Integrand size = 33, antiderivative size = 425 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx=\frac {5 b^3 (b d-a e) (3 b B d-A b e-2 a B e) x \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}-\frac {(b d-a e)^5 (B d-A e) \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x) (d+e x)^3}+\frac {(b d-a e)^4 (6 b B d-5 A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x) (d+e x)^2}-\frac {5 b (b d-a e)^3 (3 b B d-2 A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)}-\frac {b^4 (6 b B d-A b e-5 a B e) (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x)}+\frac {b^5 B (d+e x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)}-\frac {10 b^2 (b d-a e)^2 (2 b B d-A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^7 (a+b x)} \]
5*b^3*(-a*e+b*d)*(-A*b*e-2*B*a*e+3*B*b*d)*x*((b*x+a)^2)^(1/2)/e^6/(b*x+a)- 1/3*(-a*e+b*d)^5*(-A*e+B*d)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^3+1/2*(- a*e+b*d)^4*(-5*A*b*e-B*a*e+6*B*b*d)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^ 2-5*b*(-a*e+b*d)^3*(-2*A*b*e-B*a*e+3*B*b*d)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/ (e*x+d)-1/2*b^4*(-A*b*e-5*B*a*e+6*B*b*d)*(e*x+d)^2*((b*x+a)^2)^(1/2)/e^7/( b*x+a)+1/3*b^5*B*(e*x+d)^3*((b*x+a)^2)^(1/2)/e^7/(b*x+a)-10*b^2*(-a*e+b*d) ^2*(-A*b*e-B*a*e+2*B*b*d)*ln(e*x+d)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)
Time = 1.25 (sec) , antiderivative size = 504, normalized size of antiderivative = 1.19 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx=-\frac {\sqrt {(a+b x)^2} \left (a^5 e^5 (2 A e+B (d+3 e x))+5 a^4 b e^4 \left (A e (d+3 e x)+2 B \left (d^2+3 d e x+3 e^2 x^2\right )\right )+10 a^3 b^2 e^3 \left (2 A e \left (d^2+3 d e x+3 e^2 x^2\right )-B d \left (11 d^2+27 d e x+18 e^2 x^2\right )\right )-10 a^2 b^3 e^2 \left (A d e \left (11 d^2+27 d e x+18 e^2 x^2\right )-2 B \left (13 d^4+27 d^3 e x+9 d^2 e^2 x^2-9 d e^3 x^3-3 e^4 x^4\right )\right )-5 a b^4 e \left (2 A e \left (-13 d^4-27 d^3 e x-9 d^2 e^2 x^2+9 d e^3 x^3+3 e^4 x^4\right )+B \left (47 d^5+81 d^4 e x-9 d^3 e^2 x^2-63 d^2 e^3 x^3-15 d e^4 x^4+3 e^5 x^5\right )\right )+b^5 \left (A e \left (-47 d^5-81 d^4 e x+9 d^3 e^2 x^2+63 d^2 e^3 x^3+15 d e^4 x^4-3 e^5 x^5\right )+2 B \left (37 d^6+51 d^5 e x-39 d^4 e^2 x^2-73 d^3 e^3 x^3-15 d^2 e^4 x^4+3 d e^5 x^5-e^6 x^6\right )\right )+60 b^2 (b d-a e)^2 (2 b B d-A b e-a B e) (d+e x)^3 \log (d+e x)\right )}{6 e^7 (a+b x) (d+e x)^3} \]
-1/6*(Sqrt[(a + b*x)^2]*(a^5*e^5*(2*A*e + B*(d + 3*e*x)) + 5*a^4*b*e^4*(A* e*(d + 3*e*x) + 2*B*(d^2 + 3*d*e*x + 3*e^2*x^2)) + 10*a^3*b^2*e^3*(2*A*e*( d^2 + 3*d*e*x + 3*e^2*x^2) - B*d*(11*d^2 + 27*d*e*x + 18*e^2*x^2)) - 10*a^ 2*b^3*e^2*(A*d*e*(11*d^2 + 27*d*e*x + 18*e^2*x^2) - 2*B*(13*d^4 + 27*d^3*e *x + 9*d^2*e^2*x^2 - 9*d*e^3*x^3 - 3*e^4*x^4)) - 5*a*b^4*e*(2*A*e*(-13*d^4 - 27*d^3*e*x - 9*d^2*e^2*x^2 + 9*d*e^3*x^3 + 3*e^4*x^4) + B*(47*d^5 + 81* d^4*e*x - 9*d^3*e^2*x^2 - 63*d^2*e^3*x^3 - 15*d*e^4*x^4 + 3*e^5*x^5)) + b^ 5*(A*e*(-47*d^5 - 81*d^4*e*x + 9*d^3*e^2*x^2 + 63*d^2*e^3*x^3 + 15*d*e^4*x ^4 - 3*e^5*x^5) + 2*B*(37*d^6 + 51*d^5*e*x - 39*d^4*e^2*x^2 - 73*d^3*e^3*x ^3 - 15*d^2*e^4*x^4 + 3*d*e^5*x^5 - e^6*x^6)) + 60*b^2*(b*d - a*e)^2*(2*b* B*d - A*b*e - a*B*e)*(d + e*x)^3*Log[d + e*x]))/(e^7*(a + b*x)*(d + e*x)^3 )
Time = 0.58 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.62, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {1187, 27, 86, 2009}
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.
