Integrand size = 33, antiderivative size = 323 \[ \int \frac {a+b x}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {6 b^2 e^2}{(b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b^2}{3 (b d-a e)^3 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 b^2 e}{2 (b d-a e)^4 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e^3 (a+b x)}{2 (b d-a e)^4 (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {4 b e^3 (a+b x)}{(b d-a e)^5 (d+e x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {10 b^2 e^3 (a+b x) \log (a+b x)}{(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {10 b^2 e^3 (a+b x) \log (d+e x)}{(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}} \]
-6*b^2*e^2/(-a*e+b*d)^5/((b*x+a)^2)^(1/2)-1/3*b^2/(-a*e+b*d)^3/(b*x+a)^2/( (b*x+a)^2)^(1/2)+3/2*b^2*e/(-a*e+b*d)^4/(b*x+a)/((b*x+a)^2)^(1/2)-1/2*e^3* (b*x+a)/(-a*e+b*d)^4/(e*x+d)^2/((b*x+a)^2)^(1/2)-4*b*e^3*(b*x+a)/(-a*e+b*d )^5/(e*x+d)/((b*x+a)^2)^(1/2)-10*b^2*e^3*(b*x+a)*ln(b*x+a)/(-a*e+b*d)^6/(( b*x+a)^2)^(1/2)+10*b^2*e^3*(b*x+a)*ln(e*x+d)/(-a*e+b*d)^6/((b*x+a)^2)^(1/2 )
Time = 1.08 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.57 \[ \int \frac {a+b x}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {-2 b^2 (b d-a e)^3+9 b^2 e (b d-a e)^2 (a+b x)-36 b^2 e^2 (b d-a e) (a+b x)^2-\frac {3 e^3 (b d-a e)^2 (a+b x)^3}{(d+e x)^2}-\frac {24 b e^3 (b d-a e) (a+b x)^3}{d+e x}-60 b^2 e^3 (a+b x)^3 \log (a+b x)+60 b^2 e^3 (a+b x)^3 \log (d+e x)}{6 (b d-a e)^6 \left ((a+b x)^2\right )^{3/2}} \]
(-2*b^2*(b*d - a*e)^3 + 9*b^2*e*(b*d - a*e)^2*(a + b*x) - 36*b^2*e^2*(b*d - a*e)*(a + b*x)^2 - (3*e^3*(b*d - a*e)^2*(a + b*x)^3)/(d + e*x)^2 - (24*b *e^3*(b*d - a*e)*(a + b*x)^3)/(d + e*x) - 60*b^2*e^3*(a + b*x)^3*Log[a + b *x] + 60*b^2*e^3*(a + b*x)^3*Log[d + e*x])/(6*(b*d - a*e)^6*((a + b*x)^2)^ (3/2))
Time = 0.41 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.61, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {1187, 27, 54, 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.
| \(\displaystyle \int \frac {a+b x}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2} (d+e x)^3} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {b^5 (a+b x) \int \frac {1}{b^5 (a+b x)^4 (d+e x)^3}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(a+b x) \int \frac {1}{(a+b x)^4 (d+e x)^3}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \frac {(a+b x) \int \left (\frac {10 b^2 e^4}{(b d-a e)^6 (d+e x)}+\frac {4 b e^4}{(b d-a e)^5 (d+e x)^2}+\frac {e^4}{(b d-a e)^4 (d+e x)^3}-\frac {10 b^3 e^3}{(b d-a e)^6 (a+b x)}+\frac {6 b^3 e^2}{(b d-a e)^5 (a+b x)^2}-\frac {3 b^3 e}{(b d-a e)^4 (a+b x)^3}+\frac {b^3}{(b d-a e)^3 (a+b x)^4}\right )dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(a+b x) \left (-\frac {10 b^2 e^3 \log (a+b x)}{(b d-a e)^6}+\frac {10 b^2 e^3 \log (d+e x)}{(b d-a e)^6}-\frac {6 b^2 e^2}{(a+b x) (b d-a e)^5}+\frac {3 b^2 e}{2 (a+b x)^2 (b d-a e)^4}-\frac {b^2}{3 (a+b x)^3 (b d-a e)^3}-\frac {4 b e^3}{(d+e x) (b d-a e)^5}-\frac {e^3}{2 (d+e x)^2 (b d-a e)^4}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
((a + b*x)*(-1/3*b^2/((b*d - a*e)^3*(a + b*x)^3) + (3*b^2*e)/(2*(b*d - a*e )^4*(a + b*x)^2) - (6*b^2*e^2)/((b*d - a*e)^5*(a + b*x)) - e^3/(2*(b*d - a *e)^4*(d + e*x)^2) - (4*b*e^3)/((b*d - a*e)^5*(d + e*x)) - (10*b^2*e^3*Log [a + b*x])/(b*d - a*e)^6 + (10*b^2*e^3*Log[d + e*x])/(b*d - a*e)^6))/Sqrt[ a^2 + 2*a*b*x + b^2*x^2]
3.21.41.3.1 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_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
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]
Leaf count of result is larger than twice the leaf count of optimal. \(737\) vs. \(2(240)=480\).
