Integrand size = 27, antiderivative size = 436 \[ \int \frac {A+B x}{(d+e x) \left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \left (a B (2 c d-b e)-A \left (b c d-b^2 e+2 a c e\right )+c (b B d-2 A c d+A b e-2 a B e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {2 \left (4 a c e (2 c d-b e) (b B d-2 A c d+A b e-2 a B e)+\left (b c d-b^2 e+2 a c e\right ) \left (3 b^2 e (B d-A e)-4 b c d (B d+A e)+4 c \left (2 A c d^2-a B d e+3 a A e^2\right )\right )+c \left (4 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (3 b^2 e (B d-A e)-4 b c d (B d+A e)+4 c \left (2 A c d^2-a B d e+3 a A e^2\right )\right )\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \sqrt {a+b x+c x^2}}-\frac {e^3 (B d-A e) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{\left (c d^2-b d e+a e^2\right )^{5/2}} \]
2/3*(a*B*(-b*e+2*c*d)-A*(2*a*c*e-b^2*e+b*c*d)+c*(A*b*e-2*A*c*d-2*B*a*e+B*b *d)*x)/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)/(c*x^2+b*x+a)^(3/2)-e^3*(-A*e+B*d) *arctanh(1/2*(b*d-2*a*e+(-b*e+2*c*d)*x)/(a*e^2-b*d*e+c*d^2)^(1/2)/(c*x^2+b *x+a)^(1/2))/(a*e^2-b*d*e+c*d^2)^(5/2)+2/3*(4*a*c*e*(-b*e+2*c*d)*(A*b*e-2* A*c*d-2*B*a*e+B*b*d)+(2*a*c*e-b^2*e+b*c*d)*(3*b^2*e*(-A*e+B*d)-4*b*c*d*(A* e+B*d)+4*c*(3*A*a*e^2+2*A*c*d^2-B*a*d*e))+c*(4*c*e*(-2*a*e+b*d)*(A*b*e-2*A *c*d-2*B*a*e+B*b*d)+(-b*e+2*c*d)*(3*b^2*e*(-A*e+B*d)-4*b*c*d*(A*e+B*d)+4*c *(3*A*a*e^2+2*A*c*d^2-B*a*d*e)))*x)/(-4*a*c+b^2)^2/(a*e^2-b*d*e+c*d^2)^2/( c*x^2+b*x+a)^(1/2)
Time = 10.72 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.00 \[ \int \frac {A+B x}{(d+e x) \left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {4 \left (2 a c e (2 c d-b e) (b B d-2 A c d+A b e-2 a B e)+\frac {1}{2} \left (b c d-b^2 e+2 a c e\right ) \left (3 b^2 e (B d-A e)-4 b c d (B d+A e)+4 c \left (2 A c d^2-a B d e+3 a A e^2\right )\right )+c \left (2 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+\frac {1}{2} (2 c d-b e) \left (3 b^2 e (B d-A e)-4 b c d (B d+A e)+4 c \left (2 A c d^2-a B d e+3 a A e^2\right )\right )\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2+e (-b d+a e)\right )^2 \sqrt {a+x (b+c x)}}+\frac {2 \left (-A b^2 e-b B c d x+2 A c (a e+c d x)+A b c (d-e x)+a B (-2 c d+b e+2 c e x)\right )}{3 \left (b^2-4 a c\right ) \left (-c d^2+e (b d-a e)\right ) (a+x (b+c x))^{3/2}}+\frac {e^3 (B d-A e) \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{\left (c d^2+e (-b d+a e)\right )^{5/2}} \]
(4*(2*a*c*e*(2*c*d - b*e)*(b*B*d - 2*A*c*d + A*b*e - 2*a*B*e) + ((b*c*d - b^2*e + 2*a*c*e)*(3*b^2*e*(B*d - A*e) - 4*b*c*d*(B*d + A*e) + 4*c*(2*A*c*d ^2 - a*B*d*e + 3*a*A*e^2)))/2 + c*(2*c*e*(b*d - 2*a*e)*(b*B*d - 2*A*c*d + A*b*e - 2*a*B*e) + ((2*c*d - b*e)*(3*b^2*e*(B*d - A*e) - 4*b*c*d*(B*d + A* e) + 4*c*(2*A*c*d^2 - a*B*d*e + 3*a*A*e^2)))/2)*x))/(3*(b^2 - 4*a*c)^2*(c* d^2 + e*(-(b*d) + a*e))^2*Sqrt[a + x*(b + c*x)]) + (2*(-(A*b^2*e) - b*B*c* d*x + 2*A*c*(a*e + c*d*x) + A*b*c*(d - e*x) + a*B*(-2*c*d + b*e + 2*c*e*x) ))/(3*(b^2 - 4*a*c)*(-(c*d^2) + e*(b*d - a*e))*(a + x*(b + c*x))^(3/2)) + (e^3*(B*d - A*e)*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(c*d^2 + e*(-(b*d) + a*e))^(5 /2)
Time = 0.78 (sec) , antiderivative size = 475, normalized size of antiderivative = 1.09, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1235, 27, 1235, 27, 1154, 219}
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.
