Integrand size = 33, antiderivative size = 186 \[ \int \frac {d+e x+f x^2+g x^3}{x^4 \sqrt {a+b x+c x^2}} \, dx=-\frac {d \sqrt {a+b x+c x^2}}{3 a x^3}+\frac {(5 b d-6 a e) \sqrt {a+b x+c x^2}}{12 a^2 x^2}-\frac {\left (15 b^2 d-16 a c d-18 a b e+24 a^2 f\right ) \sqrt {a+b x+c x^2}}{24 a^3 x}+\frac {\left (5 b^3 d-6 a b^2 e-4 a b (3 c d-2 a f)+8 a^2 (c e-2 a g)\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{16 a^{7/2}} \]
1/16*(5*b^3*d-6*a*b^2*e-4*a*b*(-2*a*f+3*c*d)+8*a^2*(-2*a*g+c*e))*arctanh(1 /2*(b*x+2*a)/a^(1/2)/(c*x^2+b*x+a)^(1/2))/a^(7/2)-1/3*d*(c*x^2+b*x+a)^(1/2 )/a/x^3+1/12*(-6*a*e+5*b*d)*(c*x^2+b*x+a)^(1/2)/a^2/x^2-1/24*(24*a^2*f-18* a*b*e-16*a*c*d+15*b^2*d)*(c*x^2+b*x+a)^(1/2)/a^3/x
Time = 0.93 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.96 \[ \int \frac {d+e x+f x^2+g x^3}{x^4 \sqrt {a+b x+c x^2}} \, dx=\frac {\frac {\sqrt {a} \sqrt {a+x (b+c x)} \left (-15 b^2 d x^2+2 a x (5 b d+8 c d x+9 b e x)-4 a^2 (2 d+3 x (e+2 f x))\right )}{x^3}+3 \left (-5 b^3 d+16 a^3 g\right ) \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )+6 a \left (-6 b c d-3 b^2 e+4 a c e+4 a b f\right ) \text {arctanh}\left (\frac {-\sqrt {c} x+\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{24 a^{7/2}} \]
((Sqrt[a]*Sqrt[a + x*(b + c*x)]*(-15*b^2*d*x^2 + 2*a*x*(5*b*d + 8*c*d*x + 9*b*e*x) - 4*a^2*(2*d + 3*x*(e + 2*f*x))))/x^3 + 3*(-5*b^3*d + 16*a^3*g)*A rcTanh[(Sqrt[c]*x - Sqrt[a + x*(b + c*x)])/Sqrt[a]] + 6*a*(-6*b*c*d - 3*b^ 2*e + 4*a*c*e + 4*a*b*f)*ArcTanh[(-(Sqrt[c]*x) + Sqrt[a + x*(b + c*x)])/Sq rt[a]])/(24*a^(7/2))
Time = 0.52 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.08, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {2181, 27, 2181, 27, 1228, 1154, 219}
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.
