Integrand size = 23, antiderivative size = 528 \[ \int \frac {x^{7/2} (A+B x)}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {x^{5/2} \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {\sqrt {x} \left (a \left (3 b^3 B+A b^2 c-24 a b B c+20 a A c^2\right )+\left (3 b^4 B+A b^3 c-25 a b^2 B c+8 a A b c^2+28 a^2 B c^2\right ) x\right )}{4 c^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {\left (3 b^4 B+A b^3 c-27 a b^2 B c-16 a A b c^2+84 a^2 B c^2-\frac {3 b^5 B+A b^4 c-33 a b^3 B c-18 a A b^2 c^2+132 a^2 b B c^2-40 a^2 A c^3}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{4 \sqrt {2} c^{5/2} \left (b^2-4 a c\right )^2 \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\left (3 b^4 B+A b^3 c-27 a b^2 B c-16 a A b c^2+84 a^2 B c^2+\frac {3 b^5 B+A b^4 c-33 a b^3 B c-18 a A b^2 c^2+132 a^2 b B c^2-40 a^2 A c^3}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{4 \sqrt {2} c^{5/2} \left (b^2-4 a c\right )^2 \sqrt {b+\sqrt {b^2-4 a c}}} \]
-1/2*x^(5/2)*(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*(20*A*a*c^2+A*b^2*c-24*B*a*b*c+3*B*b^3)+(8*A*a*b*c^2+A* b^3*c+28*B*a^2*c^2-25*B*a*b^2*c+3*B*b^4)*x)*x^(1/2)/c^2/(-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) )*(3*b^4*B+A*b^3*c-27*a*b^2*B*c-16*a*A*b*c^2+84*a^2*B*c^2+(40*A*a^2*c^3+18 *A*a*b^2*c^2-A*b^4*c-132*B*a^2*b*c^2+33*B*a*b^3*c-3*B*b^5)/(-4*a*c+b^2)^(1 /2))/c^(5/2)/(-4*a*c+b^2)^2*2^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)+1/8*arcta n(2^(1/2)*c^(1/2)*x^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2))*(3*b^4*B+A*b^3*c-2 7*a*b^2*B*c-16*a*A*b*c^2+84*a^2*B*c^2+(-40*A*a^2*c^3-18*A*a*b^2*c^2+A*b^4* c+132*B*a^2*b*c^2-33*B*a*b^3*c+3*B*b^5)/(-4*a*c+b^2)^(1/2))/c^(5/2)/(-4*a* c+b^2)^2*2^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)
Time = 7.51 (sec) , antiderivative size = 612, normalized size of antiderivative = 1.16 \[ \int \frac {x^{7/2} (A+B x)}{\left (a+b x+c x^2\right )^3} \, dx=\frac {-\frac {2 \sqrt {c} \sqrt {x} \left (4 a^3 c (-6 b B+5 A c+7 B c x)+a b x \left (6 b^3 B+16 A c^3 x^2+b c^2 x (5 A-37 B x)+2 b^2 c (A-10 B x)\right )+b^3 x^2 \left (3 b^2 B-A c^2 x+b c (A+5 B x)\right )+a^2 \left (3 b^3 B+b^2 c (A-49 B x)-4 b c^2 x (-7 A+B x)+4 c^3 x^2 (9 A+11 B x)\right )\right )}{\left (b^2-4 a c\right )^2 (a+x (b+c x))^2}+\frac {\sqrt {2} \left (-3 b^5 B+b^3 c \left (33 a B+A \sqrt {b^2-4 a c}\right )-4 a b c^2 \left (33 a B+4 A \sqrt {b^2-4 a c}\right )+9 a b^2 c \left (2 A c-3 B \sqrt {b^2-4 a c}\right )+b^4 \left (-A c+3 B \sqrt {b^2-4 a c}\right )+4 a^2 c^2 \left (10 A c+21 B \sqrt {b^2-4 a c}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{5/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \left (3 b^5 B+4 a b c^2 \left (33 a B-4 A \sqrt {b^2-4 a c}\right )+b^4 \left (A c+3 B \sqrt {b^2-4 a c}\right )-9 a b^2 c \left (2 A c+3 B \sqrt {b^2-4 a c}\right )+4 a^2 c^2 \left (-10 A c+21 B \sqrt {b^2-4 a c}\right )+b^3 \left (-33 a B c+A c \sqrt {b^2-4 a c}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{5/2} \sqrt {b+\sqrt {b^2-4 a c}}}}{8 c^{5/2}} \]
