Integrand size = 23, antiderivative size = 414 \[ \int \frac {x^{3/2} (A+B x)}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {\sqrt {x} \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {\sqrt {x} \left (2 b^3 B-7 A b^2 c+4 a b B c+4 a A c^2+3 c \left (b^2 B-4 A b c+4 a B c\right ) x\right )}{4 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {3 \left (b^2 B-4 A b c+4 a B c-\frac {b^3 B-6 A b^2 c+12 a b B c-8 a A c^2}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{4 \sqrt {2} \sqrt {c} \left (b^2-4 a c\right )^2 \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {3 \left (b^2 B-4 A b c+4 a B c+\frac {b^3 B-6 A b^2 c+12 a b B c-8 a A c^2}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{4 \sqrt {2} \sqrt {c} \left (b^2-4 a c\right )^2 \sqrt {b+\sqrt {b^2-4 a c}}} \]
-1/2*(a*(-2*A*c+B*b)+(-A*b*c-2*B*a*c+B*b^2)*x)*x^(1/2)/c/(-4*a*c+b^2)/(c*x ^2+b*x+a)^2+1/4*(2*B*b^3-7*A*b^2*c+4*B*a*b*c+4*A*a*c^2+3*c*(-4*A*b*c+4*B*a *c+B*b^2)*x)*x^(1/2)/c/(-4*a*c+b^2)^2/(c*x^2+b*x+a)+3/8*arctan(2^(1/2)*c^( 1/2)*x^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2))*(B*b^2-4*A*b*c+4*B*a*c+(8*A*a*c ^2+6*A*b^2*c-12*B*a*b*c-B*b^3)/(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^2*2^(1/2)/ c^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)+3/8*arctan(2^(1/2)*c^(1/2)*x^(1/2)/(b +(-4*a*c+b^2)^(1/2))^(1/2))*(B*b^2-4*A*b*c+4*B*a*c+(-8*A*a*c^2-6*A*b^2*c+1 2*B*a*b*c+B*b^3)/(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^2*2^(1/2)/c^(1/2)/(b+(-4 *a*c+b^2)^(1/2))^(1/2)
Time = 6.34 (sec) , antiderivative size = 448, normalized size of antiderivative = 1.08 \[ \int \frac {x^{3/2} (A+B x)}{\left (a+b x+c x^2\right )^3} \, dx=\frac {1}{8} \left (\frac {2 \sqrt {x} \left (4 a^2 (3 b B-c (3 A+B x))+a \left (A \left (-3 b^2-16 b c x+4 c^2 x^2\right )+B x \left (19 b^2+16 b c x+12 c^2 x^2\right )\right )+b x \left (b B x (5 b+3 c x)-A \left (5 b^2+19 b c x+12 c^2 x^2\right )\right )\right )}{\left (b^2-4 a c\right )^2 (a+x (b+c x))^2}+\frac {3 \sqrt {2} \left (-b^3 B-4 b c \left (3 a B+A \sqrt {b^2-4 a c}\right )+4 a c \left (2 A c+B \sqrt {b^2-4 a c}\right )+b^2 \left (6 A c+B \sqrt {b^2-4 a c}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {c} \left (b^2-4 a c\right )^{5/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {3 \sqrt {2} \left (b^3 B+4 b c \left (3 a B-A \sqrt {b^2-4 a c}\right )+b^2 \left (-6 A c+B \sqrt {b^2-4 a c}\right )+4 a c \left (-2 A c+B \sqrt {b^2-4 a c}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {c} \left (b^2-4 a c\right )^{5/2} \sqrt {b+\sqrt {b^2-4 a c}}}\right ) \]
((2*Sqrt[x]*(4*a^2*(3*b*B - c*(3*A + B*x)) + a*(A*(-3*b^2 - 16*b*c*x + 4*c ^2*x^2) + B*x*(19*b^2 + 16*b*c*x + 12*c^2*x^2)) + b*x*(b*B*x*(5*b + 3*c*x) - A*(5*b^2 + 19*b*c*x + 12*c^2*x^2))))/((b^2 - 4*a*c)^2*(a + x*(b + c*x)) ^2) + (3*Sqrt[2]*(-(b^3*B) - 4*b*c*(3*a*B + A*Sqrt[b^2 - 4*a*c]) + 4*a*c*( 2*A*c + B*Sqrt[b^2 - 4*a*c]) + b^2*(6*A*c + B*Sqrt[b^2 - 4*a*c]))*ArcTan[( Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[c]*(b^2 - 4*a *c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (3*Sqrt[2]*(b^3*B + 4*b*c*(3*a*B - A*Sqrt[b^2 - 4*a*c]) + b^2*(-6*A*c + B*Sqrt[b^2 - 4*a*c]) + 4*a*c*(-2*A* c + B*Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt[b ^2 - 4*a*c]]])/(Sqrt[c]*(b^2 - 4*a*c)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/ 8
Time = 0.78 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {1233, 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.
