Integrand size = 23, antiderivative size = 426 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {\sqrt {x} (A b-2 a B-(b B-2 A c) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {\sqrt {x} \left (a B \left (7 b^2-4 a c\right )-A \left (b^3+8 a b c\right )+c \left (12 a b B-A \left (b^2+20 a c\right )\right ) x\right )}{4 a \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {\sqrt {c} \left (6 a B \left (3 b^2+4 a c-2 b \sqrt {b^2-4 a c}\right )+A \left (b^3-52 a b c+b^2 \sqrt {b^2-4 a c}+20 a c \sqrt {b^2-4 a c}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{4 \sqrt {2} a \left (b^2-4 a c\right )^{5/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {c} \left (12 a b B-A \left (b^2+20 a c\right )+\frac {6 a B \left (3 b^2+4 a c\right )+A \left (b^3-52 a b c\right )}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{4 \sqrt {2} a \left (b^2-4 a c\right )^2 \sqrt {b+\sqrt {b^2-4 a c}}} \]
-1/2*(A*b-2*B*a-(-2*A*c+B*b)*x)*x^(1/2)/(-4*a*c+b^2)/(c*x^2+b*x+a)^2-1/4*( a*B*(-4*a*c+7*b^2)-A*(8*a*b*c+b^3)+c*(12*a*b*B-A*(20*a*c+b^2))*x)*x^(1/2)/ a/(-4*a*c+b^2)^2/(c*x^2+b*x+a)+1/8*arctan(2^(1/2)*c^(1/2)*x^(1/2)/(b-(-4*a *c+b^2)^(1/2))^(1/2))*c^(1/2)*(6*a*B*(3*b^2+4*a*c-2*b*(-4*a*c+b^2)^(1/2))+ A*(b^3-52*a*b*c+b^2*(-4*a*c+b^2)^(1/2)+20*a*c*(-4*a*c+b^2)^(1/2)))/a/(-4*a *c+b^2)^(5/2)*2^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)-1/8*arctan(2^(1/2)*c^(1 /2)*x^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2))*c^(1/2)*(12*a*b*B-A*(20*a*c+b^2) +(6*a*B*(4*a*c+3*b^2)+A*(-52*a*b*c+b^3))/(-4*a*c+b^2)^(1/2))/a/(-4*a*c+b^2 )^2*2^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)
Time = 4.13 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.06 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a+b x+c x^2\right )^3} \, dx=\frac {\frac {2 \sqrt {x} \left (-12 a^3 B c+A b^2 x (b+c x)^2+a^2 \left (-3 b^2 B+16 b c (A-B x)+4 c^2 x (9 A+B x)\right )-a \left (b B x \left (5 b^2+19 b c x+12 c^2 x^2\right )+A \left (b^3-5 b^2 c x-28 b c^2 x^2-20 c^3 x^3\right )\right )\right )}{(a+x (b+c x))^2}+\frac {\sqrt {2} \sqrt {c} \left (6 a B \left (3 b^2+4 a c-2 b \sqrt {b^2-4 a c}\right )+A \left (b^3-52 a b c+b^2 \sqrt {b^2-4 a c}+20 a c \sqrt {b^2-4 a c}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \sqrt {c} \left (-6 a B \left (3 b^2+4 a c+2 b \sqrt {b^2-4 a c}\right )+A \left (-b^3+52 a b c+b^2 \sqrt {b^2-4 a c}+20 a c \sqrt {b^2-4 a c}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}}}{8 a \left (b^2-4 a c\right )^2} \]
((2*Sqrt[x]*(-12*a^3*B*c + A*b^2*x*(b + c*x)^2 + a^2*(-3*b^2*B + 16*b*c*(A - B*x) + 4*c^2*x*(9*A + B*x)) - a*(b*B*x*(5*b^2 + 19*b*c*x + 12*c^2*x^2) + A*(b^3 - 5*b^2*c*x - 28*b*c^2*x^2 - 20*c^3*x^3))))/(a + x*(b + c*x))^2 + (Sqrt[2]*Sqrt[c]*(6*a*B*(3*b^2 + 4*a*c - 2*b*Sqrt[b^2 - 4*a*c]) + A*(b^3 - 52*a*b*c + b^2*Sqrt[b^2 - 4*a*c] + 20*a*c*Sqrt[b^2 - 4*a*c]))*ArcTan[(Sq rt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sq rt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*Sqrt[c]*(-6*a*B*(3*b^2 + 4*a*c + 2*b *Sqrt[b^2 - 4*a*c]) + A*(-b^3 + 52*a*b*c + b^2*Sqrt[b^2 - 4*a*c] + 20*a*c* Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4 *a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(8*a*(b^2 - 4*a* c)^2)
Time = 0.90 (sec) , antiderivative size = 406, normalized size of antiderivative = 0.95, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {1234, 27, 1235, 27, 1197, 1480, 218}
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.
