Integrand size = 27, antiderivative size = 328 \[ \int \frac {1}{\sqrt {a+b x+c x^2} \left (d+b x+c x^2\right )^4} \, dx=-\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{3 (a-d) \left (b^2-4 c d\right ) \left (d+b x+c x^2\right )^3}+\frac {5 \left (b^2+4 c (a-2 d)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{12 (a-d)^2 \left (b^2-4 c d\right )^2 \left (d+b x+c x^2\right )^2}-\frac {\left (15 b^4+8 b^2 c (7 a-22 d)+16 c^2 \left (15 a^2-44 a d+44 d^2\right )\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{24 (a-d)^3 \left (b^2-4 c d\right )^3 \left (d+b x+c x^2\right )}+\frac {\left (b^2+4 c (a-2 d)\right ) \left (5 b^4-8 b^2 c (a+4 d)+16 c^2 \left (5 a^2-8 a d+8 d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {a-d} (b+2 c x)}{\sqrt {b^2-4 c d} \sqrt {a+b x+c x^2}}\right )}{8 (a-d)^{7/2} \left (b^2-4 c d\right )^{7/2}} \]
1/8*(b^2+4*c*(a-2*d))*(5*b^4-8*b^2*c*(a+4*d)+16*c^2*(5*a^2-8*a*d+8*d^2))*a rctanh((2*c*x+b)*(a-d)^(1/2)/(b^2-4*c*d)^(1/2)/(c*x^2+b*x+a)^(1/2))/(a-d)^ (7/2)/(b^2-4*c*d)^(7/2)-1/3*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/(a-d)/(b^2-4*c*d )/(c*x^2+b*x+d)^3+5/12*(b^2+4*c*(a-2*d))*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/(a- d)^2/(b^2-4*c*d)^2/(c*x^2+b*x+d)^2-1/24*(15*b^4+8*b^2*c*(7*a-22*d)+16*c^2* (15*a^2-44*a*d+44*d^2))*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/(a-d)^3/(b^2-4*c*d)^ 3/(c*x^2+b*x+d)
Leaf count is larger than twice the leaf count of optimal. \(3382\) vs. \(2(328)=656\).
Time = 16.44 (sec) , antiderivative size = 3382, normalized size of antiderivative = 10.31 \[ \int \frac {1}{\sqrt {a+b x+c x^2} \left (d+b x+c x^2\right )^4} \, dx=\text {Result too large to show} \]
(-8*c^3*(a + b*x + c*x^2))/(3*(a - d)*(b^2 - 4*c*d)^2*(b - Sqrt[b^2 - 4*c* d] + 2*c*x)^3*Sqrt[a + x*(b + c*x)]) + (8*c^3*(a + b*x + c*x^2))/((a - d)* (b^2 - 4*c*d)^(5/2)*(b - Sqrt[b^2 - 4*c*d] + 2*c*x)^2*Sqrt[a + x*(b + c*x) ]) - (20*c^3*(a + b*x + c*x^2))/((a - d)*(b^2 - 4*c*d)^3*(b - Sqrt[b^2 - 4 *c*d] + 2*c*x)*Sqrt[a + x*(b + c*x)]) - (8*c^3*(a + b*x + c*x^2))/(3*(a - d)*(b^2 - 4*c*d)^2*(b + Sqrt[b^2 - 4*c*d] + 2*c*x)^3*Sqrt[a + x*(b + c*x)] ) - (8*c^3*(a + b*x + c*x^2))/((a - d)*(b^2 - 4*c*d)^(5/2)*(b + Sqrt[b^2 - 4*c*d] + 2*c*x)^2*Sqrt[a + x*(b + c*x)]) - (20*c^3*(a + b*x + c*x^2))/((a - d)*(b^2 - 4*c*d)^3*(b + Sqrt[b^2 - 4*c*d] + 2*c*x)*Sqrt[a + x*(b + c*x) ]) - (20*c^3*Sqrt[a + b*x + c*x^2]*ArcTanh[(b^2 - 4*a*c - b*Sqrt[b^2 - 4*c *d] - 2*c*Sqrt[b^2 - 4*c*d]*x)/(4*c*Sqrt[a - d]*Sqrt[a + b*x + c*x^2])])/( Sqrt[a - d]*(b^2 - 4*c*d)^(7/2)*Sqrt[a + x*(b + c*x)]) - (5*c^2*Sqrt[a + b *x + c*x^2]*ArcTanh[(b^2 - 4*a*c - b*Sqrt[b^2 - 4*c*d] - 2*c*Sqrt[b^2 - 4* c*d]*x)/(4*c*Sqrt[a - d]*Sqrt[a + b*x + c*x^2])])/((a - d)^(3/2)*(b^2 - 4* c*d)^(5/2)*Sqrt[a + x*(b + c*x)]) - (20*c^3*Sqrt[a + b*x + c*x^2]*ArcTanh[ (4*a*c - b*(b + Sqrt[b^2 - 4*c*d]) - 2*c*Sqrt[b^2 - 4*c*d]*x)/(4*c*Sqrt[a - d]*Sqrt[a + b*x + c*x^2])])/(Sqrt[a - d]*(b^2 - 4*c*d)^(7/2)*Sqrt[a + x* (b + c*x)]) - (5*c^2*Sqrt[a + b*x + c*x^2]*ArcTanh[(4*a*c - b*(b + Sqrt[b^ 2 - 4*c*d]) - 2*c*Sqrt[b^2 - 4*c*d]*x)/(4*c*Sqrt[a - d]*Sqrt[a + b*x + c*x ^2])])/((a - d)^(3/2)*(b^2 - 4*c*d)^(5/2)*Sqrt[a + x*(b + c*x)]) - (16*...