| \(\displaystyle \int \frac {x^{3/2} (A+B x)}{\left (a+b x+c x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1233 |
| \(\displaystyle \frac {\int \frac {a (b B-2 A c)-\left (B b^2-5 A c b+6 a B c\right ) x}{2 \sqrt {x} \left (c x^2+b x+a\right )^2}dx}{2 c \left (b^2-4 a c\right )}-\frac {\sqrt {x} \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {a (b B-2 A c)-\left (B b^2-5 A c b+6 a B c\right ) x}{\sqrt {x} \left (c x^2+b x+a\right )^2}dx}{4 c \left (b^2-4 a c\right )}-\frac {\sqrt {x} \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle \frac {\frac {\sqrt {x} \left (3 c x \left (4 a B c-4 A b c+b^2 B\right )+4 a A c^2+4 a b B c-7 A b^2 c+2 b^3 B\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\int \frac {3 a c \left (4 a b B-A \left (b^2+4 a c\right )-\left (B b^2-4 A c b+4 a B c\right ) x\right )}{2 \sqrt {x} \left (c x^2+b x+a\right )}dx}{a \left (b^2-4 a c\right )}}{4 c \left (b^2-4 a c\right )}-\frac {\sqrt {x} \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\sqrt {x} \left (3 c x \left (4 a B c-4 A b c+b^2 B\right )+4 a A c^2+4 a b B c-7 A b^2 c+2 b^3 B\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {3 c \int \frac {4 a b B-A \left (b^2+4 a c\right )-\left (B b^2-4 A c b+4 a B c\right ) x}{\sqrt {x} \left (c x^2+b x+a\right )}dx}{2 \left (b^2-4 a c\right )}}{4 c \left (b^2-4 a c\right )}-\frac {\sqrt {x} \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle \frac {\frac {\sqrt {x} \left (3 c x \left (4 a B c-4 A b c+b^2 B\right )+4 a A c^2+4 a b B c-7 A b^2 c+2 b^3 B\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {3 c \int \frac {4 a b B-A \left (b^2+4 a c\right )-\left (B b^2-4 A c b+4 a B c\right ) x}{c x^2+b x+a}d\sqrt {x}}{b^2-4 a c}}{4 c \left (b^2-4 a c\right )}-\frac {\sqrt {x} \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {\frac {\sqrt {x} \left (3 c x \left (4 a B c-4 A b c+b^2 B\right )+4 a A c^2+4 a b B c-7 A b^2 c+2 b^3 B\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {3 c \left (-\frac {1}{2} \left (-\frac {-8 a A c^2+12 a b B c-6 A b^2 c+b^3 B}{\sqrt {b^2-4 a c}}+4 a B c-4 A b c+b^2 B\right ) \int \frac {1}{\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )+c x}d\sqrt {x}-\frac {1}{2} \left (\frac {-8 a A c^2+12 a b B c-6 A b^2 c+b^3 B}{\sqrt {b^2-4 a c}}+4 a B c-4 A b c+b^2 B\right ) \int \frac {1}{\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )+c x}d\sqrt {x}\right )}{b^2-4 a c}}{4 c \left (b^2-4 a c\right )}-\frac {\sqrt {x} \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\sqrt {x} \left (3 c x \left (4 a B c-4 A b c+b^2 B\right )+4 a A c^2+4 a b B c-7 A b^2 c+2 b^3 B\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {3 c \left (-\frac {\left (-\frac {-8 a A c^2+12 a b B c-6 A b^2 c+b^3 B}{\sqrt {b^2-4 a c}}+4 a B c-4 A b c+b^2 B\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\left (\frac {-8 a A c^2+12 a b B c-6 A b^2 c+b^3 B}{\sqrt {b^2-4 a c}}+4 a B c-4 A b c+b^2 B\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} \sqrt {c} \sqrt {\sqrt {b^2-4 a c}+b}}\right )}{b^2-4 a c}}{4 c \left (b^2-4 a c\right )}-\frac {\sqrt {x} \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