| \(\displaystyle \int \frac {\sqrt {x} (A+B x)}{\left (a+b x+c x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1234 |
| \(\displaystyle -\frac {\int -\frac {A b-2 a B+5 (b B-2 A c) x}{2 \sqrt {x} \left (c x^2+b x+a\right )^2}dx}{2 \left (b^2-4 a c\right )}-\frac {\sqrt {x} (-2 a B-x (b B-2 A c)+A b)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {A b-2 a B+5 (b B-2 A c) x}{\sqrt {x} \left (c x^2+b x+a\right )^2}dx}{4 \left (b^2-4 a c\right )}-\frac {\sqrt {x} (-2 a B-x (b B-2 A c)+A b)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle \frac {-\frac {\int -\frac {3 a B \left (b^2+4 a c\right )+2 A \left (\frac {b^3}{2}-8 a b c\right )-c \left (12 a b B-A \left (b^2+20 a c\right )\right ) x}{2 \sqrt {x} \left (c x^2+b x+a\right )}dx}{a \left (b^2-4 a c\right )}-\frac {\sqrt {x} \left (-A \left (8 a b c+b^3\right )+c x \left (12 a b B-A \left (20 a c+b^2\right )\right )+a B \left (7 b^2-4 a c\right )\right )}{a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{4 \left (b^2-4 a c\right )}-\frac {\sqrt {x} (-2 a B-x (b B-2 A c)+A b)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {3 a B \left (b^2+4 a c\right )+A \left (b^3-16 a b c\right )-c \left (12 a b B-A \left (b^2+20 a c\right )\right ) x}{\sqrt {x} \left (c x^2+b x+a\right )}dx}{2 a \left (b^2-4 a c\right )}-\frac {\sqrt {x} \left (-A \left (8 a b c+b^3\right )+c x \left (12 a b B-A \left (20 a c+b^2\right )\right )+a B \left (7 b^2-4 a c\right )\right )}{a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{4 \left (b^2-4 a c\right )}-\frac {\sqrt {x} (-2 a B-x (b B-2 A c)+A b)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle \frac {\frac {\int \frac {3 a B \left (b^2+4 a c\right )+A \left (b^3-16 a b c\right )-c \left (12 a b B-A \left (b^2+20 a c\right )\right ) x}{c x^2+b x+a}d\sqrt {x}}{a \left (b^2-4 a c\right )}-\frac {\sqrt {x} \left (-A \left (8 a b c+b^3\right )+c x \left (12 a b B-A \left (20 a c+b^2\right )\right )+a B \left (7 b^2-4 a c\right )\right )}{a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{4 \left (b^2-4 a c\right )}-\frac {\sqrt {x} (-2 a B-x (b B-2 A c)+A b)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {\frac {-\frac {1}{2} c \left (-A \left (20 a c+b^2\right )-\frac {A \left (b^3-52 a b c\right )+6 a B \left (4 a c+3 b^2\right )}{\sqrt {b^2-4 a c}}+12 a b B\right ) \int \frac {1}{\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )+c x}d\sqrt {x}-\frac {1}{2} c \left (-A \left (20 a c+b^2\right )+\frac {A \left (b^3-52 a b c\right )+6 a B \left (4 a c+3 b^2\right )}{\sqrt {b^2-4 a c}}+12 a b B\right ) \int \frac {1}{\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )+c x}d\sqrt {x}}{a \left (b^2-4 a c\right )}-\frac {\sqrt {x} \left (-A \left (8 a b c+b^3\right )+c x \left (12 a b B-A \left (20 a c+b^2\right )\right )+a B \left (7 b^2-4 a c\right )\right )}{a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{4 \left (b^2-4 a c\right )}-\frac {\sqrt {x} (-2 a B-x (b B-2 A c)+A b)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {-\frac {\sqrt {c} \left (-A \left (20 a c+b^2\right )-\frac {A \left (b^3-52 a b c\right )+6 a B \left (4 a c+3 b^2\right )}{\sqrt {b^2-4 a c}}+12 a b B\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {c} \left (-A \left (20 a c+b^2\right )+\frac {A \left (b^3-52 a b c\right )+6 a B \left (4 a c+3 b^2\right )}{\sqrt {b^2-4 a c}}+12 a b B\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} \sqrt {\sqrt {b^2-4 a c}+b}}}{a \left (b^2-4 a c\right )}-\frac {\sqrt {x} \left (-A \left (8 a b c+b^3\right )+c x \left (12 a b B-A \left (20 a c+b^2\right )\right )+a B \left (7 b^2-4 a c\right )\right )}{a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{4 \left (b^2-4 a c\right )}-\frac {\sqrt {x} (-2 a B-x (b B-2 A c)+A b)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