Time = 0.52 (sec) , antiderivative size = 738, normalized size of antiderivative = 2.28
| method | result | size |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\frac {10 b^{4} e^{4} x^{4}}{e^{5} a^{5}-5 b d \,e^{4} a^{4}+10 b^{2} d^{2} e^{3} a^{3}-10 b^{3} d^{3} e^{2} a^{2}+5 b^{4} d^{4} e a -b^{5} d^{5}}+\frac {5 b^{3} e^{3} \left (5 a e +3 b d \right ) x^{3}}{e^{5} a^{5}-5 b d \,e^{4} a^{4}+10 b^{2} d^{2} e^{3} a^{3}-10 b^{3} d^{3} e^{2} a^{2}+5 b^{4} d^{4} e a -b^{5} d^{5}}+\frac {5 b^{2} e^{2} \left (11 e^{2} a^{2}+23 a b d e +2 b^{2} d^{2}\right ) x^{2}}{3 \left (e^{5} a^{5}-5 b d \,e^{4} a^{4}+10 b^{2} d^{2} e^{3} a^{3}-10 b^{3} d^{3} e^{2} a^{2}+5 b^{4} d^{4} e a -b^{5} d^{5}\right )}+\frac {5 \left (3 a^{3} e^{3}+35 a^{2} b d \,e^{2}+11 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) b e x}{6 \left (e^{5} a^{5}-5 b d \,e^{4} a^{4}+10 b^{2} d^{2} e^{3} a^{3}-10 b^{3} d^{3} e^{2} a^{2}+5 b^{4} d^{4} e a -b^{5} d^{5}\right )}-\frac {3 e^{4} a^{4}-27 b d \,e^{3} a^{3}-47 b^{2} d^{2} e^{2} a^{2}+13 b^{3} d^{3} e a -2 b^{4} d^{4}}{6 \left (e^{5} a^{5}-5 b d \,e^{4} a^{4}+10 b^{2} d^{2} e^{3} a^{3}-10 b^{3} d^{3} e^{2} a^{2}+5 b^{4} d^{4} e a -b^{5} d^{5}\right )}\right )}{\left (b x +a \right )^{4} \left (e x +d \right )^{2}}+\frac {10 \sqrt {\left (b x +a \right )^{2}}\, b^{2} e^{3} \ln \left (-e x -d \right )}{\left (b x +a \right ) \left (e^{6} a^{6}-6 b d \,e^{5} a^{5}+15 b^{2} d^{2} e^{4} a^{4}-20 b^{3} d^{3} e^{3} a^{3}+15 b^{4} d^{4} e^{2} a^{2}-6 b^{5} d^{5} e a +b^{6} d^{6}\right )}-\frac {10 \sqrt {\left (b x +a \right )^{2}}\, b^{2} e^{3} \ln \left (b x +a \right )}{\left (b x +a \right ) \left (e^{6} a^{6}-6 b d \,e^{5} a^{5}+15 b^{2} d^{2} e^{4} a^{4}-20 b^{3} d^{3} e^{3} a^{3}+15 b^{4} d^{4} e^{2} a^{2}-6 b^{5} d^{5} e a +b^{6} d^{6}\right )}\) | \(738\) |