((-2*Sqrt[c]*Sqrt[x]*(4*a^3*c*(-6*b*B + 5*A*c + 7*B*c*x) + a*b*x*(6*b^3*B + 16*A*c^3*x^2 + b*c^2*x*(5*A - 37*B*x) + 2*b^2*c*(A - 10*B*x)) + b^3*x^2* (3*b^2*B - A*c^2*x + b*c*(A + 5*B*x)) + a^2*(3*b^3*B + b^2*c*(A - 49*B*x) - 4*b*c^2*x*(-7*A + B*x) + 4*c^3*x^2*(9*A + 11*B*x))))/((b^2 - 4*a*c)^2*(a + x*(b + c*x))^2) + (Sqrt[2]*(-3*b^5*B + b^3*c*(33*a*B + A*Sqrt[b^2 - 4*a *c]) - 4*a*b*c^2*(33*a*B + 4*A*Sqrt[b^2 - 4*a*c]) + 9*a*b^2*c*(2*A*c - 3*B *Sqrt[b^2 - 4*a*c]) + b^4*(-(A*c) + 3*B*Sqrt[b^2 - 4*a*c]) + 4*a^2*c^2*(10 *A*c + 21*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]*(3*b^5*B + 4*a*b*c^2*(33*a*B - 4*A*Sqrt[b^2 - 4*a*c]) + b^4*(A*c + 3*B*Sqrt[b^2 - 4*a*c]) - 9*a*b^2*c*(2*A*c + 3*B*Sqrt[b^2 - 4*a*c]) + 4*a^ 2*c^2*(-10*A*c + 21*B*Sqrt[b^2 - 4*a*c]) + b^3*(-33*a*B*c + A*c*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]]))/(8*c^(5/2))
Time = 0.95 (sec) , antiderivative size = 533, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {1233, 27, 1233, 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.
| \(\displaystyle \int \frac {x^{7/2} (A+B x)}{\left (a+b x+c x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1233 |
| \(\displaystyle \frac {\int \frac {x^{3/2} \left (5 a (b B-2 A c)+\left (3 B b^2+A c b-14 a B c\right ) x\right )}{2 \left (c x^2+b x+a\right )^2}dx}{2 c \left (b^2-4 a c\right )}-\frac {x^{5/2} \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {x^{3/2} \left (5 a (b B-2 A c)+\left (3 B b^2+A c b-14 a B c\right ) x\right )}{\left (c x^2+b x+a\right )^2}dx}{4 c \left (b^2-4 a c\right )}-\frac {x^{5/2} \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1233 |
| \(\displaystyle \frac {\frac {\int \frac {a \left (3 B b^3+A c b^2-24 a B c b+20 a A c^2\right )+\left (3 B b^4+A c b^3-27 a B c b^2-16 a A c^2 b+84 a^2 B c^2\right ) x}{2 \sqrt {x} \left (c x^2+b x+a\right )}dx}{c \left (b^2-4 a c\right )}-\frac {\sqrt {x} \left (x \left (28 a^2 B c^2+8 a A b c^2-25 a b^2 B c+A b^3 c+3 b^4 B\right )+a \left (20 a A c^2-24 a b B c+A b^2 c+3 b^3 B\right )\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{4 c \left (b^2-4 a c\right )}-\frac {x^{5/2} \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {a \left (3 B b^3+A c b^2-24 a B c b+20 a A c^2\right )+\left (3 B b^4+A c b^3-27 a B c b^2-16 a A c^2 b+84 a^2 B c^2\right ) x}{\sqrt {x} \left (c x^2+b x+a\right )}dx}{2 c \left (b^2-4 a c\right )}-\frac {\sqrt {x} \left (x \left (28 a^2 B c^2+8 a A b c^2-25 a b^2 B c+A b^3 c+3 b^4 B\right )+a \left (20 a A c^2-24 a b B c+A b^2 c+3 b^3 B\right )\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{4 c \left (b^2-4 a c\right )}-\frac {x^{5/2} \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle \frac {\frac {\int \frac {a \left (3 B