| \(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2} (A+B x)}{(d+e x)^4} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {b^5 (a+b x)^5 (A+B x)}{(d+e x)^4}dx}{b^5 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x)^5 (A+B x)}{(d+e x)^4}dx}{a+b x}\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {B (d+e x)^2 b^5}{e^6}+\frac {(-6 b B d+A b e+5 a B e) (d+e x) b^4}{e^6}-\frac {5 (b d-a e) (-3 b B d+A b e+2 a B e) b^3}{e^6}+\frac {10 (b d-a e)^2 (-2 b B d+A b e+a B e) b^2}{e^6 (d+e x)}-\frac {5 (b d-a e)^3 (-3 b B d+2 A b e+a B e) b}{e^6 (d+e x)^2}+\frac {(a e-b d)^4 (-6 b B d+5 A b e+a B e)}{e^6 (d+e x)^3}+\frac {(a e-b d)^5 (A e-B d)}{e^6 (d+e x)^4}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-\frac {b^4 (d+e x)^2 (-5 a B e-A b e+6 b B d)}{2 e^7}+\frac {5 b^3 x (b d-a e) (-2 a B e-A b e+3 b B d)}{e^6}-\frac {10 b^2 (b d-a e)^2 \log (d+e x) (-a B e-A b e+2 b B d)}{e^7}-\frac {5 b (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{e^7 (d+e x)}+\frac {(b d-a e)^4 (-a B e-5 A b e+6 b B d)}{2 e^7 (d+e x)^2}-\frac {(b d-a e)^5 (B d-A e)}{3 e^7 (d+e x)^3}+\frac {b^5 B (d+e x)^3}{3 e^7}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*((5*b^3*(b*d - a*e)*(3*b*B*d - A*b*e - 2*a* B*e)*x)/e^6 - ((b*d - a*e)^5*(B*d - A*e))/(3*e^7*(d + e*x)^3) + ((b*d - a* e)^4*(6*b*B*d - 5*A*b*e - a*B*e))/(2*e^7*(d + e*x)^2) - (5*b*(b*d - a*e)^3 *(3*b*B*d - 2*A*b*e - a*B*e))/(e^7*(d + e*x)) - (b^4*(6*b*B*d - A*b*e - 5* a*B*e)*(d + e*x)^2)/(2*e^7) + (b^5*B*(d + e*x)^3)/(3*e^7) - (10*b^2*(b*d - a*e)^2*(2*b*B*d - A*b*e - a*B*e)*Log[d + e*x])/e^7))/(a + b*x)
3.18.48.3.1 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_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x + c*x^2)^FracPart[p]/(c^ IntPart[p]*(b/2 + c*x)^(2*FracPart[p])) Int[(d + e*x)^m*(f + g*x)^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
Time = 0.65 (sec) , antiderivative size = 646, normalized size of antiderivative = 1.52
| method | result | size |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{3} \left (\frac {1}{3} b^{2} B \,x^{3} e^{2}+\frac {1}{2} A \,b^{2} e^{2} x^{2}+\frac {5}{2} B a b \,e^{2} x^{2}-2 B \,b^{2} d e \,x^{2}+5 A a b \,e^{2} x -4 A \,b^{2} d e x +10 B \,a^{2} e^{2} x -20 B a b d e x +10 B \,b^{2} d^{2} x \right )}{\left (b x +a \right ) e^{6}}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\left (-10 A \,a^{3} b^{2} e^{5}+30 