Integrand size = 47, antiderivative size = 214 \[ \int \frac {d+e x+f x^2}{(g+h x) \left (-c g^2+b g h+b h^2 x+c h^2 x^2\right )^{3/2}} \, dx=\frac {2 \left (f g^2-e g h+d h^2\right )}{3 h^3 (2 c g-b h) (g+h x) \sqrt {-g (c g-b h)+b h^2 x+c h^2 x^2}}-\frac {2 \left (12 c^2 f g^3+3 b^2 h^2 (3 f g-e h)-2 b c h \left (10 f g^2-h (e g+2 d h)\right )-h \left (6 b c e h^2-3 b^2 f h^2+4 c^2 \left (f g^2-h (e g+2 d h)\right )\right ) x\right )}{3 h^3 (2 c g-b h)^3 \sqrt {-g (c g-b h)+b h^2 x+c h^2 x^2}} \] Output:
2/3*(d*h^2-e*g*h+f*g^2)/h^3/(-b*h+2*c*g)/(h*x+g)/(-g*(-b*h+c*g)+b*h^2*x+c* h^2*x^2)^(1/2)-2/3*(12*c^2*f*g^3+3*b^2*h^2*(-e*h+3*f*g)-2*b*c*h*(10*f*g^2- h*(2*d*h+e*g))-h*(6*b*c*e*h^2-3*b^2*f*h^2+4*c^2*(f*g^2-h*(2*d*h+e*g)))*x)/ h^3/(-b*h+2*c*g)^3/(-g*(-b*h+c*g)+b*h^2*x+c*h^2*x^2)^(1/2)
Time = 0.47 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.02 \[ \int \frac {d+e x+f x^2}{(g+h x) \left (-c g^2+b g h+b h^2 x+c h^2 x^2\right )^{3/2}} \, dx=\frac {2 b^2 h^2 \left (-h (2 e g+d h+3 e h x)+f \left (8 g^2+12 g h x+3 h^2 x^2\right )\right )+8 c^2 \left (f g^2 \left (2 g^2+2 g h x-h^2 x^2\right )+h \left (e g \left (g^2+g h x+h^2 x^2\right )+d h \left (-g^2+2 g h x+2 h^2 x^2\right )\right )\right )-4 b c h \left (2 f g^2 (4 g+5 h x)+h \left (-2 d h (2 g+h x)+e \left (g^2+2 g h x+3 h^2 x^2\right )\right )\right )}{3 h^3 (-2 c g+b h)^3 (g+h x) \sqrt {(g+h x) (-c g+b h+c h x)}} \] Input:
Integrate[(d + e*x + f*x^2)/((g + h*x)*(-(c*g^2) + b*g*h + b*h^2*x + c*h^2 *x^2)^(3/2)),x]
Output:
(2*b^2*h^2*(-(h*(2*e*g + d*h + 3*e*h*x)) + f*(8*g^2 + 12*g*h*x + 3*h^2*x^2 )) + 8*c^2*(f*g^2*(2*g^2 + 2*g*h*x - h^2*x^2) + h*(e*g*(g^2 + g*h*x + h^2* x^2) + d*h*(-g^2 + 2*g*h*x + 2*h^2*x^2))) - 4*b*c*h*(2*f*g^2*(4*g + 5*h*x) + h*(-2*d*h*(2*g + h*x) + e*(g^2 + 2*g*h*x + 3*h^2*x^2))))/(3*h^3*(-2*c*g + b*h)^3*(g + h*x)*Sqrt[(g + h*x)*(-(c*g) + b*h + c*h*x)])
Time = 0.58 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.085, Rules used = {2169, 27, 1220, 1088}
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.