| \(\displaystyle \int \frac {d+e x+f x^2+g x^3}{x^4 \sqrt {a+b x+c x^2}} \, dx\) |
| \(\Big \downarrow \) 2181 |
| \(\displaystyle -\frac {\int \frac {-6 a g x^2+2 (2 c d-3 a f) x+5 b d-6 a e}{2 x^3 \sqrt {c x^2+b x+a}}dx}{3 a}-\frac {d \sqrt {a+b x+c x^2}}{3 a x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {-6 a g x^2+2 (2 c d-3 a f) x+5 b d-6 a e}{x^3 \sqrt {c x^2+b x+a}}dx}{6 a}-\frac {d \sqrt {a+b x+c x^2}}{3 a x^3}\) |
| \(\Big \downarrow \) 2181 |
| \(\displaystyle -\frac {-\frac {\int \frac {15 d b^2-18 a e b-8 a (2 c d-3 a f)+2 \left (12 g a^2-6 c e a+5 b c d\right ) x}{2 x^2 \sqrt {c x^2+b x+a}}dx}{2 a}-\frac {\sqrt {a+b x+c x^2} (5 b d-6 a e)}{2 a x^2}}{6 a}-\frac {d \sqrt {a+b x+c x^2}}{3 a x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\int \frac {24 f a^2-16 c d a-18 b e a+15 b^2 d+2 \left (12 g a^2-6 c e a+5 b c d\right ) x}{x^2 \sqrt {c x^2+b x+a}}dx}{4 a}-\frac {\sqrt {a+b x+c x^2} (5 b d-6 a e)}{2 a x^2}}{6 a}-\frac {d \sqrt {a+b x+c x^2}}{3 a x^3}\) |
| \(\Big \downarrow \) 1228 |
| \(\displaystyle -\frac {-\frac {-\frac {3 \left (8 a^2 (c e-2 a g)-6 a b^2 e-4 a b (3 c d-2 a f)+5 b^3 d\right ) \int \frac {1}{x \sqrt {c x^2+b x+a}}dx}{2 a}-\frac {\sqrt {a+b x+c x^2} \left (-18 a b e-8 a (2 c d-3 a f)+15 b^2 d\right )}{a x}}{4 a}-\frac {\sqrt {a+b x+c x^2} (5 b d-6 a e)}{2 a x^2}}{6 a}-\frac {d \sqrt {a+b x+c x^2}}{3 a x^3}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle -\frac {-\frac {\frac {3 \left (8 a^2 (c e-2 a g)-6 a b^2 e-4 a b (3 c d-2 a f)+5 b^3 d\right ) \int \frac {1}{4 a-\frac {(2 a+b x)^2}{c x^2+b x+a}}d\frac {2 a+b x}{\sqrt {c x^2+b x+a}}}{a}-\frac {\sqrt {a+b x+c x^2} \left (-18 a b e-8 a (2 c d-3 a f)+15 b^2 d\right )}{a x}}{4 a}-\frac {\sqrt {a+b x+c x^2} (5 b d-6 a e)}{2 a x^2}}{6 a}-\frac {d \sqrt {a+b x+c x^2}}{3 a x^3}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {-\frac {\frac {3 \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right ) \left (8 a^2 (c e-2 a g)-6 a b^2 e-4 a b (3 c d-2 a f)+5 b^3 d\right )}{2 a^{3/2}}-\frac {\sqrt {a+b x+c x^2} \left (-18 a b e-8 a (2 c d-3 a f)+15 b^2 d\right )}{a x}}{4 a}-\frac {\sqrt {a+b x+c x^2} (5 b d-6 a e)}{2 a x^2}}{6 a}-\frac {d \sqrt {a+b x+c x^2}}{3 a x^3}\) |
-1/3*(d*Sqrt[a + b*x + c*x^2])/(a*x^3) - (-1/2*((5*b*d - 6*a*e)*Sqrt[a + b *x + c*x^2])/(a*x^2) - (-(((15*b^2*d - 18*a*b*e - 8*a*(2*c*d - 3*a*f))*Sqr t[a + b*x + c*x^2])/(a*x)) + (3*(5*b^3*d - 6*a*b^2*e - 4*a*b*(3*c*d - 2*a* f) + 8*a^2*(c*e - 2*a*g))*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c* x^2])])/(2*a^(3/2)))/(4*a))/(6*a)
3.3.86.3.1 Defintions of rubi rules used
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_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e *f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)) 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[Simplify[m + 2*p + 3], 0]
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_ ), x_Symbol] :> With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = Polynomi alRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + b*x + c*x^2) ^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Simp[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*ExpandToSum[(m + 1)*(c*d^2 - b*d*e + a*e^2)*Qx + c*d*R*(m + 1) - b*e*R*(m + p + 2) - c*e*R *(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, e, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1]
Time = 0.82 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.81
| method | result | size |
| risch | \(-\frac {\sqrt {c \,x^{2}+b x +a}\, \left (24 a^{2} f \,x^{2}-18 a b e \,x^{2}-16 a c d \,x^{2}+15 b^{2} d \,x^{2}+12 a^{2} e x -10 a b d x +8 a^{2} d \right )}{24 a^{3} x^{3}}-\frac {\left (16 a^{3} g -8 a^{2} b f -8 a^{2} c e +6 a \,b^{2} e +12 a b c d -5 b^{3} d \right ) \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{16 a^{\frac {7}{2}}}\) | \(150\) |