b^3+A c b^2-24 a B c b+20 a A c^2\right )+\left (3 B b^4+A c b^3-27 a B c b^2-16 a A c^2 b+84 a^2 B c^2\right ) x}{c x^2+b x+a}d\sqrt {x}}{c \left (b^2-4 a c\right )}-\frac {\sqrt {x} \left (x \left (28 a^2 B c^2+8 a A b c^2-25 a b^2 B c+A b^3 c+3 b^4 B\right )+a \left (20 a A c^2-24 a b B c+A b^2 c+3 b^3 B\right )\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{4 c \left (b^2-4 a c\right )}-\frac {x^{5/2} \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {\frac {\frac {1}{2} \left (-\frac {-40 a^2 A c^3+132 a^2 b B c^2-18 a A b^2 c^2-33 a b^3 B c+A b^4 c+3 b^5 B}{\sqrt {b^2-4 a c}}+84 a^2 B c^2-16 a A b c^2-27 a b^2 B c+A b^3 c+3 b^4 B\right ) \int \frac {1}{\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )+c x}d\sqrt {x}+\frac {1}{2} \left (\frac {-40 a^2 A c^3+132 a^2 b B c^2-18 a A b^2 c^2-33 a b^3 B c+A b^4 c+3 b^5 B}{\sqrt {b^2-4 a c}}+84 a^2 B c^2-16 a A b c^2-27 a b^2 B c+A b^3 c+3 b^4 B\right ) \int \frac {1}{\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )+c x}d\sqrt {x}}{c \left (b^2-4 a c\right )}-\frac {\sqrt {x} \left (x \left (28 a^2 B c^2+8 a A b c^2-25 a b^2 B c+A b^3 c+3 b^4 B\right )+a \left (20 a A c^2-24 a b B c+A b^2 c+3 b^3 B\right )\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{4 c \left (b^2-4 a c\right )}-\frac {x^{5/2} \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\frac {\left (-\frac {-40 a^2 A c^3+132 a^2 b B c^2-18 a A b^2 c^2-33 a b^3 B c+A b^4 c+3 b^5 B}{\sqrt {b^2-4 a c}}+84 a^2 B c^2-16 a A b c^2-27 a b^2 B c+A b^3 c+3 b^4 B\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\left (\frac {-40 a^2 A c^3+132 a^2 b B c^2-18 a A b^2 c^2-33 a b^3 B c+A b^4 c+3 b^5 B}{\sqrt {b^2-4 a c}}+84 a^2 B c^2-16 a A b c^2-27 a b^2 B c+A b^3 c+3 b^4 B\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} \sqrt {c} \sqrt {\sqrt {b^2-4 a c}+b}}}{c \left (b^2-4 a c\right )}-\frac {\sqrt {x} \left (x \left (28 a^2 B c^2+8 a A b c^2-25 a b^2 B c+A b^3 c+3 b^4 B\right )+a \left (20 a A c^2-24 a b B c+A b^2 c+3 b^3 B\right )\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{4 c \left (b^2-4 a c\right )}-\frac {x^{5/2} \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
-1/2*(x^(5/2)*(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*(3*b^3*B + A*b^2*c - 24*a*b*B* c + 20*a*A*c^2) + (3*b^4*B + A*b^3*c - 25*a*b^2*B*c + 8*a*A*b*c^2 + 28*a^2 *B*c^2)*x))/(c*(b^2 - 4*a*c)*(a + b*x + c*x^2))) + (((3*b^4*B + A*b^3*c - 27*a*b^2*B*c - 16*a*A*b*c^2 + 84*a^2*B*c^2 - (3*b^5*B + A*b^4*c - 33*a*b^3 *B*c - 18*a*A*b^2*c^2 + 132*a^2*b*B*c^2 - 40*a^2*A*c^3)/Sqrt[b^2 - 4*a*c]) *ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*S qrt[c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + ((3*b^4*B + A*b^3*c - 27*a*b^2*B*c - 16*a*A*b*c^2 + 84*a^2*B*c^2 + (3*b^5*B + A*b^4*c - 33*a*b^3*B*c - 18*a*A* b^2*c^2 + 132*a^2*b*B*c^2 - 40*a^2*A*c^3)/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]]))/(c*(b^2 - 4*a*c)))/(4*c*(b^2 - 4*a*c))