| \(\displaystyle \int \frac {d+e x+f x^2}{(g+h x) \left (b g h+b h^2 x-c g^2+c h^2 x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2169 |
| \(\displaystyle -\frac {\int \frac {h^3 (b f g-2 c d h+(2 c f g-2 c e h+b f h) x)}{2 (g+h x) \left (c x^2 h^2+b x h^2-g (c g-b h)\right )^{3/2}}dx}{c h^4}-\frac {f}{c h^3 \sqrt {-g (c g-b h)+b h^2 x+c h^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {b f g-2 c d h+(2 c f g-2 c e h+b f h) x}{(g+h x) \left (c x^2 h^2+b x h^2-g (c g-b h)\right )^{3/2}}dx}{2 c h}-\frac {f}{c h^3 \sqrt {-g (c g-b h)+b h^2 x+c h^2 x^2}}\) |
| \(\Big \downarrow \) 1220 |
| \(\displaystyle -\frac {\frac {\left (-3 b^2 f h^2+6 b c e h^2+c^2 \left (4 f g^2-4 h (2 d h+e g)\right )\right ) \int \frac {1}{\left (c x^2 h^2+b x h^2-g (c g-b h)\right )^{3/2}}dx}{3 h (2 c g-b h)}-\frac {4 c \left (d h^2-e g h+f g^2\right )}{3 h^2 (g+h x) (2 c g-b h) \sqrt {-g (c g-b h)+b h^2 x+c h^2 x^2}}}{2 c h}-\frac {f}{c h^3 \sqrt {-g (c g-b h)+b h^2 x+c h^2 x^2}}\) |
| \(\Big \downarrow \) 1088 |
| \(\displaystyle -\frac {-\frac {2 (b+2 c x) \left (-3 b^2 f h^2+6 b c e h^2+c^2 \left (4 f g^2-4 h (2 d h+e g)\right )\right )}{3 h (2 c g-b h)^3 \sqrt {-g (c g-b h)+b h^2 x+c h^2 x^2}}-\frac {4 c \left (d h^2-e g h+f g^2\right )}{3 h^2 (g+h x) (2 c g-b h) \sqrt {-g (c g-b h)+b h^2 x+c h^2 x^2}}}{2 c h}-\frac {f}{c h^3 \sqrt {-g (c g-b h)+b h^2 x+c h^2 x^2}}\) |
Input:
Int[(d + e*x + f*x^2)/((g + h*x)*(-(c*g^2) + b*g*h + b*h^2*x + c*h^2*x^2)^ (3/2)),x]
Output:
-(f/(c*h^3*Sqrt[-(g*(c*g - b*h)) + b*h^2*x + c*h^2*x^2])) - ((-2*(6*b*c*e* h^2 - 3*b^2*f*h^2 + c^2*(4*f*g^2 - 4*h*(e*g + 2*d*h)))*(b + 2*c*x))/(3*h*( 2*c*g - b*h)^3*Sqrt[-(g*(c*g - b*h)) + b*h^2*x + c*h^2*x^2]) - (4*c*(f*g^2 - e*g*h + d*h^2))/(3*h^2*(2*c*g - b*h)*(g + h*x)*Sqrt[-(g*(c*g - b*h)) + b*h^2*x + c*h^2*x^2]))/(2*c*h)
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_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[-2*((b + 2*c*x)/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d*g - e*f)*(d + e*x)^m*((a + b*x + c*x ^2)^(p + 1)/((2*c*d - b*e)*(m + p + 1))), x] + Simp[(m*(g*(c*d - b*e) + c*e *f) + e*(p + 1)*(2*c*f - b*g))/(e*(2*c*d - b*e)*(m + p + 1)) Int[(d + e*x )^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ((LtQ[m, -1] && !IGtQ[m + p + 1, 0 ]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0 ]
