Integrand size = 33, antiderivative size = 371 \[ \int \frac {d+e x+f x^2+g x^3}{x^6 \sqrt {a+b x+c x^2}} \, dx=-\frac {d \sqrt {a+b x+c x^2}}{5 a x^5}+\frac {(9 b d-10 a e) \sqrt {a+b x+c x^2}}{40 a^2 x^4}-\frac {\left (63 b^2 d-64 a c d-70 a b e+80 a^2 f\right ) \sqrt {a+b x+c x^2}}{240 a^3 x^3}+\frac {\left (315 b^3 d-350 a b^2 e-4 a b (161 c d-100 a f)+120 a^2 (3 c e-4 a g)\right ) \sqrt {a+b x+c x^2}}{960 a^4 x^2}-\frac {\left (945 b^4 d-1050 a b^3 e-60 a b^2 (49 c d-20 a f)+256 a^2 c (4 c d-5 a f)+40 a^2 b (55 c e-36 a g)\right ) \sqrt {a+b x+c x^2}}{1920 a^5 x}+\frac {\left (63 b^5 d-70 a b^4 e+48 a^2 b c (5 c d-4 a f)-40 a b^3 (7 c d-2 a f)-32 a^3 c (3 c e-4 a g)+48 a^2 b^2 (5 c e-2 a g)\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{256 a^{11/2}} \]
1/256*(63*b^5*d-70*a*b^4*e+48*a^2*b*c*(-4*a*f+5*c*d)-40*a*b^3*(-2*a*f+7*c* d)-32*a^3*c*(-4*a*g+3*c*e)+48*a^2*b^2*(-2*a*g+5*c*e))*arctanh(1/2*(b*x+2*a )/a^(1/2)/(c*x^2+b*x+a)^(1/2))/a^(11/2)-1/5*d*(c*x^2+b*x+a)^(1/2)/a/x^5+1/ 40*(-10*a*e+9*b*d)*(c*x^2+b*x+a)^(1/2)/a^2/x^4-1/240*(80*a^2*f-70*a*b*e-64 *a*c*d+63*b^2*d)*(c*x^2+b*x+a)^(1/2)/a^3/x^3+1/960*(315*b^3*d-350*a*b^2*e- 4*a*b*(-100*a*f+161*c*d)+120*a^2*(-4*a*g+3*c*e))*(c*x^2+b*x+a)^(1/2)/a^4/x ^2-1/1920*(945*b^4*d-1050*a*b^3*e-60*a*b^2*(-20*a*f+49*c*d)+256*a^2*c*(-5* a*f+4*c*d)+40*a^2*b*(-36*a*g+55*c*e))*(c*x^2+b*x+a)^(1/2)/a^5/x
Time = 2.14 (sec) , antiderivative size = 328, normalized size of antiderivative = 0.88 \[ \int \frac {d+e x+f x^2+g x^3}{x^6 \sqrt {a+b x+c x^2}} \, dx=\frac {\frac {\sqrt {a} \sqrt {a+x (b+c x)} \left (-945 b^4 d x^4+210 a b^2 x^3 (3 b d+14 c d x+5 b e x)-32 a^4 \left (12 d+5 x \left (3 e+4 f x+6 g x^2\right )\right )-4 a^2 x^2 \left (256 c^2 d x^2+2 b c x (161 d+275 e x)+b^2 (126 d+25 x (7 e+12 f x))\right )+16 a^3 x (c x (32 d+5 x (9 e+16 f x))+b (27 d+5 x (7 e+2 x (5 f+9 g x))))\right )}{x^5}-15 \left (63 b^5 d+128 a^4 c g\right ) \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )-30 a \left (35 b^4 e+48 a^2 c^2 e+20 b^3 (7 c d-2 a f)+24 a b c (-5 c d+4 a f)+24 a b^2 (-5 c e+2 a g)\right ) \text {arctanh}\left (\frac {-\sqrt {c} x+\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{1920 a^{11/2}} \]
