Integrand size = 23, antiderivative size = 406 \[ \int \frac {A+B x}{x^{3/2} \left (a+b x+c x^2\right )^2} \, dx=-\frac {3 A b^2-a b B-10 a A c}{a^2 \left (b^2-4 a c\right ) \sqrt {x}}+\frac {A b^2-a b B-2 a A c+(A b-2 a B) c x}{a \left (b^2-4 a c\right ) \sqrt {x} \left (a+b x+c x^2\right )}+\frac {\sqrt {c} \left (a B \left (b^2-12 a c+b \sqrt {b^2-4 a c}\right )-A \left (3 b^3-16 a b c+3 b^2 \sqrt {b^2-4 a c}-10 a c \sqrt {b^2-4 a c}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {c} \left (a B \left (b^2-12 a c-b \sqrt {b^2-4 a c}\right )-A \left (3 b^3-16 a b c-3 b^2 \sqrt {b^2-4 a c}+10 a c \sqrt {b^2-4 a c}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}}} \]
(10*A*a*c-3*A*b^2+B*a*b)/a^2/(-4*a*c+b^2)/x^(1/2)+(A*b^2-a*b*B-2*A*a*c+(A* b-2*B*a)*c*x)/a/(-4*a*c+b^2)/(c*x^2+b*x+a)/x^(1/2)+1/2*arctan(2^(1/2)*c^(1 /2)*x^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2))*c^(1/2)*(a*B*(b^2-12*a*c+b*(-4*a *c+b^2)^(1/2))-A*(3*b^3-16*a*b*c+3*b^2*(-4*a*c+b^2)^(1/2)-10*a*c*(-4*a*c+b ^2)^(1/2)))/a^2/(-4*a*c+b^2)^(3/2)*2^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)-1/ 2*arctan(2^(1/2)*c^(1/2)*x^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2))*c^(1/2)*(a* B*(b^2-12*a*c-b*(-4*a*c+b^2)^(1/2))-A*(3*b^3-16*a*b*c-3*b^2*(-4*a*c+b^2)^( 1/2)+10*a*c*(-4*a*c+b^2)^(1/2)))/a^2/(-4*a*c+b^2)^(3/2)*2^(1/2)/(b+(-4*a*c +b^2)^(1/2))^(1/2)
Time = 2.35 (sec) , antiderivative size = 399, normalized size of antiderivative = 0.98 \[ \int \frac {A+B x}{x^{3/2} \left (a+b x+c x^2\right )^2} \, dx=\frac {-\frac {2 \left (-3 A b^2 x (b+c x)+a b B x (b+c x)+a^2 (8 A c-2 B c x)+a A \left (-2 b^2+11 b c x+10 c^2 x^2\right )\right )}{\sqrt {x} (a+x (b+c x))}-\frac {\sqrt {2} \sqrt {c} \left (a B \left (b^2-12 a c+b \sqrt {b^2-4 a c}\right )+A \left (-3 b^3+16 a b c-3 b^2 \sqrt {b^2-4 a c}+10 a c \sqrt {b^2-4 a c}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \sqrt {c} \left (a B \left (b^2-12 a c-b \sqrt {b^2-4 a c}\right )+A \left (-3 b^3+16 a b c+3 b^2 \sqrt {b^2-4 a c}-10 a c \sqrt {b^2-4 a c}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}}}{2 a^2 \left (-b^2+4 a c\right )} \]
((-2*(-3*A*b^2*x*(b + c*x) + a*b*B*x*(b + c*x) + a^2*(8*A*c - 2*B*c*x) + a *A*(-2*b^2 + 11*b*c*x + 10*c^2*x^2)))/(Sqrt[x]*(a + x*(b + c*x))) - (Sqrt[ 2]*Sqrt[c]*(a*B*(b^2 - 12*a*c + b*Sqrt[b^2 - 4*a*c]) + A*(-3*b^3 + 16*a*b* c - 3*b^2*Sqrt[b^2 - 4*a*c] + 10*a*c*Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[2]*S qrt[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]]) + (Sqrt[2]*Sqrt[c]*(a*B*(b^2 - 12*a*c - b*Sqrt[b^2 - 4 *a*c]) + A*(-3*b^3 + 16*a*b*c + 3*b^2*Sqrt[b^2 - 4*a*c] - 10*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]]))/(2*a^2*(-b^2 + 4*a*c))
Time = 0.72 (sec) , antiderivative size = 384, normalized size of antiderivative = 0.95, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {1235, 27, 1198, 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.
