^{2} e g \right ) \sqrt {f x +e}}{8}}{\left (\left (f x +e \right ) d +c f -d e \right )^{2}}-\frac {\left (3 a^{2} c \,d^{2} f^{2} h -4 a^{2} d^{3} e f h +a^{2} d^{3} f^{2} g -30 a b \,c^{2} d \,f^{2} h +48 a b c \,d^{2} e f h +6 a b c \,d^{2} f^{2} g -16 e^{2} h b a \,d^{3}-8 a b \,d^{3} e f g +35 b^{2} c^{3} f^{2} h -60 b^{2} c^{2} d e f h -15 b^{2} c^{2} d \,f^{2} g +24 b^{2} c \,d^{2} e^{2} h +24 b^{2} c \,d^{2} e f g -8 b^{2} d^{3} e^{2} g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{8 \left (c f -d e \right ) \sqrt {\left (c f -d e \right ) d}}\right )}{d^{4}}}{f}\) | \(559\) |
Input:
int((b*x+a)^2*(f*x+e)^(1/2)*(h*x+g)/(d*x+c)^3,x,method=_RETURNVERBOSE)
Output:
-3/4/((c*f-d*e)*d)^(1/2)*(-(d*x+c)^2*f*(1/3*(a^2*f^2*g-4*a*e*(a*h+2*b*g)*f -16*b*e^2*(a*h+1/2*b*g))*d^3+c*((a^2*h+2*a*b*g)*f^2+8*(2*a*b*h+b^2*g)*e*f+ 8*b^2*e^2*h)*d^2-10*c^2*((a*h+1/2*b*g)*f+2*e*h*b)*b*f*d+35/3*b^2*c^3*f^2*h )*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*(-a^2*f^2*g*x-2*(4*(-1/3*h*x^3-g*x^2)*b^2+4*a*x*(-2*h*x+g)*b+a^ 2*(2*h*x+g))*e*f+8/3*b^2*e^2*h*x^2)*d^4-2/3*c*((4*(1/3*h*x^3+g*x^2)*b^2-5* a*x*(-8/5*h*x+g)*b-1/2*a^2*(5*h*x+g))*f^2+(-12*x*(-8/9*h*x+g)*b^2+2*a*(-12 *h*x+g)*b+a^2*h)*e*f-8/3*b^2*e^2*h*x)*d^3+((1/3*(56/3*h*x^2-25*g*x)*b^2+2* a*(-25/3*h*x+g)*b+a^2*h)*f^2+28/3*((-47/21*h*x+1/2*g)*b+a*h)*b*e*f+8/9*b^2 *e^2*h)*c^2*d^2-10*c^3*((1/2*(-35/9*h*x+g)*b+a*h)*f+11/9*e*h*b)*b*f*d+35/3 *b^2*c^4*f^2*h))/(d*x+c)^2/d^4/(c*f-d*e)/f
Leaf count of result is larger than twice the leaf count of optimal. 1337 vs. \(2 (327) = 654\).
Time = 0.31 (sec) , antiderivative size = 2688, normalized size of antiderivative = 7.61 \[ \int \frac {(a+b x)^2 \sqrt {e+f x} (g+h x)}{(c+d x)^3} \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^2*(f*x+e)^(1/2)*(h*x+g)/(d*x+c)^3,x, algorithm="fricas")
Output:
[1/24*(3*sqrt(d^2*e - c*d*f)*(((8*b^2*d^5*e^2*f - 8*(3*b^2*c*d^4 - a*b*d^5 )*e*f^2 + (15*b^2*c^2*d^3 - 6*a*b*c*d^4 - a^2*d^5)*f^3)*g - (8*(3*b^2*c*d^ 4 - 2*a*b*d^5)*e^2*f - 4*(15*b^2*c^2*d^3 - 12*a*b*c*d^4 + a^2*d^5)*e*f^2 + (35*b^2*c^3*d^2 - 30*a*b*c^2*d^3 + 3*a^2*c*d^4)*f^3)*h)*x^2 + (8*b^2*c^2* d^3*e^2*f - 8*(3*b^2*c^3*d^2 - a*b*c^2*d^3)*e*f^2 + (15*b^2*c^4*d - 6*a*b* c^3*d^2 - a^2*c^2*d^3)*f^3)*g - (8*(3*b^2*c^3*d^2 - 2*a*b*c^2*d^3)*e^2*f - 4*(15*b^2*c^4*d - 12*a*b*c^3*d^2 + a^2*c^2*d^3)*e*f^2 + (35*b^2*c^5 - 30* a*b*c^4*d + 3*a^2*c^3*d^2)*f^3)*h + 2*((8*b^2*c*d^4*e^2*f - 8*(3*b^2*c^2*d ^3 - a*b*c*d^4)*e*f^2 + (15*b^2*c^3*d^2 - 6*a*b*c^2*d^3 - a^2*c*d^4)*f^3)* g - (8*(3*b^2*c^2*d^3 - 