| default | \(-\frac {\left (-360 \ln \left (e x +d \right ) x^{2} a^{2} b^{3} d \,e^{4}-180 \ln \left (e x +d \right ) x^{2} a \,b^{4} d^{2} e^{3}+180 \ln \left (b x +a \right ) a^{2} b^{3} e^{5} x^{3}+60 \ln \left (b x +a \right ) b^{5} d^{2} e^{3} x^{3}+60 \ln \left (b x +a \right ) a^{3} b^{2} e^{5} x^{2}+60 \ln \left (b x +a \right ) a^{3} b^{2} d^{2} e^{3}-30 b d \,e^{4} a^{4}-20 b^{2} d^{2} e^{3} a^{3}+60 b^{3} d^{3} e^{2} a^{2}-15 b^{4} d^{4} e a -60 x^{4} a \,b^{4} e^{5}+60 x^{4} b^{5} d \,e^{4}-150 x^{3} a^{2} b^{3} e^{5}+90 x^{3} b^{5} d^{2} e^{3}-110 x^{2} a^{3} b^{2} e^{5}+20 x^{2} b^{5} d^{3} e^{2}-15 x \,a^{4} b \,e^{5}-5 x \,b^{5} d^{4} e +360 \ln \left (b x +a \right ) a \,b^{4} d \,e^{4} x^{3}+360 \ln \left (b x +a \right ) a^{2} b^{3} d \,e^{4} x^{2}+180 \ln \left (b x +a \right ) a \,b^{4} d^{2} e^{3} x^{2}+120 \ln \left (b x +a \right ) a^{3} b^{2} d \,e^{4} x +180 \ln \left (b x +a \right ) a^{2} b^{3} d^{2} e^{3} x -60 \ln \left (e x +d \right ) x^{2} a^{3} b^{2} e^{5}-60 \ln \left (e x +d \right ) a^{3} b^{2} d^{2} e^{3}-160 x \,a^{3} b^{2} d \,e^{4}+120 x \,a^{2} b^{3} d^{2} e^{3}+60 x a \,b^{4} d^{3} e^{2}+60 x^{3} a \,b^{4} d \,e^{4}-120 x^{2} a^{2} b^{3} d \,e^{4}+210 x^{2} a \,b^{4} d^{2} e^{3}-180 \ln \left (e x +d \right ) x^{3} a^{2} b^{3} e^{5}-60 \ln \left (e x +d \right ) x^{3} b^{5} d^{2} e^{3}-180 \ln \left (e x +d \right ) x \,a^{2} b^{3} d^{2} e^{3}-120 \ln \left (e x +d \right ) x \,a^{3} b^{2} d \,e^{4}+2 b^{5} d^{5}-360 \ln \left (e x +d \right ) x^{3} a \,b^{4} d \,e^{4}+3 e^{5} a^{5}+180 \ln \left (b x +a \right ) a \,b^{4} e^{5} x^{4}+120 \ln \left (b x +a \right ) b^{5} d \,e^{4} x^{4}-180 \ln \left (e x +d \right ) a \,b^{4} e^{5} x^{4}-120 \ln \left (e x +d \right ) b^{5} d \,e^{4} x^{4}+60 \ln \left (b x +a \right ) b^{5} e^{5} x^{5}-60 \ln \left (e x +d \right ) b^{5} e^{5} x^{5}\right ) \left (b x +a \right )^{2}}{6 \left (e x +d \right )^{2} \left (a e -b d \right )^{6} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\) | \(753\) |