Time = 1.09 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.14, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {1305, 27, 2135, 27, 2135, 27, 1313, 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.
| \(\displaystyle \int \frac {1}{\sqrt {a+b x+c x^2} \left (b x+c x^2+d\right )^4} \, dx\) |
| \(\Big \downarrow \) 1305 |
| \(\displaystyle \frac {\int -\frac {16 (a-d) x^2 c^4+16 b (a-d) x c^3+(a-d) \left (5 b^2+20 a c-24 c d\right ) c^2}{2 \sqrt {c x^2+b x+a} \left (c x^2+b x+d\right )^3}dx}{3 c^2 (a-d)^2 \left (b^2-4 c d\right )}-\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{3 (a-d) \left (b^2-4 c d\right ) \left (b x+c x^2+d\right )^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {16 (a-d) x^2 c^4+16 b (a-d) x c^3+(a-d) \left (5 b^2+20 a c-24 c d\right ) c^2}{\sqrt {c x^2+b x+a} \left (c x^2+b x+d\right )^3}dx}{6 c^2 (a-d)^2 \left (b^2-4 c d\right )}-\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{3 (a-d) \left (b^2-4 c d\right ) \left (b x+c x^2+d\right )^3}\) |
| \(\Big \downarrow \) 2135 |
| \(\displaystyle -\frac {-\frac {\int \frac {40 \left (b^2+4 c (a-2 d)\right ) (a-d)^2 x^2 c^6+40 b \left (b^2+4 c (a-2 d)\right ) (a-d)^2 x c^5+(a-d)^2 \left (15 b^4+8 c (7 a-17 d) b^2+16 c^2 \left (15 a^2-34 d a+24 d^2\right )\right ) c^4}{2 \sqrt {c x^2+b x+a} \left (c x^2+b x+d\right )^2}dx}{2 c^2 (a-d)^2 \left (b^2-4 c d\right )}-\frac {5 c^2 (b+2 c x) \left (4 c (a-2 d)+b^2\right ) \sqrt {a+b x+c x^2}}{2 \left (b^2-4 c d\right ) \left (b x+c x^2+d\right )^2}}{6 c^2 (a-d)^2 \left (b^2-4 c d\right )}-\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{3 (a-d) \left (b^2-4 c d\right ) \left (b x+c x^2+d\right )^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\int \frac {40 \left (b^2+4 c (a-2 d)\right ) (a-d)^2 x^2 c^6+40 b \left (b^2+4 c (a-2 d)\right ) (a-d)^2 x c^5+(a-d)^2 \left (15 b^4+8 c (7 a-17 d) b^2+16 c^2 \left (15 a^2-34 d a+24 d^2\right )\right ) c^4}{\sqrt {c x^2+b x+a} \left (c x^2+b x+d\right )^2}dx}{4 c^2 (a-d)^2 \left (b^2-4 c d\right )}-\frac {5 c^2 (b+2 c x) \left (4 c (a-2 d)+b^2\right ) \sqrt {a+b x+c x^2}}{2 \left (b^2-4 c d\right ) \left (b x+c x^2+d\right )^2}}{6 c^2 (a-d)^2 \left (b^2-4 c d\right )}-\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{3 (a-d) \left (b^2-4 c d\right ) \left (b x+c x^2+d\right )^3}\) |
| \(\Big \downarrow \) 2135 |