2*a*b*c*d^4)*e^2*f - 4*(15*b^2*c^3*d^2 - 12*a*b*c^ 2*d^3 + a^2*c*d^4)*e*f^2 + (35*b^2*c^4*d - 30*a*b*c^3*d^2 + 3*a^2*c^2*d^3) *f^3)*h)*x)*log((d*f*x + 2*d*e - c*f - 2*sqrt(d^2*e - c*d*f)*sqrt(f*x + e) )/(d*x + c)) + 2*(8*(b^2*d^6*e^2*f - 2*b^2*c*d^5*e*f^2 + b^2*c^2*d^4*f^3)* h*x^3 + 8*(3*(b^2*d^6*e^2*f - 2*b^2*c*d^5*e*f^2 + b^2*c^2*d^4*f^3)*g + (b^ 2*d^6*e^3 - 3*(3*b^2*c*d^5 - 2*a*b*d^6)*e^2*f + 3*(5*b^2*c^2*d^4 - 4*a*b*c *d^5)*e*f^2 - (7*b^2*c^3*d^3 - 6*a*b*c^2*d^4)*f^3)*h)*x^2 + 3*(2*(7*b^2*c^ 2*d^4 - 2*a*b*c*d^5 - a^2*d^6)*e^2*f - (29*b^2*c^3*d^3 - 10*a*b*c^2*d^4 - 3*a^2*c*d^5)*e*f^2 + (15*b^2*c^4*d^2 - 6*a*b*c^3*d^3 - a^2*c^2*d^4)*f^3)*g + (8*b^2*c^2*d^4*e^3 - 2*(59*b^2*c^3*d^3 - 42*a*b*c^2*d^4 + 3*a^2*c*d^5)* e^2*f + (215*b^2*c^4*d^2 - 174*a*b*c^3*d^3 + 15*a^2*c^2*d^4)*e*f^2 - 3*...
Timed out. \[ \int \frac {(a+b x)^2 \sqrt {e+f x} (g+h x)}{(c+d x)^3} \, dx=\text {Timed out} \] Input:
integrate((b*x+a)**2*(f*x+e)**(1/2)*(h*x+g)/(d*x+c)**3,x)
Output:
Timed out
Exception generated. \[ \int \frac {(a+b x)^2 \sqrt {e+f x} (g+h x)}{(c+d x)^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((b*x+a)^2*(f*x+e)^(1/2)*(h*x+g)/(d*x+c)^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(c*f-d*e>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 904 vs. \(2 (327) = 654\).
Time = 0.16 (sec) , antiderivative size = 904, normalized size of antiderivative = 2.56 \[ \int \frac {(a+b x)^2 \sqrt {e+f x} (g+h x)}{(c+d x)^3} \, dx =\text {Too large to display} \] Input:
integrate((b*x+a)^2*(f*x+e)^(1/2)*(h*x+g)/(d*x+c)^3,x, algorithm="giac")
Output:
1/4*(8*b^2*d^3*e^2*g - 24*b^2*c*d^2*e*f*g + 8*a*b*d^3*e*f*g + 15*b^2*c^2*d *f^2*g - 6*a*b*c*d^2*f^2*g - a^2*d^3*f^2*g - 24*b^2*c*d^2*e^2*h + 16*a*b*d ^3*e^2*h + 60*b^2*c^2*d*e*f*h - 48*a*b*c*d^2*e*f*h + 4*a^2*d^3*e*f*h - 35* b^2*c^3*f^2*h + 30*a*b*c^2*d*f^2*h - 3*a^2*c*d^2*f^2*h)*arctan(sqrt(f*x + e)*d/sqrt(-d^2*e + c*d*f))/((d^5*e - c*d^4*f)*sqrt(-d^2*e + c*d*f)) + 1/4* (8*(f*x + e)^(3/2)*b^2*c*d^3*e*f*g - 8*(f*x + e)^(3/2)*a*b*d^4*e*f*g - 8*s qrt(f*x + e)*b^2*c*d^3*e^2*f*g + 8*sqrt(f*x + e)*a*b*d^4*e^2*f*g - 9*(f*x + e)^(3/2)*b^2*c^2*d^2*f^2*g + 10*(f*x + e)^(3/2)*a*b*c*d^3*f^2*g - (f*x + e)^(3/2)*a^2*d^4*f^2*g + 15*sqrt(f*x + e)*b^2*c^2*d^2*e*f^2*g - 14*sqrt(f *x + e)*a*b*c*d^3*e*f^2*g - sqrt(f*x + e)*a^2*d^4*e*f^2*g - 7*sqrt(f*x + e )*b^2*c^3*d*f^3*g + 6*sqrt(f*x + e)*a*b*c^2*d^2*f^3*g + sqrt(f*x + e)*a^2* c*d^3*f^3*g - 12*(f*x + e)^(3/2)*b^2*c^2*d^2*e*f*h + 16*(f*x + e)^(3/2)*a* b*c*d^3*e*f*h - 4*(f*x + e)^(3/2)*a^2*d^4*e*f*h + 