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p _), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, S imp[f*(d + e*x)^(m + q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q + 2*p + 1))), x] + Simp[1/(c*e^q*(m + q + 2*p + 1)) Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 1)*(d + e*x)^q + e*f*(m + p + q)*(d + e*x)^(q - 2)*(b*d - 2*a*e + (2*c*d - b*e)*x), x], x], x] /; NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2 , 0]
Time = 0.49 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.51
| method | result | size |
| gosper | \(-\frac {2 \left (c h x +b h -c g \right ) \left (-3 b^{2} f \,h^{4} x^{2}+6 b c e \,h^{4} x^{2}-8 c^{2} d \,h^{4} x^{2}-4 c^{2} e g \,h^{3} x^{2}+4 c^{2} f \,g^{2} h^{2} x^{2}+3 b^{2} e \,h^{4} x -12 b^{2} f g \,h^{3} x -4 b c d \,h^{4} x +4 b c e g \,h^{3} x +20 b c f \,g^{2} h^{2} x -8 c^{2} d g \,h^{3} x -4 c^{2} e \,g^{2} h^{2} x -8 c^{2} f \,g^{3} h x +b^{2} d \,h^{4}+2 b^{2} e g \,h^{3}-8 b^{2} f \,g^{2} h^{2}-8 b c d g \,h^{3}+2 b c e \,g^{2} h^{2}+16 b c f \,g^{3} h +4 c^{2} d \,g^{2} h^{2}-4 c^{2} e \,g^{3} h -8 c^{2} f \,g^{4}\right )}{3 \left (b^{3} h^{3}-6 b^{2} c g \,h^{2}+12 b \,c^{2} g^{2} h -8 c^{3} g^{3}\right ) h^{3} \left (c \,h^{2} x^{2}+b \,h^{2} x +b g h -c \,g^{2}\right )^{\frac {3}{2}}}\) | \(324\) |
| orering | \(-\frac {2 \left (c h x +b h -c g \right ) \left (-3 b^{2} f \,h^{4} x^{2}+6 b c e \,h^{4} x^{2}-8 c^{2} d \,h^{4} x^{2}-4 c^{2} e g \,h^{3} x^{2}+4 c^{2} f \,g^{2} h^{2} x^{2}+3 b^{2} e \,h^{4} x -12 b^{2} f g \,h^{3} x -4 b c d \,h^{4} x +4 b c e g \,h^{3} x +20 b c f \,g^{2} h^{2} x -8 c^{2} d g \,h^{3} x -4 c^{2} e \,g^{2} h^{2} x -8 c^{2} f \,g^{3} h x +b^{2} d \,h^{4}+2 b^{2} e g \,h^{3}-8 b^{2} f \,g^{2} h^{2}-8 b c d g \,h^{3}+2 b c e \,g^{2} h^{2}+16 b c f \,g^{3} h +4 c^{2} d \,g^{2} h^{2}-4 c^{2} e \,g^{3} h -8 c^{2} f \,g^{4}\right )}{3 \left (b^{3} h^{3}-6 b^{2} c g \,h^{2}+12 b \,c^{2} g^{2} h -8 c^{3} g^{3}\right ) h^{3} \left (c \,h^{2} x^{2}+b \,h^{2} x +b g h -c \,g^{2}\right )^{\frac {3}{2}}}\) | \(324\) |
