Integrand size = 33, antiderivative size = 365 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^7} \, dx=-\frac {(B d-A e) (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 e (b d-a e) (d+e x)^6}+\frac {B (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x) (d+e x)^5}-\frac {5 b B (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x) (d+e x)^4}+\frac {10 b^2 B (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x) (d+e x)^3}-\frac {5 b^3 B (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)^2}+\frac {5 b^4 B (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)}+\frac {b^5 B \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^7 (a+b x)} \]
-1/6*(-A*e+B*d)*(b*x+a)^5*((b*x+a)^2)^(1/2)/e/(-a*e+b*d)/(e*x+d)^6+1/5*B*( -a*e+b*d)^5*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^5-5/4*b*B*(-a*e+b*d)^4*( (b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^4+10/3*b^2*B*(-a*e+b*d)^3*((b*x+a)^2) ^(1/2)/e^7/(b*x+a)/(e*x+d)^3-5*b^3*B*(-a*e+b*d)^2*((b*x+a)^2)^(1/2)/e^7/(b *x+a)/(e*x+d)^2+5*b^4*B*(-a*e+b*d)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)+b ^5*B*ln(e*x+d)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)
Time = 1.21 (sec) , antiderivative size = 477, normalized size of antiderivative = 1.31 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^7} \, dx=-\frac {\sqrt {(a+b x)^2} \left (2 a^5 e^5 (5 A e+B (d+6 e x))+5 a^4 b e^4 \left (2 A e (d+6 e x)+B \left (d^2+6 d e x+15 e^2 x^2\right )\right )+10 a^3 b^2 e^3 \left (A e \left (d^2+6 d e x+15 e^2 x^2\right )+B \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )\right )+10 a^2 b^3 e^2 \left (A e \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+2 B \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )\right )+10 a b^4 e \left (A e \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )+5 B \left (d^5+6 d^4 e x+15 d^3 e^2 x^2+20 d^2 e^3 x^3+15 d e^4 x^4+6 e^5 x^5\right )\right )+b^5 \left (10 A e \left (d^5+6 d^4 e x+15 d^3 e^2 x^2+20 d^2 e^3 x^3+15 d e^4 x^4+6 e^5 x^5\right )-B d \left (147 d^5+822 d^4 e x+1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+1350 d e^4 x^4+360 e^5 x^5\right )\right )-60 b^5 B (d+e x)^6 \log (d+e x)\right )}{60 e^7 (a+b x) (d+e x)^6} \]
-1/60*(Sqrt[(a + b*x)^2]*(2*a^5*e^5*(5*A*e + B*(d + 6*e*x)) + 5*a^4*b*e^4* (2*A*e*(d + 6*e*x) + B*(d^2 + 6*d*e*x + 15*e^2*x^2)) + 10*a^3*b^2*e^3*(A*e *(d^2 + 6*d*e*x + 15*e^2*x^2) + B*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3 *x^3)) + 10*a^2*b^3*e^2*(A*e*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3) + 2*B*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4)) + 1 0*a*b^4*e*(A*e*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x ^4) + 5*B*(d^5 + 6*d^4*e*x + 15*d^3*e^2*x^2 + 20*d^2*e^3*x^3 + 15*d*e^4*x^ 4 + 6*e^5*x^5)) + b^5*(10*A*e*(d^5 + 6*d^4*e*x + 15*d^3*e^2*x^2 + 20*d^2*e ^3*x^3 + 15*d*e^4*x^4 + 6*e^5*x^5) - B*d*(147*d^5 + 822*d^4*e*x + 1875*d^3 *e^2*x^2 + 2200*d^2*e^3*x^3 + 1350*d*e^4*x^4 + 360*e^5*x^5)) - 60*b^5*B*(d + e*x)^6*Log[d + e*x]))/(e^7*(a + b*x)*(d + e*x)^6)
Time = 0.44 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.58, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {1187, 27, 87, 49, 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.