| \(\displaystyle -\frac {-\frac {-\frac {\int \frac {3 c^6 \left (b^2+4 c (a-2 d)\right ) (a-d)^3 \left (5 b^4-8 c (a+4 d) b^2+16 c^2 \left (5 a^2-8 d a+8 d^2\right )\right )}{2 \sqrt {c x^2+b x+a} \left (c x^2+b x+d\right )}dx}{c^2 (a-d)^2 \left (b^2-4 c d\right )}-\frac {c^4 (a-d) (b+2 c x) \left (16 c^2 \left (15 a^2-44 a d+44 d^2\right )+8 b^2 c (7 a-22 d)+15 b^4\right ) \sqrt {a+b x+c x^2}}{\left (b^2-4 c d\right ) \left (b x+c x^2+d\right )}}{4 c^2 (a-d)^2 \left (b^2-4 c d\right )}-\frac {5 c^2 (b+2 c x) \left (4 c (a-2 d)+b^2\right ) \sqrt {a+b x+c x^2}}{2 \left (b^2-4 c d\right ) \left (b x+c x^2+d\right )^2}}{6 c^2 (a-d)^2 \left (b^2-4 c d\right )}-\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{3 (a-d) \left (b^2-4 c d\right ) \left (b x+c x^2+d\right )^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {-\frac {3 c^4 (a-d) \left (4 c (a-2 d)+b^2\right ) \left (16 c^2 \left (5 a^2-8 a d+8 d^2\right )-8 b^2 c (a+4 d)+5 b^4\right ) \int \frac {1}{\sqrt {c x^2+b x+a} \left (c x^2+b x+d\right )}dx}{2 \left (b^2-4 c d\right )}-\frac {c^4 (a-d) (b+2 c x) \left (16 c^2 \left (15 a^2-44 a d+44 d^2\right )+8 b^2 c (7 a-22 d)+15 b^4\right ) \sqrt {a+b x+c x^2}}{\left (b^2-4 c d\right ) \left (b x+c x^2+d\right )}}{4 c^2 (a-d)^2 \left (b^2-4 c d\right )}-\frac {5 c^2 (b+2 c x) \left (4 c (a-2 d)+b^2\right ) \sqrt {a+b x+c x^2}}{2 \left (b^2-4 c d\right ) \left (b x+c x^2+d\right )^2}}{6 c^2 (a-d)^2 \left (b^2-4 c d\right )}-\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{3 (a-d) \left (b^2-4 c d\right ) \left (b x+c x^2+d\right )^3}\) |
| \(\Big \downarrow \) 1313 |
| \(\displaystyle -\frac {-\frac {\frac {3 b c^4 (a-d) \left (4 c (a-2 d)+b^2\right ) \left (16 c^2 \left (5 a^2-8 a d+8 d^2\right )-8 b^2 c (a+4 d)+5 b^4\right ) \int \frac {1}{b \left (b^2-4 c d\right )-\frac {b (a-d) (b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{b^2-4 c d}-\frac {c^4 (a-d) (b+2 c x) \left (16 c^2 \left (15 a^2-44 a d+44 d^2\right )+8 b^2 c (7 a-22 d)+15 b^4\right ) \sqrt {a+b x+c x^2}}{\left (b^2-4 c d\right ) \left (b x+c x^2+d\right )}}{4 c^2 (a-d)^2 \left (b^2-4 c d\right )}-\frac {5 c^2 (b+2 c x) \left (4 c (a-2 d)+b^2\right ) \sqrt {a+b x+c x^2}}{2 \left (b^2-4 c d\right ) \left (b x+c x^2+d\right )^2}}{6 c^2 (a-d)^2 \left (b^2-4 c d\right )}-\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{3 (a-d) \left (b^2-4 c d\right ) \left (b x+c x^2+d\right )^3}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {-\frac {\frac {3 c^4 \sqrt {a-d} \left (4 c (a-2 d)+b^2\right ) \left (16 c^2 \left (5 a^2-8 a d+8 d^2\right )-8 b^2 c (a+4 d)+5 b^4\right ) \text {arctanh}\left (\frac {\sqrt {a-d} (b+2 c x)}{\sqrt {b^2-4 c d} \sqrt {a+b x+c x^2}}\right )}{\left (b^2-4 c d\right )^{3/2}}-\frac {c^4 (a-d) (b+2 c x) \left (16 c^2 \left (15 a^2-44 a d+44 d^2\right )+8 b^2 c (7 a-22 d)+15 b^4\right ) \sqrt {a+b x+c x^2}}{\left (b^2-4 c d\right ) \left (b x+c x^2+d\right )}}{4 c^2 (a-d)^2 \left (b^2-4 c d\right )}-\frac {5 c^2 (b+2 c x) \left (4 c (a-2 d)+b^2\right ) \sqrt {a+b x+c x^2}}{2 \left (b^2-4 c d\right ) \left (b x+c x^2+d\right )^2}}{6 c^2 (a-d)^2 \left (b^2-4 c d\right )}-\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{3 (a-d) \left (b^2-4 c d\right ) \left (b x+c x^2+d\right )^3}\) |
-1/3*((b + 2*c*x)*Sqrt[a + b*x + c*x^2])/((a - d)*(b^2 - 4*c*d)*(d + b*x + c*x^2)^3) - ((-5*c^2*(b^2 + 4*c*(a - 2*d))*(b + 2*c*x)*Sqrt[a + b*x + c*x ^2])/(2*(b^2 - 4*c*d)*(d + b*x + c*x^2)^2) - (-((c^4*(a - d)*(15*b^4 + 8*b ^2*c*(7*a - 22*d) + 16*c^2*(15*a^2 - 44*a*d + 44*d^2))*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/((b^2 - 4*c*d)*(d + b*x + c*x^2))) + (3*c^4*(b^2 + 4*c*(a - 2*d))*Sqrt[a - d]*(5*b^4 - 8*b^2*c*(a + 4*d) + 16*c^2*(5*a^2 - 8*a*d + 8 *d^2))*ArcTanh[(Sqrt[a - d]*(b + 2*c*x))/(Sqrt[b^2 - 4*c*d]*Sqrt[a + b*x + c*x^2])])/(b^2 - 4*c*d)^(3/2))/(4*c^2*(a - d)^2*(b^2 - 4*c*d)))/(6*c^2*(a - d)^2*(b^2 - 4*c*d))
3.1.6.3.1 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[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_.) + (e_.)*(x_) + (f_.)*(x _)^2)^(q_), x_Symbol] :> Simp[(2*a*c^2*e - b^2*c*e + b^3*f + b*c*(c*d - 3*a *f) + c*(2*c^2*d + b^2*f - c*(b*e + 2*a*f))*x)*(a + b*x + c*x^2)^(p + 1)*(( d + e*x + f*x^2)^(q + 1)/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1))), x] - Simp[1/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*( c*e - b*f))*(p + 1)) Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q*Si mp[2*c*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1) - (2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(a*f*(p + 1) - c*d*(p + 2)) - e*(b^2*c*e - 2*a*c^2*e - b^3*f - b*c*(c*d - 3*a*f))*(p + q + 2) + (2*f*(2*a*c^2*e - b^2*c*e + b^3*f + b*c*(c*d - 3*a*f))*(p + q + 2) - (2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(b*f *(p + 1) - c*e*(2*p + q + 4)))*x + c*f*(2*c^2*d + b^2*f - c*(b*e + 2*a*f))* (2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, q}, x] && NeQ[b ^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 0] && !( !IntegerQ[p] && ILtQ[q, -1]) && !IGtQ[q , 0]
Int[1/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*( x_)^2]), x_Symbol] :> Simp[-2*e Subst[Int[1/(e*(b*e - 4*a*f) - (b*d - a*e )*x^2), x], x, (e + 2*f*x)/Sqrt[d + e*x + f*x^2]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && EqQ[c*e - b*f, 0]
Int[(Px_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f_. )*(x_)^2)^(q_), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1] , C = Coeff[Px, x, 2]}, Simp[(a + b*x + c*x^2)^(p + 1)*((d + e*x + f*x^2)^( q + 1)/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)))*( (A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c*(b *e + 2*a*f)) + c*(A*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) - B*(b*c*d - 2*a*c* e + a*b*f) + C*(b^2*d - a*b*e - 2*a*(c*d - a*f)))*x), x] + Simp[1/((b^2 - 4 *a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)) Int[(a + b*x + c *x^2)^(p + 1)*(d + e*x + f*x^2)^q*Simp[(b*B - 2*A*c - 2*a*C)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1) + (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(a*f*(p + 1) - c*d*(p + 2)) - e*((A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B )*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)))*(p + q + 2) - (2*f*((A*c - a*C)*(2*a *c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)))*(p + q + 2) - (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*( c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(b*f*(p + 1) - c*e*(2*p + q + 4))) *x - c*f*(b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(2*p + 2*q + 5)*x^2, x], x], x]] /; F reeQ[{a, b, c, d, e, f, q}, x] && PolyQ[Px, x, 2] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 0] && !( !IntegerQ[p] && ILtQ[q, -1]) && !IGtQ[q, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(3648\) vs. \(2(304)=608\).
Time = 1.95 (sec) , antiderivative size = 3649, normalized size of antiderivative = 11.12
1/(b^2-4*c*d)^2*(-1/3/(a-d)/(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)^3*((x-1/2*(-b +(b^2-4*c*d)^(1/2))/c)^2*c+(b^2-4*c*d)^(1/2)*(x-1/2*(-b+(b^2-4*c*d)^(1/2)) /c)+a-d)^(1/2)-5/6*(b^2-4*c*d)^(1/2)/(a-d)*(-1/2/(a-d)/(x-1/2*(-b+(b^2-4*c *d)^(1/2))/c)^2*((x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)^2*c+(b^2-4*c*d)^(1/2)*(x -1/2*(-b+(b^2-4*c*d)^(1/2))/c)+a-d)^(1/2)-3/4*(b^2-4*c*d)^(1/2)/(a-d)*(-1/ (a-d)/(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)*((x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)^2 *c+(b^2-4*c*d)^(1/2)*(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)+a-d)^(1/2)+1/2*(b^2- 4*c*d)^(1/2)/(a-d)^(3/2)*ln((2*a-2*d+(b^2-4*c*d)^(1/2)*(x-1/2*(-b+(b^2-4*c *d)^(1/2))/c)+2*(a-d)^(1/2)*((x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)^2*c+(b^2-4*c *d)^(1/2)*(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)+a-d)^(1/2))/(x-1/2*(-b+(b^2-4*c *d)^(1/2))/c)))+1/2*c/(a-d)^(3/2)*ln((2*a-2*d+(b^2-4*c*d)^(1/2)*(x-1/2*(-b +(b^2-4*c*d)^(1/2))/c)+2*(a-d)^(1/2)*((x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)^2*c +(b^2-4*c*d)^(1/2)*(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)+a-d)^(1/2))/(x-1/2*(-b +(b^2-4*c*d)^(1/2))/c)))-2/3*c/(a-d)*(-1/(a-d)/(x-1/2*(-b+(b^2-4*c*d)^(1/2 ))/c)*((x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)^2*c+(b^2-4*c*d)^(1/2)*(x-1/2*(-b+( b^2-4*c*d)^(1/2))/c)+a-d)^(1/2)+1/2*(b^2-4*c*d)^(1/2)/(a-d)^(3/2)*ln((2*a- 2*d+(b^2-4*c*d)^(1/2)*(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)+2*(a-d)^(1/2)*((x-1 /2*(-b+(b^2-4*c*d)^(1/2))/c)^2*c+(b^2-4*c*d)^(1/2)*(x-1/2*(-b+(b^2-4*c*d)^ (1/2))/c)+a-d)^(1/2))/(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c))))+10*c^2/(b^2-4*c* d)^3*(-1/(a-d)/(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)*((x-1/2*(-b+(b^2-4*c*d)...
Leaf count of result is larger than twice the leaf count of optimal. 3962 vs. \(2 (304) = 608\).
Time = 11.83 (sec) , antiderivative size = 8134, normalized size of antiderivative = 24.80 \[ \int \frac {1}{\sqrt {a+b x+c x^2} \left (d+b x+c x^2\right )^4} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {1}{\sqrt {a+b x+c x^2} \left (d+b x+c x^2\right )^4} \, dx=\text {Timed out} \]
\[ \int \frac {1}{\sqrt {a+b x+c x^2} \left (d+b x+c x^2\right )^4} \, dx=\int { \frac {1}{\sqrt {c x^{2} + b x + a} {\left (c x^{2} + b x + d\right )}^{4}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 30280 vs. \(2 (304) = 608\).
Time = 4.51 (sec) , antiderivative size = 30280, normalized size of antiderivative = 92.32 \[ \int \frac {1}{\sqrt {a+b x+c x^2} \left (d+b x+c x^2\right )^4} \, dx=\text {Too large to display} \]
-1/16*((5*b^6 + 12*a*b^4*c + 48*a^2*b^2*c^2 + 320*a^3*c^3 - 72*b^4*c*d - 1 92*a*b^2*c^2*d - 1152*a^2*c^3*d + 384*b^2*c^2*d^2 + 1536*a*c^3*d^2 - 1024* c^3*d^3)*log(abs((sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^2*c + 4*(sqrt(c)* x - sqrt(c*x^2 + b*x + a))^2*a*c^2 - 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)) ^2*c^2*d + (sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^3*sqrt(c) + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b*c^(3/2) - 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a ))*b*c^(3/2)*d + 3*a*b^2*c + 4*sqrt(a*b^2 - b^2*d - 4*a*c*d + 4*c*d^2)*(sq rt(c)*x - sqrt(c*x^2 + b*x + a))^2*c^(3/2) - 4*a^2*c^2 - 2*b^2*c*d + 4*sqr t(a*b^2 - b^2*d - 4*a*c*d + 4*c*d^2)*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b *c + sqrt(a*b^2 - b^2*d - 4*a*c*d + 4*c*d^2)*b^2*sqrt(c)))/sqrt(a*b^2 - b^ 2*d - 4*a*c*d + 4*c*d^2) - (5*b^6 + 12*a*b^4*c + 48*a^2*b^2*c^2 + 320*a^3* c^3 - 72*b^4*c*d - 192*a*b^2*c^2*d - 1152*a^2*c^3*d + 384*b^2*c^2*d^2 + 15 36*a*c^3*d^2 - 1024*c^3*d^3)*log(abs((sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2 *b^2*c + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*c^2 - 8*(sqrt(c)*x - sq rt(c*x^2 + b*x + a))^2*c^2*d + (sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^3*sqr t(c) + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b*c^(3/2) - 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b*c^(3/2)*d + 3*a*b^2*c - 4*sqrt(a*b^2 - b^2*d - 4* a*c*d + 4*c*d^2)*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*c^(3/2) - 4*a^2*c^2 - 2*b^2*c*d - 4*sqrt(a*b^2 - b^2*d - 4*a*c*d + 4*c*d^2)*(sqrt(c)*x - sqrt (c*x^2 + b*x + a))*b*c - sqrt(a*b^2 - b^2*d - 4*a*c*d + 4*c*d^2)*b^2*sq...
Timed out. \[ \int \frac {1}{\sqrt {a+b x+c x^2} \left (d+b x+c x^2\right )^4} \, dx=\int \frac {1}{\sqrt {c\,x^2+b\,x+a}\,{\left (c\,x^2+b\,x+d\right )}^4} \,d x \]