| trager | \(-\frac {2 \left (-3 b^{2} f \,h^{4} x^{2}+6 b c e \,h^{4} x^{2}-8 c^{2} d \,h^{4} x^{2}-4 c^{2} e g \,h^{3} x^{2}+4 c^{2} f \,g^{2} h^{2} x^{2}+3 b^{2} e \,h^{4} x -12 b^{2} f g \,h^{3} x -4 b c d \,h^{4} x +4 b c e g \,h^{3} x +20 b c f \,g^{2} h^{2} x -8 c^{2} d g \,h^{3} x -4 c^{2} e \,g^{2} h^{2} x -8 c^{2} f \,g^{3} h x +b^{2} d \,h^{4}+2 b^{2} e g \,h^{3}-8 b^{2} f \,g^{2} h^{2}-8 b c d g \,h^{3}+2 b c e \,g^{2} h^{2}+16 b c f \,g^{3} h +4 c^{2} d \,g^{2} h^{2}-4 c^{2} e \,g^{3} h -8 c^{2} f \,g^{4}\right ) \sqrt {c \,h^{2} x^{2}+b \,h^{2} x +b g h -c \,g^{2}}}{3 h^{3} \left (b^{2} h^{2}-4 b c g h +4 c^{2} g^{2}\right ) \left (h x +g \right )^{2} \left (b h -2 c g \right ) \left (c h x +b h -c g \right )}\) | \(329\) |
| default | \(\frac {\frac {2 e h \left (2 c \,h^{2} x +b \,h^{2}\right )}{\left (4 c \,h^{2} \left (b g h -c \,g^{2}\right )-b^{2} h^{4}\right ) \sqrt {c \,h^{2} x^{2}+b \,h^{2} x +b g h -c \,g^{2}}}+f h \left (-\frac {1}{c \,h^{2} \sqrt {c \,h^{2} x^{2}+b \,h^{2} x +b g h -c \,g^{2}}}-\frac {b \left (2 c \,h^{2} x +b \,h^{2}\right )}{c \left (4 c \,h^{2} \left (b g h -c \,g^{2}\right )-b^{2} h^{4}\right ) \sqrt {c \,h^{2} x^{2}+b \,h^{2} x +b g h -c \,g^{2}}}\right )-\frac {2 f g \left (2 c \,h^{2} x +b \,h^{2}\right )}{\left (4 c \,h^{2} \left (b g h -c \,g^{2}\right )-b^{2} h^{4}\right ) \sqrt {c \,h^{2} x^{2}+b \,h^{2} x +b g h -c \,g^{2}}}}{h^{2}}+\frac {\left (d \,h^{2}-e g h +f \,g^{2}\right ) \left (-\frac {2}{3 \left (b \,h^{2}-2 c g h \right ) \left (x +\frac {g}{h}\right ) \sqrt {c \,h^{2} \left (x +\frac {g}{h}\right )^{2}+\left (b \,h^{2}-2 c g h \right ) \left (x +\frac {g}{h}\right )}}+\frac {8 c \,h^{2} \left (2 c \,h^{2} \left (x +\frac {g}{h}\right )+b \,h^{2}-2 c g h \right )}{3 \left (b \,h^{2}-2 c g h \right )^{3} \sqrt {c \,h^{2} \left (x +\frac {g}{h}\right )^{2}+\left (b \,h^{2}-2 c g h \right ) \left (x +\frac {g}{h}\right )}}\right )}{h^{3}}\) | \(424\) |
Input:
int((f*x^2+e*x+d)/(h*x+g)/(c*h^2*x^2+b*h^2*x+b*g*h-c*g^2)^(3/2),x,method=_ RETURNVERBOSE)
Output:
-2/3*(c*h*x+b*h-c*g)*(-3*b^2*f*h^4*x^2+6*b*c*e*h^4*x^2-8*c^2*d*h^4*x^2-4*c ^2*e*g*h^3*x^2+4*c^2*f*g^2*h^2*x^2+3*b^2*e*h^4*x-12*b^2*f*g*h^3*x-4*b*c*d* h^4*x+4*b*c*e*g*h^3*x+20*b*c*f*g^2*h^2*x-8*c^2*d*g*h^3*x-4*c^2*e*g^2*h^2*x -8*c^2*f*g^3*h*x+b^2*d*h^4+2*b^2*e*g*h^3-8*b^2*f*g^2*h^2-8*b*c*d*g*h^3+2*b *c*e*g^2*h^2+16*b*c*f*g^3*h+4*c^2*d*g^2*h^2-4*c^2*e*g^3*h-8*c^2*f*g^4)/(b^ 3*h^3-6*b^2*c*g*h^2+12*b*c^2*g^2*h-8*c^3*g^3)/h^3/(c*h^2*x^2+b*h^2*x+b*g*h -c*g^2)^(3/2)
Leaf count of result is larger than twice the leaf count of optimal. 465 vs. \(2 (206) = 412\).
