Integrand size = 33, antiderivative size = 340 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{d+e x} \, dx=-\frac {b (b d-a e)^4 (B d-A e) x \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}+\frac {(b d-a e)^3 (B d-A e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^5}-\frac {(b d-a e)^2 (B d-A e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^4}+\frac {(b d-a e) (B d-A e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^3}-\frac {(B d-A e) (a+b x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^2}+\frac {B (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 b e}+\frac {(b d-a e)^5 (B d-A e) \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^7 (a+b x)} \]
-b*(-a*e+b*d)^4*(-A*e+B*d)*x*((b*x+a)^2)^(1/2)/e^6/(b*x+a)+1/2*(-a*e+b*d)^ 3*(-A*e+B*d)*(b*x+a)*((b*x+a)^2)^(1/2)/e^5-1/3*(-a*e+b*d)^2*(-A*e+B*d)*(b* x+a)^2*((b*x+a)^2)^(1/2)/e^4+1/4*(-a*e+b*d)*(-A*e+B*d)*(b*x+a)^3*((b*x+a)^ 2)^(1/2)/e^3-1/5*(-A*e+B*d)*(b*x+a)^4*((b*x+a)^2)^(1/2)/e^2+1/6*B*(b*x+a)^ 5*((b*x+a)^2)^(1/2)/b/e+(-a*e+b*d)^5*(-A*e+B*d)*ln(e*x+d)*((b*x+a)^2)^(1/2 )/e^7/(b*x+a)
Time = 1.15 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.14 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{d+e x} \, dx=\frac {\sqrt {(a+b x)^2} \left (e x \left (60 a^5 B e^5+150 a^4 b e^4 (-2 B d+2 A e+B e x)+100 a^3 b^2 e^3 \left (3 A e (-2 d+e x)+B \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )+50 a^2 b^3 e^2 \left (2 A e \left (6 d^2-3 d e x+2 e^2 x^2\right )+B \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )\right )+5 a b^4 e \left (5 A e \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+B \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )\right )+b^5 \left (A e \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )+B \left (-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right )\right )\right )+60 (b d-a e)^5 (B d-A e) \log (d+e x)\right )}{60 e^7 (a+b x)} \]
(Sqrt[(a + b*x)^2]*(e*x*(60*a^5*B*e^5 + 150*a^4*b*e^4*(-2*B*d + 2*A*e + B* e*x) + 100*a^3*b^2*e^3*(3*A*e*(-2*d + e*x) + B*(6*d^2 - 3*d*e*x + 2*e^2*x^ 2)) + 50*a^2*b^3*e^2*(2*A*e*(6*d^2 - 3*d*e*x + 2*e^2*x^2) + B*(-12*d^3 + 6 *d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3)) + 5*a*b^4*e*(5*A*e*(-12*d^3 + 6*d^2*e *x - 4*d*e^2*x^2 + 3*e^3*x^3) + B*(60*d^4 - 30*d^3*e*x + 20*d^2*e^2*x^2 - 15*d*e^3*x^3 + 12*e^4*x^4)) + b^5*(A*e*(60*d^4 - 30*d^3*e*x + 20*d^2*e^2*x ^2 - 15*d*e^3*x^3 + 12*e^4*x^4) + B*(-60*d^5 + 30*d^4*e*x - 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 - 12*d*e^4*x^4 + 10*e^5*x^5))) + 60*(b*d - a*e)^5*(B*d - A*e)*Log[d + e*x]))/(60*e^7*(a + b*x))
Time = 0.44 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.64, 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.