3.11.23.3.1 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_)^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 = 0.46 (sec) , antiderivative size = 714, normalized size of antiderivative = 1.35
| method | result | size |
| derivativedivides | \(\frac {-\frac {\left (16 a A b \,c^{2}-A \,b^{3} c +44 a^{2} B \,c^{2}-37 a \,b^{2} B c +5 b^{4} B \right ) x^{\frac {7}{2}}}{4 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) c}-\frac {\left (36 A \,a^{2} c^{3}+5 A a \,b^{2} c^{2}+A \,b^{4} c -4 B \,a^{2} b \,c^{2}-20 B a \,b^{3} c +3 B \,b^{5}\right ) x^{\frac {5}{2}}}{4 c^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {a \left (28 a A b \,c^{2}+2 A \,b^{3} c +28 a^{2} B \,c^{2}-49 a \,b^{2} B c +6 b^{4} B \right ) x^{\frac {3}{2}}}{4 c^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {a^{2} \left (20 A a \,c^{2}+A \,b^{2} c -24 B a b c +3 B \,b^{3}\right ) \sqrt {x}}{4 c^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {-\frac {\left (-16 a A b \,c^{2} \sqrt {-4 a c +b^{2}}+A \sqrt {-4 a c +b^{2}}\, b^{3} c +40 A \,a^{2} c^{3}+18 A a \,b^{2} c^{2}-A \,b^{4} c +84 a^{2} B \,c^{2} \sqrt {-4 a c +b^{2}}-27 a \,b^{2} B c \sqrt {-4 a c +b^{2}}+3 B \sqrt {-4 a c +b^{2}}\, b^{4}-132 B \,a^{2} b \,c^{2}+33 B a \,b^{3} c -3 B \,b^{5}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (-16 a A b \,c^{2} \sqrt {-4 a c +b^{2}}+A \sqrt {-4 a c +b^{2}}\, b^{3} c -40 A \,a^{2} c^{3}-18 A a \,b^{2} c^{2}+A \,b^{4} c +84 a^{2} B \,c^{2} \sqrt {-4 a c +b^{2}}-27 a \,b^{2} B c \sqrt {-4 a c +b^{2}}+3 B \sqrt {-4 a c +b^{2}}\, b^{4}+132 B \,a^{2} b \,c^{2}-33 B a \,b^{3} c +3 B \,b^{5}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}}{c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}\) | \(714\) |
| default | \(\frac {-\frac {\left (16 a A b \,c^{2}-A \,b^{3} c +44 a^{2} B \,c^{2}-37 a \,b^{2} B c +5 b^{4} B \right ) x^{\frac {7}{2}}}{4 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) c}-\frac {\left (36 A \,a^{2} c^{3}+5 A a \,b^{2} c^{2}+A \,b^{4} c -4 B \,a^{2} b \,c^{2}-20 B a \,b^{3} c +3 B \,b^{5}\right ) x^{\frac {5}{2}}}{4 c^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {a \left (28 a A b \,c^{2}+2 A \,b^{3} c +28 a^{2} B \,c^{2}-49 a \,b^{2} B c +6 b^{4} B \right ) x^{\frac {3}{2}}}{4 c^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {a^{2} \left (20 A a \,c^{2}+A \,b^{2} c -24 B a b c +3 B \,b^{3}\right ) \sqrt {x}}{4 c^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {-\frac {\left (-16 a A b \,c^{2} \sqrt {-4 a c +b^{2}}+A \sqrt {-4 a c +b^{2}}\, b^{3} c +40 A \,a^{2} c^{3}+18 A a \,b^{2} c^{2}-A \,b^{4} c +84 a^{2} B \,c^{2} \sqrt {-4 a c +b^{2}}-27 a \,b^{2} B c \sqrt {-4 a c +b^{2}}+3 B \sqrt {-4 a c +b^{2}}\, b^{4}-132 B \,a^{2} b \,c^{2}+33 B a \,b^{3} c -3 B \,b^{5}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (-16 a A b \,c^{2} \sqrt {-4 a c +b^{2}}+A \sqrt {-4 a c +b^{2}}\, b^{3} c -40 A \,a^{2} c^{3}-18 A a \,b^{2} c^{2}+A \,b^{4} c +84 a^{2} B \,c^{2} \sqrt {-4 a c +b^{2}}-27 a \,b^{2} B c \sqrt {-4 a c +b^{2}}+3 B \sqrt {-4 a c +b^{2}}\, b^{4}+132 B \,a^{2} b \,c^{2}-33 B a \,b^{3} c +3 B \,b^{5}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}}{c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}\) | \(714\) |