-1/2*(Sqrt[x]*(A*b - 2*a*B - (b*B - 2*A*c)*x))/((b^2 - 4*a*c)*(a + b*x + c *x^2)^2) + (-((Sqrt[x]*(a*B*(7*b^2 - 4*a*c) - A*(b^3 + 8*a*b*c) + c*(12*a* b*B - A*(b^2 + 20*a*c))*x))/(a*(b^2 - 4*a*c)*(a + b*x + c*x^2))) + (-((Sqr t[c]*(12*a*b*B - A*(b^2 + 20*a*c) - (6*a*B*(3*b^2 + 4*a*c) + A*(b^3 - 52*a *b*c))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b ^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b - Sqrt[b^2 - 4*a*c]])) - (Sqrt[c]*(12*a*b*B - A*(b^2 + 20*a*c) + (6*a*B*(3*b^2 + 4*a*c) + A*(b^3 - 52*a*b*c))/Sqrt[b^ 2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]]) /(Sqrt[2]*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(a*(b^2 - 4*a*c)))/(4*(b^2 - 4*a*c ))
3.11.26.3.1 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)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)), x_Symbol] :> Simp[2 Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; Fr eeQ[{a, b, c, d, e, f, g}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*( (f*b - 2*a*g + (2*c*f - b*g)*x)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)) Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*Simp[g *(2*a*e*m + b*d*(2*p + 3)) - f*(b*e*m + 2*c*d*(2*p + 3)) - e*(2*c*f - b*g)* (m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, -1 ] && GtQ[m, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 *a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^m *(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p] )
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
Time = 0.34 (sec) , antiderivative size = 528, normalized size of antiderivative = 1.24
| method | result | size |
| derivativedivides | \(\frac {\frac {c^{2} \left (20 A a c +A \,b^{2}-12 a b B \right ) x^{\frac {7}{2}}}{4 a \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {c \left (28 A a b c +2 A \,b^{3}+4 B \,a^{2} c -19 B a \,b^{2}\right ) x^{\frac {5}{2}}}{4 a \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {\left (36 A \,a^{2} c^{2}+5 A a \,b^{2} c +A \,b^{4}-16 a^{2} b B c -5 B \,b^{3} a \right ) x^{\frac {3}{2}}}{4 a \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {2 \left (16 A a b c -A \,b^{3}-12 B \,a^{2} c -3 B a \,b^{2}\right ) \sqrt {x}}{128 a^{2} c^{2}-64 a \,b^{2} c +8 b^{4}}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {c \left (-\frac {\left (20 A \sqrt {-4 a c +b^{2}}\, a c +A \sqrt {-4 a c +b^{2}}\, b^{2}-52 A a b c +A \,b^{3}-12 a b B \sqrt {-4 a c +b^{2}}+24 B \,a^{2} c +18 B a \,b^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (20 A \sqrt {-4 a c +b^{2}}\, a c +A \sqrt {-4 a c +b^{2}}\, b^{2}+52 A a b c -A \,b^{3}-12 a b B \sqrt {-4 a c +b^{2}}-24 B \,a^{2} c -18 B a \,b^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{a \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}\) | \(528\) |