| \(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2} (A+B x)}{d+e x} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {b^5 (a+b x)^5 (A+B x)}{d+e x}dx}{b^5 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x)^5 (A+B x)}{d+e x}dx}{a+b x}\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {(A e-B d) (a e-b d)^5}{e^6 (d+e x)}+\frac {B (a+b x)^5}{e}+\frac {b (A e-B d) (a+b x)^4}{e^2}-\frac {b (b d-a e) (A e-B d) (a+b x)^3}{e^3}+\frac {b (b d-a e)^2 (A e-B d) (a+b x)^2}{e^4}+\frac {b (b d-a e)^4 (A e-B d)}{e^6}-\frac {b (b d-a e)^3 (A e-B d) (a+b x)}{e^5}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {(b d-a e)^5 (B d-A e) \log (d+e x)}{e^7}-\frac {b x (b d-a e)^4 (B d-A e)}{e^6}+\frac {(a+b x)^2 (b d-a e)^3 (B d-A e)}{2 e^5}-\frac {(a+b x)^3 (b d-a e)^2 (B d-A e)}{3 e^4}+\frac {(a+b x)^4 (b d-a e) (B d-A e)}{4 e^3}-\frac {(a+b x)^5 (B d-A e)}{5 e^2}+\frac {B (a+b x)^6}{6 b e}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-((b*(b*d - a*e)^4*(B*d - A*e)*x)/e^6) + ( (b*d - a*e)^3*(B*d - A*e)*(a + b*x)^2)/(2*e^5) - ((b*d - a*e)^2*(B*d - A*e )*(a + b*x)^3)/(3*e^4) + ((b*d - a*e)*(B*d - A*e)*(a + b*x)^4)/(4*e^3) - ( (B*d - A*e)*(a + b*x)^5)/(5*e^2) + (B*(a + b*x)^6)/(6*b*e) + ((b*d - a*e)^ 5*(B*d - A*e)*Log[d + e*x])/e^7))/(a + b*x)
3.18.45.3.1 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]
Leaf count of result is larger than twice the leaf count of optimal. \(696\) vs. \(2(253)=506\).
Time = 0.30 (sec) , antiderivative size = 697, normalized size of antiderivative = 2.05
| method | result | size |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (5 A \,a^{4} b \,e^{5} x +A \,b^{5} d^{4} e x +B a \,b^{4} e^{5} x^{5}-\frac {5}{2} B a \,b^{4} d^{3} e^{2} x^{2}+5 B \,a^{2} b^{3} d^{2} e^{3} x^{2}-5 B \,a^{3} b^{2} d \,e^{4} x^{2}+\frac {5}{2} A a \,b^{4} d^{2} e^{3} x^{2}-\frac {5}{4} B a \,b^{4} d \,e^{4} x^{4}-\frac {5}{3} A a \,b^{4} d \,e^{4} x^{3}-\frac {10}{3} B \,a^{2} b^{3} d \,e^{4} x^{3}+\frac {5}{3} B a \,b^{4} d^{2} e^{3} x^{3}-5 A \,a^{2} b^{3} d \,e^{4} x^{2}+\frac {1}{6} B \,x^{6} b^{5} e^{5}+\frac {1}{5} A \,b^{5} e^{5} x^{5}+B \,a^{5} e^{5} x -B \,b^{5} d^{5} x -10 B \,a^{2} b^{3} d^{3} e^{2} x +5 B a \,b^{4} d^{4} e x -10 A \,a^{3} b^{2} d \,e^{4} x +10 A \,a^{2} b^{3} d^{2} e^{3} x -5 A a \,b^{4} d^{3} e^{2} x -5 B \,a^{4} b d \,e^{4} x +10 B \,a^{3} b^{2} d^{2} e^{3} x -\frac {1}{3} B \,b^{5} d^{3} e^{2} x^{3}+5 A \,a^{3} b^{2} e^{5} x^{2}-\frac {1}{2} A \,b^{5} d^{3} e^{2} x^{2}+\frac {5}{2} B \,a^{4} b \,e^{5} x^{2}+\frac {1}{2} B \,b^{5} d^{4} e \,x^{2}-\frac {1}{5} B \,b^{5} d \,e^{4} x^{5}+\frac {5}{4} A a \,b^{4} e^{5} x^{4}-\frac {1}{4} A \,b^{5} d \,e^{4} x^{4}+\frac {5}{2} B \,a^{2} b^{3} e^{5} x^{4}+\frac {1}{4} B \,b^{5} d^{2} e^{3} x^{4}+\frac {10}{3} A \,a^{2} b^{3} e^{5} x^{3}+\frac {1}{3} A \,b^{5} d^{2} e^{3} x^{3}+\frac {10}{3} B \,a^{3} b^{2} e^{5} x^{3}\right )}{\left (b x +a \right ) e^{6}}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (A \,a^{5} e^{6}-5 A \,a^{4} b d \,e^{5}+10 A \,a^{3} b^{2} d^{2} e^{4}-10 A \,a^{2} b^{3} d^{3} e^{3}+5 A a \,b^{4} d^{4} e^{2}-A \,b^{5} d^{5} e -B \,a^{5} d \,e^{5}+5 B \,a^{4} b \,d^{2} e^{4}-10 B \,a^{3} b^{2} d^{3} e^{3}+10 B \,a^{2} b^{3} d^{4} e^{2}-5 B a \,b^{4} d^{5} e +B \,b^{5} d^{6}\right ) \ln \left (e x +d \right )}{\left (b x +a \right ) e^{7}}\) | \(697\) |