| \(\displaystyle \int \frac {A+B x}{(d+e x) \left (a+b x+c x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle \frac {2 \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac {2 \int \frac {3 e (B d-A e) b^2-4 c d (B d+A e) b+4 c \left (2 A c d^2-a B e d+3 a A e^2\right )-4 c e (b B d-2 A c d+A b e-2 a B e) x}{2 (d+e x) \left (c x^2+b x+a\right )^{3/2}}dx}{3 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac {\int \frac {3 e (B d-A e) b^2-4 c d (B d+A e) b+4 c \left (2 A c d^2-a B e d+3 a A e^2\right )-4 c e (b B d-2 A c d+A b e-2 a B e) x}{(d+e x) \left (c x^2+b x+a\right )^{3/2}}dx}{3 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle \frac {2 \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac {-\frac {2 \int -\frac {3 \left (b^2-4 a c\right )^2 e^3 (B d-A e)}{2 (d+e x) \sqrt {c x^2+b x+a}}dx}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (c x \left ((2 c d-b e) \left (4 c \left (3 a A e^2-a B d e+2 A c d^2\right )+3 b^2 e (B d-A e)-4 b c d (A e+B d)\right )+4 c e (b d-2 a e) (-2 a B e+A b e-2 A c d+b B d)\right )+\left (2 a c e+b^2 (-e)+b c d\right ) \left (4 c \left (3 a A e^2-a B d e+2 A c d^2\right )+3 b^2 e (B d-A e)-4 b c d (A e+B d)\right )+4 a c e (2 c d-b e) (-2 a B e+A b e-2 A c d+b B d)\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}}{3 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac {\frac {3 e^3 \left (b^2-4 a c\right ) (B d-A e) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{a e^2-b d e+c d^2}-\frac {2 \left (c x \left ((2 c d-b e) \left (4 c \left (3 a A e^2-a B d e+2 A c d^2\right )+3 b^2 e (B d-A e)-4 b c d (A e+B d)\right )+4 c e (b d-2 a e) (-2 a B e+A b e-2 A c d+b B d)\right )+\left (2 a c e+b^2 (-e)+b c d\right ) \left (4 c \left (3 a A e^2-a B d e+2 A c d^2\right )+3 b^2 e (B d-A e)-4 b c d (A e+B d)\right )+4 a c e (2 c d-b e) (-2 a B e+A b e-2 A c d+b B d)\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}}{3 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {2 \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac {-\frac {6 e^3 \left (b^2-4 a c\right ) (B d-A e) \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right )}{a e^2-b d e+c d^2}-\frac {2 \left (c x \left ((2 c d-b e) \left (4 c \left (3 a A e^2-a B d e+2 A c d^2\right )+3 b^2 e (B d-A e)-4 b c d (A e+B d)\right )+4 c e (b d-2 a e) (-2 a B e+A b e-2 A c d+b B d)\right )+\left (2 a c e+b^2 (-e)+b c d\right ) \left (4 c \left (3 a A e^2-a B d e+2 A c d^2\right )+3 b^2 e (B d-A e)-4 b c d (A e+B d)\right )+4 a c e (2 c d-b e) (-2 a B e+A b e-2 A c d+b B d)\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}}{3 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {2 \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac {\frac {3 e^3 \left (b^2-4 a c\right ) (B d-A e) \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{\left (a e^2-b d e+c d^2\right )^{3/2}}-\frac {2 \left (c x \left ((2 c d-b e) \left (4 c \left (3 a A e^2-a B d e+2 A c d^2\right )+3 b^2 e (B d-A e)-4 b c d (A e+B d)\right )+4 c e (b d-2 a e) (-2 a B e+A b e-2 A c d+b B d)\right )+\left (2 a c e+b^2 (-e)+b c d\right ) \left (4 c \left (3 a A e^2-a B d e+2 A c d^2\right )+3 b^2 e (B d-A e)-4 b c d (A e+B d)\right )+4 a c e (2 c d-b e) (-2 a B e+A b e-2 A c d+b B d)\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}}{3 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}\) |