A \,a^{2} b^{3} d \,e^{4}-30 A a \,b^{4} d^{2} e^{3}+10 A \,b^{5} d^{3} e^{2}-5 B \,a^{4} b \,e^{5}+30 B \,a^{3} b^{2} d \,e^{4}-60 B \,a^{2} b^{3} d^{2} e^{3}+50 B a \,b^{4} d^{3} e^{2}-15 B \,b^{5} d^{4} e \right ) x^{2}+\left (-\frac {5}{2} A \,a^{4} b \,e^{5}-10 A \,a^{3} b^{2} d \,e^{4}+45 A \,a^{2} b^{3} d^{2} e^{3}-50 A a \,b^{4} d^{3} e^{2}+\frac {35}{2} A \,b^{5} d^{4} e -\frac {1}{2} B \,a^{5} e^{5}-5 B \,a^{4} b d \,e^{4}+45 B \,a^{3} b^{2} d^{2} e^{3}-100 B \,a^{2} b^{3} d^{3} e^{2}+\frac {175}{2} B a \,b^{4} d^{4} e -27 B \,b^{5} d^{5}\right ) x -\frac {2 A \,a^{5} e^{6}+5 A \,a^{4} b d \,e^{5}+20 A \,a^{3} b^{2} d^{2} e^{4}-110 A \,a^{2} b^{3} d^{3} e^{3}+130 A a \,b^{4} d^{4} e^{2}-47 A \,b^{5} d^{5} e +B \,a^{5} d \,e^{5}+10 B \,a^{4} b \,d^{2} e^{4}-110 B \,a^{3} b^{2} d^{3} e^{3}+260 B \,a^{2} b^{3} d^{4} e^{2}-235 B a \,b^{4} d^{5} e +74 B \,b^{5} d^{6}}{6 e}\right )}{\left (b x +a \right ) e^{6} \left (e x +d \right )^{3}}+\frac {10 \sqrt {\left (b x +a \right )^{2}}\, b^{2} \left (A \,a^{2} b \,e^{3}-2 A a \,b^{2} d \,e^{2}+A \,b^{3} d^{2} e +B \,e^{3} a^{3}-4 B \,a^{2} b d \,e^{2}+5 B a \,b^{2} d^{2} e -2 B \,b^{3} d^{3}\right ) \ln \left (e x +d \right )}{\left (b x +a \right ) e^{7}}\) | \(646\) |
| default | \(\text {Expression too large to display}\) | \(1233\) |
((b*x+a)^2)^(1/2)/(b*x+a)*b^3/e^6*(1/3*b^2*B*x^3*e^2+1/2*A*b^2*e^2*x^2+5/2 *B*a*b*e^2*x^2-2*B*b^2*d*e*x^2+5*A*a*b*e^2*x-4*A*b^2*d*e*x+10*B*a^2*e^2*x- 20*B*a*b*d*e*x+10*B*b^2*d^2*x)+((b*x+a)^2)^(1/2)/(b*x+a)*((-10*A*a^3*b^2*e ^5+30*A*a^2*b^3*d*e^4-30*A*a*b^4*d^2*e^3+10*A*b^5*d^3*e^2-5*B*a^4*b*e^5+30 *B*a^3*b^2*d*e^4-60*B*a^2*b^3*d^2*e^3+50*B*a*b^4*d^3*e^2-15*B*b^5*d^4*e)*x ^2+(-5/2*A*a^4*b*e^5-10*A*a^3*b^2*d*e^4+45*A*a^2*b^3*d^2*e^3-50*A*a*b^4*d^ 3*e^2+35/2*A*b^5*d^4*e-1/2*B*a^5*e^5-5*B*a^4*b*d*e^4+45*B*a^3*b^2*d^2*e^3- 100*B*a^2*b^3*d^3*e^2+175/2*B*a*b^4*d^4*e-27*B*b^5*d^5)*x-1/6/e*(2*A*a^5*e ^6+5*A*a^4*b*d*e^5+20*A*a^3*b^2*d^2*e^4-110*A*a^2*b^3*d^3*e^3+130*A*a*b^4* d^4*e^2-47*A*b^5*d^5*e+B*a^5*d*e^5+10*B*a^4*b*d^2*e^4-110*B*a^3*b^2*d^3*e^ 3+260*B*a^2*b^3*d^4*e^2-235*B*a*b^4*d^5*e+74*B*b^5*d^6))/e^6/(e*x+d)^3+10* ((b*x+a)^2)^(1/2)/(b*x+a)*b^2/e^7*(A*a^2*b*e^3-2*A*a*b^2*d*e^2+A*b^3*d^2*e +B*a^3*e^3-4*B*a^2*b*d*e^2+5*B*a*b^2*d^2*e-2*B*b^3*d^3)*ln(e*x+d)
Leaf count of result is larger than twice the leaf count of optimal. 899 vs. \(2 (340) = 680\).