((Sqrt[a]*Sqrt[a + x*(b + c*x)]*(-945*b^4*d*x^4 + 210*a*b^2*x^3*(3*b*d + 1 4*c*d*x + 5*b*e*x) - 32*a^4*(12*d + 5*x*(3*e + 4*f*x + 6*g*x^2)) - 4*a^2*x ^2*(256*c^2*d*x^2 + 2*b*c*x*(161*d + 275*e*x) + b^2*(126*d + 25*x*(7*e + 1 2*f*x))) + 16*a^3*x*(c*x*(32*d + 5*x*(9*e + 16*f*x)) + b*(27*d + 5*x*(7*e + 2*x*(5*f + 9*g*x))))))/x^5 - 15*(63*b^5*d + 128*a^4*c*g)*ArcTanh[(Sqrt[c ]*x - Sqrt[a + x*(b + c*x)])/Sqrt[a]] - 30*a*(35*b^4*e + 48*a^2*c^2*e + 20 *b^3*(7*c*d - 2*a*f) + 24*a*b*c*(-5*c*d + 4*a*f) + 24*a*b^2*(-5*c*e + 2*a* g))*ArcTanh[(-(Sqrt[c]*x) + Sqrt[a + x*(b + c*x)])/Sqrt[a]])/(1920*a^(11/2 ))
Time = 0.91 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.08, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2181, 27, 2181, 27, 1237, 27, 1237, 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.
| \(\displaystyle \int \frac {d+e x+f x^2+g x^3}{x^6 \sqrt {a+b x+c x^2}} \, dx\) |
| \(\Big \downarrow \) 2181 |
| \(\displaystyle -\frac {\int \frac {-10 a g x^2+2 (4 c d-5 a f) x+9 b d-10 a e}{2 x^5 \sqrt {c x^2+b x+a}}dx}{5 a}-\frac {d \sqrt {a+b x+c x^2}}{5 a x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {-10 a g x^2+2 (4 c d-5 a f) x+9 b d-10 a e}{x^5 \sqrt {c x^2+b x+a}}dx}{10 a}-\frac {d \sqrt {a+b x+c x^2}}{5 a x^5}\) |
| \(\Big \downarrow \) 2181 |
| \(\displaystyle -\frac {-\frac {\int \frac {63 d b^2-70 a e b-16 a (4 c d-5 a f)+2 \left (40 g a^2-30 c e a+27 b c d\right ) x}{2 x^4 \sqrt {c x^2+b x+a}}dx}{4 a}-\frac {\sqrt {a+b x+c x^2} (9 b d-10 a e)}{4 a x^4}}{10 a}-\frac {d \sqrt {a+b x+c x^2}}{5 a x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\int \frac {80 f a^2-64 c d a-70 b e a+63 b^2 d+2 \left (40 g a^2-30 c e a+27 b c d\right ) x}{x^4 \sqrt {c x^2+b x+a}}dx}{8 a}-\frac {\sqrt {a+b x+c x^2} (9 b d-10 a e)}{4 a x^4}}{10 a}-\frac {d \sqrt {a+b x+c x^2}}{5 a x^5}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle -\frac {-\frac {-\frac {\int \frac {315 d b^3-350 a e b^2-4 a (161 c d-100 a f) b+120 a^2 (3 c e-4 a g)+4 c \left (63 d b^2-70 a e b-16 a (4 c d-5 a f)\right ) x}{2 x^3 \sqrt {c x^2+b x+a}}dx}{3 a}-\frac {\sqrt {a+b x+c x^2} \left (80 a^2 f-70 a b e-64 a c d+63 b^2 d\right )}{3 a x^3}}{8 a}-\frac {\sqrt {a+b x+c x^2} (9 b d-10 a e)}{4 a x^4}}{10 a}-\frac {d \sqrt {a+b x+c x^2}}{5 a x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {-\frac {\int \frac {-480 g a^3+360 c e a^2+400 b f a^2-644 b c d a-350 b^2 e a+315 b^3 d+4 c \left (63 d b^2-70 a e b-16 a (4 c d-5 a f)\right ) x}{x^3 \sqrt {c x^2+b x+a}}dx}{6 a}-\frac {\sqrt {a+b x+c x^2} \left (80 a^2 f-70 a b e-64 a c d+63 b^2 d\right )}{3 a x^3}}{8 a}-\frac {\sqrt {a+b x+c x^2} (9 b d-10 a e)}{4 a x^4}}{10 a}-\frac {d \sqrt {a+b x+c x^2}}{5 a x^5}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle -\frac {-\frac {-\frac {-\frac {\int \frac {945 d b^4-1050 a e b^3-60 a (49 c d-20 a f) b^2+40 a^2 (55 c e-36 a g) b+256 a^2 c (4 c d-5 a f)+2 c \left (315 d b^3-350 a e b^2-4 a (161 c d-100 a f) b+120 a^2 (3 c e-4 a g)\right ) x}{2 x^2 \sqrt {c x^2+b x+a}}dx}{2 a}-\frac {\sqrt {a+b x+c x^2} \left (120 a^2 (3 c e-4 a g)-350 a b^2 e-4 a b (161 c d-100 a f)+315 b^3 d\right )}{2 a x^2}}{6 a}-\frac {\sqrt {a+b x+c x^2} \left (80 a^2 f-70 a b e-64 a c d+63 b^2 d\right )}{3 a x^3}}{8 a}-\frac {\sqrt {a+b x+c x^2} (9 b d-10 a e)}{4 a x^4}}{10 a}-\frac {d \sqrt {a+b x+c x^2}}{5 a x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {-\frac {-\frac {\int \frac {945 d b^4-1050 a e b^3-60 a (49 c d-20 a f) b^2+40 a^2 (55 c e-36 a g) b+256 a^2 c (4 c d-5 a f)+2 c \left (315 d b^3-350 a e b^2-4 a (161 c d-100 a f) b+120 a^2 (3 c e-4 a g)\right ) x}{x^2 \sqrt {c x^2+b x+a}}dx}{4 a}-\frac {\sqrt {a+b x+c x^2} \left (120 a^2 (3 c e-4 a g)-350 a b^2 e-4 a b (161 c d-100 a f)+315 b^3 d\right )}{2 a x^2}}{6 a}-\frac {\sqrt {a+b x+c x^2} \left (80 a^2 f-70 a b e-64 a c d+63 b^2 d\right )}{3 a x^3}}{8 a}-\frac {\sqrt {a+b x+c x^2} (9 b d-10 a e)}{4 a x^4}}{10 a}-\frac {d \sqrt {a+b x+c x^2}}{5 a x^5}\) |
| \(\Big \downarrow \) 1228 |
| \(\displaystyle -\frac {-\frac {-\frac {-\frac {-\frac {15 \left (-32 a^3 c (3 c e-4 a g)+48 a^2 b^2 (5 c e-2 a g)+48 a^2 b c (5 c d-4 a f)-70 a b^4 e-40 a b^3 (7 c d-2 a f)+63 b^5 d\right ) \int \frac {1}{x \sqrt {c x^2+b x+a}}dx}{2 a}-\frac {\sqrt {a+b x+c x^2} \left (40 a^2 b (55 c e-36 a g)+256 a^2 c (4 c d-5 a f)-1050 a b^3 e-60 a b^2 (49 c d-20 a f)+945 b^4 d\right )}{a x}}{4 a}-\frac {\sqrt {a+b x+c x^2} \left (120 a^2 (3 c e-4 a g)-350 a b^2 e-4 a b (161 c d-100 a f)+315 b^3 d\right )}{2 a x^2}}{6 a}-\frac {\sqrt {a+b x+c x^2} \left (80 a^2 f-70 a b e-64 a c d+63 b^2 d\right )}{3 a x^3}}{8 a}-\frac {\sqrt {a+b x+c x^2} (9 b d-10 a e)}{4 a x^4}}{10 a}-\frac {d \sqrt {a+b x+c x^2}}{5 a x^5}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle -\frac {-\frac {-\frac {-\frac {\frac {15 \left (-32 a^3 c (3 c e-4 a g)+48 a^2 b^2 (5 c e-2 a g)+48 a^2 b c (5 c d-4 a f)-70 a b^4 e-40 a b^3 (7 c d-2 a f)+63 b^5 d\right ) \int \frac {1}{4 a-\frac {(2 a+b x)^2}{c x^2+b x+a}}d\frac {2 a+b x}{\sqrt {c x^2+b x+a}}}{a}-\frac {\sqrt {a+b x+c x^2} \left (40 a^2 b (55 c e-36 a g)+256 a^2 c (4 c d-5 