((b*x+a)^2)^(1/2)/(b*x+a)^4*(10*b^4*e^4/(a^5*e^5-5*a^4*b*d*e^4+10*a^3*b^2* d^2*e^3-10*a^2*b^3*d^3*e^2+5*a*b^4*d^4*e-b^5*d^5)*x^4+5*b^3*e^3*(5*a*e+3*b *d)/(a^5*e^5-5*a^4*b*d*e^4+10*a^3*b^2*d^2*e^3-10*a^2*b^3*d^3*e^2+5*a*b^4*d ^4*e-b^5*d^5)*x^3+5/3*b^2*e^2*(11*a^2*e^2+23*a*b*d*e+2*b^2*d^2)/(a^5*e^5-5 *a^4*b*d*e^4+10*a^3*b^2*d^2*e^3-10*a^2*b^3*d^3*e^2+5*a*b^4*d^4*e-b^5*d^5)* x^2+5/6*(3*a^3*e^3+35*a^2*b*d*e^2+11*a*b^2*d^2*e-b^3*d^3)*b*e/(a^5*e^5-5*a ^4*b*d*e^4+10*a^3*b^2*d^2*e^3-10*a^2*b^3*d^3*e^2+5*a*b^4*d^4*e-b^5*d^5)*x- 1/6*(3*a^4*e^4-27*a^3*b*d*e^3-47*a^2*b^2*d^2*e^2+13*a*b^3*d^3*e-2*b^4*d^4) /(a^5*e^5-5*a^4*b*d*e^4+10*a^3*b^2*d^2*e^3-10*a^2*b^3*d^3*e^2+5*a*b^4*d^4* e-b^5*d^5))/(e*x+d)^2+10*((b*x+a)^2)^(1/2)/(b*x+a)*b^2*e^3/(a^6*e^6-6*a^5* b*d*e^5+15*a^4*b^2*d^2*e^4-20*a^3*b^3*d^3*e^3+15*a^2*b^4*d^4*e^2-6*a*b^5*d ^5*e+b^6*d^6)*ln(-e*x-d)-10*((b*x+a)^2)^(1/2)/(b*x+a)*b^2*e^3/(a^6*e^6-6*a ^5*b*d*e^5+15*a^4*b^2*d^2*e^4-20*a^3*b^3*d^3*e^3+15*a^2*b^4*d^4*e^2-6*a*b^ 5*d^5*e+b^6*d^6)*ln(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 1151 vs. \(2 (240) = 480\).
Time = 0.31 (sec) , antiderivative size = 1151, normalized size of antiderivative = 3.56 \[ \int \frac {a+b x}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {2 \, b^{5} d^{5} - 15 \, a b^{4} d^{4} e + 60 \, a^{2} b^{3} d^{3} e^{2} - 20 \, a^{3} b^{2} d^{2} e^{3} - 30 \, a^{4} b d e^{4} + 3 \, a^{5} e^{5} + 60 \, {\left (b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 30 \, {\left (3 \, b^{5} d^{2} e^{3} + 2 \, a b^{4} d e^{4} - 5 \, a^{2} b^{3} e^{5}\right )} x^{3} + 10 \, {\left (2 \, b^{5} d^{3} e^{2} + 21 \, a b^{4} d^{2} e^{3} - 12 \, a^{2} b^{3} d e^{4} - 11 \, a^{3} b^{2} e^{5}\right )} x^{2} - 5 \, {\left (b^{5} d^{4} e - 12 \, a b^{4} d^{3} e^{2} - 24 \, a^{2} b^{3} d^{2} e^{3} + 32 \, a^{3} b^{2} d e^{4} + 3 \, a^{4} b e^{5}\right )} x + 60 \, {\left (b^{5} e^{5} x^{5} + a^{3} b^{2} d^{2} e^{3} + {\left (2 \, b^{5} d e^{4} + 3 \, a b^{4} e^{5}\right )} x^{4} + {\left (b^{5} d^{2} e^{3} + 6 \, a b^{4} d e^{4} + 3 \, a^{2} b^{3} e^{5}\right )} x^{3} + {\left (3 \, a b^{4} d^{2} e^{3} + 6 \, a^{2} b^{3} d e^{4} + a^{3} b^{2} e^{5}\right )} x^{2} + {\left (3 \, a^{2} b^{3} d^{2} e^{3} + 2 \, a^{3} b^{2} d e^{4}\right )} x\right )} \log \left (b x + a\right ) - 60 \, {\left (b^{5} e^{5} x^{5} + a^{3} b^{2} d^{2} e^{3} + {\left (2 \, b^{5} d e^{4} + 3 \, a b^{4} e^{5}\right )} x^{4} + {\left (b^{5} d^{2} e^{3} + 6 \, a b^{4} d e^{4} + 3 \, a^{2} b^{3} e^{5}\right )} x^{3} + {\left (3 \, a b^{4} d^{2} e^{3} + 6 \, a^{2} b^{3} d e^{4} + a^{3} b^{2} e^{5}\right )} x^{2} + {\left (3 \, a^{2} b^{3} d^{2} e^{3} + 2 \, a^{3} b^{2} d e^{4}\right )} x\right )} \log \left (e x + d\right )}{6 \, {\left (a^{3} b^{6} d^{8} - 6 \, a^{4} b^{5} d^{7} e + 15 \, a^{5} b^{4} d^{6} e^{2} - 20 \, a^{6} b^{3} d^{5} e^{3} + 15 \, a^{7} b^{2} d^{4} e^{4} - 6 \, a^{8} b d^{3} e^{5} + a^{9} d^{2} e^{6} + {\left (b^{9} d^{6} e^{2} - 6 \, a b^{8} d^{5} e^{3} + 15 \, a^{2} b^{7} d^{4} e^{4} - 20 \, a^{3} b^{6} d^{3} e^{5} + 15 \, a^{4} b^{5} d^{2} e^{6} - 6 \, a^{5} b^{4} d e^{7} + a^{6} b^{3} e^{8}\right )} x^{5} + {\left (2 \, b^{9} d^{7} e - 9 \, a b^{8} d^{6} e^{2} + 12 \, a^{2} b^{7} d^{5} e^{3} + 5 \, a^{3} b^{6} d^{4} e^{4} - 30 \, a^{4} b^{5} d^{3} e^{5} + 33 \, a^{5} b^{4} d^{2} e^{6} - 16 \, a^{6} b^{3} d e^{7} + 3 \, a^{7} b^{2} e^{8}\right )} x^{4} + {\left (b^{9} d^{8} - 18 \, a^{2} b^{7} d^{6} e^{2} + 52 \, a^{3} b^{6} d^{5} e^{3} - 60 \, a^{4} b^{5} d^{4} e^{4} + 24 \, a^{5} b^{4} d^{3} e^{5} + 10 \, a^{6} b^{3} d^{2} e^{6} - 12 \, a^{7} b^{2} d e^{7} + 3 \, a^{8} b e^{8}\right )} x^{3} + {\left (3 \, a b^{8} d^{8} - 12 \, a^{2} b^{7} d^{7} e + 10 \, a^{3} b^{6} d^{6} e^{2} + 24 \, a^{4} b^{5} d^{5} e^{3} - 60 \, a^{5} b^{4} d^{4} e^{4} + 52 \, a^{6} b^{3} d^{3} e^{5} - 18 \, a^{7} b^{2} d^{2} e^{6} + a^{9} e^{8}\right )} x^{2} + {\left (3 \, a^{2} b^{7} d^{8} - 16 \, a^{3} b^{6} d^{7} e + 33 \, a^{4} b^{5} d^{6} e^{2} - 30 \, a^{5} b^{4} d^{5} e^{3} + 5 \, a^{6} b^{3} d^{4} e^{4} + 12 \, a^{7} b^{2} d^{3} e^{5} - 9 \, a^{8} b d^{2} e^{6} + 2 \, a^{9} d e^{7}\right )} x\right )}} \]