-1/2*(Sqrt[x]*(a*(b*B - 2*A*c) + (b^2*B - A*b*c - 2*a*B*c)*x))/(c*(b^2 - 4 *a*c)*(a + b*x + c*x^2)^2) + ((Sqrt[x]*(2*b^3*B - 7*A*b^2*c + 4*a*b*B*c + 4*a*A*c^2 + 3*c*(b^2*B - 4*A*b*c + 4*a*B*c)*x))/((b^2 - 4*a*c)*(a + b*x + c*x^2)) - (3*c*(-(((b^2*B - 4*A*b*c + 4*a*B*c - (b^3*B - 6*A*b^2*c + 12*a* b*B*c - 8*a*A*c^2)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqr t[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[c]*Sqrt[b - Sqrt[b^2 - 4*a*c]])) - ((b^2*B - 4*A*b*c + 4*a*B*c + (b^3*B - 6*A*b^2*c + 12*a*b*B*c - 8*a*A*c^ 2)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[c]*Sqrt[b + Sqrt[b^2 - 4*a*c]])))/(b^2 - 4*a*c))/ (4*c*(b^2 - 4*a*c))
3.11.25.3.1 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 - 1))*(a + b*x + c*x^2) ^(p + 1)*((2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g - c *(b*e*f + b*d*g + 2*a*e*g))*x)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Simp[1/(c*( p + 1)*(b^2 - 4*a*c)) Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Sim p[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a*e*(e*f *(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*( m + p + 1) + 2*c^2*d*f*(m + 2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2* p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, b, c, d, e, f, g]) | | !ILtQ[m + 2*p + 3, 0])
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 = 0.35 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.21
| method | result | size |
| derivativedivides | \(\frac {-\frac {3 c \left (4 A b c -4 B a c -B \,b^{2}\right ) x^{\frac {7}{2}}}{4 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {2 \left (4 A a \,c^{2}-19 A \,b^{2} c +16 B a b c +5 B \,b^{3}\right ) x^{\frac {5}{2}}}{128 a^{2} c^{2}-64 a \,b^{2} c +8 b^{4}}-\frac {\left (16 A a b c +5 A \,b^{3}+4 B \,a^{2} c -19 B a \,b^{2}\right ) x^{\frac {3}{2}}}{4 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {3 a \left (4 A a c +A \,b^{2}-4 a b B \right ) \sqrt {x}}{4 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {3 c \left (-\frac {\left (-4 A b c \sqrt {-4 a c +b^{2}}+8 A a \,c^{2}+6 A \,b^{2} c +4 B a c \sqrt {-4 a c +b^{2}}+B \,b^{2} \sqrt {-4 a c +b^{2}}-12 B a b c -B \,b^{3}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (-4 A b c \sqrt {-4 a c +b^{2}}-8 A a \,c^{2}-6 A \,b^{2} c +4 B a c \sqrt {-4 a c +b^{2}}+B \,b^{2} \sqrt {-4 a c +b^{2}}+12 B a b c +B \,b^{3}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}\) | \(500\) |