| default | \(\frac {\left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \left (-300 B \ln \left (e x +d \right ) a \,b^{4} d^{5} e -600 B \ln \left (e x +d \right ) a^{3} b^{2} d^{3} e^{3}+600 B \ln \left (e x +d \right ) a^{2} b^{3} d^{4} e^{2}-300 A \ln \left (e x +d \right ) a^{4} b d \,e^{5}+600 A \ln \left (e x +d \right ) a^{3} b^{2} d^{2} e^{4}-600 A \ln \left (e x +d \right ) a^{2} b^{3} d^{3} e^{3}+300 A \ln \left (e x +d \right ) a \,b^{4} d^{4} e^{2}+300 B \ln \left (e x +d \right ) a^{4} b \,d^{2} e^{4}+300 B a \,b^{4} d^{4} e^{2} x -300 B \,a^{4} b d \,e^{5} x +600 B \,a^{3} b^{2} d^{2} e^{4} x -600 B \,a^{2} b^{3} d^{3} e^{3} x -150 B a \,b^{4} d^{3} e^{3} x^{2}-600 A \,a^{3} b^{2} d \,e^{5} x +600 A \,a^{2} b^{3} d^{2} e^{4} x -300 A a \,b^{4} d^{3} e^{3} x -75 B a \,b^{4} d \,e^{5} x^{4}-100 A a \,b^{4} d \,e^{5} x^{3}-200 B \,a^{2} b^{3} d \,e^{5} x^{3}+100 B a \,b^{4} d^{2} e^{4} x^{3}-300 A \,a^{2} b^{3} d \,e^{5} x^{2}+150 A a \,b^{4} d^{2} e^{4} x^{2}-300 B \,a^{3} b^{2} d \,e^{5} x^{2}+300 B \,a^{2} b^{3} d^{2} e^{4} x^{2}+60 A \ln \left (e x +d \right ) a^{5} e^{6}+60 B \ln \left (e x +d \right ) b^{5} d^{6}+10 B \,b^{5} e^{6} x^{6}+12 A \,b^{5} e^{6} x^{5}+60 B \,a^{5} e^{6} x +60 A \,b^{5} d^{4} e^{2} x -60 B \,b^{5} d^{5} e x -60 A \ln \left (e x +d \right ) b^{5} d^{5} e -60 B \ln \left (e x +d \right ) a^{5} d \,e^{5}+60 B a \,b^{4} e^{6} x^{5}-12 B \,b^{5} d \,e^{5} x^{5}+75 A a \,b^{4} e^{6} x^{4}-15 A \,b^{5} d \,e^{5} x^{4}+150 B \,a^{2} b^{3} e^{6} x^{4}+15 B \,b^{5} d^{2} e^{4} x^{4}+200 A \,a^{2} b^{3} e^{6} x^{3}+20 A \,b^{5} d^{2} e^{4} x^{3}+200 B \,a^{3} b^{2} e^{6} x^{3}-20 B \,b^{5} d^{3} e^{3} x^{3}+300 A \,a^{3} b^{2} e^{6} x^{2}-30 A \,b^{5} d^{3} e^{3} x^{2}+150 B \,a^{4} b \,e^{6} x^{2}+30 B \,b^{5} d^{4} e^{2} x^{2}+300 A \,a^{4} b \,e^{6} x \right )}{60 \left (b x +a \right )^{5} e^{7}}\) | \(754\) |
((b*x+a)^2)^(1/2)/(b*x+a)/e^6*(5*A*a^4*b*e^5*x+A*b^5*d^4*e*x+B*a*b^4*e^5*x ^5-5/2*B*a*b^4*d^3*e^2*x^2+5*B*a^2*b^3*d^2*e^3*x^2-5*B*a^3*b^2*d*e^4*x^2+5 /2*A*a*b^4*d^2*e^3*x^2-5/4*B*a*b^4*d*e^4*x^4-5/3*A*a*b^4*d*e^4*x^3-10/3*B* a^2*b^3*d*e^4*x^3+5/3*B*a*b^4*d^2*e^3*x^3-5*A*a^2*b^3*d*e^4*x^2+1/6*B*x^6* b^5*e^5+1/5*A*b^5*e^5*x^5+B*a^5*e^5*x-B*b^5*d^5*x-10*B*a^2*b^3*d^3*e^2*x+5 *B*a*b^4*d^4*e*x-10*A*a^3*b^2*d*e^4*x+10*A*a^2*b^3*d^2*e^3*x-5*A*a*b^4*d^3 *e^2*x-5*B*a^4*b*d*e^4*x+10*B*a^3*b^2*d^2*e^3*x-1/3*B*b^5*d^3*e^2*x^3+5*A* a^3*b^2*e^5*x^2-1/2*A*b^5*d^3*e^2*x^2+5/2*B*a^4*b*e^5*x^2+1/2*B*b^5*d^4*e* x^2-1/5*B*b^5*d*e^4*x^5+5/4*A*a*b^4*e^5*x^4-1/4*A*b^5*d*e^4*x^4+5/2*B*a^2* b^3*e^5*x^4+1/4*B*b^5*d^2*e^3*x^4+10/3*A*a^2*b^3*e^5*x^3+1/3*A*b^5*d^2*e^3 *x^3+10/3*B*a^3*b^2*e^5*x^3)+((b*x+a)^2)^(1/2)/(b*x+a)*(A*a^5*e^6-5*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+5*A*a*b^4*d^4*e^2-A*b^5* d^5*e-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+10*B*a^2*b^3*d^4* e^2-5*B*a*b^4*d^5*e+B*b^5*d^6)/e^7*ln(e*x+d)