12*sqrt(f*x + e)*b^2*c^2 *d^2*e^2*f*h - 16*sqrt(f*x + e)*a*b*c*d^3*e^2*f*h + 4*sqrt(f*x + e)*a^2*d^ 4*e^2*f*h + 13*(f*x + e)^(3/2)*b^2*c^3*d*f^2*h - 18*(f*x + e)^(3/2)*a*b*c^ 2*d^2*f^2*h + 5*(f*x + e)^(3/2)*a^2*c*d^3*f^2*h - 23*sqrt(f*x + e)*b^2*c^3 *d*e*f^2*h + 30*sqrt(f*x + e)*a*b*c^2*d^2*e*f^2*h - 7*sqrt(f*x + e)*a^2*c* d^3*e*f^2*h + 11*sqrt(f*x + e)*b^2*c^4*f^3*h - 14*sqrt(f*x + e)*a*b*c^3*d* f^3*h + 3*sqrt(f*x + e)*a^2*c^2*d^2*f^3*h)/((d^5*e - c*d^4*f)*((f*x + e)*d - d*e + c*f)^2) + 2/3*(3*sqrt(f*x + e)*b^2*d^6*f^3*g + (f*x + e)^(3/2)...
Time = 0.39 (sec) , antiderivative size = 639, normalized size of antiderivative = 1.81 \[ \int \frac {(a+b x)^2 \sqrt {e+f x} (g+h x)}{(c+d x)^3} \, dx=\sqrt {e+f\,x}\,\left (\frac {2\,b^2\,f\,g-6\,b^2\,e\,h+4\,a\,b\,f\,h}{d^3\,f}-\frac {6\,b^2\,h\,\left (c\,f-d\,e\right )}{d^4\,f}\right )-\frac {\sqrt {e+f\,x}\,\left (\frac {3\,h\,a^2\,c\,d^2\,f^2}{4}+\frac {g\,a^2\,d^3\,f^2}{4}-e\,h\,a^2\,d^3\,f-\frac {7\,h\,a\,b\,c^2\,d\,f^2}{2}+\frac {3\,g\,a\,b\,c\,d^2\,f^2}{2}+4\,e\,h\,a\,b\,c\,d^2\,f-2\,e\,g\,a\,b\,d^3\,f+\frac {11\,h\,b^2\,c^3\,f^2}{4}-\frac {7\,g\,b^2\,c^2\,d\,f^2}{4}-3\,e\,h\,b^2\,c^2\,d\,f+2\,e\,g\,b^2\,c\,d^2\,f\right )-\frac {{\left (e+f\,x\right )}^{3/2}\,\left (-5\,h\,a^2\,c\,d^3\,f^2+g\,a^2\,d^4\,f^2+4\,e\,h\,a^2\,d^4\,f+18\,h\,a\,b\,c^2\,d^2\,f^2-10\,g\,a\,b\,c\,d^3\,f^2-16\,e\,h\,a\,b\,c\,d^3\,f+8\,e\,g\,a\,b\,d^4\,f-13\,h\,b^2\,c^3\,d\,f^2+9\,g\,b^2\,c^2\,d^2\,f^2+12\,e\,h\,b^2\,c^2\,d^2\,f-8\,e\,g\,b^2\,c\,d^3\,f\right )}{4\,\left (c\,f-d\,e\right )}}{d^6\,{\left (e+f\,x\right )}^2-\left (e+f\,x\right )\,\left (2\,d^6\,e-2\,c\,d^5\,f\right )+d^6\,e^2+c^2\,d^4\,f^2-2\,c\,d^5\,e\,f}+\frac {\mathrm {atan}\left (\frac {\sqrt {d}\,\sqrt {e+f\,x}}{\sqrt {c\,f-d\,e}}\right )\,\left (3\,h\,a^2\,c\,d^2\,f^2-4\,h\,a^2\,d^3\,e\,f+g\,a^2\,d^3\,f^2-30\,h\,a\,b\,c^2\,d\,f^2+48\,h\,a\,b\,c\,d^2\,e\,f+6\,g\,a\,b\,c\,d^2\,f^2-16\,h\,a\,b\,d^3\,e^2-8\,g\,a\,b\,d^3\,e\,f+35\,h\,b^2\,c^3\,f^2-60\,h\,b^2\,c^2\,d\,e\,f-15\,g\,b^2\,c^2\,d\,f^2+24\,h\,b^2\,c\,d^2\,e^2+24\,g\,b^2\,c\,d^2\,e\,f-8\,g\,b^2\,d^3\,e^2\right )}{4\,d^{9/2}\,{\left (c\,f-d\,e\right )}^{3/2}}+\frac {2\,b^2\,h\,{\left (e+f\,x\right )}^{3/2}}{3\,d^3\,f} \] Input:
int(((e + f*x)^(1/2)*(g + h*x)*(a + b*x)^2)/(c + d*x)^3,x)
Output:
(e + f*x)^(1/2)*((2*b^2*f*g - 6*b^2*e*h + 4*a*b*f*h)/(d^3*f) - (6*b^2*h*(c *f - d*e))/(d^4*f)) - ((e + f*x)^(1/2)*((a^2*d^3*f^2*g)/4 + (11*b^2*c^3*f^ 2*h)/4 - a^2*d^3*e*f*h + (3*a^2*c*d^2*f^2*h)/4 - (7*b^2*c^2*d*f^2*g)/4 - 2 *a*b*d^3*e*f*g + (3*a*b*c*d^2*f^2*g)/2 - (7*a*b*c^2*d*f^2*h)/2 + 2*b^2*c*d ^2*e*f*g - 3*b^2*c^2*d*e*f*h + 4*a*b*c*d^2*e*f*h) - ((e + f*x)^(3/2)*(a^2* d^4*f^2*g + 4*a^2*d^4*e*f*h - 5*a^2*c*d^3*f^2*h - 13*b^2*c^3*d*f^2*h + 9*b ^2*c^2*d^2*f^2*g + 8*a*b*d^4*e*f*g - 10*a*b*c*d^3*f^2*g - 8*b^2*c*d^3*e*f* g + 18*a*b*c^2*d^2*f^2*h + 12*b^2*c^2*d^2*e*f*h - 16*a*b*c*d^3*e*f*h))/(4* (c*f - d*e)))/(d^6*(e + f*x)^2 - (e + f*x)*(2*d^6*e - 2*c*d^5*f) + d^6*e^2 + c^2*d^4*f^2 - 2*c*d^5*e*f) + (atan((d^(1/2)*(e + f*x)^(1/2))/(c*f - d*e )^(1/2))*(a^2*d^3*f^2*g - 8*b^2*d^3*e^2*g + 35*b^2*c^3*f^2*h - 16*a*b*d^3* e^2*h - 4*a^2*d^3*e*f*h + 3*a^2*c*d^2*f^2*h + 24*b^2*c*d^2*e^2*h - 15*b^2* c^2*d*f^2*g - 8*a*b*d^3*e*f*g + 6*a*b*c*d^2*f^2*g - 30*a*b*c^2*d*f^2*h + 2 4*b^2*c*d^2*e*f*g - 60*b^2*c^2*d*e*f*h + 48*a*b*c*d^2*e*f*h))/(4*d^(9/2)*( c*f - d*e)^(3/2)) + (2*b^2*h*(e + f*x)^(3/2))/(3*d^3*f)
Time = 0.18 (sec) , antiderivative size = 3292, normalized size of antiderivative = 9.33 \[ \int \frac {(a+b x)^2 \sqrt {e+f x} (g+h x)}{(c+d x)^3} \, dx =\text {Too large to display} \] Input:
int((b*x+a)^2*(f*x+e)^(1/2)*(h*x+g)/(d*x+c)^3,x)
Output:
(9*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e) ))*a**2*c**3*d**2*f**3*h - 12*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**3*e*f**2*h + 3*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**3*f **3*g + 18*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**3*f**3*h*x - 24*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt (e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a**2*c*d**4*e*f**2*h*x + 6*sqrt(d) *sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a**2*c* d**4*f**3*g*x + 9*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)* sqrt(c*f - d*e)))*a**2*c*d**4*f**3*h*x**2 - 12*sqrt(d)*sqrt(c*f - d*e)*ata n((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a**2*d**5*e*f**2*h*x**2 + 3 *sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e))) *a**2*d**5*f**3*g*x**2 - 90*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d) /(sqrt(d)*sqrt(c*f - d*e)))*a*b*c**4*d*f**3*h + 144*sqrt(d)*sqrt(c*f - d*e )*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a*b*c**3*d**2*e*f**2*h + 18*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d *e)))*a*b*c**3*d**2*f**3*g - 180*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f* x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a*b*c**3*d**2*f**3*h*x - 48*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**3 *e**2*f*h - 24*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*...