| \(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2} (A+B x)}{(d+e x)^7} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {b^5 (a+b x)^5 (A+B x)}{(d+e x)^7}dx}{b^5 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x)^5 (A+B x)}{(d+e x)^7}dx}{a+b x}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {B \int \frac {(a+b x)^5}{(d+e x)^6}dx}{e}-\frac {(a+b x)^6 (B d-A e)}{6 e (d+e x)^6 (b d-a e)}\right )}{a+b x}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {B \int \left (\frac {b^5}{e^5 (d+e x)}-\frac {5 (b d-a e) b^4}{e^5 (d+e x)^2}+\frac {10 (b d-a e)^2 b^3}{e^5 (d+e x)^3}-\frac {10 (b d-a e)^3 b^2}{e^5 (d+e x)^4}+\frac {5 (b d-a e)^4 b}{e^5 (d+e x)^5}+\frac {(a e-b d)^5}{e^5 (d+e x)^6}\right )dx}{e}-\frac {(a+b x)^6 (B d-A e)}{6 e (d+e x)^6 (b d-a e)}\right )}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {B \left (\frac {5 b^4 (b d-a e)}{e^6 (d+e x)}-\frac {5 b^3 (b d-a e)^2}{e^6 (d+e x)^2}+\frac {10 b^2 (b d-a e)^3}{3 e^6 (d+e x)^3}-\frac {5 b (b d-a e)^4}{4 e^6 (d+e x)^4}+\frac {(b d-a e)^5}{5 e^6 (d+e x)^5}+\frac {b^5 \log (d+e x)}{e^6}\right )}{e}-\frac {(a+b x)^6 (B d-A e)}{6 e (d+e x)^6 (b d-a e)}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-1/6*((B*d - A*e)*(a + b*x)^6)/(e*(b*d - a *e)*(d + e*x)^6) + (B*((b*d - a*e)^5/(5*e^6*(d + e*x)^5) - (5*b*(b*d - a*e )^4)/(4*e^6*(d + e*x)^4) + (10*b^2*(b*d - a*e)^3)/(3*e^6*(d + e*x)^3) - (5 *b^3*(b*d - a*e)^2)/(e^6*(d + e*x)^2) + (5*b^4*(b*d - a*e))/(e^6*(d + e*x) ) + (b^5*Log[d + e*x])/e^6))/e))/(a + b*x)
3.18.51.3.1 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 [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
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. \(612\) vs. \(2(280)=560\).
Time = 1.56 (sec) , antiderivative size = 613, normalized size of antiderivative = 1.68
| method | result | size |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {b^{4} \left (A b e +5 B a e -6 B b d \right ) x^{5}}{e^{2}}-\frac {5 b^{3} \left (A a b \,e^{2}+A \,b^{2} d e +2 a^{2} B \,e^{2}+5 B a b d e -9 B \,b^{2} d^{2}\right ) x^{4}}{2 e^{3}}-\frac {10 b^{2} \left (A \,a^{2} b \,e^{3}+A a \,b^{2} d \,e^{2}+A \,b^{3} d^{2} e +B \,e^{3} a^{3}+2 B \,a^{2} b d \,e^{2}+5 B a \,b^{2} d^{2} e -11 B \,b^{3} d^{3}\right ) x^{3}}{3 e^{4}}-\frac {5 b \left (2 A \,a^{3} b \,e^{4}+2 A \,a^{2} b^{2} d \,e^{3}+2 A a \,b^{3} d^{2} e^{2}+2 A \,b^{4} d^{3} e +B \,a^{4} e^{4}+2 B \,a^{3} b d \,e^{3}+4 B \,a^{2} b^{2} d^{2} e^{2}+10 B a \,b^{3} d^{3} e -25 b^{4} B \,d^{4}\right ) x^{2}}{4 e^{5}}-\frac {\left (10 A \,a^{4} b \,e^{5}+10 A \,a^{3} b^{2} d \,e^{4}+10 A \,a^{2} b^{3} d^{2} e^{3}+10 A a \,b^{4} d^{3} e^{2}+10 A \,b^{5} d^{4} e +2 B \,a^{5} e^{5}+5 B \,a^{4} b d \,e^{4}+10 B \,a^{3} b^{2} d^{2} e^{3}+20 B \,a^{2} b^{3} d^{3} e^{2}+50 B a \,b^{4} d^{4} e -137 B \,b^{5} d^{5}\right ) x}{10 e^{6}}-\frac {10 A \,a^{5} e^{6}+10 A \,a^{4} b d \,e^{5}+10 A \,a^{3} b^{2} d^{2} e^{4}+10 A \,a^{2} b^{3} d^{3} e^{3}+10 A a \,b^{4} d^{4} e^{2}+10 A \,b^{5} d^{5} e +2 B \,a^{5} d \,e^{5}+5 B \,a^{4} b \,d^{2} e^{4}+10 B \,a^{3} b^{2} d^{3} e^{3}+20 B \,a^{2} b^{3} d^{4} e^{2}+50 B a \,b^{4} d^{5} e -147 B \,b^{5} d^{6}}{60 e^{7}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{6}}+\frac {b^{5} B \ln \left (e x +d \right ) \sqrt {\left (b x +a \right )^{2}}}{e^{7} \left (b x +a \right )}\) | \(613\) |
