Integrand size = 23, antiderivative size = 459 \[ \int \frac {x^{5/2} (A+B x)}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {x^{3/2} \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {\sqrt {x} \left (a \left (b^2 B-12 A b c+20 a B c\right )-\left (b^3 B+3 A b^2 c-16 a b B c+12 a A c^2\right ) x\right )}{4 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {\left (b^3 B+3 A b^2 c-16 a b B c+12 a A c^2-\frac {b^4 B+3 A b^3 c-18 a b^2 B c+36 a A b c^2-40 a^2 B c^2}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{4 \sqrt {2} c^{3/2} \left (b^2-4 a c\right )^2 \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\left (b^3 B+3 A b^2 c-16 a b B c+12 a A c^2+\frac {b^4 B+3 A b^3 c-18 a b^2 B c+36 a A b c^2-40 a^2 B c^2}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{4 \sqrt {2} c^{3/2} \left (b^2-4 a c\right )^2 \sqrt {b+\sqrt {b^2-4 a c}}} \]
-1/2*x^(3/2)*(a*(-2*A*c+B*b)+(-A*b*c-2*B*a*c+B*b^2)*x)/c/(-4*a*c+b^2)/(c*x ^2+b*x+a)^2-1/4*(a*(-12*A*b*c+20*B*a*c+B*b^2)-(12*A*a*c^2+3*A*b^2*c-16*B*a *b*c+B*b^3)*x)*x^(1/2)/c/(-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))*(B*b^3+3*A*b^2*c-16*B*a*b*c+1 2*A*a*c^2+(-36*A*a*b*c^2-3*A*b^3*c+40*B*a^2*c^2+18*B*a*b^2*c-B*b^4)/(-4*a* c+b^2)^(1/2))/c^(3/2)/(-4*a*c+b^2)^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))*(B*b^3+3* A*b^2*c-16*B*a*b*c+12*A*a*c^2+(36*A*a*b*c^2+3*A*b^3*c-40*B*a^2*c^2-18*B*a* b^2*c+B*b^4)/(-4*a*c+b^2)^(1/2))/c^(3/2)/(-4*a*c+b^2)^2*2^(1/2)/(b+(-4*a*c +b^2)^(1/2))^(1/2)
Time = 6.08 (sec) , antiderivative size = 519, normalized size of antiderivative = 1.13 \[ \int \frac {x^{5/2} (A+B x)}{\left (a+b x+c x^2\right )^3} \, dx=\frac {\frac {2 \sqrt {c} \sqrt {x} \left (-20 a^3 B c+a x \left (-2 b^3 B+12 A c^3 x^2+b^2 c (19 A-5 B x)+16 b c^2 x (A-B x)\right )+b^2 x^2 \left (-b^2 B+3 A c^2 x+b c (5 A+B x)\right )-a^2 \left (b^2 B-4 b c (3 A-7 B x)+4 c^2 x (A+9 B x)\right )\right )}{\left (b^2-4 a c\right )^2 (a+x (b+c x))^2}+\frac {\sqrt {2} \left (-b^4 B+3 b^2 c \left (6 a B+A \sqrt {b^2-4 a c}\right )+4 a c^2 \left (10 a B+3 A \sqrt {b^2-4 a c}\right )+b^3 \left (-3 A c+B \sqrt {b^2-4 a c}\right )-4 a b c \left (9 A c+4 B \sqrt {b^2-4 a c}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{5/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \left (b^4 B+3 b^2 c \left (-6 a B+A \sqrt {b^2-4 a c}\right )+4 a c^2 \left (-10 a B+3 A \sqrt {b^2-4 a c}\right )+4 a b c \left (9 A c-4 B \sqrt {b^2-4 a c}\right )+b^3 \left (3 A c+B \sqrt {b^2-4 a c}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{5/2} \sqrt {b+\sqrt {b^2-4 a c}}}}{8 c^{3/2}} \]
((2*Sqrt[c]*Sqrt[x]*(-20*a^3*B*c + a*x*(-2*b^3*B + 12*A*c^3*x^2 + b^2*c*(1 9*A - 5*B*x) + 16*b*c^2*x*(A - B*x)) + b^2*x^2*(-(b^2*B) + 3*A*c^2*x + b*c *(5*A + B*x)) - a^2*(b^2*B - 4*b*c*(3*A - 7*B*x) + 4*c^2*x*(A + 9*B*x))))/ ((b^2 - 4*a*c)^2*(a + x*(b + c*x))^2) + (Sqrt[2]*(-(b^4*B) + 3*b^2*c*(6*a* B + A*Sqrt[b^2 - 4*a*c]) + 4*a*c^2*(10*a*B + 3*A*Sqrt[b^2 - 4*a*c]) + b^3* (-3*A*c + B*Sqrt[b^2 - 4*a*c]) - 4*a*b*c*(9*A*c + 4*B*Sqrt[b^2 - 4*a*c]))* ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a *c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*(b^4*B + 3*b^2*c*(-6*a*B + A*Sqrt[b^2 - 4*a*c]) + 4*a*c^2*(-10*a*B + 3*A*Sqrt[b^2 - 4*a*c]) + 4*a* b*c*(9*A*c - 4*B*Sqrt[b^2 - 4*a*c]) + b^3*(3*A*c + B*Sqrt[b^2 - 4*a*c]))*A rcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a* c)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(8*c^(3/2))
Time = 0.81 (sec) , antiderivative size = 458, 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, 1234, 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.