| \(\displaystyle \int \frac {A+B x}{x^{3/2} \left (a+b x+c x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle \frac {c x (A b-2 a B)-2 a A c-a b B+A b^2}{a \sqrt {x} \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\int -\frac {3 A b^2-a B b-10 a A c+3 (A b-2 a B) c x}{2 x^{3/2} \left (c x^2+b x+a\right )}dx}{a \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {3 A b^2-a B b-10 a A c+3 (A b-2 a B) c x}{x^{3/2} \left (c x^2+b x+a\right )}dx}{2 a \left (b^2-4 a c\right )}+\frac {c x (A b-2 a B)-2 a A c-a b B+A b^2}{a \sqrt {x} \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\) |
| \(\Big \downarrow \) 1198 |
| \(\displaystyle \frac {\frac {\int \frac {a B \left (b^2-6 a c\right )-A \left (3 b^3-13 a b c\right )-c \left (3 A b^2-a B b-10 a A c\right ) x}{\sqrt {x} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 \left (-10 a A c-a b B+3 A b^2\right )}{a \sqrt {x}}}{2 a \left (b^2-4 a c\right )}+\frac {c x (A b-2 a B)-2 a A c-a b B+A b^2}{a \sqrt {x} \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle \frac {\frac {2 \int \frac {a B \left (b^2-6 a c\right )-A \left (3 b^3-13 a b c\right )-c \left (3 A b^2-a B b-10 a A c\right ) x}{c x^2+b x+a}d\sqrt {x}}{a}-\frac {2 \left (-10 a A c-a b B+3 A b^2\right )}{a \sqrt {x}}}{2 a \left (b^2-4 a c\right )}+\frac {c x (A b-2 a B)-2 a A c-a b B+A b^2}{a \sqrt {x} \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {\frac {2 \left (\frac {c \left (a B \left (b \sqrt {b^2-4 a c}-12 a c+b^2\right )-A \left (3 b^2 \sqrt {b^2-4 a c}-10 a c \sqrt {b^2-4 a c}-16 a b c+3 b^3\right )\right ) \int \frac {1}{\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )+c x}d\sqrt {x}}{2 \sqrt {b^2-4 a c}}-\frac {1}{2} c \left (\frac {a B \left (b^2-12 a c\right )-A \left (3 b^3-16 a b c\right )}{\sqrt {b^2-4 a c}}-10 a A c-a b B+3 A b^2\right ) \int \frac {1}{\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )+c x}d\sqrt {x}\right )}{a}-\frac {2 \left (-10 a A c-a b B+3 A b^2\right )}{a \sqrt {x}}}{2 a \left (b^2-4 a c\right )}+\frac {c x (A b-2 a B)-2 a A c-a b B+A b^2}{a \sqrt {x} \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {2 \left (\frac {\sqrt {c} \left (a B \left (b \sqrt {b^2-4 a c}-12 a c+b^2\right )-A \left (3 b^2 \sqrt {b^2-4 a c}-10 a c \sqrt {b^2-4 a c}-16 a b c+3 b^3\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {c} \left (\frac {a B \left (b^2-12 a c\right )-A \left (3 b^3-16 a b c\right )}{\sqrt {b^2-4 a c}}-10 a A c-a b B+3 A b^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} \sqrt {\sqrt {b^2-4 a c}+b}}\right )}{a}-\frac {2 \left (-10 a A c-a b B+3 A b^2\right )}{a \sqrt {x}}}{2 a \left (b^2-4 a c\right )}+\frac {c x (A b-2 a B)-2 a A c-a b B+A b^2}{a \sqrt {x} \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\) |
(A*b^2 - a*b*B - 2*a*A*c + (A*b - 2*a*B)*c*x)/(a*(b^2 - 4*a*c)*Sqrt[x]*(a + b*x + c*x^2)) + ((-2*(3*A*b^2 - a*b*B - 10*a*A*c))/(a*Sqrt[x]) + (2*((Sq rt[c]*(a*B*(b^2 - 12*a*c + b*Sqrt[b^2 - 4*a*c]) - A*(3*b^3 - 16*a*b*c + 3* b^2*Sqrt[b^2 - 4*a*c] - 10*a*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^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[c]*(3*A*b^2 - a*b*B - 10*a*A*c + (a*B*(b^2 - 12*a*c) - A*(3*b^3 - 16*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)/(2*a*(b^2 - 4*a*c))
3.11.21.3.1 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), x_Symbol] :> Simp[(e*f - d*g)*((d + e*x)^(m + 1)/((m + 1)*(c *d^2 - b*d*e + a*e^2))), x] + Simp[1/(c*d^2 - b*d*e + a*e^2) Int[(d + e*x )^(m + 1)*(Simp[c*d*f - f*b*e + a*e*g - c*(e*f - d*g)*x, x]/(a + b*x + c*x^ 2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && FractionQ[m] && LtQ[m, -1 ]
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 = 0.41 (sec) , antiderivative size = 383, normalized size of antiderivative = 0.94
| method | result | size |
| derivativedivides | \(-\frac {2 \left (\frac {\frac {c \left (2 A a c -A \,b^{2}+a b B \right ) x^{\frac {3}{2}}}{8 a c -2 b^{2}}+\frac {\left (3 A a b c -A \,b^{3}-2 B \,a^{2} c +B a \,b^{2}\right ) \sqrt {x}}{8 a c -2 b^{2}}}{c \,x^{2}+b x +a}+\frac {2 c \left (-\frac {\left (10 A \sqrt {-4 a c +b^{2}}\, a c -3 A \sqrt {-4 a c +b^{2}}\, b^{2}+16 A a b c -3 A \,b^{3}+a b B \sqrt {-4 a c +b^{2}}-12 B \,a^{2} c +B a \,b^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (10 A \sqrt {-4 a c +b^{2}}\, a c -3 A \sqrt {-4 a c +b^{2}}\, b^{2}-16 A a b c +3 A \,b^{3}+a b B \sqrt {-4 a c +b^{2}}+12 B \,a^{2} c -B a \,b^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{4 a c -b^{2}}\right )}{a^{2}}-\frac {2 A}{a^{2} \sqrt {x}}\) | \(383\) |
| default | \(-\frac {2 \left (\frac {\frac {c \left (2 A a c -A \,b^{2}+a b B \right ) x^{\frac {3}{2}}}{8 a c -2 b^{2}}+\frac {\left (3 A a b c -A \,b^{3}-2 B \,a^{2} c +B a \,b^{2}\right ) \sqrt {x}}{8 a c -2 b^{2}}}{c \,x^{2}+b x +a}+\frac {2 c \left (-\frac {\left (10 A \sqrt {-4 a c +b^{2}}\, a c -3 A \sqrt {-4 a c +b^{2}}\, b^{2}+16 A a b c -3 A \,b^{3}+a b B \sqrt {-4 a c +b^{2}}-12 B \,a^{2} c +B a \,b^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (10 A \sqrt {-4 a c +b^{2}}\, a c -3 A \sqrt {-4 a c +b^{2}}\, b^{2}-16 A a b c +3 A \,b^{3}+a b B \sqrt {-4 a c +b^{2}}+12 B \,a^{2} c -B a \,b^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{4 a c -b^{2}}\right )}{a^{2}}-\frac {2 A}{a^{2} \sqrt {x}}\) | \(383\) |
| risch | \(-\frac {2 A}{a^{2} \sqrt {x}}-\frac {\frac {\frac {2 c \left (2 A a c -A \,b^{2}+a b B \right ) x^{\frac {3}{2}}}{8 a c -2 b^{2}}+\frac {2 \left (3 A a b c -A \,b^{3}-2 B \,a^{2} c +B a \,b^{2}\right ) \sqrt {x}}{8 a c -2 b^{2}}}{c \,x^{2}+b x +a}+\frac {4 c \left (-\frac {\left (10 A \sqrt {-4 a c +b^{2}}\, a c -3 A \sqrt {-4 a c +b^{2}}\, b^{2}+16 A a b c -3 A \,b^{3}+a b B \sqrt {-4 a c +b^{2}}-12 B \,a^{2} c +B a \,b^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (10 A \sqrt {-4 a c +b^{2}}\, a c -3 A \sqrt {-4 a c +b^{2}}\, b^{2}-16 A a b c +3 A \,b^{3}+a b B \sqrt {-4 a c +b^{2}}+12 B \,a^{2} c -B a \,b^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{4 a c -b^{2}}}{a^{2}}\) | \(384\) |
-2/a^2*((1/2*c*(2*A*a*c-A*b^2+B*a*b)/(4*a*c-b^2)*x^(3/2)+1/2*(3*A*a*b*c-A* b^3-2*B*a^2*c+B*a*b^2)/(4*a*c-b^2)*x^(1/2))/(c*x^2+b*x+a)+2/(4*a*c-b^2)*c* (-1/8*(10*A*(-4*a*c+b^2)^(1/2)*a*c-3*A*(-4*a*c+b^2)^(1/2)*b^2+16*A*a*b*c-3 *A*b^3+a*b*B*(-4*a*c+b^2)^(1/2)-12*B*a^2*c+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*(10*A*(-4*a*c+b^2)^(1/2)*a*c-3*A*(-4*a*c+b^2 )^(1/2)*b^2-16*A*a*b*c+3*A*b^3+a*b*B*(-4*a*c+b^2)^(1/2)+12*B*a^2*c-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))))-2*A/a^2/x^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 7597 vs. \(2 (351) = 702\).
Time = 26.17 (sec) , antiderivative size = 7597, normalized size of antiderivative = 18.71 \[ \int \frac {A+B x}{x^{3/2} \left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {A+B x}{x^{3/2} \left (a+b x+c x^2\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {A+B x}{x^{3/2} \left (a+b x+c x^2\right )^2} \, dx=\int { \frac {B x + A}{{\left (c x^{2} + b x + a\right )}^{2} x^{\frac {3}{2}}} \,d x } \]
-(((3*b^3*c - 13*a*b*c^2)*A - (a*b^2*c - 6*a^2*c^2)*B)*x^(5/2) + ((3*b^4 - 10*a*b^2*c - 10*a^2*c^2)*A - (a*b^3 - 5*a^2*b*c)*B)*x^(3/2) + 2*(a^2*b^2 - 4*a^3*c)*A/sqrt(x) + 2*(3*(a*b^3 - 4*a^2*b*c)*A - (a^2*b^2 - 4*a^3*c)*B) *sqrt(x))/(a^4*b^2 - 4*a^5*c + (a^3*b^2*c - 4*a^4*c^2)*x^2 + (a^3*b^3 - 4* a^4*b*c)*x) + integrate(1/2*(((3*b^3*c - 13*a*b*c^2)*A - (a*b^2*c - 6*a^2* c^2)*B)*x^(3/2) + ((3*b^4 - 16*a*b^2*c + 10*a^2*c^2)*A - (a*b^3 - 7*a^2*b* c)*B)*sqrt(x))/(a^4*b^2 - 4*a^5*c + (a^3*b^2*c - 4*a^4*c^2)*x^2 + (a^3*b^3 - 4*a^4*b*c)*x), x)
Leaf count of result is larger than twice the leaf count of optimal. 5405 vs. \(2 (351) = 702\).
Time = 1.36 (sec) , antiderivative size = 5405, normalized size of antiderivative = 13.31 \[ \int \frac {A+B x}{x^{3/2} \left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
(B*a*b*c*x^2 - 3*A*b^2*c*x^2 + 10*A*a*c^2*x^2 + B*a*b^2*x - 3*A*b^3*x - 2* B*a^2*c*x + 11*A*a*b*c*x - 2*A*a*b^2 + 8*A*a^2*c)/((a^2*b^2 - 4*a^3*c)*(c* x^(5/2) + b*x^(3/2) + a*sqrt(x))) - 1/8*((6*b^4*c^2 - 44*a*b^2*c^3 + 80*a^ 2*c^4 - 3*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4 + 22*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c + 6*s qrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c - 40*sqrt(2 )*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*c^2 - 20*sqrt(2)*s qrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^2 - 3*sqrt(2)*sqrt( b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^2*c^2 + 10*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*c^3 - 6*(b^2 - 4*a*c)*b^2*c^2 + 20*(b^2 - 4*a*c)*a*c^3)*(a^2*b^2 - 4*a^3*c)^2*A - (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*sqr t(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c - sqrt(2)*s qrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^2 - 2*(b^2 - 4*a*c) *a*b*c^2)*(a^2*b^2 - 4*a^3*c)^2*B + 2*(3*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a *c)*c)*a^2*b^7 - 37*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^5*c - 6* sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^6*c - 6*a^2*b^7*c + 152*sqrt (2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^3*c^2 + 50*sqrt(2)*sqrt(b*c + sq rt(b^2 - 4*a*c)*c)*a^3*b^4*c^2 + 3*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)...
Time = 15.45 (sec) , antiderivative size = 17623, normalized size of antiderivative = 43.41 \[ \int \frac {A+B x}{x^{3/2} \left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
- ((2*A)/a - (x*(3*A*b^3 - B*a*b^2 + 2*B*a^2*c - 11*A*a*b*c))/(a^2*(4*a*c - b^2)) + (c*x^2*(10*A*a*c - 3*A*b^2 + B*a*b))/(a^2*(4*a*c - b^2)))/(a*x^( 1/2) + b*x^(3/2) + c*x^(5/2)) - atan(((x^(1/2)*(25600*A^2*a^12*c^9 - 9216* B^2*a^13*c^8 + 18*A^2*a^6*b^12*c^3 - 408*A^2*a^7*b^10*c^4 + 3764*A^2*a^8*b ^8*c^5 - 17920*A^2*a^9*b^6*c^6 + 45696*A^2*a^10*b^4*c^7 - 57344*A^2*a^11*b ^2*c^8 + 2*B^2*a^8*b^10*c^3 - 52*B^2*a^9*b^8*c^4 + 576*B^2*a^10*b^6*c^5 - 3200*B^2*a^11*b^4*c^6 + 8704*B^2*a^12*b^2*c^7 - 12*A*B*a^7*b^11*c^3 + 292* A*B*a^8*b^9*c^4 - 2816*A*B*a^9*b^7*c^5 + 13440*A*B*a^10*b^5*c^6 - 31744*A* B*a^11*b^3*c^7 + 29696*A*B*a^12*b*c^8) + (-(9*A^2*b^13 + B^2*a^2*b^11 + 9* A^2*b^4*(-(4*a*c - b^2)^9)^(1/2) - 6*A*B*a*b^12 + 2077*A^2*a^2*b^9*c^2 - 1 0656*A^2*a^3*b^7*c^3 + 30240*A^2*a^4*b^5*c^4 - 44800*A^2*a^5*b^3*c^5 + 25* A^2*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2) + B^2*a^2*b^2*(-(4*a*c - b^2)^9)^(1/2 ) + 288*B^2*a^4*b^7*c^2 - 1504*B^2*a^5*b^5*c^3 + 3840*B^2*a^6*b^3*c^4 - 15 360*A*B*a^7*c^6 - 213*A^2*a*b^11*c + 26880*A^2*a^6*b*c^6 - 27*B^2*a^3*b^9* c - 3840*B^2*a^7*b*c^5 - 9*B^2*a^3*c*(-(4*a*c - b^2)^9)^(1/2) - 1548*A*B*a ^3*b^8*c^2 + 8064*A*B*a^4*b^6*c^3 - 22400*A*B*a^5*b^4*c^4 + 30720*A*B*a^6* b^2*c^5 - 51*A^2*a*b^2*c*(-(4*a*c - b^2)^9)^(1/2) - 6*A*B*a*b^3*(-(4*a*c - b^2)^9)^(1/2) + 152*A*B*a^2*b^10*c + 44*A*B*a^2*b*c*(-(4*a*c - b^2)^9)^(1 /2))/(8*(a^5*b^12 + 4096*a^11*c^6 - 24*a^6*b^10*c + 240*a^7*b^8*c^2 - 1280 *a^8*b^6*c^3 + 3840*a^9*b^4*c^4 - 6144*a^10*b^2*c^5)))^(1/2)*(x^(1/2)*(...