| default | \(-\frac {g \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{\sqrt {a}}+e \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{2 a \,x^{2}}-\frac {3 b \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{a x}+\frac {b \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{4 a}+\frac {c \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )+f \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{a x}+\frac {b \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )+d \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{3 a \,x^{3}}-\frac {5 b \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{2 a \,x^{2}}-\frac {3 b \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{a x}+\frac {b \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{4 a}+\frac {c \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{6 a}-\frac {2 c \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{a x}+\frac {b \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{3 a}\right )\) | \(424\) |
-1/24*(c*x^2+b*x+a)^(1/2)*(24*a^2*f*x^2-18*a*b*e*x^2-16*a*c*d*x^2+15*b^2*d *x^2+12*a^2*e*x-10*a*b*d*x+8*a^2*d)/a^3/x^3-1/16*(16*a^3*g-8*a^2*b*f-8*a^2 *c*e+6*a*b^2*e+12*a*b*c*d-5*b^3*d)/a^(7/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b* x+a)^(1/2))/x)
Time = 1.89 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.96 \[ \int \frac {d+e x+f x^2+g x^3}{x^4 \sqrt {a+b x+c x^2}} \, dx=\left [-\frac {3 \, {\left (8 \, a^{2} b f - 16 \, a^{3} g + {\left (5 \, b^{3} - 12 \, a b c\right )} d - 2 \, {\left (3 \, a b^{2} - 4 \, a^{2} c\right )} e\right )} \sqrt {a} x^{3} \log \left (-\frac {8 \, a b x + {\left (b^{2} + 4 \, a c\right )} x^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{2}}\right ) + 4 \, {\left (8 \, a^{3} d - {\left (18 \, a^{2} b e - 24 \, a^{3} f - {\left (15 \, a b^{2} - 16 \, a^{2} c\right )} d\right )} x^{2} - 2 \, {\left (5 \, a^{2} b d - 6 \, a^{3} e\right )} x\right )} \sqrt {c x^{2} + b x + a}}{96 \, a^{4} x^{3}}, -\frac {3 \, {\left (8 \, a^{2} b f - 16 \, a^{3} g + {\left (5 \, b^{3} - 12 \, a b c\right )} d - 2 \, {\left (3 \, a b^{2} - 4 \, a^{2} c\right )} e\right )} \sqrt {-a} x^{3} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{2} + a b x + a^{2}\right )}}\right ) + 2 \, {\left (8 \, a^{3} d - {\left (18 \, a^{2} b e - 24 \, a^{3} f - {\left (15 \, a b^{2} - 16 \, a^{2} c\right )} d\right )} x^{2} - 2 \, {\left (5 \, a^{2} b d - 6 \, a^{3} e\right )} x\right )} \sqrt {c x^{2} + b x + a}}{48 \, a^{4} x^{3}}\right ] \]
[-1/96*(3*(8*a^2*b*f - 16*a^3*g + (5*b^3 - 12*a*b*c)*d - 2*(3*a*b^2 - 4*a^ 2*c)*e)*sqrt(a)*x^3*log(-(8*a*b*x + (b^2 + 4*a*c)*x^2 - 4*sqrt(c*x^2 + b*x + a)*(b*x + 2*a)*sqrt(a) + 8*a^2)/x^2) + 4*(8*a^3*d - (18*a^2*b*e - 24*a^ 3*f - (15*a*b^2 - 16*a^2*c)*d)*x^2 - 2*(5*a^2*b*d - 6*a^3*e)*x)*sqrt(c*x^2 + b*x + a))/(a^4*x^3), -1/48*(3*(8*a^2*b*f - 16*a^3*g + (5*b^3 - 12*a*b*c )*d - 2*(3*a*b^2 - 4*a^2*c)*e)*sqrt(-a)*x^3*arctan(1/2*sqrt(c*x^2 + b*x + a)*(b*x + 2*a)*sqrt(-a)/(a*c*x^2 + a*b*x + a^2)) + 2*(8*a^3*d - (18*a^2*b* e - 24*a^3*f - (15*a*b^2 - 16*a^2*c)*d)*x^2 - 2*(5*a^2*b*d - 6*a^3*e)*x)*s qrt(c*x^2 + b*x + a))/(a^4*x^3)]
\[ \int \frac {d+e x+f x^2+g x^3}{x^4 \sqrt {a+b x+c x^2}} \, dx=\int \frac {d + e x + f x^{2} + g x^{3}}{x^{4} \sqrt {a + b x + c x^{2}}}\, dx \]
Exception generated. \[ \int \frac {d+e x+f x^2+g x^3}{x^4 \sqrt {a+b x+c x^2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
Leaf count of result is larger than twice the leaf count of optimal. 680 vs. \(2 (164) = 328\).