-1/6*(2*b^5*d^5 - 15*a*b^4*d^4*e + 60*a^2*b^3*d^3*e^2 - 20*a^3*b^2*d^2*e^3 - 30*a^4*b*d*e^4 + 3*a^5*e^5 + 60*(b^5*d*e^4 - a*b^4*e^5)*x^4 + 30*(3*b^5 *d^2*e^3 + 2*a*b^4*d*e^4 - 5*a^2*b^3*e^5)*x^3 + 10*(2*b^5*d^3*e^2 + 21*a*b ^4*d^2*e^3 - 12*a^2*b^3*d*e^4 - 11*a^3*b^2*e^5)*x^2 - 5*(b^5*d^4*e - 12*a* b^4*d^3*e^2 - 24*a^2*b^3*d^2*e^3 + 32*a^3*b^2*d*e^4 + 3*a^4*b*e^5)*x + 60* (b^5*e^5*x^5 + a^3*b^2*d^2*e^3 + (2*b^5*d*e^4 + 3*a*b^4*e^5)*x^4 + (b^5*d^ 2*e^3 + 6*a*b^4*d*e^4 + 3*a^2*b^3*e^5)*x^3 + (3*a*b^4*d^2*e^3 + 6*a^2*b^3* d*e^4 + a^3*b^2*e^5)*x^2 + (3*a^2*b^3*d^2*e^3 + 2*a^3*b^2*d*e^4)*x)*log(b* x + a) - 60*(b^5*e^5*x^5 + a^3*b^2*d^2*e^3 + (2*b^5*d*e^4 + 3*a*b^4*e^5)*x ^4 + (b^5*d^2*e^3 + 6*a*b^4*d*e^4 + 3*a^2*b^3*e^5)*x^3 + (3*a*b^4*d^2*e^3 + 6*a^2*b^3*d*e^4 + a^3*b^2*e^5)*x^2 + (3*a^2*b^3*d^2*e^3 + 2*a^3*b^2*d*e^ 4)*x)*log(e*x + d))/(a^3*b^6*d^8 - 6*a^4*b^5*d^7*e + 15*a^5*b^4*d^6*e^2 - 20*a^6*b^3*d^5*e^3 + 15*a^7*b^2*d^4*e^4 - 6*a^8*b*d^3*e^5 + a^9*d^2*e^6 + (b^9*d^6*e^2 - 6*a*b^8*d^5*e^3 + 15*a^2*b^7*d^4*e^4 - 20*a^3*b^6*d^3*e^5 + 15*a^4*b^5*d^2*e^6 - 6*a^5*b^4*d*e^7 + a^6*b^3*e^8)*x^5 + (2*b^9*d^7*e - 9*a*b^8*d^6*e^2 + 12*a^2*b^7*d^5*e^3 + 5*a^3*b^6*d^4*e^4 - 30*a^4*b^5*d^3* e^5 + 33*a^5*b^4*d^2*e^6 - 16*a^6*b^3*d*e^7 + 3*a^7*b^2*e^8)*x^4 + (b^9*d^ 8 - 18*a^2*b^7*d^6*e^2 + 52*a^3*b^6*d^5*e^3 - 60*a^4*b^5*d^4*e^4 + 24*a^5* b^4*d^3*e^5 + 10*a^6*b^3*d^2*e^6 - 12*a^7*b^2*d*e^7 + 3*a^8*b*e^8)*x^3 + ( 3*a*b^8*d^8 - 12*a^2*b^7*d^7*e + 10*a^3*b^6*d^6*e^2 + 24*a^4*b^5*d^5*e^...
Timed out. \[ \int \frac {a+b x}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {a+b x}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*e-b*d>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 550 vs. \(2 (240) = 480\).