| default | \(-\frac {\left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \left (10 A \,a^{2} b^{3} d^{3} e^{3}+10 A a \,b^{4} d^{4} e^{2}+10 A \,a^{5} e^{6}-147 B \,b^{5} d^{6}-1200 B \ln \left (e x +d \right ) b^{5} d^{3} e^{3} x^{3}-900 B \ln \left (e x +d \right ) b^{5} d^{4} e^{2} x^{2}-360 B \ln \left (e x +d \right ) b^{5} d^{5} e x -900 B \ln \left (e x +d \right ) b^{5} d^{2} e^{4} x^{4}-360 B \ln \left (e x +d \right ) b^{5} d \,e^{5} x^{5}+300 B a \,b^{4} d^{4} e^{2} x +30 B \,a^{4} b d \,e^{5} x +60 B \,a^{3} b^{2} d^{2} e^{4} x +120 B \,a^{2} b^{3} d^{3} e^{3} x +750 B a \,b^{4} d^{3} e^{3} x^{2}+60 A \,a^{3} b^{2} d \,e^{5} x +60 A \,a^{2} b^{3} d^{2} e^{4} x +60 A a \,b^{4} d^{3} e^{3} x +750 B a \,b^{4} d \,e^{5} x^{4}+200 A a \,b^{4} d \,e^{5} x^{3}+400 B \,a^{2} b^{3} d \,e^{5} x^{3}+1000 B a \,b^{4} d^{2} e^{4} x^{3}+150 A \,a^{2} b^{3} d \,e^{5} x^{2}+150 A a \,b^{4} d^{2} e^{4} x^{2}+150 B \,a^{3} b^{2} d \,e^{5} x^{2}+300 B \,a^{2} b^{3} d^{2} e^{4} x^{2}-60 B \ln \left (e x +d \right ) b^{5} d^{6}+60 A \,b^{5} e^{6} x^{5}+12 B \,a^{5} e^{6} x +10 A \,b^{5} d^{5} e +2 B \,a^{5} d \,e^{5}-60 B \ln \left (e x +d \right ) b^{5} e^{6} x^{6}+60 A \,b^{5} d^{4} e^{2} x -822 B \,b^{5} d^{5} e x +10 A \,a^{3} b^{2} d^{2} e^{4}+300 B a \,b^{4} e^{6} x^{5}-360 B \,b^{5} d \,e^{5} x^{5}+50 B a \,b^{4} d^{5} e +5 B \,a^{4} b \,d^{2} e^{4}+10 B \,a^{3} b^{2} d^{3} e^{3}+20 B \,a^{2} b^{3} d^{4} e^{2}+10 A \,a^{4} b d \,e^{5}+150 A a \,b^{4} e^{6} x^{4}+150 A \,b^{5} d \,e^{5} x^{4}+300 B \,a^{2} b^{3} e^{6} x^{4}-1350 B \,b^{5} d^{2} e^{4} x^{4}+200 A \,a^{2} b^{3} e^{6} x^{3}+200 A \,b^{5} d^{2} e^{4} x^{3}+200 B \,a^{3} b^{2} e^{6} x^{3}-2200 B \,b^{5} d^{3} e^{3} x^{3}+150 A \,a^{3} b^{2} e^{6} x^{2}+150 A \,b^{5} d^{3} e^{3} x^{2}+75 B \,a^{4} b \,e^{6} x^{2}-1875 B \,b^{5} d^{4} e^{2} x^{2}+60 A \,a^{4} b \,e^{6} x \right )}{60 \left (b x +a \right )^{5} e^{7} \left (e x +d \right )^{6}}\) | \(809\) |
((b*x+a)^2)^(1/2)/(b*x+a)*(-b^4*(A*b*e+5*B*a*e-6*B*b*d)/e^2*x^5-5/2*b^3*(A *a*b*e^2+A*b^2*d*e+2*B*a^2*e^2+5*B*a*b*d*e-9*B*b^2*d^2)/e^3*x^4-10/3*b^2*( A*a^2*b*e^3+A*a*b^2*d*e^2+A*b^3*d^2*e+B*a^3*e^3+2*B*a^2*b*d*e^2+5*B*a*b^2* d^2*e-11*B*b^3*d^3)/e^4*x^3-5/4*b*(2*A*a^3*b*e^4+2*A*a^2*b^2*d*e^3+2*A*a*b ^3*d^2*e^2+2*A*b^4*d^3*e+B*a^4*e^4+2*B*a^3*b*d*e^3+4*B*a^2*b^2*d^2*e^2+10* B*a*b^3*d^3*e-25*B*b^4*d^4)/e^5*x^2-1/10*(10*A*a^4*b*e^5+10*A*a^3*b^2*d*e^ 4+10*A*a^2*b^3*d^2*e^3+10*A*a*b^4*d^3*e^2+10*A*b^5*d^4*e+2*B*a^5*e^5+5*B*a ^4*b*d*e^4+10*B*a^3*b^2*d^2*e^3+20*B*a^2*b^3*d^3*e^2+50*B*a*b^4*d^4*e-137* B*b^5*d^5)/e^6*x-1/60*(10*A*a^5*e^6+10*A*a^4*b*d*e^5+10*A*a^3*b^2*d^2*e^4+ 10*A*a^2*b^3*d^3*e^3+10*A*a*b^4*d^4*e^2+10*A*b^5*d^5*e+2*B*a^5*d*e^5+5*B*a ^4*b*d^2*e^4+10*B*a^3*b^2*d^3*e^3+20*B*a^2*b^3*d^4*e^2+50*B*a*b^4*d^5*e-14 7*B*b^5*d^6)/e^7)/(e*x+d)^6+b^5*B*ln(e*x+d)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 703 vs. \(2 (280) = 560\).