Leaf count of result is larger than twice the leaf count of optimal. 555 vs. \(2 (253) = 506\).
Time = 0.32 (sec) , antiderivative size = 555, normalized size of antiderivative = 1.63 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{d+e x} \, dx=\frac {10 \, B b^{5} e^{6} x^{6} - 12 \, {\left (B b^{5} d e^{5} - {\left (5 \, B a b^{4} + A b^{5}\right )} e^{6}\right )} x^{5} + 15 \, {\left (B b^{5} d^{2} e^{4} - {\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} + 5 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{4} - 20 \, {\left (B b^{5} d^{3} e^{3} - {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + 5 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5} - 10 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{6}\right )} x^{3} + 30 \, {\left (B b^{5} d^{4} e^{2} - {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + 5 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} - 10 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5} + 5 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{6}\right )} x^{2} - 60 \, {\left (B b^{5} d^{5} e - {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 5 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} - 10 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} + 5 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{5} - {\left (B a^{5} + 5 \, A a^{4} b\right )} e^{6}\right )} x + 60 \, {\left (B b^{5} d^{6} + A a^{5} e^{6} - {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e + 5 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} - 10 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} + 5 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{4} - {\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{5}\right )} \log \left (e x + d\right )}{60 \, e^{7}} \]
1/60*(10*B*b^5*e^6*x^6 - 12*(B*b^5*d*e^5 - (5*B*a*b^4 + A*b^5)*e^6)*x^5 + 15*(B*b^5*d^2*e^4 - (5*B*a*b^4 + A*b^5)*d*e^5 + 5*(2*B*a^2*b^3 + A*a*b^4)* e^6)*x^4 - 20*(B*b^5*d^3*e^3 - (5*B*a*b^4 + A*b^5)*d^2*e^4 + 5*(2*B*a^2*b^ 3 + A*a*b^4)*d*e^5 - 10*(B*a^3*b^2 + A*a^2*b^3)*e^6)*x^3 + 30*(B*b^5*d^4*e ^2 - (5*B*a*b^4 + A*b^5)*d^3*e^3 + 5*(2*B*a^2*b^3 + A*a*b^4)*d^2*e^4 - 10* (B*a^3*b^2 + A*a^2*b^3)*d*e^5 + 5*(B*a^4*b + 2*A*a^3*b^2)*e^6)*x^2 - 60*(B *b^5*d^5*e - (5*B*a*b^4 + A*b^5)*d^4*e^2 + 5*(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 - (B*a^5 + 5*A*a^4*b)*e^6)*x + 60*(B*b^5*d^6 + A*a^5*e^6 - (5*B*a*b^4 + A* b^5)*d^5*e + 5*(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 - (B*a^5 + 5*A*a^4*b)*d*e^5) *log(e*x + d))/e^7
\[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{d+e x} \, dx=\int \frac {\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{d + e x}\, dx \]
Exception generated. \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{d+e x} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Leaf count of result is larger than twice the leaf count of optimal. 957 vs. \(2 (253) = 506\).