Time = 0.34 (sec) , antiderivative size = 899, normalized size of antiderivative = 2.12 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx=\frac {2 \, B b^{5} e^{6} x^{6} - 74 \, B b^{5} d^{6} - 2 \, A a^{5} e^{6} + 47 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e - 130 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} + 110 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} - 10 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{4} - {\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{5} - 3 \, {\left (2 \, B b^{5} d e^{5} - {\left (5 \, B a b^{4} + A b^{5}\right )} e^{6}\right )} x^{5} + 15 \, {\left (2 \, B b^{5} d^{2} e^{4} - {\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} + 2 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{4} + {\left (146 \, B b^{5} d^{3} e^{3} - 63 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + 90 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5}\right )} x^{3} + 3 \, {\left (26 \, B b^{5} d^{4} e^{2} - 3 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} - 30 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} + 60 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5} - 10 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{6}\right )} x^{2} - 3 \, {\left (34 \, B b^{5} d^{5} e - 27 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 90 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} - 90 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} + 10 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{5} + {\left (B a^{5} + 5 \, A a^{4} b\right )} e^{6}\right )} x - 60 \, {\left (2 \, B b^{5} d^{6} - {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e + 2 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} - {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} + {\left (2 \, B b^{5} d^{3} e^{3} - {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + 2 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5} - {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{6}\right )} x^{3} + 3 \, {\left (2 \, B b^{5} d^{4} e^{2} - {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + 2 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} - {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5}\right )} x^{2} + 3 \, {\left (2 \, B b^{5} d^{5} e - {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 2 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} - {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4}\right )} x\right )} \log \left (e x + d\right )}{6 \, {\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )}} \]
1/6*(2*B*b^5*e^6*x^6 - 74*B*b^5*d^6 - 2*A*a^5*e^6 + 47*(5*B*a*b^4 + A*b^5) *d^5*e - 130*(2*B*a^2*b^3 + A*a*b^4)*d^4*e^2 + 110*(B*a^3*b^2 + A*a^2*b^3) *d^3*e^3 - 10*(B*a^4*b + 2*A*a^3*b^2)*d^2*e^4 - (B*a^5 + 5*A*a^4*b)*d*e^5 - 3*(2*B*b^5*d*e^5 - (5*B*a*b^4 + A*b^5)*e^6)*x^5 + 15*(2*B*b^5*d^2*e^4 - (5*B*a*b^4 + A*b^5)*d*e^5 + 2*(2*B*a^2*b^3 + A*a*b^4)*e^6)*x^4 + (146*B*b^ 5*d^3*e^3 - 63*(5*B*a*b^4 + A*b^5)*d^2*e^4 + 90*(2*B*a^2*b^3 + A*a*b^4)*d* e^5)*x^3 + 3*(26*B*b^5*d^4*e^2 - 3*(5*B*a*b^4 + A*b^5)*d^3*e^3 - 30*(2*B*a ^2*b^3 + A*a*b^4)*d^2*e^4 + 60*(B*a^3*b^2 + A*a^2*b^3)*d*e^5 - 10*(B*a^4*b + 2*A*a^3*b^2)*e^6)*x^2 - 3*(34*B*b^5*d^5*e - 27*(5*B*a*b^4 + A*b^5)*d^4* e^2 + 90*(2*B*a^2*b^3 + A*a*b^4)*d^3*e^3 - 90*(B*a^3*b^2 + A*a^2*b^3)*d^2* e^4 + 10*(B*a^4*b + 2*A*a^3*b^2)*d*e^5 + (B*a^5 + 5*A*a^4*b)*e^6)*x - 60*( 2*B*b^5*d^6 - (5*B*a*b^4 + A*b^5)*d^5*e + 2*(2*B*a^2*b^3 + A*a*b^4)*d^4*e^ 2 - (B*a^3*b^2 + A*a^2*b^3)*d^3*e^3 + (2*B*b^5*d^3*e^3 - (5*B*a*b^4 + A*b^ 5)*d^2*e^4 + 2*(2*B*a^2*b^3 + A*a*b^4)*d*e^5 - (B*a^3*b^2 + A*a^2*b^3)*e^6 )*x^3 + 3*(2*B*b^5*d^4*e^2 - (5*B*a*b^4 + A*b^5)*d^3*e^3 + 2*(2*B*a^2*b^3 + A*a*b^4)*d^2*e^4 - (B*a^3*b^2 + A*a^2*b^3)*d*e^5)*x^2 + 3*(2*B*b^5*d^5*e - (5*B*a*b^4 + A*b^5)*d^4*e^2 + 2*(2*B*a^2*b^3 + A*a*b^4)*d^3*e^3 - (B*a^ 3*b^2 + A*a^2*b^3)*d^2*e^4)*x)*log(e*x + d))/(e^10*x^3 + 3*d*e^9*x^2 + 3*d ^2*e^8*x + d^3*e^7)
\[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx=\int \frac {\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{4}}\, dx \]
Exception generated. \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*e-b*d>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 911 vs. \(2 (340) = 680\).