2*(-1/8*(16*A*a*b*c^2-A*b^3*c+44*B*a^2*c^2-37*B*a*b^2*c+5*B*b^4)/(16*a^2*c ^2-8*a*b^2*c+b^4)/c*x^(7/2)-1/8*(36*A*a^2*c^3+5*A*a*b^2*c^2+A*b^4*c-4*B*a^ 2*b*c^2-20*B*a*b^3*c+3*B*b^5)/c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(5/2)-1/8*a /c^2*(28*A*a*b*c^2+2*A*b^3*c+28*B*a^2*c^2-49*B*a*b^2*c+6*B*b^4)/(16*a^2*c^ 2-8*a*b^2*c+b^4)*x^(3/2)-1/8*a^2*(20*A*a*c^2+A*b^2*c-24*B*a*b*c+3*B*b^3)/c ^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(1/2))/(c*x^2+b*x+a)^2+1/c/(16*a^2*c^2-8*a *b^2*c+b^4)*(-1/8*(-16*a*A*b*c^2*(-4*a*c+b^2)^(1/2)+A*(-4*a*c+b^2)^(1/2)*b ^3*c+40*A*a^2*c^3+18*A*a*b^2*c^2-A*b^4*c+84*a^2*B*c^2*(-4*a*c+b^2)^(1/2)-2 7*a*b^2*B*c*(-4*a*c+b^2)^(1/2)+3*B*(-4*a*c+b^2)^(1/2)*b^4-132*B*a^2*b*c^2+ 33*B*a*b^3*c-3*B*b^5)/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*(-16*a*A*b*c^2*(-4*a*c+b^2)^(1/2)+A*(-4*a*c+b^2)^(1/2)*b^3*c-40*A*a^2*c^ 3-18*A*a*b^2*c^2+A*b^4*c+84*a^2*B*c^2*(-4*a*c+b^2)^(1/2)-27*a*b^2*B*c*(-4* a*c+b^2)^(1/2)+3*B*(-4*a*c+b^2)^(1/2)*b^4+132*B*a^2*b*c^2-33*B*a*b^3*c+3*B *b^5)/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. 9631 vs. \(2 (476) = 952\).
Time = 52.97 (sec) , antiderivative size = 9631, normalized size of antiderivative = 18.24 \[ \int \frac {x^{7/2} (A+B x)}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {x^{7/2} (A+B x)}{\left (a+b x+c x^2\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {x^{7/2} (A+B x)}{\left (a+b x+c x^2\right )^3} \, dx=\int { \frac {{\left (B x + A\right )} x^{\frac {7}{2}}}{{\left (c x^{2} + b x + a\right )}^{3}} \,d x } \]
1/4*(((b^2*c^2 + 20*a*c^3)*A + 3*(b^3*c - 8*a*b*c^2)*B)*x^(9/2) + (3*(b^3* c + 8*a*b*c^2)*A + (b^4 - 11*a*b^2*c - 44*a^2*c^2)*B)*x^(7/2) + ((17*a*b^2 *c + 4*a^2*c^2)*A + 2*(a*b^3 - 22*a^2*b*c)*B)*x^(5/2) + (12*A*a^2*b*c + (a ^2*b^2 - 28*a^3*c)*B)*x^(3/2))/(a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3 + ( b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*x^4 + 2*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^ 2*b*c^4)*x^3 + (b^6*c - 6*a*b^4*c^2 + 32*a^3*c^4)*x^2 + 2*(a*b^5*c - 8*a^2 *b^3*c^2 + 16*a^3*b*c^3)*x) - integrate(1/8*(((b^2*c + 20*a*c^2)*A + 3*(b^ 3 - 8*a*b*c)*B)*x^(3/2) + 3*(12*A*a*b*c + (a*b^2 - 28*a^2*c)*B)*sqrt(x))/( a*b^4*c - 8*a^2*b^2*c^2 + 16*a^3*c^3 + (b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4 )*x^2 + (b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x), x)
Leaf count of result is larger than twice the leaf count of optimal. 3993 vs. \(2 (476) = 952\).