a f)-1050 a b^3 e-60 a b^2 (49 c d-20 a f)+945 b^4 d\right )}{a x}}{4 a}-\frac {\sqrt {a+b x+c x^2} \left (120 a^2 (3 c e-4 a g)-350 a b^2 e-4 a b (161 c d-100 a f)+315 b^3 d\right )}{2 a x^2}}{6 a}-\frac {\sqrt {a+b x+c x^2} \left (80 a^2 f-70 a b e-64 a c d+63 b^2 d\right )}{3 a x^3}}{8 a}-\frac {\sqrt {a+b x+c x^2} (9 b d-10 a e)}{4 a x^4}}{10 a}-\frac {d \sqrt {a+b x+c x^2}}{5 a x^5}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {-\frac {-\frac {\sqrt {a+b x+c x^2} \left (80 a^2 f-70 a b e-64 a c d+63 b^2 d\right )}{3 a x^3}-\frac {-\frac {\sqrt {a+b x+c x^2} \left (120 a^2 (3 c e-4 a g)-350 a b^2 e-4 a b (161 c d-100 a f)+315 b^3 d\right )}{2 a x^2}-\frac {\frac {15 \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right ) \left (-32 a^3 c (3 c e-4 a g)+48 a^2 b^2 (5 c e-2 a g)+48 a^2 b c (5 c d-4 a f)-70 a b^4 e-40 a b^3 (7 c d-2 a f)+63 b^5 d\right )}{2 a^{3/2}}-\frac {\sqrt {a+b x+c x^2} \left (40 a^2 b (55 c e-36 a g)+256 a^2 c (4 c d-5 a f)-1050 a b^3 e-60 a b^2 (49 c d-20 a f)+945 b^4 d\right )}{a x}}{4 a}}{6 a}}{8 a}-\frac {\sqrt {a+b x+c x^2} (9 b d-10 a e)}{4 a x^4}}{10 a}-\frac {d \sqrt {a+b x+c x^2}}{5 a x^5}\) |
-1/5*(d*Sqrt[a + b*x + c*x^2])/(a*x^5) - (-1/4*((9*b*d - 10*a*e)*Sqrt[a + b*x + c*x^2])/(a*x^4) - (-1/3*((63*b^2*d - 64*a*c*d - 70*a*b*e + 80*a^2*f) *Sqrt[a + b*x + c*x^2])/(a*x^3) - (-1/2*((315*b^3*d - 350*a*b^2*e - 4*a*b* (161*c*d - 100*a*f) + 120*a^2*(3*c*e - 4*a*g))*Sqrt[a + b*x + c*x^2])/(a*x ^2) - (-(((945*b^4*d - 1050*a*b^3*e - 60*a*b^2*(49*c*d - 20*a*f) + 256*a^2 *c*(4*c*d - 5*a*f) + 40*a^2*b*(55*c*e - 36*a*g))*Sqrt[a + b*x + c*x^2])/(a *x)) + (15*(63*b^5*d - 70*a*b^4*e + 48*a^2*b*c*(5*c*d - 4*a*f) - 40*a*b^3* (7*c*d - 2*a*f) - 32*a^3*c*(3*c*e - 4*a*g) + 48*a^2*b^2*(5*c*e - 2*a*g))*A rcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(2*a^(3/2)))/(4*a)) /(6*a))/(8*a))/(10*a)
3.3.88.3.1 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[((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)/((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*Simp[ (c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 ] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
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 = 0.99 (sec) , antiderivative size = 350, normalized size of antiderivative = 0.94
| method | result | size |