Time = 14.24 (sec) , antiderivative size = 465, normalized size of antiderivative = 2.17 \[ \int \frac {d+e x+f x^2}{(g+h x) \left (-c g^2+b g h+b h^2 x+c h^2 x^2\right )^{3/2}} \, dx=\frac {2 \, {\left (8 \, c^{2} f g^{4} - b^{2} d h^{4} + 4 \, {\left (c^{2} e - 4 \, b c f\right )} g^{3} h - 2 \, {\left (2 \, c^{2} d + b c e - 4 \, b^{2} f\right )} g^{2} h^{2} + 2 \, {\left (4 \, b c d - b^{2} e\right )} g h^{3} - {\left (4 \, c^{2} f g^{2} h^{2} - 4 \, c^{2} e g h^{3} - {\left (8 \, c^{2} d - 6 \, b c e + 3 \, b^{2} f\right )} h^{4}\right )} x^{2} + {\left (8 \, c^{2} f g^{3} h + 4 \, {\left (c^{2} e - 5 \, b c f\right )} g^{2} h^{2} + 4 \, {\left (2 \, c^{2} d - b c e + 3 \, b^{2} f\right )} g h^{3} + {\left (4 \, b c d - 3 \, b^{2} e\right )} h^{4}\right )} x\right )} \sqrt {c h^{2} x^{2} + b h^{2} x - c g^{2} + b g h}}{3 \, {\left (8 \, c^{4} g^{6} h^{3} - 20 \, b c^{3} g^{5} h^{4} + 18 \, b^{2} c^{2} g^{4} h^{5} - 7 \, b^{3} c g^{3} h^{6} + b^{4} g^{2} h^{7} - {\left (8 \, c^{4} g^{3} h^{6} - 12 \, b c^{3} g^{2} h^{7} + 6 \, b^{2} c^{2} g h^{8} - b^{3} c h^{9}\right )} x^{3} - {\left (8 \, c^{4} g^{4} h^{5} - 4 \, b c^{3} g^{3} h^{6} - 6 \, b^{2} c^{2} g^{2} h^{7} + 5 \, b^{3} c g h^{8} - b^{4} h^{9}\right )} x^{2} + {\left (8 \, c^{4} g^{5} h^{4} - 28 \, b c^{3} g^{4} h^{5} + 30 \, b^{2} c^{2} g^{3} h^{6} - 13 \, b^{3} c g^{2} h^{7} + 2 \, b^{4} g h^{8}\right )} x\right )}} \] Input:
integrate((f*x^2+e*x+d)/(h*x+g)/(c*h^2*x^2+b*h^2*x+b*g*h-c*g^2)^(3/2),x, a lgorithm="fricas")
Output:
2/3*(8*c^2*f*g^4 - b^2*d*h^4 + 4*(c^2*e - 4*b*c*f)*g^3*h - 2*(2*c^2*d + b* c*e - 4*b^2*f)*g^2*h^2 + 2*(4*b*c*d - b^2*e)*g*h^3 - (4*c^2*f*g^2*h^2 - 4* c^2*e*g*h^3 - (8*c^2*d - 6*b*c*e + 3*b^2*f)*h^4)*x^2 + (8*c^2*f*g^3*h + 4* (c^2*e - 5*b*c*f)*g^2*h^2 + 4*(2*c^2*d - b*c*e + 3*b^2*f)*g*h^3 + (4*b*c*d - 3*b^2*e)*h^4)*x)*sqrt(c*h^2*x^2 + b*h^2*x - c*g^2 + b*g*h)/(8*c^4*g^6*h ^3 - 20*b*c^3*g^5*h^4 + 18*b^2*c^2*g^4*h^5 - 7*b^3*c*g^3*h^6 + b^4*g^2*h^7 - (8*c^4*g^3*h^6 - 12*b*c^3*g^2*h^7 + 6*b^2*c^2*g*h^8 - b^3*c*h^9)*x^3 - (8*c^4*g^4*h^5 - 4*b*c^3*g^3*h^6 - 6*b^2*c^2*g^2*h^7 + 5*b^3*c*g*h^8 - b^4 *h^9)*x^2 + (8*c^4*g^5*h^4 - 28*b*c^3*g^4*h^5 + 30*b^2*c^2*g^3*h^6 - 13*b^ 3*c*g^2*h^7 + 2*b^4*g*h^8)*x)
\[ \int \frac {d+e x+f x^2}{(g+h x) \left (-c g^2+b g h+b h^2 x+c h^2 x^2\right )^{3/2}} \, dx=\int \frac {d + e x + f x^{2}}{\left (\left (g + h x\right ) \left (b h - c g + c h x\right )\right )^{\frac {3}{2}} \left (g + h x\right )}\, dx \] Input:
integrate((f*x**2+e*x+d)/(h*x+g)/(c*h**2*x**2+b*h**2*x+b*g*h-c*g**2)**(3/2 ),x)
Output:
Integral((d + e*x + f*x**2)/(((g + h*x)*(b*h - c*g + c*h*x))**(3/2)*(g + h *x)), x)
Exception generated. \[ \int \frac {d+e x+f x^2}{(g+h x) \left (-c g^2+b g h+b h^2 x+c h^2 x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((f*x^2+e*x+d)/(h*x+g)/(c*h^2*x^2+b*h^2*x+b*g*h-c*g^2)^(3/2),x, a lgorithm="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(b*h-2*c*g>0)', see `assume?` for more deta
\[ \int \frac {d+e x+f x^2}{(g+h x) \left (-c g^2+b g h+b h^2 x+c h^2 x^2\right )^{3/2}} \, dx=\int { \frac {f x^{2} + e x + d}{{\left (c h^{2} x^{2} + b h^{2} x - c g^{2} + b g h\right )}^{\frac {3}{2}} {\left (h x + g\right )}} \,d x } \] Input:
integrate((f*x^2+e*x+d)/(h*x+g)/(c*h^2*x^2+b*h^2*x+b*g*h-c*g^2)^(3/2),x, a lgorithm="giac")
Output:
integrate((f*x^2 + e*x + d)/((c*h^2*x^2 + b*h^2*x - c*g^2 + b*g*h)^(3/2)*( h*x + g)), x)
Time = 18.84 (sec) , antiderivative size = 1089, normalized size of antiderivative = 5.09 \[ \int \frac {d+e x+f x^2}{(g+h x) \left (-c g^2+b g h+b h^2 x+c h^2 x^2\right )^{3/2}} \, dx=\frac {16\,c^2\,f\,g^4\,\sqrt {-c\,g^2+b\,g\,h+c\,h^2\,x^2+b\,h^2\,x}-2\,b^2\,d\,h^4\,\sqrt {-c\,g^2+b\,g\,h+c\,h^2\,x^2+b\,h^2\,x}-8\,c^2\,d\,g^2\,h^2\,\sqrt {-c\,g^2+b\,g\,h+c\,h^2\,x^2+b\,h^2\,x}+16\,b^2\,f\,g^2\,h^2\,\sqrt {-c\,g^2+b\,g\,h+c\,h^2\,x^2+b\,h^2\,x}+16\,c^2\,d\,h^4\,x^2\,\sqrt {-c\,g^2+b\,g\,h+c\,h^2\,x^2+b\,h^2\,x}+6\,b^2\,f\,h^4\,x^2\,\sqrt {-c\,g^2+b\,g\,h+c\,h^2\,x^2+b\,h^2\,x}-4\,b^2\,e\,g\,h^3\,\sqrt {-c\,g^2+b\,g\,h+c\,h^2\,x^2+b\,h^2\,x}+8\,c^2\,e\,g^3\,h\,\sqrt {-c\,g^2+b\,g\,h+c\,h^2\,x^2+b\,h^2\,x}-6\,b^2\,e\,h^4\,x\,\sqrt {-c\,g^2+b\,g\,h+c\,h^2\,x^2+b\,h^2\,x}+8\,b\,c\,d\,h^4\,x\,\sqrt {-c\,g^2+b\,g\,h+c\,h^2\,x^2+b\,h^2\,x}-8\,c^2\,f\,g^2\,h^2\,x^2\,\sqrt {-c\,g^2+b\,g\,h+c\,h^2\,x^2+b\,h^2\,x}-4\,b\,c\,e\,g^2\,h^2\,\sqrt {-c\,g^2+b\,g\,h+c\,h^2\,x^2+b\,h^2\,x}-12\,b\,c\,e\,h^4\,x^2\,\sqrt {-c\,g^2+b\,g\,h+c\,h^2\,x^2+b\,h^2\,x}+16\,c^2\,d\,g\,h^3\,x\,\sqrt {-c\,g^2+b\,g\,h+c\,h^2\,x^2+b\,h^2\,x}+24\,b^2\,f\,g\,h^3\,x\,\sqrt {-c\,g^2+b\,g\,h+c\,h^2\,x^2+b\,h^2\,x}+16\,c^2\,f\,g^3\,h\,x\,\sqrt {-c\,g^2+b\,g\,h+c\,h^2\,x^2+b\,h^2\,x}+8\,c^2\,e\,g^2\,h^2\,x\,\sqrt {-c\,g^2+b\,g\,h+c\,h^2\,x^2+b\,h^2\,x}+8\,c^2\,e\,g\,h^3\,x^2\,\sqrt {-c\,g^2+b\,g\,h+c\,h^2\,x^2+b\,h^2\,x}+16\,b\,c\,d\,g\,h^3\,\sqrt {-c\,g^2+b\,g\,h+c\,h^2\,x^2+b\,h^2\,x}-32\,b\,c\,f\,g^3\,h\,\sqrt {-c\,g^2+b\,g\,h+c\,h^2\,x^2+b\,h^2\,x}-8\,b\,c\,e\,g\,h^3\,x\,\sqrt {-c\,g^2+b\,g\,h+c\,h^2\,x^2+b\,h^2\,x}-40\,b\,c\,f\,g^2\,h^2\,x\,\sqrt {-c\,g^2+b\,g\,h+c\,h^2\,x^2+b\,h^2\,x}}{3\,b^4\,g^2\,h^7+6\,b^4\,g\,h^8\,x+3\,b^4\,h^9\,x^2-21\,b^3\,c\,g^3\,h^6-39\,b^3\,c\,g^2\,h^7\,x-15\,b^3\,c\,g\,h^8\,x^2+3\,b^3\,c\,h^9\,x^3+54\,b^2\,c^2\,g^4\,h^5+90\,b^2\,c^2\,g^3\,h^6\,x+18\,b^2\,c^2\,g^2\,h^7\,x^2-18\,b^2\,c^2\,g\,h^8\,x^3-60\,b\,c^3\,g^5\,h^4-84\,b\,c^3\,g^4\,h^5\,x+12\,b\,c^3\,g^3\,h^6\,x^2+36\,b\,c^3\,g^2\,h^7\,x^3+24\,c^4\,g^6\,h^3+24\,c^4\,g^5\,h^4\,x-24\,c^4\,g^4\,h^5\,x^2-24\,c^4\,g^3\,h^6\,x^3} \] Input:
int((d + e*x + f*x^2)/((g + h*x)*(b*h^2*x - c*g^2 + c*h^2*x^2 + b*g*h)^(3/ 2)),x)
Output:
(16*c^2*f*g^4*(b*h^2*x - c*g^2 + c*h^2*x^2 + b*g*h)^(1/2) - 2*b^2*d*h^4*(b *h^2*x - c*g^2 + c*h^2*x^2 + b*g*h)^(1/2) - 8*c^2*d*g^2*h^2*(b*h^2*x - c*g ^2 + c*h^2*x^2 + b*g*h)^(1/2) + 16*b^2*f*g^2*h^2*(b*h^2*x - c*g^2 + c*h^2* x^2 + b*g*h)^(1/2) + 16*c^2*d*h^4*x^2*(b*h^2*x - c*g^2 + c*h^2*x^2 + b*g*h )^(1/2) + 6*b^2*f*h^4*x^2*(b*h^2*x - c*g^2 + c*h^2*x^2 + b*g*h)^(1/2) - 4* b^2*e*g*h^3*(b*h^2*x - c*g^2 + c*h^2*x^2 + b*g*h)^(1/2) + 8*c^2*e*g^3*h*(b *h^2*x - c*g^2 + c*h^2*x^2 + b*g*h)^(1/2) - 6*b^2*e*h^4*x*(b*h^2*x - c*g^2 + c*h^2*x^2 + b*g*h)^(1/2) + 8*b*c*d*h^4*x*(b*h^2*x - c*g^2 + c*h^2*x^2 + b*g*h)^(1/2) - 8*c^2*f*g^2*h^2*x^2*(b*h^2*x - c*g^2 + c*h^2*x^2 + b*g*h)^ (1/2) - 4*b*c*e*g^2*h^2*(b*h^2*x - c*g^2 + c*h^2*x^2 + b*g*h)^(1/2) - 12*b *c*e*h^4*x^2*(b*h^2*x - c*g^2 + c*h^2*x^2 + b*g*h)^(1/2) + 16*c^2*d*g*h^3* x*(b*h^2*x - c*g^2 + c*h^2*x^2 + b*g*h)^(1/2) + 24*b^2*f*g*h^3*x*(b*h^2*x - c*g^2 + c*h^2*x^2 + b*g*h)^(1/2) + 16*c^2*f*g^3*h*x*(b*h^2*x - c*g^2 + c *h^2*x^2 + b*g*h)^(1/2) + 8*c^2*e*g^2*h^2*x*(b*h^2*x - c*g^2 + c*h^2*x^2 + b*g*h)^(1/2) + 8*c^2*e*g*h^3*x^2*(b*h^2*x - c*g^2 + c*h^2*x^2 + b*g*h)^(1 /2) + 16*b*c*d*g*h^3*(b*h^2*x - c*g^2 + c*h^2*x^2 + b*g*h)^(1/2) - 32*b*c* f*g^3*h*(b*h^2*x - c*g^2 + c*h^2*x^2 + b*g*h)^(1/2) - 8*b*c*e*g*h^3*x*(b*h ^2*x - c*g^2 + c*h^2*x^2 + b*g*h)^(1/2) - 40*b*c*f*g^2*h^2*x*(b*h^2*x - c* g^2 + c*h^2*x^2 + b*g*h)^(1/2))/(3*b^4*g^2*h^7 + 24*c^4*g^6*h^3 + 3*b^4*h^ 9*x^2 - 60*b*c^3*g^5*h^4 - 21*b^3*c*g^3*h^6 + 3*b^3*c*h^9*x^3 + 24*c^4*...