Time = 1.52 (sec) , antiderivative size = 3993, normalized size of antiderivative = 7.56 \[ \int \frac {x^{7/2} (A+B x)}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]
1/16*((sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^6*c + 12*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^4*c^2 - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c) *c)*b^5*c^2 - 2*b^6*c^2 - 144*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2* b^2*c^3 - 32*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c^3 + sqrt(2)*s qrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c^3 - 24*a*b^4*c^3 - 2*b^5*c^3 + 320*sq rt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*c^4 + 160*sqrt(2)*sqrt(b*c + sqr t(b^2 - 4*a*c)*c)*a^2*b*c^4 + 16*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a *b^2*c^4 + 288*a^2*b^2*c^4 + 112*a*b^3*c^4 - 80*sqrt(2)*sqrt(b*c + sqrt(b^ 2 - 4*a*c)*c)*a^2*c^5 - 640*a^3*c^5 - 416*a^2*b*c^5 + sqrt(2)*sqrt(b^2 - 4 *a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^5*c - 56*sqrt(2)*sqrt(b^2 - 4*a*c) *sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c^2 - 2*sqrt(2)*sqrt(b^2 - 4*a*c)*s qrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c^2 + 208*sqrt(2)*sqrt(b^2 - 4*a*c)*sqr t(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^3 + 104*sqrt(2)*sqrt(b^2 - 4*a*c)*sqr t(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^3 + sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b* c + sqrt(b^2 - 4*a*c)*c)*b^3*c^3 - 52*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^4 + 2*(b^2 - 4*a*c)*b^4*c^2 + 32*(b^2 - 4*a*c) *a*b^2*c^3 + 2*(b^2 - 4*a*c)*b^3*c^3 - 160*(b^2 - 4*a*c)*a^2*c^4 - 104*(b^ 2 - 4*a*c)*a*b*c^4)*A + 3*(sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^7 - 1 6*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^5*c - 2*sqrt(2)*sqrt(b*c + s qrt(b^2 - 4*a*c)*c)*b^6*c - 2*b^7*c + 80*sqrt(2)*sqrt(b*c + sqrt(b^2 - ...
Time = 14.73 (sec) , antiderivative size = 22943, normalized size of antiderivative = 43.45 \[ \int \frac {x^{7/2} (A+B x)}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]
- ((x^(5/2)*(3*B*b^5 + 36*A*a^2*c^3 + A*b^4*c - 20*B*a*b^3*c + 5*A*a*b^2*c ^2 - 4*B*a^2*b*c^2))/(4*c^2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (x^(7/2)*(5* B*b^4 + 44*B*a^2*c^2 - A*b^3*c + 16*A*a*b*c^2 - 37*B*a*b^2*c))/(4*c*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (x^(3/2)*(28*B*a^3*c^2 + 6*B*a*b^4 + 2*A*a*b^3 *c + 28*A*a^2*b*c^2 - 49*B*a^2*b^2*c))/(4*c^2*(b^4 + 16*a^2*c^2 - 8*a*b^2* c)) + (a^2*x^(1/2)*(3*B*b^3 + 20*A*a*c^2 + A*b^2*c - 24*B*a*b*c))/(4*c^2*( 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^12*c^4 - 1310720*A*a^7*c^10 + 192*B*a* b^13*c^3 + 1572864*B*a^7*b*c^9 - 15360*A*a^3*b^8*c^6 + 163840*A*a^4*b^6*c^ 7 - 737280*A*a^5*b^4*c^8 + 1572864*A*a^6*b^2*c^9 - 5376*B*a^2*b^11*c^4 + 6 1440*B*a^3*b^9*c^5 - 368640*B*a^4*b^7*c^6 + 1228800*B*a^5*b^5*c^7 - 216268 8*B*a^6*b^3*c^8)/(64*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^ 8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)) - (x^(1/2 )*(-(9*B^2*b^19 + A^2*b^17*c^2 + 9*B^2*b^4*(-(4*a*c - b^2)^15)^(1/2) + 6*A *B*b^18*c + 1140*A^2*a^2*b^13*c^4 - 10160*A^2*a^3*b^11*c^5 + 34880*A^2*a^4 *b^9*c^6 + 43776*A^2*a^5*b^7*c^7 - 680960*A^2*a^6*b^5*c^8 + 1863680*A^2*a^ 7*b^3*c^9 + 6921*B^2*a^2*b^15*c^2 - 77580*B^2*a^3*b^13*c^3 + 570960*B^2*a^ 4*b^11*c^4 - 2851776*B^2*a^5*b^9*c^5 + 9628416*B^2*a^6*b^7*c^6 - 21095424* B^2*a^7*b^5*c^7 + 27095040*B^2*a^8*b^3*c^8 + A^2*b^2*c^2*(-(4*a*c - b^2)^1 5)^(1/2) + 441*B^2*a^2*c^2*(-(4*a*c - b^2)^15)^(1/2) + 6881280*A*B*a^9*...