(2*(a*B*(2*c*d - b*e) - A*(b*c*d - b^2*e + 2*a*c*e) + c*(b*B*d - 2*A*c*d + A*b*e - 2*a*B*e)*x))/(3*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)^(3/2)) - ((-2*(4*a*c*e*(2*c*d - b*e)*(b*B*d - 2*A*c*d + A*b*e - 2*a *B*e) + (b*c*d - b^2*e + 2*a*c*e)*(3*b^2*e*(B*d - A*e) - 4*b*c*d*(B*d + A* e) + 4*c*(2*A*c*d^2 - a*B*d*e + 3*a*A*e^2)) + c*(4*c*e*(b*d - 2*a*e)*(b*B* d - 2*A*c*d + A*b*e - 2*a*B*e) + (2*c*d - b*e)*(3*b^2*e*(B*d - A*e) - 4*b* c*d*(B*d + A*e) + 4*c*(2*A*c*d^2 - a*B*d*e + 3*a*A*e^2)))*x))/((b^2 - 4*a* c)*(c*d^2 - b*d*e + a*e^2)*Sqrt[a + b*x + c*x^2]) + (3*(b^2 - 4*a*c)*e^3*( B*d - A*e)*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/(c*d^2 - b*d*e + a*e^2)^(3/2))/(3*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))
3.25.85.3.1 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[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
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] )
Leaf count of result is larger than twice the leaf count of optimal. \(845\) vs. \(2(420)=840\).
Time = 0.50 (sec) , antiderivative size = 846, normalized size of antiderivative = 1.94
| method | result | size |
| default | \(\frac {B \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{e}+\frac {\left (A e -B d \right ) \left (\frac {e^{2}}{3 \left (e^{2} a -b d e +c \,d^{2}\right ) \left (\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}\right )^{\frac {3}{2}}}-\frac {\left (b e -2 c d \right ) e \left (\frac {\frac {4 c \left (x +\frac {d}{e}\right )}{3}+\frac {2 \left (b e -2 c d \right )}{3 e}}{\left (\frac {4 c \left (e^{2} a -b d e +c \,d^{2}\right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \left (\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}\right )^{\frac {3}{2}}}+\frac {16 c \left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right )}{3 {\left (\frac {4 c \left (e^{2} a -b d e +c \,d^{2}\right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right )}^{2} \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}\right )}{2 \left (e^{2} a -b d e +c \,d^{2}\right )}+\frac {e^{2} \left (\frac {e^{2}}{\left (e^{2} a -b d e +c \,d^{2}\right ) \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}-\frac {\left (b e -2 c d \right ) e \left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right )}{\left (e^{2} a -b d e +c \,d^{2}\right ) \left (\frac {4 c \left (e^{2} a -b d e +c \,d^{2}\right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}-\frac {e^{2} \ln \left (\frac {\frac {2 e^{2} a -2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (e^{2} a -b d e +c \,d^{2}\right ) \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}\right )}{e^{2} a -b d e +c \,d^{2}}\right )}{e^{2}}\) | \(846\) |
B/e*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2 *c*x+b)/(c*x^2+b*x+a)^(1/2))+(A*e-B*d)/e^2*(1/3/(a*e^2-b*d*e+c*d^2)*e^2/(( x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)-1/2*(b*e-2 *c*d)*e/(a*e^2-b*d*e+c*d^2)*(2/3*(2*c*(x+d/e)+(b*e-2*c*d)/e)/(4*c*(a*e^2-b *d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e ^2-b*d*e+c*d^2)/e^2)^(3/2)+16/3*c/(4*c*(a*e^2-b*d*e+c*d^2)/e^2-(b*e-2*c*d) ^2/e^2)^2*(2*c*(x+d/e)+(b*e-2*c*d)/e)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+( a*e^2-b*d*e+c*d^2)/e^2)^(1/2))+1/(a*e^2-b*d*e+c*d^2)*e^2*(1/(a*e^2-b*d*e+c *d^2)*e^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2 )-(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*(2*c*(x+d/e)+(b*e-2*c*d)/e)/(4*c*(a*e^ 2-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+( a*e^2-b*d*e+c*d^2)/e^2)^(1/2)-1/(a*e^2-b*d*e+c*d^2)*e^2/((a*e^2-b*d*e+c*d^ 2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^ 2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+ c*d^2)/e^2)^(1/2))/(x+d/e))))
Leaf count of result is larger than twice the leaf count of optimal. 3013 vs. \(2 (422) = 844\).