| default | \(\frac {\frac {c^{2} \left (20 A a c +A \,b^{2}-12 a b B \right ) x^{\frac {7}{2}}}{4 a \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {c \left (28 A a b c +2 A \,b^{3}+4 B \,a^{2} c -19 B a \,b^{2}\right ) x^{\frac {5}{2}}}{4 a \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {\left (36 A \,a^{2} c^{2}+5 A a \,b^{2} c +A \,b^{4}-16 a^{2} b B c -5 B \,b^{3} a \right ) x^{\frac {3}{2}}}{4 a \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {2 \left (16 A a b c -A \,b^{3}-12 B \,a^{2} c -3 B a \,b^{2}\right ) \sqrt {x}}{128 a^{2} c^{2}-64 a \,b^{2} c +8 b^{4}}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {c \left (-\frac {\left (20 A \sqrt {-4 a c +b^{2}}\, a c +A \sqrt {-4 a c +b^{2}}\, b^{2}-52 A a b c +A \,b^{3}-12 a b B \sqrt {-4 a c +b^{2}}+24 B \,a^{2} c +18 B a \,b^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (20 A \sqrt {-4 a c +b^{2}}\, a c +A \sqrt {-4 a c +b^{2}}\, b^{2}+52 A a b c -A \,b^{3}-12 a b B \sqrt {-4 a c +b^{2}}-24 B \,a^{2} c -18 B a \,b^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{a \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}\) | \(528\) |
2*(1/8*c^2*(20*A*a*c+A*b^2-12*B*a*b)/a/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(7/2)+ 1/8/a*c*(28*A*a*b*c+2*A*b^3+4*B*a^2*c-19*B*a*b^2)/(16*a^2*c^2-8*a*b^2*c+b^ 4)*x^(5/2)+1/8*(36*A*a^2*c^2+5*A*a*b^2*c+A*b^4-16*B*a^2*b*c-5*B*a*b^3)/a/( 16*a^2*c^2-8*a*b^2*c+b^4)*x^(3/2)+1/8*(16*A*a*b*c-A*b^3-12*B*a^2*c-3*B*a*b ^2)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(1/2))/(c*x^2+b*x+a)^2+1/a/(16*a^2*c^2-8* a*b^2*c+b^4)*c*(-1/8*(20*A*(-4*a*c+b^2)^(1/2)*a*c+A*(-4*a*c+b^2)^(1/2)*b^2 -52*A*a*b*c+A*b^3-12*a*b*B*(-4*a*c+b^2)^(1/2)+24*B*a^2*c+18*B*a*b^2)/(-4*a *c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x^(1/2)* 2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))+1/8*(20*A*(-4*a*c+b^2)^(1/2)*a* c+A*(-4*a*c+b^2)^(1/2)*b^2+52*A*a*b*c-A*b^3-12*a*b*B*(-4*a*c+b^2)^(1/2)-24 *B*a^2*c-18*B*a*b^2)/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c) ^(1/2)*arctan(c*x^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 7267 vs. \(2 (368) = 736\).
Time = 21.71 (sec) , antiderivative size = 7267, normalized size of antiderivative = 17.06 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {\sqrt {x} (A+B x)}{\left (a+b x+c x^2\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {x} (A+B x)}{\left (a+b x+c x^2\right )^3} \, dx=\int { \frac {{\left (B x + A\right )} \sqrt {x}}{{\left (c x^{2} + b x + a\right )}^{3}} \,d x } \]
1/4*(((b^3*c^2 - 16*a*b*c^3)*A + 3*(a*b^2*c^2 + 4*a^2*c^3)*B)*x^(9/2) + (( 2*b^4*c - 31*a*b^2*c^2 + 20*a^2*c^3)*A + 6*(a*b^3*c + 2*a^2*b*c^2)*B)*x^(7 /2) + ((b^5 - 12*a*b^3*c - 4*a^2*b*c^2)*A + (3*a*b^4 - a^2*b^2*c + 28*a^3* c^2)*B)*x^(5/2) + (3*(a*b^4 - 9*a^2*b^2*c + 12*a^3*c^2)*A + (a^2*b^3 + 8*a ^3*b*c)*B)*x^(3/2))/(a^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2 + (a^2*b^4*c^2 - 8 *a^3*b^2*c^3 + 16*a^4*c^4)*x^4 + 2*(a^2*b^5*c - 8*a^3*b^3*c^2 + 16*a^4*b*c ^3)*x^3 + (a^2*b^6 - 6*a^3*b^4*c + 32*a^5*c^3)*x^2 + 2*(a^3*b^5 - 8*a^4*b^ 3*c + 16*a^5*b*c^2)*x) + integrate(-1/8*(((b^3*c - 16*a*b*c^2)*A + 3*(a*b^ 2*c + 4*a^2*c^2)*B)*x^(3/2) + ((b^4 - 17*a*b^2*c - 20*a^2*c^2)*A + 3*(a*b^ 3 + 8*a^2*b*c)*B)*sqrt(x))/(a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2 + (a^2*b^4* c - 8*a^3*b^2*c^2 + 16*a^4*c^3)*x^2 + (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^ 2)*x), x)
Leaf count of result is larger than twice the leaf count of optimal. 7277 vs. \(2 (368) = 736\).