Time = 0.30 (sec) , antiderivative size = 957, normalized size of antiderivative = 2.81 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{d+e x} \, dx=\text {Too large to display} \]
1/60*(10*B*b^5*e^5*x^6*sgn(b*x + a) - 12*B*b^5*d*e^4*x^5*sgn(b*x + a) + 60 *B*a*b^4*e^5*x^5*sgn(b*x + a) + 12*A*b^5*e^5*x^5*sgn(b*x + a) + 15*B*b^5*d ^2*e^3*x^4*sgn(b*x + a) - 75*B*a*b^4*d*e^4*x^4*sgn(b*x + a) - 15*A*b^5*d*e ^4*x^4*sgn(b*x + a) + 150*B*a^2*b^3*e^5*x^4*sgn(b*x + a) + 75*A*a*b^4*e^5* x^4*sgn(b*x + a) - 20*B*b^5*d^3*e^2*x^3*sgn(b*x + a) + 100*B*a*b^4*d^2*e^3 *x^3*sgn(b*x + a) + 20*A*b^5*d^2*e^3*x^3*sgn(b*x + a) - 200*B*a^2*b^3*d*e^ 4*x^3*sgn(b*x + a) - 100*A*a*b^4*d*e^4*x^3*sgn(b*x + a) + 200*B*a^3*b^2*e^ 5*x^3*sgn(b*x + a) + 200*A*a^2*b^3*e^5*x^3*sgn(b*x + a) + 30*B*b^5*d^4*e*x ^2*sgn(b*x + a) - 150*B*a*b^4*d^3*e^2*x^2*sgn(b*x + a) - 30*A*b^5*d^3*e^2* x^2*sgn(b*x + a) + 300*B*a^2*b^3*d^2*e^3*x^2*sgn(b*x + a) + 150*A*a*b^4*d^ 2*e^3*x^2*sgn(b*x + a) - 300*B*a^3*b^2*d*e^4*x^2*sgn(b*x + a) - 300*A*a^2* b^3*d*e^4*x^2*sgn(b*x + a) + 150*B*a^4*b*e^5*x^2*sgn(b*x + a) + 300*A*a^3* b^2*e^5*x^2*sgn(b*x + a) - 60*B*b^5*d^5*x*sgn(b*x + a) + 300*B*a*b^4*d^4*e *x*sgn(b*x + a) + 60*A*b^5*d^4*e*x*sgn(b*x + a) - 600*B*a^2*b^3*d^3*e^2*x* sgn(b*x + a) - 300*A*a*b^4*d^3*e^2*x*sgn(b*x + a) + 600*B*a^3*b^2*d^2*e^3* x*sgn(b*x + a) + 600*A*a^2*b^3*d^2*e^3*x*sgn(b*x + a) - 300*B*a^4*b*d*e^4* x*sgn(b*x + a) - 600*A*a^3*b^2*d*e^4*x*sgn(b*x + a) + 60*B*a^5*e^5*x*sgn(b *x + a) + 300*A*a^4*b*e^5*x*sgn(b*x + a))/e^6 + (B*b^5*d^6*sgn(b*x + a) - 5*B*a*b^4*d^5*e*sgn(b*x + a) - A*b^5*d^5*e*sgn(b*x + a) + 10*B*a^2*b^3*d^4 *e^2*sgn(b*x + a) + 5*A*a*b^4*d^4*e^2*sgn(b*x + a) - 10*B*a^3*b^2*d^3*e...
Timed out. \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{d+e x} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}}{d+e\,x} \,d x \]