Time = 0.31 (sec) , antiderivative size = 680, normalized size of antiderivative = 3.66 \[ \int \frac {d+e x+f x^2+g x^3}{x^4 \sqrt {a+b x+c x^2}} \, dx=-\frac {{\left (5 \, b^{3} d - 12 \, a b c d - 6 \, a b^{2} e + 8 \, a^{2} c e + 8 \, a^{2} b f - 16 \, a^{3} g\right )} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + b x + a}}{\sqrt {-a}}\right )}{8 \, \sqrt {-a} a^{3}} + \frac {15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} b^{3} d - 36 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} a b c d - 18 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} a b^{2} e + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} a^{2} c e + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} a^{2} b f + 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} a^{3} \sqrt {c} f - 40 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} a b^{3} d + 96 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} a^{2} b c d + 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} a^{2} b^{2} e - 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} a^{3} b f + 96 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a^{3} c^{\frac {3}{2}} d + 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a^{3} b \sqrt {c} e - 96 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a^{4} \sqrt {c} f + 33 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a^{2} b^{3} d + 36 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a^{3} b c d - 30 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a^{3} b^{2} e - 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a^{4} c e + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a^{4} b f + 48 \, a^{3} b^{2} \sqrt {c} d - 32 \, a^{4} c^{\frac {3}{2}} d - 48 \, a^{4} b \sqrt {c} e + 48 \, a^{5} \sqrt {c} f}{24 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} - a\right )}^{3} a^{3}} \]
-1/8*(5*b^3*d - 12*a*b*c*d - 6*a*b^2*e + 8*a^2*c*e + 8*a^2*b*f - 16*a^3*g) *arctan(-(sqrt(c)*x - sqrt(c*x^2 + b*x + a))/sqrt(-a))/(sqrt(-a)*a^3) + 1/ 24*(15*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^3*d - 36*(sqrt(c)*x - sqrt( c*x^2 + b*x + a))^5*a*b*c*d - 18*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b ^2*e + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^2*c*e + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^2*b*f + 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^ 4*a^3*sqrt(c)*f - 40*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^3*d + 96*(s qrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*b*c*d + 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*b^2*e - 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^3*b* f + 96*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*c^(3/2)*d + 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*b*sqrt(c)*e - 96*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^4*sqrt(c)*f + 33*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^ 3*d + 36*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*b*c*d - 30*(sqrt(c)*x - s qrt(c*x^2 + b*x + a))*a^3*b^2*e - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a ^4*c*e + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^4*b*f + 48*a^3*b^2*sqrt( c)*d - 32*a^4*c^(3/2)*d - 48*a^4*b*sqrt(c)*e + 48*a^5*sqrt(c)*f)/(((sqrt(c )*x - sqrt(c*x^2 + b*x + a))^2 - a)^3*a^3)
Timed out. \[ \int \frac {d+e x+f x^2+g x^3}{x^4 \sqrt {a+b x+c x^2}} \, dx=\int \frac {g\,x^3+f\,x^2+e\,x+d}{x^4\,\sqrt {c\,x^2+b\,x+a}} \,d x \]