Time = 1.80 (sec) , antiderivative size = 7277, normalized size of antiderivative = 17.08 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]
1/32*((2*b^4*c^2 + 32*a*b^2*c^3 - 160*a^2*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)* sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4 - 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b* c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c + 80*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt( b^2 - 4*a*c)*c)*a^2*c^2 + 40*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a *c)*c)*b^2*c^2 - 20*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c) *c)*a*c^3 - 2*(b^2 - 4*a*c)*b^2*c^2 - 40*(b^2 - 4*a*c)*a*c^3)*(a*b^4 - 8*a ^2*b^2*c + 16*a^3*c^2)^2*A - 12*(2*a*b^3*c^2 - 8*a^2*b*c^3 - sqrt(2)*sqrt( b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c + 2*sqrt(2)*sqrt(b^2 - 4*a* c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c - sqrt(2)*sqrt(b^2 - 4*a*c)*sqr t(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^2 - 2*(b^2 - 4*a*c)*a*b*c^2)*(a*b^4 - 8 *a^2*b^2*c + 16*a^3*c^2)^2*B + 2*(sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)* a*b^9 - 28*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^7*c - 2*sqrt(2)*s qrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^8*c - 2*a*b^9*c + 240*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^5*c^2 + 48*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a* c)*c)*a^2*b^6*c^2 + sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^7*c^2 + 56 *a^2*b^7*c^2 - 832*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^3*c^3 - 2 88*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^4*c^3 - 24*sqrt(2)*sqr...
Time = 13.81 (sec) , antiderivative size = 19024, normalized size of antiderivative = 44.66 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]
((x^(3/2)*(A*b^4 + 36*A*a^2*c^2 - 5*B*a*b^3 + 5*A*a*b^2*c - 16*B*a^2*b*c)) /(4*a*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) - (x^(1/2)*(A*b^3 + 3*B*a*b^2 + 12*B *a^2*c - 16*A*a*b*c))/(4*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (x^(5/2)*(4*B*a ^2*c^2 + 2*A*b^3*c + 28*A*a*b*c^2 - 19*B*a*b^2*c))/(4*a*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (c*x^(7/2)*(20*A*a*c^2 + A*b^2*c - 12*B*a*b*c))/(4*a*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))/(x^2*(2*a*c + b^2) + a^2 + c^2*x^4 + 2*a*b*x + 2*b*c*x^3) + atan(((((64*A*a*b^13*c^2 - 786432*B*a^8*c^8 + 1048576*A*a^7* b*c^8 - 2304*A*a^2*b^11*c^3 + 30720*A*a^3*b^9*c^4 - 204800*A*a^4*b^7*c^5 + 737280*A*a^5*b^5*c^6 - 1376256*A*a^6*b^3*c^7 + 192*B*a^2*b^12*c^2 - 3072* B*a^3*b^10*c^3 + 15360*B*a^4*b^8*c^4 - 245760*B*a^6*b^4*c^6 + 786432*B*a^7 *b^2*c^7)/(64*(a^2*b^12 + 4096*a^8*c^6 - 24*a^3*b^10*c + 240*a^4*b^8*c^2 - 1280*a^5*b^6*c^3 + 3840*a^6*b^4*c^4 - 6144*a^7*b^2*c^5)) - (x^(1/2)*(-(A^ 2*b^17 + 9*B^2*a^2*b^15 + A^2*b^2*(-(4*a*c - b^2)^15)^(1/2) + 9*B^2*a^2*(- (4*a*c - b^2)^15)^(1/2) + 6*A*B*a*b^16 + 1140*A^2*a^2*b^13*c^2 - 10160*A^2 *a^3*b^11*c^3 + 34880*A^2*a^4*b^9*c^4 + 43776*A^2*a^5*b^7*c^5 - 680960*A^2 *a^6*b^5*c^6 + 1863680*A^2*a^7*b^3*c^7 - 5040*B^2*a^4*b^11*c^2 + 37440*B^2 *a^5*b^9*c^3 - 103680*B^2*a^6*b^7*c^4 - 9216*B^2*a^7*b^5*c^5 + 552960*B^2* a^8*b^3*c^6 + 983040*A*B*a^9*c^8 - 55*A^2*a*b^15*c - 25*A^2*a*c*(-(4*a*c - b^2)^15)^(1/2) - 1720320*A^2*a^8*b*c^8 + 180*B^2*a^3*b^13*c - 737280*B^2* a^9*b*c^7 + 240*A*B*a^3*b^12*c^2 + 24000*A*B*a^4*b^10*c^3 - 241920*A*B*...