Time = 0.21 (sec) , antiderivative size = 1034, normalized size of antiderivative = 4.83 \[ \int \frac {d+e x+f x^2}{(g+h x) \left (-c g^2+b g h+b h^2 x+c h^2 x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
int((f*x^2+e*x+d)/(h*x+g)/(c*h^2*x^2+b*h^2*x+b*g*h-c*g^2)^(3/2),x)
Output:
(2*( - 6*sqrt(c)*sqrt(b*h - c*g + c*h*x)*b**2*f*g**2*h**2 - 12*sqrt(c)*sqr t(b*h - c*g + c*h*x)*b**2*f*g*h**3*x - 6*sqrt(c)*sqrt(b*h - c*g + c*h*x)*b **2*f*h**4*x**2 + 6*sqrt(c)*sqrt(b*h - c*g + c*h*x)*b*c*e*g**2*h**2 + 12*s qrt(c)*sqrt(b*h - c*g + c*h*x)*b*c*e*g*h**3*x + 6*sqrt(c)*sqrt(b*h - c*g + c*h*x)*b*c*e*h**4*x**2 + 12*sqrt(c)*sqrt(b*h - c*g + c*h*x)*b*c*f*g**3*h + 24*sqrt(c)*sqrt(b*h - c*g + c*h*x)*b*c*f*g**2*h**2*x + 12*sqrt(c)*sqrt(b *h - c*g + c*h*x)*b*c*f*g*h**3*x**2 - 8*sqrt(c)*sqrt(b*h - c*g + c*h*x)*c* *2*d*g**2*h**2 - 16*sqrt(c)*sqrt(b*h - c*g + c*h*x)*c**2*d*g*h**3*x - 8*sq rt(c)*sqrt(b*h - c*g + c*h*x)*c**2*d*h**4*x**2 - 4*sqrt(c)*sqrt(b*h - c*g + c*h*x)*c**2*e*g**3*h - 8*sqrt(c)*sqrt(b*h - c*g + c*h*x)*c**2*e*g**2*h** 2*x - 4*sqrt(c)*sqrt(b*h - c*g + c*h*x)*c**2*e*g*h**3*x**2 - 8*sqrt(c)*sqr t(b*h - c*g + c*h*x)*c**2*f*g**4 - 16*sqrt(c)*sqrt(b*h - c*g + c*h*x)*c**2 *f*g**3*h*x - 8*sqrt(c)*sqrt(b*h - c*g + c*h*x)*c**2*f*g**2*h**2*x**2 - sq rt(g + h*x)*b**2*c*d*h**4 - 2*sqrt(g + h*x)*b**2*c*e*g*h**3 - 3*sqrt(g + h *x)*b**2*c*e*h**4*x + 8*sqrt(g + h*x)*b**2*c*f*g**2*h**2 + 12*sqrt(g + h*x )*b**2*c*f*g*h**3*x + 3*sqrt(g + h*x)*b**2*c*f*h**4*x**2 + 8*sqrt(g + h*x) *b*c**2*d*g*h**3 + 4*sqrt(g + h*x)*b*c**2*d*h**4*x - 2*sqrt(g + h*x)*b*c** 2*e*g**2*h**2 - 4*sqrt(g + h*x)*b*c**2*e*g*h**3*x - 6*sqrt(g + h*x)*b*c**2 *e*h**4*x**2 - 16*sqrt(g + h*x)*b*c**2*f*g**3*h - 20*sqrt(g + h*x)*b*c**2* f*g**2*h**2*x - 4*sqrt(g + h*x)*c**3*d*g**2*h**2 + 8*sqrt(g + h*x)*c**3...