| risch | \(-\frac {\sqrt {c \,x^{2}+b x +a}\, \left (-1440 a^{3} b g \,x^{4}-1280 a^{3} c f \,x^{4}+1200 a^{2} b^{2} f \,x^{4}+2200 x^{4} a^{2} b c e +1024 x^{4} a^{2} c^{2} d -1050 x^{4} a \,b^{3} e -2940 x^{4} a \,b^{2} c d +945 x^{4} b^{4} d +960 a^{4} g \,x^{3}-800 a^{3} b f \,x^{3}-720 x^{3} c e \,a^{3}+700 x^{3} a^{2} b^{2} e +1288 x^{3} a^{2} b c d -630 x^{3} a \,b^{3} d +640 a^{4} f \,x^{2}-560 x^{2} a^{3} b e -512 x^{2} a^{3} c d +504 x^{2} b^{2} d \,a^{2}+480 x \,a^{4} e -432 a^{3} b d x +384 a^{4} d \right )}{1920 a^{5} x^{5}}+\frac {\left (128 a^{4} c g -96 a^{3} b^{2} g -192 a^{3} b c f -96 a^{3} c^{2} e +80 a^{2} b^{3} f +240 a^{2} b^{2} c e +240 a^{2} b \,c^{2} d -70 e \,b^{4} a -280 a \,b^{3} c d +63 d \,b^{5}\right ) \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{256 a^{\frac {11}{2}}}\) | \(350\) |
| default | \(\text {Expression too large to display}\) | \(1291\) |
-1/1920*(c*x^2+b*x+a)^(1/2)*(-1440*a^3*b*g*x^4-1280*a^3*c*f*x^4+1200*a^2*b ^2*f*x^4+2200*a^2*b*c*e*x^4+1024*a^2*c^2*d*x^4-1050*a*b^3*e*x^4-2940*a*b^2 *c*d*x^4+945*b^4*d*x^4+960*a^4*g*x^3-800*a^3*b*f*x^3-720*a^3*c*e*x^3+700*a ^2*b^2*e*x^3+1288*a^2*b*c*d*x^3-630*a*b^3*d*x^3+640*a^4*f*x^2-560*a^3*b*e* x^2-512*a^3*c*d*x^2+504*a^2*b^2*d*x^2+480*a^4*e*x-432*a^3*b*d*x+384*a^4*d) /a^5/x^5+1/256*(128*a^4*c*g-96*a^3*b^2*g-192*a^3*b*c*f-96*a^3*c^2*e+80*a^2 *b^3*f+240*a^2*b^2*c*e+240*a^2*b*c^2*d-70*a*b^4*e-280*a*b^3*c*d+63*b^5*d)/ a^(11/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)
Time = 11.94 (sec) , antiderivative size = 727, normalized size of antiderivative = 1.96 \[ \int \frac {d+e x+f x^2+g x^3}{x^6 \sqrt {a+b x+c x^2}} \, dx=\left [\frac {15 \, {\left ({\left (63 \, b^{5} - 280 \, a b^{3} c + 240 \, a^{2} b c^{2}\right )} d - 2 \, {\left (35 \, a b^{4} - 120 \, a^{2} b^{2} c + 48 \, a^{3} c^{2}\right )} e + 16 \, {\left (5 \, a^{2} b^{3} - 12 \, a^{3} b c\right )} f - 32 \, {\left (3 \, a^{3} b^{2} - 4 \, a^{4} c\right )} g\right )} \sqrt {a} x^{5} \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 (384 \, a^{5} d - {\left (1440 \, a^{4} b g - {\left (945 \, a b^{4} - 2940 \, a^{2} b^{2} c + 1024 \, a^{3} c^{2}\right )} d + 50 \, {\left (21 \, a^{2} b^{3} - 44 \, a^{3} b c\right )} e - 80 \, {\left (15 \, a^{3} b^{2} - 16 \, a^{4} c\right )} f\right )} x^{4} - 2 \, {\left (400 \, a^{4} b f - 480 \, a^{5} g + 7 \, {\left (45 \, a^{2} b^{3} - 92 \, a^{3} b c\right )} d - 10 \, {\left (35 \, a^{3} b^{2} - 36 \, a^{4} c\right )} e\right )} x^{3} - 8 \, {\left (70 \, a^{4} b e - 80 \, a^{5} f - {\left (63 \, a^{3} b^{2} - 64 \, a^{4} c\right )} d\right )} x^{2} - 48 \, {\left (9 \, a^{4} b d - 10 \, a^{5} e\right )} x\right )} \sqrt {c x^{2} + b x + a}}{7680 \, a^{6} x^{5}}, -\frac {15 \, {\left ({\left (63 \, b^{5} - 280 \, a b^{3} c + 240 \, a^{2} b c^{2}\right )} d - 2 \, {\left (35 \, a b^{4} - 120 \, a^{2} b^{2} c + 48 \, a^{3} c^{2}\right )} e + 16 \, {\left (5 \, a^{2} b^{3} - 12 \, a^{3} b c\right )} f - 32 \, {\left (3 \, a^{3} b^{2} - 4 \, a^{4} c\right )} g\right )} \sqrt {-a} x^{5} \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 (384 \, a^{5} d - {\left (1440 \, a^{4} b g - {\left (945 \, a b^{4} - 2940 \, a^{2} b^{2} c + 1024 \, a^{3} c^{2}\right )} d + 50 \, {\left (21 \, a^{2} b^{3} - 44 \, a^{3} b c\right )} e - 80 \, {\left (15 \, a^{3} b^{2} - 16 \, a^{4} c\right )} f\right )} x^{4} - 2 \, {\left (400 \, a^{4} b f - 480 \, a^{5} g + 7 \, {\left (45 \, a^{2} b^{3} - 92 \, a^{3} b c\right )} d - 10 \, {\left (35 \, a^{3} b^{2} - 36 \, a^{4} c\right )} e\right )} x^{3} - 8 \, {\left (70 \, a^{4} b e - 80 \, a^{5} f - {\left (63 \, a^{3} b^{2} - 64 \, a^{4} c\right )} d\right )} x^{2} - 48 \, {\left (9 \, a^{4} b d - 10 \, a^{5} e\right )} x\right )} \sqrt {c x^{2} + b x + a}}{3840 \, a^{6} x^{5}}\right ] \]
[1/7680*(15*((63*b^5 - 280*a*b^3*c + 240*a^2*b*c^2)*d - 2*(35*a*b^4 - 120* a^2*b^2*c + 48*a^3*c^2)*e + 16*(5*a^2*b^3 - 12*a^3*b*c)*f - 32*(3*a^3*b^2 - 4*a^4*c)*g)*sqrt(a)*x^5*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*(384*a^5*d - (1440*a^4*b *g - (945*a*b^4 - 2940*a^2*b^2*c + 1024*a^3*c^2)*d + 50*(21*a^2*b^3 - 44*a ^3*b*c)*e - 80*(15*a^3*b^2 - 16*a^4*c)*f)*x^4 - 2*(400*a^4*b*f - 480*a^5*g + 7*(45*a^2*b^3 - 92*a^3*b*c)*d - 10*(35*a^3*b^2 - 36*a^4*c)*e)*x^3 - 8*( 70*a^4*b*e - 80*a^5*f - (63*a^3*b^2 - 64*a^4*c)*d)*x^2 - 48*(9*a^4*b*d - 1 0*a^5*e)*x)*sqrt(c*x^2 + b*x + a))/(a^6*x^5), -1/3840*(15*((63*b^5 - 280*a *b^3*c + 240*a^2*b*c^2)*d - 2*(35*a*b^4 - 120*a^2*b^2*c + 48*a^3*c^2)*e + 16*(5*a^2*b^3 - 12*a^3*b*c)*f - 32*(3*a^3*b^2 - 4*a^4*c)*g)*sqrt(-a)*x^5*a rctan(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*(384*a^5*d - (1440*a^4*b*g - (945*a*b^4 - 2940*a^2*b^2*c + 1024*a^ 3*c^2)*d + 50*(21*a^2*b^3 - 44*a^3*b*c)*e - 80*(15*a^3*b^2 - 16*a^4*c)*f)* x^4 - 2*(400*a^4*b*f - 480*a^5*g + 7*(45*a^2*b^3 - 92*a^3*b*c)*d - 10*(35* a^3*b^2 - 36*a^4*c)*e)*x^3 - 8*(70*a^4*b*e - 80*a^5*f - (63*a^3*b^2 - 64*a ^4*c)*d)*x^2 - 48*(9*a^4*b*d - 10*a^5*e)*x)*sqrt(c*x^2 + b*x + a))/(a^6*x^ 5)]