| \(\displaystyle \int \frac {x^{5/2} (A+B x)}{\left (a+b x+c x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1233 |
| \(\displaystyle \frac {\int \frac {\sqrt {x} \left (3 a (b B-2 A c)+\left (B b^2+3 A c b-10 a B c\right ) x\right )}{2 \left (c x^2+b x+a\right )^2}dx}{2 c \left (b^2-4 a c\right )}-\frac {x^{3/2} \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {x} \left (3 a (b B-2 A c)+\left (B b^2+3 A c b-10 a B c\right ) x\right )}{\left (c x^2+b x+a\right )^2}dx}{4 c \left (b^2-4 a c\right )}-\frac {x^{3/2} \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1234 |
| \(\displaystyle \frac {-\frac {\int -\frac {a \left (B b^2-12 A c b+20 a B c\right )+\left (B b^3+3 A c b^2-16 a B c b+12 a A c^2\right ) x}{2 \sqrt {x} \left (c x^2+b x+a\right )}dx}{b^2-4 a c}-\frac {\sqrt {x} \left (a \left (20 a B c-12 A b c+b^2 B\right )-x \left (12 a A c^2-16 a b B c+3 A b^2 c+b^3 B\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{4 c \left (b^2-4 a c\right )}-\frac {x^{3/2} \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {a \left (B b^2-12 A c b+20 a B c\right )+\left (B b^3+3 A c b^2-16 a B c b+12 a A c^2\right ) x}{\sqrt {x} \left (c x^2+b x+a\right )}dx}{2 \left (b^2-4 a c\right )}-\frac {\sqrt {x} \left (a \left (20 a B c-12 A b c+b^2 B\right )-x \left (12 a A c^2-16 a b B c+3 A b^2 c+b^3 B\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{4 c \left (b^2-4 a c\right )}-\frac {x^{3/2} \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle \frac {\frac {\int \frac {a \left (B b^2-12 A c b+20 a B c\right )+\left (B b^3+3 A c b^2-16 a B c b+12 a A c^2\right ) x}{c x^2+b x+a}d\sqrt {x}}{b^2-4 a c}-\frac {\sqrt {x} \left (a \left (20 a B c-12 A b c+b^2 B\right )-x \left (12 a A c^2-16 a b B c+3 A b^2 c+b^3 B\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{4 c \left (b^2-4 a c\right )}-\frac {x^{3/2} \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {\frac {\frac {1}{2} \left (-\frac {-40 a^2 B c^2+36 a A b c^2-18 a b^2 B c+3 A b^3 c+b^4 B}{\sqrt {b^2-4 a c}}+12 a A c^2-16 a b B c+3 A b^2 c+b^3 B\right ) \int \frac {1}{\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )+c x}d\sqrt {x}+\frac {1}{2} \left (\frac {-40 a^2 B c^2+36 a A b c^2-18 a b^2 B c+3 A b^3 c+b^4 B}{\sqrt {b^2-4 a c}}+12 a A c^2-16 a b B c+3 A b^2 c+b^3 B\right ) \int \frac {1}{\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )+c x}d\sqrt {x}}{b^2-4 a c}-\frac {\sqrt {x} \left (a \left (20 a B c-12 A b c+b^2 B\right )-x \left (12 a A c^2-16 a b B c+3 A b^2 c+b^3 B\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{4 c \left (b^2-4 a c\right )}-\frac {x^{3/2} \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\frac {\left (-\frac {-40 a^2 B c^2+36 a A b c^2-18 a b^2 B c+3 A b^3 c+b^4 B}{\sqrt {b^2-4 a c}}+12 a A c^2-16 a b B c+3 A b^2 c+b^3 B\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\left (\frac {-40 a^2 B c^2+36 a A b c^2-18 a b^2 B c+3 