Time = 0.30 (sec) , antiderivative size = 703, normalized size of antiderivative = 1.93 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^7} \, dx=\frac {147 \, B b^{5} d^{6} - 10 \, A a^{5} e^{6} - 10 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e - 10 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} - 10 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} - 5 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{4} - 2 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{5} + 60 \, {\left (6 \, B b^{5} d e^{5} - {\left (5 \, B a b^{4} + A b^{5}\right )} e^{6}\right )} x^{5} + 150 \, {\left (9 \, B b^{5} d^{2} e^{4} - {\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} - {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{4} + 200 \, {\left (11 \, B b^{5} d^{3} e^{3} - {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} - {\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} + 75 \, {\left (25 \, B b^{5} d^{4} e^{2} - 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} - 2 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5} - {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{6}\right )} x^{2} + 6 \, {\left (137 \, B b^{5} d^{5} e - 10 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} - 10 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} - 10 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} - 5 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{5} - 2 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} e^{6}\right )} x + 60 \, {\left (B b^{5} e^{6} x^{6} + 6 \, B b^{5} d e^{5} x^{5} + 15 \, B b^{5} d^{2} e^{4} x^{4} + 20 \, B b^{5} d^{3} e^{3} x^{3} + 15 \, B b^{5} d^{4} e^{2} x^{2} + 6 \, B b^{5} d^{5} e x + B b^{5} d^{6}\right )} \log \left (e x + d\right )}{60 \, {\left (e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} e^{7}\right )}} \]
1/60*(147*B*b^5*d^6 - 10*A*a^5*e^6 - 10*(5*B*a*b^4 + A*b^5)*d^5*e - 10*(2* B*a^2*b^3 + A*a*b^4)*d^4*e^2 - 10*(B*a^3*b^2 + A*a^2*b^3)*d^3*e^3 - 5*(B*a ^4*b + 2*A*a^3*b^2)*d^2*e^4 - 2*(B*a^5 + 5*A*a^4*b)*d*e^5 + 60*(6*B*b^5*d* e^5 - (5*B*a*b^4 + A*b^5)*e^6)*x^5 + 150*(9*B*b^5*d^2*e^4 - (5*B*a*b^4 + A *b^5)*d*e^5 - (2*B*a^2*b^3 + A*a*b^4)*e^6)*x^4 + 200*(11*B*b^5*d^3*e^3 - ( 5*B*a*b^4 + A*b^5)*d^2*e^4 - (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 + 75*(25*B*b^5*d^4*e^2 - 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 - 2*(B*a^3*b^2 + A*a^2*b^3)*d*e^5 - ( B*a^4*b + 2*A*a^3*b^2)*e^6)*x^2 + 6*(137*B*b^5*d^5*e - 10*(5*B*a*b^4 + A*b ^5)*d^4*e^2 - 10*(2*B*a^2*b^3 + A*a*b^4)*d^3*e^3 - 10*(B*a^3*b^2 + A*a^2*b ^3)*d^2*e^4 - 5*(B*a^4*b + 2*A*a^3*b^2)*d*e^5 - 2*(B*a^5 + 5*A*a^4*b)*e^6) *x + 60*(B*b^5*e^6*x^6 + 6*B*b^5*d*e^5*x^5 + 15*B*b^5*d^2*e^4*x^4 + 20*B*b ^5*d^3*e^3*x^3 + 15*B*b^5*d^4*e^2*x^2 + 6*B*b^5*d^5*e*x + B*b^5*d^6)*log(e *x + d))/(e^13*x^6 + 6*d*e^12*x^5 + 15*d^2*e^11*x^4 + 20*d^3*e^10*x^3 + 15 *d^4*e^9*x^2 + 6*d^5*e^8*x + d^6*e^7)
Timed out. \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^7} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^7} \, dx=\text {Exception raised: ValueError} \]
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. 907 vs. \(2 (280) = 560\).