Time = 0.30 (sec) , antiderivative size = 911, normalized size of antiderivative = 2.14 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx=\text {Too large to display} \]
-10*(2*B*b^5*d^3*sgn(b*x + a) - 5*B*a*b^4*d^2*e*sgn(b*x + a) - A*b^5*d^2*e *sgn(b*x + a) + 4*B*a^2*b^3*d*e^2*sgn(b*x + a) + 2*A*a*b^4*d*e^2*sgn(b*x + a) - B*a^3*b^2*e^3*sgn(b*x + a) - A*a^2*b^3*e^3*sgn(b*x + a))*log(abs(e*x + d))/e^7 - 1/6*(74*B*b^5*d^6*sgn(b*x + a) - 235*B*a*b^4*d^5*e*sgn(b*x + a) - 47*A*b^5*d^5*e*sgn(b*x + a) + 260*B*a^2*b^3*d^4*e^2*sgn(b*x + a) + 13 0*A*a*b^4*d^4*e^2*sgn(b*x + a) - 110*B*a^3*b^2*d^3*e^3*sgn(b*x + a) - 110* A*a^2*b^3*d^3*e^3*sgn(b*x + a) + 10*B*a^4*b*d^2*e^4*sgn(b*x + a) + 20*A*a^ 3*b^2*d^2*e^4*sgn(b*x + a) + B*a^5*d*e^5*sgn(b*x + a) + 5*A*a^4*b*d*e^5*sg n(b*x + a) + 2*A*a^5*e^6*sgn(b*x + a) + 30*(3*B*b^5*d^4*e^2*sgn(b*x + a) - 10*B*a*b^4*d^3*e^3*sgn(b*x + a) - 2*A*b^5*d^3*e^3*sgn(b*x + a) + 12*B*a^2 *b^3*d^2*e^4*sgn(b*x + a) + 6*A*a*b^4*d^2*e^4*sgn(b*x + a) - 6*B*a^3*b^2*d *e^5*sgn(b*x + a) - 6*A*a^2*b^3*d*e^5*sgn(b*x + a) + B*a^4*b*e^6*sgn(b*x + a) + 2*A*a^3*b^2*e^6*sgn(b*x + a))*x^2 + 3*(54*B*b^5*d^5*e*sgn(b*x + a) - 175*B*a*b^4*d^4*e^2*sgn(b*x + a) - 35*A*b^5*d^4*e^2*sgn(b*x + a) + 200*B* a^2*b^3*d^3*e^3*sgn(b*x + a) + 100*A*a*b^4*d^3*e^3*sgn(b*x + a) - 90*B*a^3 *b^2*d^2*e^4*sgn(b*x + a) - 90*A*a^2*b^3*d^2*e^4*sgn(b*x + a) + 10*B*a^4*b *d*e^5*sgn(b*x + a) + 20*A*a^3*b^2*d*e^5*sgn(b*x + a) + B*a^5*e^6*sgn(b*x + a) + 5*A*a^4*b*e^6*sgn(b*x + a))*x)/((e*x + d)^3*e^7) + 1/6*(2*B*b^5*e^8 *x^3*sgn(b*x + a) - 12*B*b^5*d*e^7*x^2*sgn(b*x + a) + 15*B*a*b^4*e^8*x^2*s gn(b*x + a) + 3*A*b^5*e^8*x^2*sgn(b*x + a) + 60*B*b^5*d^2*e^6*x*sgn(b*x...
Timed out. \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}}{{\left (d+e\,x\right )}^4} \,d x \]