A b^3 c+b^4 B}{\sqrt {b^2-4 a c}}+12 a A c^2-16 a b B c+3 A b^2 c+b^3 B\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} \sqrt {c} \sqrt {\sqrt {b^2-4 a c}+b}}}{b^2-4 a c}-\frac {\sqrt {x} \left (a \left (20 a B c-12 A b c+b^2 B\right )-x \left (12 a A c^2-16 a b B c+3 A b^2 c+b^3 B\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{4 c \left (b^2-4 a c\right )}-\frac {x^{3/2} \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
-1/2*(x^(3/2)*(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]*(a*(b^2*B - 12*A*b*c + 20*a*B*c) - (b^3*B + 3*A*b^2*c - 16*a*b*B*c + 12*a*A*c^2)*x))/((b^2 - 4*a*c)*(a + b* x + c*x^2))) + (((b^3*B + 3*A*b^2*c - 16*a*b*B*c + 12*a*A*c^2 - (b^4*B + 3 *A*b^3*c - 18*a*b^2*B*c + 36*a*A*b*c^2 - 40*a^2*B*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]*Sq rt[c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + ((b^3*B + 3*A*b^2*c - 16*a*b*B*c + 12 *a*A*c^2 + (b^4*B + 3*A*b^3*c - 18*a*b^2*B*c + 36*a*A*b*c^2 - 40*a^2*B*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))
3.11.24.3.1 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*(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_)^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 = 0.44 (sec) , antiderivative size = 595, normalized size of antiderivative = 1.30
| method | result | size |
| derivativedivides | \(\frac {\frac {2 \left (12 A a \,c^{2}+3 A \,b^{2} c -16 B a b c +B \,b^{3}\right ) x^{\frac {7}{2}}}{128 a^{2} c^{2}-64 a \,b^{2} c +8 b^{4}}+\frac {\left (16 a A b \,c^{2}+5 A \,b^{3} c -36 a^{2} B \,c^{2}-5 a \,b^{2} B c -b^{4} B \right ) x^{\frac {5}{2}}}{4 c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {a \left (4 A a \,c^{2}-19 A \,b^{2} c +28 B a b c +2 B \,b^{3}\right ) x^{\frac {3}{2}}}{4 c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {a^{2} \left (12 A b c -20 B a c -B \,b^{2}\right ) \sqrt {x}}{4 c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {-\frac {\left (12 A a \,c^{2} \sqrt {-4 a c +b^{2}}+3 A \,b^{2} c \sqrt {-4 a c +b^{2}}-36 a A b \,c^{2}-3 A \,b^{3} c -16 B a b c \sqrt {-4 a c +b^{2}}+B \,b^{3} \sqrt {-4 a c +b^{2}}+40 a^{2} B \,c^{2}+18 a \,b^{2} B c -b^{4} B \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (12 A a \,c^{2} \sqrt {-4 a c +b^{2}}+3 A \,b^{2} c \sqrt {-4 a c +b^{2}}+36 a A b \,c^{2}+3 A \,b^{3} c -16 B a b c \sqrt {-4 a c +b^{2}}+B \,b^{3} \sqrt {-4 a c +b^{2}}-40 a^{2} B \,c^{2}-18 a \,b^{2} B c +b^{4} B \right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}\) | \(595\) |