\[ \int \frac {d+e x+f x^2+g x^3}{x^6 \sqrt {a+b x+c x^2}} \, dx=\int \frac {d + e x + f x^{2} + g x^{3}}{x^{6} \sqrt {a + b x + c x^{2}}}\, dx \]
Exception generated. \[ \int \frac {d+e x+f x^2+g x^3}{x^6 \sqrt {a+b x+c x^2}} \, dx=\text {Exception raised: ValueError} \]
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. 2155 vs. \(2 (341) = 682\).
Time = 0.31 (sec) , antiderivative size = 2155, normalized size of antiderivative = 5.81 \[ \int \frac {d+e x+f x^2+g x^3}{x^6 \sqrt {a+b x+c x^2}} \, dx=\text {Too large to display} \]
-1/128*(63*b^5*d - 280*a*b^3*c*d + 240*a^2*b*c^2*d - 70*a*b^4*e + 240*a^2* b^2*c*e - 96*a^3*c^2*e + 80*a^2*b^3*f - 192*a^3*b*c*f - 96*a^3*b^2*g + 128 *a^4*c*g)*arctan(-(sqrt(c)*x - sqrt(c*x^2 + b*x + a))/sqrt(-a))/(sqrt(-a)* a^5) + 1/1920*(945*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*b^5*d - 4200*(sqr t(c)*x - sqrt(c*x^2 + b*x + a))^9*a*b^3*c*d + 3600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a^2*b*c^2*d - 1050*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a* b^4*e + 3600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a^2*b^2*c*e - 1440*(sqr t(c)*x - sqrt(c*x^2 + b*x + a))^9*a^3*c^2*e + 1200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a^2*b^3*f - 2880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a^3* b*c*f - 1440*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a^3*b^2*g + 1920*(sqrt( c)*x - sqrt(c*x^2 + b*x + a))^9*a^4*c*g - 4410*(sqrt(c)*x - sqrt(c*x^2 + b *x + a))^7*a*b^5*d + 19600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a^2*b^3*c *d - 16800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a^3*b*c^2*d + 4900*(sqrt( c)*x - sqrt(c*x^2 + b*x + a))^7*a^2*b^4*e - 16800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a^3*b^2*c*e + 6720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a^4 *c^2*e - 5600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a^3*b^3*f + 13440*(sqr t(c)*x - sqrt(c*x^2 + b*x + a))^7*a^4*b*c*f + 6720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a^4*b^2*g - 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a^5* c*g + 7680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^5*c^(3/2)*f + 3840*(sqr t(c)*x - sqrt(c*x^2 + b*x + a))^6*a^5*b*sqrt(c)*g + 8064*(sqrt(c)*x - s...
Timed out. \[ \int \frac {d+e x+f x^2+g x^3}{x^6 \sqrt {a+b x+c x^2}} \, dx=\int \frac {g\,x^3+f\,x^2+e\,x+d}{x^6\,\sqrt {c\,x^2+b\,x+a}} \,d x \]