| default | \(\frac {-\frac {3 c \left (4 A b c -4 B a c -B \,b^{2}\right ) x^{\frac {7}{2}}}{4 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {2 \left (4 A a \,c^{2}-19 A \,b^{2} c +16 B a b c +5 B \,b^{3}\right ) x^{\frac {5}{2}}}{128 a^{2} c^{2}-64 a \,b^{2} c +8 b^{4}}-\frac {\left (16 A a b c +5 A \,b^{3}+4 B \,a^{2} c -19 B a \,b^{2}\right ) x^{\frac {3}{2}}}{4 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {3 a \left (4 A a c +A \,b^{2}-4 a b B \right ) \sqrt {x}}{4 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {3 c \left (-\frac {\left (-4 A b c \sqrt {-4 a c +b^{2}}+8 A a \,c^{2}+6 A \,b^{2} c +4 B a c \sqrt {-4 a c +b^{2}}+B \,b^{2} \sqrt {-4 a c +b^{2}}-12 B a b c -B \,b^{3}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (-4 A b c \sqrt {-4 a c +b^{2}}-8 A a \,c^{2}-6 A \,b^{2} c +4 B a c \sqrt {-4 a c +b^{2}}+B \,b^{2} \sqrt {-4 a c +b^{2}}+12 B a b c +B \,b^{3}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}\) | \(500\) |
2*(-3/8*c*(4*A*b*c-4*B*a*c-B*b^2)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(7/2)+1/8*( 4*A*a*c^2-19*A*b^2*c+16*B*a*b*c+5*B*b^3)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(5/2 )-1/8*(16*A*a*b*c+5*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^(3/2)-3/8*a*(4*A*a*c+A*b^2-4*B*a*b)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(1/2)) /(c*x^2+b*x+a)^2+3/(16*a^2*c^2-8*a*b^2*c+b^4)*c*(-1/8*(-4*A*b*c*(-4*a*c+b^ 2)^(1/2)+8*A*a*c^2+6*A*b^2*c+4*B*a*c*(-4*a*c+b^2)^(1/2)+B*b^2*(-4*a*c+b^2) ^(1/2)-12*B*a*b*c-B*b^3)/c/(-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*(-4*A*b*c*(-4*a*c+b^2)^(1/2)-8*A*a*c^2-6*A*b^2*c+4*B*a*c*(-4*a*c+b^2) ^(1/2)+B*b^2*(-4*a*c+b^2)^(1/2)+12*B*a*b*c+B*b^3)/c/(-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. 5646 vs. \(2 (362) = 724\).
Time = 10.49 (sec) , antiderivative size = 5646, normalized size of antiderivative = 13.64 \[ \int \frac {x^{3/2} (A+B x)}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {x^{3/2} (A+B x)}{\left (a+b x+c x^2\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {x^{3/2} (A+B x)}{\left (a+b x+c x^2\right )^3} \, dx=\int { \frac {{\left (B x + A\right )} x^{\frac {3}{2}}}{{\left (c x^{2} + b x + a\right )}^{3}} \,d x } \]
-1/4*(3*(4*B*a*b*c^2 - (b^2*c^2 + 4*a*c^3)*A)*x^(9/2) - 3*(2*(b^3*c + 2*a* b*c^2)*A - (7*a*b^2*c - 4*a^2*c^2)*B)*x^(7/2) - ((3*b^4 - a*b^2*c + 28*a^2 *c^2)*A - (7*a*b^3 + 8*a^2*b*c)*B)*x^(5/2) - ((a*b^3 + 8*a^2*b*c)*A - (5*a ^2*b^2 + 4*a^3*c)*B)*x^(3/2))/(a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2 + (a*b^4 *c^2 - 8*a^2*b^2*c^3 + 16*a^3*c^4)*x^4 + 2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a ^3*b*c^3)*x^3 + (a*b^6 - 6*a^2*b^4*c + 32*a^4*c^3)*x^2 + 2*(a^2*b^5 - 8*a^ 3*b^3*c + 16*a^4*b*c^2)*x) - integrate(-3/8*((4*B*a*b*c - (b^2*c + 4*a*c^2 )*A)*x^(3/2) - ((b^3 + 8*a*b*c)*A - (5*a*b^2 + 4*a^2*c)*B)*sqrt(x))/(a^2*b ^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (a*b^4*c - 8*a^2*b^2*c^2 + 16*a^3*c^3)*x^2 + (a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*x), x)
Leaf count of result is larger than twice the leaf count of optimal. 3170 vs. \(2 (362) = 724\).
Time = 1.22 (sec) , antiderivative size = 3170, normalized size of antiderivative = 7.66 \[ \int \frac {x^{3/2} (A+B x)}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]
3/16*((sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^6 - 4*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^4*c - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b ^5*c - 2*b^6*c - 16*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^2 + sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c^2 + 8*a*b^4*c^2 + 2*b^5*c^2 + 64*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*c^3 + 32*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^3 + 32*a^2*b^2*c^3 + 16*a*b^3*c^3 - 16*sqr t(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*c^4 - 128*a^3*c^4 - 96*a^2*b*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^5 - 8*sqrt(2 )*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c + 2*sqrt(2)*sq rt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c + 48*sqrt(2)*sqrt(b^ 2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^2 + 24*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^2 - 12*sqrt(2)*sqrt(b^2 - 4*a* c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^3 + 2*(b^2 - 4*a*c)*b^4*c - 2*(b^ 2 - 4*a*c)*b^3*c^2 - 32*(b^2 - 4*a*c)*a^2*c^3 - 24*(b^2 - 4*a*c)*a*b*c^3)* A - 2*(2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^5 - 16*sqrt(2)*sqrt(b *c + sqrt(b^2 - 4*a*c)*c)*a^2*b^3*c - 4*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a* c)*c)*a*b^4*c - 4*a*b^5*c + 32*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3 *b*c^2 + 16*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^2 + 2*sqrt(2 )*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c^2 + 32*a^2*b^3*c^2 + 6*a*b^4*...
Time = 13.39 (sec) , antiderivative size = 16720, normalized size of antiderivative = 40.39 \[ \int \frac {x^{3/2} (A+B x)}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]
atan(((((3*(262144*A*a^6*c^8 - 64*A*b^12*c^2 + 1024*A*a*b^10*c^3 + 256*B*a *b^11*c^2 - 262144*B*a^6*b*c^7 - 5120*A*a^2*b^8*c^4 + 81920*A*a^4*b^4*c^6 - 262144*A*a^5*b^2*c^7 - 5120*B*a^2*b^9*c^3 + 40960*B*a^3*b^7*c^4 - 163840 *B*a^4*b^5*c^5 + 327680*B*a^5*b^3*c^6))/(64*(b^12 + 4096*a^6*c^6 + 240*a^2 *b^8*c^2 - 1280*a^3*b^6*c^3 + 3840*a^4*b^4*c^4 - 6144*a^5*b^2*c^5 - 24*a*b ^10*c)) - (x^(1/2)*(-(9*(B^2*a*b^15 + B^2*a*(-(4*a*c - b^2)^15)^(1/2) + A^ 2*b^15*c - A^2*c*(-(4*a*c - b^2)^15)^(1/2) - 560*A^2*a^2*b^11*c^3 + 4160*A ^2*a^3*b^9*c^4 - 11520*A^2*a^4*b^7*c^5 - 1024*A^2*a^5*b^5*c^6 + 61440*A^2* a^6*b^3*c^7 - 560*B^2*a^3*b^11*c^2 + 4160*B^2*a^4*b^9*c^3 - 11520*B^2*a^5* b^7*c^4 - 1024*B^2*a^6*b^5*c^5 + 61440*B^2*a^7*b^3*c^6 + 65536*A*B*a^8*c^8 + 20*A^2*a*b^13*c^2 - 81920*A^2*a^7*b*c^8 + 20*B^2*a^2*b^13*c - 81920*B^2 *a^8*b*c^7 + 240*A*B*a^2*b^12*c^2 - 64*A*B*a^3*b^10*c^3 - 11520*A*B*a^4*b^ 8*c^4 + 66560*A*B*a^5*b^6*c^5 - 143360*A*B*a^6*b^4*c^6 + 81920*A*B*a^7*b^2 *c^7 - 20*A*B*a*b^14*c))/(128*(1048576*a^11*c^11 - 40*a^2*b^18*c^2 + 720*a ^3*b^16*c^3 - 7680*a^4*b^14*c^4 + 53760*a^5*b^12*c^5 - 258048*a^6*b^10*c^6 + 860160*a^7*b^8*c^7 - 1966080*a^8*b^6*c^8 + 2949120*a^9*b^4*c^9 - 262144 0*a^10*b^2*c^10 + a*b^20*c)))^(1/2)*(64*b^11*c^2 - 1280*a*b^9*c^3 - 65536* a^5*b*c^7 + 10240*a^2*b^7*c^4 - 40960*a^3*b^5*c^5 + 81920*a^4*b^3*c^6))/(8 *(b^8 + 256*a^4*c^4 + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 - 16*a*b^6*c)))*(-( 9*(B^2*a*b^15 + B^2*a*(-(4*a*c - b^2)^15)^(1/2) + A^2*b^15*c - A^2*c*(-...