| default | \(\frac {\frac {2 \left (12 A a \,c^{2}+3 A \,b^{2} c -16 B a b c +B \,b^{3}\right ) x^{\frac {7}{2}}}{128 a^{2} c^{2}-64 a \,b^{2} c +8 b^{4}}+\frac {\left (16 a A b \,c^{2}+5 A \,b^{3} c -36 a^{2} B \,c^{2}-5 a \,b^{2} B c -b^{4} B \right ) x^{\frac {5}{2}}}{4 c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {a \left (4 A a \,c^{2}-19 A \,b^{2} c +28 B a b c +2 B \,b^{3}\right ) x^{\frac {3}{2}}}{4 c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {a^{2} \left (12 A b c -20 B a c -B \,b^{2}\right ) \sqrt {x}}{4 c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {-\frac {\left (12 A a \,c^{2} \sqrt {-4 a c +b^{2}}+3 A \,b^{2} c \sqrt {-4 a c +b^{2}}-36 a A b \,c^{2}-3 A \,b^{3} c -16 B a b c \sqrt {-4 a c +b^{2}}+B \,b^{3} \sqrt {-4 a c +b^{2}}+40 a^{2} B \,c^{2}+18 a \,b^{2} B c -b^{4} B \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (12 A a \,c^{2} \sqrt {-4 a c +b^{2}}+3 A \,b^{2} c \sqrt {-4 a c +b^{2}}+36 a A b \,c^{2}+3 A \,b^{3} c -16 B a b c \sqrt {-4 a c +b^{2}}+B \,b^{3} \sqrt {-4 a c +b^{2}}-40 a^{2} B \,c^{2}-18 a \,b^{2} B c +b^{4} B \right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}\) | \(595\) |
2*(1/8*(12*A*a*c^2+3*A*b^2*c-16*B*a*b*c+B*b^3)/(16*a^2*c^2-8*a*b^2*c+b^4)* x^(7/2)+1/8*(16*A*a*b*c^2+5*A*b^3*c-36*B*a^2*c^2-5*B*a*b^2*c-B*b^4)/c/(16* a^2*c^2-8*a*b^2*c+b^4)*x^(5/2)-1/8/c*a*(4*A*a*c^2-19*A*b^2*c+28*B*a*b*c+2* B*b^3)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(3/2)+1/8*a^2*(12*A*b*c-20*B*a*c-B*b^2 )/c/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(1/2))/(c*x^2+b*x+a)^2+1/(16*a^2*c^2-8*a* b^2*c+b^4)*(-1/8*(12*A*a*c^2*(-4*a*c+b^2)^(1/2)+3*A*b^2*c*(-4*a*c+b^2)^(1/ 2)-36*a*A*b*c^2-3*A*b^3*c-16*B*a*b*c*(-4*a*c+b^2)^(1/2)+B*b^3*(-4*a*c+b^2) ^(1/2)+40*a^2*B*c^2+18*a*b^2*B*c-b^4*B)/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*(12*A*a*c^2*(-4*a*c+b^2)^(1/2)+3*A*b^2*c*(-4*a*c+b^2)^ (1/2)+36*a*A*b*c^2+3*A*b^3*c-16*B*a*b*c*(-4*a*c+b^2)^(1/2)+B*b^3*(-4*a*c+b ^2)^(1/2)-40*a^2*B*c^2-18*a*b^2*B*c+b^4*B)/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. 7056 vs. \(2 (407) = 814\).
Time = 13.64 (sec) , antiderivative size = 7056, normalized size of antiderivative = 15.37 \[ \int \frac {x^{5/2} (A+B x)}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {x^{5/2} (A+B x)}{\left (a+b x+c x^2\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {x^{5/2} (A+B x)}{\left (a+b x+c x^2\right )^3} \, dx=\int { \frac {{\left (B x + A\right )} x^{\frac {5}{2}}}{{\left (c x^{2} + b x + a\right )}^{3}} \,d x } \]
-1/4*((12*A*b*c^2 - (b^2*c + 20*a*c^2)*B)*x^(9/2) + 3*((7*b^2*c - 4*a*c^2) *A - (b^3 + 8*a*b*c)*B)*x^(7/2) + ((7*b^3 + 8*a*b*c)*A - (17*a*b^2 + 4*a^2 *c)*B)*x^(5/2) - (12*B*a^2*b - (5*a*b^2 + 4*a^2*c)*A)*x^(3/2))/(a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^4 + 2*(b ^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^3 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^ 2 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*x) + integrate(1/8*((12*A*b*c - (b^2 + 20*a*c)*B)*x^(3/2) - 3*(12*B*a*b - (5*b^2 + 4*a*c)*A)*sqrt(x))/(a* b^4 - 8*a^2*b^2*c + 16*a^3*c^2 + (b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*x^2 + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*x), x)
Leaf count of result is larger than twice the leaf count of optimal. 7586 vs. \(2 (407) = 814\).
Time = 1.67 (sec) , antiderivative size = 7586, normalized size of antiderivative = 16.53 \[ \int \frac {x^{5/2} (A+B x)}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]
1/32*(3*(2*b^4*c^3 - 32*a^2*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqr t(b^2 - 4*a*c)*c)*b^4*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c^2 + 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4 *a*c)*c)*a^2*c^3 + 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c )*c)*a*b*c^3 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b ^2*c^3 - 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*c^4 - 2*(b^2 - 4*a*c)*b^2*c^3 - 8*(b^2 - 4*a*c)*a*c^4)*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)^2*A + (2*b^5*c^2 - 40*a*b^3*c^3 + 128*a^2*b*c^4 - sqrt(2)*sqr t(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^5 + 20*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)*sqrt(b^2 - 4*a *c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c - 64*sqrt(2)*sqrt(b^2 - 4*a*c)*s qrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^2 - 32*sqrt(2)*sqrt(b^2 - 4*a*c)*sq rt(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 + 16*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^3*c^2 + 32*(b^2 - 4*a*c )*a*b*c^3)*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)^2*B - 24*(sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^7*c^3 - 12*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)* c)*a^2*b^5*c^4 - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^6*c^4 - 2*a *b^7*c^4 + 48*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^3*c^5 + 16*sqr t(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^4*c^5 + sqrt(2)*sqrt(b*c + s...
Time = 13.68 (sec) , antiderivative size = 19073, normalized size of antiderivative = 41.55 \[ \int \frac {x^{5/2} (A+B x)}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]
atan(((((1310720*B*a^7*c^8 + 768*A*a*b^11*c^3 - 786432*A*a^6*b*c^8 - 64*B* a*b^12*c^2 - 15360*A*a^2*b^9*c^4 + 122880*A*a^3*b^7*c^5 - 491520*A*a^4*b^5 *c^6 + 983040*A*a^5*b^3*c^7 + 15360*B*a^3*b^8*c^4 - 163840*B*a^4*b^6*c^5 + 737280*B*a^5*b^4*c^6 - 1572864*B*a^6*b^2*c^7)/(64*(b^12*c + 4096*a^6*c^7 - 24*a*b^10*c^2 + 240*a^2*b^8*c^3 - 1280*a^3*b^6*c^4 + 3840*a^4*b^4*c^5 - 6144*a^5*b^2*c^6)) - (x^(1/2)*(-(B^2*b^17 + 9*A^2*b^15*c^2 + 9*A^2*c^2*(-( 4*a*c - b^2)^15)^(1/2) + B^2*b^2*(-(4*a*c - b^2)^15)^(1/2) + 6*A*B*b^16*c - 5040*A^2*a^2*b^11*c^4 + 37440*A^2*a^3*b^9*c^5 - 103680*A^2*a^4*b^7*c^6 - 9216*A^2*a^5*b^5*c^7 + 552960*A^2*a^6*b^3*c^8 + 1140*B^2*a^2*b^13*c^2 - 1 0160*B^2*a^3*b^11*c^3 + 34880*B^2*a^4*b^9*c^4 + 43776*B^2*a^5*b^7*c^5 - 68 0960*B^2*a^6*b^5*c^6 + 1863680*B^2*a^7*b^3*c^7 + 983040*A*B*a^8*c^9 - 55*B ^2*a*b^15*c - 25*B^2*a*c*(-(4*a*c - b^2)^15)^(1/2) + 180*A^2*a*b^13*c^3 - 737280*A^2*a^7*b*c^9 - 1720320*B^2*a^8*b*c^8 + 240*A*B*a^2*b^12*c^3 + 2400 0*A*B*a^3*b^10*c^4 - 241920*A*B*a^4*b^8*c^5 + 992256*A*B*a^5*b^6*c^6 - 178 1760*A*B*a^6*b^4*c^7 + 737280*A*B*a^7*b^2*c^8 + 6*A*B*b*c*(-(4*a*c - b^2)^ 15)^(1/2) - 180*A*B*a*b^14*c^2)/(128*(1048576*a^10*c^13 + b^20*c^3 - 40*a* b^18*c^4 + 720*a^2*b^16*c^5 - 7680*a^3*b^14*c^6 + 53760*a^4*b^12*c^7 - 258 048*a^5*b^10*c^8 + 860160*a^6*b^8*c^9 - 1966080*a^7*b^6*c^10 + 2949120*a^8 *b^4*c^11 - 2621440*a^9*b^2*c^12)))^(1/2)*(64*b^11*c^3 - 1280*a*b^9*c^4 - 65536*a^5*b*c^8 + 10240*a^2*b^7*c^5 - 40960*a^3*b^5*c^6 + 81920*a^4*b^3...