Time = 0.29 (sec) , antiderivative size = 550, normalized size of antiderivative = 1.70 \[ \int \frac {a+b x}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {10 \, b^{3} e^{3} \log \left ({\left | b x + a \right |}\right )}{b^{7} d^{6} \mathrm {sgn}\left (b x + a\right ) - 6 \, a b^{6} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{2} b^{5} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) - 20 \, a^{3} b^{4} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{4} b^{3} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - 6 \, a^{5} b^{2} d e^{5} \mathrm {sgn}\left (b x + a\right ) + a^{6} b e^{6} \mathrm {sgn}\left (b x + a\right )} + \frac {10 \, b^{2} e^{4} \log \left ({\left | e x + d \right |}\right )}{b^{6} d^{6} e \mathrm {sgn}\left (b x + a\right ) - 6 \, a b^{5} d^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{2} b^{4} d^{4} e^{3} \mathrm {sgn}\left (b x + a\right ) - 20 \, a^{3} b^{3} d^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{4} b^{2} d^{2} e^{5} \mathrm {sgn}\left (b x + a\right ) - 6 \, a^{5} b d e^{6} \mathrm {sgn}\left (b x + a\right ) + a^{6} e^{7} \mathrm {sgn}\left (b x + a\right )} - \frac {2 \, b^{5} d^{5} - 15 \, a b^{4} d^{4} e + 60 \, a^{2} b^{3} d^{3} e^{2} - 20 \, a^{3} b^{2} d^{2} e^{3} - 30 \, a^{4} b d e^{4} + 3 \, a^{5} e^{5} + 60 \, {\left (b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 30 \, {\left (3 \, b^{5} d^{2} e^{3} + 2 \, a b^{4} d e^{4} - 5 \, a^{2} b^{3} e^{5}\right )} x^{3} + 10 \, {\left (2 \, b^{5} d^{3} e^{2} + 21 \, a b^{4} d^{2} e^{3} - 12 \, a^{2} b^{3} d e^{4} - 11 \, a^{3} b^{2} e^{5}\right )} x^{2} - 5 \, {\left (b^{5} d^{4} e - 12 \, a b^{4} d^{3} e^{2} - 24 \, a^{2} b^{3} d^{2} e^{3} + 32 \, a^{3} b^{2} d e^{4} + 3 \, a^{4} b e^{5}\right )} x}{6 \, {\left (b d - a e\right )}^{6} {\left (b x + a\right )}^{3} {\left (e x + d\right )}^{2} \mathrm {sgn}\left (b x + a\right )} \]
-10*b^3*e^3*log(abs(b*x + a))/(b^7*d^6*sgn(b*x + a) - 6*a*b^6*d^5*e*sgn(b* x + a) + 15*a^2*b^5*d^4*e^2*sgn(b*x + a) - 20*a^3*b^4*d^3*e^3*sgn(b*x + a) + 15*a^4*b^3*d^2*e^4*sgn(b*x + a) - 6*a^5*b^2*d*e^5*sgn(b*x + a) + a^6*b* e^6*sgn(b*x + a)) + 10*b^2*e^4*log(abs(e*x + d))/(b^6*d^6*e*sgn(b*x + a) - 6*a*b^5*d^5*e^2*sgn(b*x + a) + 15*a^2*b^4*d^4*e^3*sgn(b*x + a) - 20*a^3*b ^3*d^3*e^4*sgn(b*x + a) + 15*a^4*b^2*d^2*e^5*sgn(b*x + a) - 6*a^5*b*d*e^6* sgn(b*x + a) + a^6*e^7*sgn(b*x + a)) - 1/6*(2*b^5*d^5 - 15*a*b^4*d^4*e + 6 0*a^2*b^3*d^3*e^2 - 20*a^3*b^2*d^2*e^3 - 30*a^4*b*d*e^4 + 3*a^5*e^5 + 60*( b^5*d*e^4 - a*b^4*e^5)*x^4 + 30*(3*b^5*d^2*e^3 + 2*a*b^4*d*e^4 - 5*a^2*b^3 *e^5)*x^3 + 10*(2*b^5*d^3*e^2 + 21*a*b^4*d^2*e^3 - 12*a^2*b^3*d*e^4 - 11*a ^3*b^2*e^5)*x^2 - 5*(b^5*d^4*e - 12*a*b^4*d^3*e^2 - 24*a^2*b^3*d^2*e^3 + 3 2*a^3*b^2*d*e^4 + 3*a^4*b*e^5)*x)/((b*d - a*e)^6*(b*x + a)^3*(e*x + d)^2*s gn(b*x + a))
Timed out. \[ \int \frac {a+b x}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {a+b\,x}{{\left (d+e\,x\right )}^3\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]