Time = 0.29 (sec) , antiderivative size = 907, normalized size of antiderivative = 2.48 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^7} \, dx=\text {Too large to display} \]
B*b^5*log(abs(e*x + d))*sgn(b*x + a)/e^7 + 1/60*(60*(6*B*b^5*d*e^4*sgn(b*x + a) - 5*B*a*b^4*e^5*sgn(b*x + a) - A*b^5*e^5*sgn(b*x + a))*x^5 + 150*(9* B*b^5*d^2*e^3*sgn(b*x + a) - 5*B*a*b^4*d*e^4*sgn(b*x + a) - A*b^5*d*e^4*sg n(b*x + a) - 2*B*a^2*b^3*e^5*sgn(b*x + a) - A*a*b^4*e^5*sgn(b*x + a))*x^4 + 200*(11*B*b^5*d^3*e^2*sgn(b*x + a) - 5*B*a*b^4*d^2*e^3*sgn(b*x + a) - A* b^5*d^2*e^3*sgn(b*x + a) - 2*B*a^2*b^3*d*e^4*sgn(b*x + a) - A*a*b^4*d*e^4* sgn(b*x + a) - B*a^3*b^2*e^5*sgn(b*x + a) - A*a^2*b^3*e^5*sgn(b*x + a))*x^ 3 + 75*(25*B*b^5*d^4*e*sgn(b*x + a) - 10*B*a*b^4*d^3*e^2*sgn(b*x + a) - 2* A*b^5*d^3*e^2*sgn(b*x + a) - 4*B*a^2*b^3*d^2*e^3*sgn(b*x + a) - 2*A*a*b^4* d^2*e^3*sgn(b*x + a) - 2*B*a^3*b^2*d*e^4*sgn(b*x + a) - 2*A*a^2*b^3*d*e^4* sgn(b*x + a) - B*a^4*b*e^5*sgn(b*x + a) - 2*A*a^3*b^2*e^5*sgn(b*x + a))*x^ 2 + 6*(137*B*b^5*d^5*sgn(b*x + a) - 50*B*a*b^4*d^4*e*sgn(b*x + a) - 10*A*b ^5*d^4*e*sgn(b*x + a) - 20*B*a^2*b^3*d^3*e^2*sgn(b*x + a) - 10*A*a*b^4*d^3 *e^2*sgn(b*x + a) - 10*B*a^3*b^2*d^2*e^3*sgn(b*x + a) - 10*A*a^2*b^3*d^2*e ^3*sgn(b*x + a) - 5*B*a^4*b*d*e^4*sgn(b*x + a) - 10*A*a^3*b^2*d*e^4*sgn(b* x + a) - 2*B*a^5*e^5*sgn(b*x + a) - 10*A*a^4*b*e^5*sgn(b*x + a))*x + (147* B*b^5*d^6*sgn(b*x + a) - 50*B*a*b^4*d^5*e*sgn(b*x + a) - 10*A*b^5*d^5*e*sg n(b*x + a) - 20*B*a^2*b^3*d^4*e^2*sgn(b*x + a) - 10*A*a*b^4*d^4*e^2*sgn(b* x + a) - 10*B*a^3*b^2*d^3*e^3*sgn(b*x + a) - 10*A*a^2*b^3*d^3*e^3*sgn(b*x + a) - 5*B*a^4*b*d^2*e^4*sgn(b*x + a) - 10*A*a^3*b^2*d^2*e^4*sgn(b*x + ...
Timed out. \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^7} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}}{{\left (d+e\,x\right )}^7} \,d x \]