Time = 15.32 (sec) , antiderivative size = 6068, normalized size of antiderivative = 13.92 \[ \int \frac {A+B x}{(d+e x) \left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {A+B x}{(d+e x) \left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {A+B x}{(d+e x) \left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume((b/e-(2*c*d)/e^2)^2>0)', see `as sume?` for
Leaf count of result is larger than twice the leaf count of optimal. 12448 vs. \(2 (422) = 844\).
Time = 0.52 (sec) , antiderivative size = 12448, normalized size of antiderivative = 28.55 \[ \int \frac {A+B x}{(d+e x) \left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Too large to display} \]
-2*(B*d*e^3 - A*e^4)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x + a))*e + sqrt (c)*d)/sqrt(-c*d^2 + b*d*e - a*e^2))/((c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2 + 2*a*c*d^2*e^2 - 2*a*b*d*e^3 + a^2*e^4)*sqrt(-c*d^2 + b*d*e - a*e^2)) - 2/3*((((8*B*b*c^10*d^15 - 16*A*c^11*d^15 - 62*B*b^2*c^9*d^14*e + 8*B*a*c^1 0*d^14*e + 120*A*b*c^10*d^14*e + 207*B*b^3*c^8*d^13*e^2 + 12*B*a*b*c^9*d^1 3*e^2 - 386*A*b^2*c^9*d^13*e^2 - 136*A*a*c^10*d^13*e^2 - 388*B*b^4*c^7*d^1 2*e^3 - 276*B*a*b^2*c^8*d^12*e^3 + 689*A*b^3*c^8*d^12*e^3 + 32*B*a^2*c^9*d ^12*e^3 + 884*A*a*b*c^9*d^12*e^3 + 445*B*b^5*c^6*d^11*e^4 + 938*B*a*b^3*c^ 7*d^11*e^4 - 732*A*b^4*c^7*d^11*e^4 + 48*B*a^2*b*c^8*d^11*e^4 - 2412*A*a*b ^2*c^8*d^11*e^4 - 480*A*a^2*c^9*d^11*e^4 - 318*B*b^6*c^5*d^10*e^5 - 1530*B *a*b^4*c^6*d^10*e^5 + 451*A*b^5*c^6*d^10*e^5 - 810*B*a^2*b^2*c^7*d^10*e^5 + 3542*A*a*b^3*c^7*d^10*e^5 + 24*B*a^3*c^8*d^10*e^5 + 2640*A*a^2*b*c^8*d^1 0*e^5 + 137*B*b^7*c^4*d^9*e^6 + 1392*B*a*b^5*c^5*d^9*e^6 - 130*A*b^6*c^5*d ^9*e^6 + 2165*B*a^2*b^3*c^6*d^9*e^6 - 2950*A*a*b^4*c^6*d^9*e^6 + 340*B*a^3 *b*c^7*d^9*e^6 - 5910*A*a^2*b^2*c^7*d^9*e^6 - 920*A*a^3*c^8*d^9*e^6 - 32*B *b^8*c^3*d^8*e^7 - 712*B*a*b^6*c^4*d^8*e^7 - 9*A*b^7*c^4*d^8*e^7 - 2640*B* a^2*b^4*c^5*d^8*e^7 + 1296*A*a*b^5*c^5*d^8*e^7 - 1720*B*a^3*b^2*c^6*d^8*e^ 7 + 6795*A*a^2*b^3*c^6*d^8*e^7 - 80*B*a^4*c^7*d^8*e^7 + 4140*A*a^3*b*c^7*d ^8*e^7 + 3*B*b^9*c^2*d^7*e^8 + 186*B*a*b^7*c^3*d^7*e^8 + 16*A*b^8*c^3*d^7* e^8 + 1638*B*a^2*b^5*c^4*d^7*e^8 - 184*A*a*b^6*c^4*d^7*e^8 + 2940*B*a^3...
Timed out. \[ \int \frac {A+B x}{(d+e x) \left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {A+B\,x}{\left (d+e\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \]