Integrand size = 33, antiderivative size = 172 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11}} \, dx=\frac {\left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{10 (b d-a e) (d+e x)^{10}}+\frac {b \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{30 (b d-a e)^2 (d+e x)^9}+\frac {b^2 \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{120 (b d-a e)^3 (d+e x)^8}+\frac {b^3 \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{840 (b d-a e)^4 (d+e x)^7} \] Output:
1/10*(b^2*x^2+2*a*b*x+a^2)^(7/2)/(-a*e+b*d)/(e*x+d)^10+1/30*b*(b^2*x^2+2*a *b*x+a^2)^(7/2)/(-a*e+b*d)^2/(e*x+d)^9+1/120*b^2*(b^2*x^2+2*a*b*x+a^2)^(7/ 2)/(-a*e+b*d)^3/(e*x+d)^8+1/840*b^3*(b^2*x^2+2*a*b*x+a^2)^(7/2)/(-a*e+b*d) ^4/(e*x+d)^7
Time = 1.12 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.72 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11}} \, dx=-\frac {\sqrt {(a+b x)^2} \left (84 a^6 e^6+56 a^5 b e^5 (d+10 e x)+35 a^4 b^2 e^4 \left (d^2+10 d e x+45 e^2 x^2\right )+20 a^3 b^3 e^3 \left (d^3+10 d^2 e x+45 d e^2 x^2+120 e^3 x^3\right )+10 a^2 b^4 e^2 \left (d^4+10 d^3 e x+45 d^2 e^2 x^2+120 d e^3 x^3+210 e^4 x^4\right )+4 a b^5 e \left (d^5+10 d^4 e x+45 d^3 e^2 x^2+120 d^2 e^3 x^3+210 d e^4 x^4+252 e^5 x^5\right )+b^6 \left (d^6+10 d^5 e x+45 d^4 e^2 x^2+120 d^3 e^3 x^3+210 d^2 e^4 x^4+252 d e^5 x^5+210 e^6 x^6\right )\right )}{840 e^7 (a+b x) (d+e x)^{10}} \] Input:
Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^11,x]
Output:
-1/840*(Sqrt[(a + b*x)^2]*(84*a^6*e^6 + 56*a^5*b*e^5*(d + 10*e*x) + 35*a^4 *b^2*e^4*(d^2 + 10*d*e*x + 45*e^2*x^2) + 20*a^3*b^3*e^3*(d^3 + 10*d^2*e*x + 45*d*e^2*x^2 + 120*e^3*x^3) + 10*a^2*b^4*e^2*(d^4 + 10*d^3*e*x + 45*d^2* e^2*x^2 + 120*d*e^3*x^3 + 210*e^4*x^4) + 4*a*b^5*e*(d^5 + 10*d^4*e*x + 45* d^3*e^2*x^2 + 120*d^2*e^3*x^3 + 210*d*e^4*x^4 + 252*e^5*x^5) + b^6*(d^6 + 10*d^5*e*x + 45*d^4*e^2*x^2 + 120*d^3*e^3*x^3 + 210*d^2*e^4*x^4 + 252*d*e^ 5*x^5 + 210*e^6*x^6)))/(e^7*(a + b*x)*(d + e*x)^10)
Time = 0.44 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1187, 27, 55, 55, 55, 48}
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.
| \(\displaystyle \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11}} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {b^5 (a+b x)^6}{(d+e x)^{11}}dx}{b^5 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x)^6}{(d+e x)^{11}}dx}{a+b x}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {3 b \int \frac {(a+b x)^6}{(d+e x)^{10}}dx}{10 (b d-a e)}+\frac {(a+b x)^7}{10 (d+e x)^{10} (b d-a e)}\right )}{a+b x}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {3 b \left (\frac {2 b \int \frac {(a+b x)^6}{(d+e x)^9}dx}{9 (b d-a e)}+\frac {(a+b x)^7}{9 (d+e x)^9 (b d-a e)}\right )}{10 (b d-a e)}+\frac {(a+b x)^7}{10 (d+e x)^{10} (b d-a e)}\right )}{a+b x}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {3 b \left (\frac {2 b \left (\frac {b \int \frac {(a+b x)^6}{(d+e x)^8}dx}{8 (b d-a e)}+\frac {(a+b x)^7}{8 (d+e x)^8 (b d-a e)}\right )}{9 (b d-a e)}+\frac {(a+b x)^7}{9 (d+e x)^9 (b d-a e)}\right )}{10 (b d-a e)}+\frac {(a+b x)^7}{10 (d+e x)^{10} (b d-a e)}\right )}{a+b x}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {(a+b x)^7}{10 (d+e x)^{10} (b d-a e)}+\frac {3 b \left (\frac {(a+b x)^7}{9 (d+e x)^9 (b d-a e)}+\frac {2 b \left (\frac {b (a+b x)^7}{56 (d+e x)^7 (b d-a e)^2}+\frac {(a+b x)^7}{8 (d+e x)^8 (b d-a e)}\right )}{9 (b d-a e)}\right )}{10 (b d-a e)}\right )}{a+b x}\) |
Input:
Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^11,x]
Output:
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*((a + b*x)^7/(10*(b*d - a*e)*(d + e*x)^10) + (3*b*((a + b*x)^7/(9*(b*d - a*e)*(d + e*x)^9) + (2*b*((a + b*x)^7/(8*(b* d - a*e)*(d + e*x)^8) + (b*(a + b*x)^7)/(56*(b*d - a*e)^2*(d + e*x)^7)))/( 9*(b*d - a*e))))/(10*(b*d - a*e))))/(a + b*x)
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] :> Simp [(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S implify[m + n + 2]/((b*c - a*d)*(m + 1))) Int[(a + b*x)^Simplify[m + 1]*( c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] || !SumSimp lerQ[n, 1])
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. \(350\) vs. \(2(156)=312\).
Time = 7.08 (sec) , antiderivative size = 351, normalized size of antiderivative = 2.04
| method | result | size |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {b^{6} x^{6}}{4 e}-\frac {3 b^{5} \left (4 a e +b d \right ) x^{5}}{10 e^{2}}-\frac {b^{4} \left (10 e^{2} a^{2}+4 a b d e +b^{2} d^{2}\right ) x^{4}}{4 e^{3}}-\frac {b^{3} \left (20 e^{3} a^{3}+10 a^{2} b d \,e^{2}+4 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x^{3}}{7 e^{4}}-\frac {3 b^{2} \left (35 a^{4} e^{4}+20 a^{3} b d \,e^{3}+10 a^{2} b^{2} d^{2} e^{2}+4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) x^{2}}{56 e^{5}}-\frac {b \left (56 e^{5} a^{5}+35 a^{4} b d \,e^{4}+20 a^{3} b^{2} d^{2} e^{3}+10 a^{2} b^{3} d^{3} e^{2}+4 a \,b^{4} d^{4} e +b^{5} d^{5}\right ) x}{84 e^{6}}-\frac {84 a^{6} e^{6}+56 a^{5} b d \,e^{5}+35 a^{4} b^{2} d^{2} e^{4}+20 a^{3} b^{3} d^{3} e^{3}+10 a^{2} b^{4} d^{4} e^{2}+4 a \,b^{5} d^{5} e +b^{6} d^{6}}{840 e^{7}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{10}}\) | \(351\) |
| gosper | \(-\frac {\left (210 b^{6} x^{6} e^{6}+1008 x^{5} a \,b^{5} e^{6}+252 x^{5} b^{6} d \,e^{5}+2100 x^{4} a^{2} b^{4} e^{6}+840 x^{4} a \,b^{5} d \,e^{5}+210 x^{4} b^{6} d^{2} e^{4}+2400 x^{3} a^{3} b^{3} e^{6}+1200 x^{3} a^{2} b^{4} d \,e^{5}+480 x^{3} a \,b^{5} d^{2} e^{4}+120 x^{3} b^{6} d^{3} e^{3}+1575 x^{2} a^{4} b^{2} e^{6}+900 x^{2} a^{3} b^{3} d \,e^{5}+450 x^{2} a^{2} b^{4} d^{2} e^{4}+180 x^{2} a \,b^{5} d^{3} e^{3}+45 x^{2} b^{6} d^{4} e^{2}+560 x \,a^{5} b \,e^{6}+350 x \,a^{4} b^{2} d \,e^{5}+200 x \,a^{3} b^{3} d^{2} e^{4}+100 x \,a^{2} b^{4} d^{3} e^{3}+40 x a \,b^{5} d^{4} e^{2}+10 x \,b^{6} d^{5} e +84 a^{6} e^{6}+56 a^{5} b d \,e^{5}+35 a^{4} b^{2} d^{2} e^{4}+20 a^{3} b^{3} d^{3} e^{3}+10 a^{2} b^{4} d^{4} e^{2}+4 a \,b^{5} d^{5} e +b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{840 e^{7} \left (e x +d \right )^{10} \left (b x +a \right )^{5}}\) | \(392\) |
| default | \(-\frac {\left (210 b^{6} x^{6} e^{6}+1008 x^{5} a \,b^{5} e^{6}+252 x^{5} b^{6} d \,e^{5}+2100 x^{4} a^{2} b^{4} e^{6}+840 x^{4} a \,b^{5} d \,e^{5}+210 x^{4} b^{6} d^{2} e^{4}+2400 x^{3} a^{3} b^{3} e^{6}+1200 x^{3} a^{2} b^{4} d \,e^{5}+480 x^{3} a \,b^{5} d^{2} e^{4}+120 x^{3} b^{6} d^{3} e^{3}+1575 x^{2} a^{4} b^{2} e^{6}+900 x^{2} a^{3} b^{3} d \,e^{5}+450 x^{2} a^{2} b^{4} d^{2} e^{4}+180 x^{2} a \,b^{5} d^{3} e^{3}+45 x^{2} b^{6} d^{4} e^{2}+560 x \,a^{5} b \,e^{6}+350 x \,a^{4} b^{2} d \,e^{5}+200 x \,a^{3} b^{3} d^{2} e^{4}+100 x \,a^{2} b^{4} d^{3} e^{3}+40 x a \,b^{5} d^{4} e^{2}+10 x \,b^{6} d^{5} e +84 a^{6} e^{6}+56 a^{5} b d \,e^{5}+35 a^{4} b^{2} d^{2} e^{4}+20 a^{3} b^{3} d^{3} e^{3}+10 a^{2} b^{4} d^{4} e^{2}+4 a \,b^{5} d^{5} e +b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{840 e^{7} \left (e x +d \right )^{10} \left (b x +a \right )^{5}}\) | \(392\) |
| orering | \(-\frac {\left (210 b^{6} x^{6} e^{6}+1008 x^{5} a \,b^{5} e^{6}+252 x^{5} b^{6} d \,e^{5}+2100 x^{4} a^{2} b^{4} e^{6}+840 x^{4} a \,b^{5} d \,e^{5}+210 x^{4} b^{6} d^{2} e^{4}+2400 x^{3} a^{3} b^{3} e^{6}+1200 x^{3} a^{2} b^{4} d \,e^{5}+480 x^{3} a \,b^{5} d^{2} e^{4}+120 x^{3} b^{6} d^{3} e^{3}+1575 x^{2} a^{4} b^{2} e^{6}+900 x^{2} a^{3} b^{3} d \,e^{5}+450 x^{2} a^{2} b^{4} d^{2} e^{4}+180 x^{2} a \,b^{5} d^{3} e^{3}+45 x^{2} b^{6} d^{4} e^{2}+560 x \,a^{5} b \,e^{6}+350 x \,a^{4} b^{2} d \,e^{5}+200 x \,a^{3} b^{3} d^{2} e^{4}+100 x \,a^{2} b^{4} d^{3} e^{3}+40 x a \,b^{5} d^{4} e^{2}+10 x \,b^{6} d^{5} e +84 a^{6} e^{6}+56 a^{5} b d \,e^{5}+35 a^{4} b^{2} d^{2} e^{4}+20 a^{3} b^{3} d^{3} e^{3}+10 a^{2} b^{4} d^{4} e^{2}+4 a \,b^{5} d^{5} e +b^{6} d^{6}\right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{\frac {5}{2}}}{840 e^{7} \left (b x +a \right )^{5} \left (e x +d \right )^{10}}\) | \(401\) |
Input:
int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^11,x,method=_RETURNVERBOSE )
Output:
((b*x+a)^2)^(1/2)/(b*x+a)*(-1/4/e*b^6*x^6-3/10*b^5/e^2*(4*a*e+b*d)*x^5-1/4 /e^3*b^4*(10*a^2*e^2+4*a*b*d*e+b^2*d^2)*x^4-1/7*b^3/e^4*(20*a^3*e^3+10*a^2 *b*d*e^2+4*a*b^2*d^2*e+b^3*d^3)*x^3-3/56*b^2/e^5*(35*a^4*e^4+20*a^3*b*d*e^ 3+10*a^2*b^2*d^2*e^2+4*a*b^3*d^3*e+b^4*d^4)*x^2-1/84*b/e^6*(56*a^5*e^5+35* a^4*b*d*e^4+20*a^3*b^2*d^2*e^3+10*a^2*b^3*d^3*e^2+4*a*b^4*d^4*e+b^5*d^5)*x -1/840/e^7*(84*a^6*e^6+56*a^5*b*d*e^5+35*a^4*b^2*d^2*e^4+20*a^3*b^3*d^3*e^ 3+10*a^2*b^4*d^4*e^2+4*a*b^5*d^5*e+b^6*d^6))/(e*x+d)^10
Leaf count of result is larger than twice the leaf count of optimal. 452 vs. \(2 (156) = 312\).
Time = 0.08 (sec) , antiderivative size = 452, normalized size of antiderivative = 2.63 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11}} \, dx=-\frac {210 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 4 \, a b^{5} d^{5} e + 10 \, a^{2} b^{4} d^{4} e^{2} + 20 \, a^{3} b^{3} d^{3} e^{3} + 35 \, a^{4} b^{2} d^{2} e^{4} + 56 \, a^{5} b d e^{5} + 84 \, a^{6} e^{6} + 252 \, {\left (b^{6} d e^{5} + 4 \, a b^{5} e^{6}\right )} x^{5} + 210 \, {\left (b^{6} d^{2} e^{4} + 4 \, a b^{5} d e^{5} + 10 \, a^{2} b^{4} e^{6}\right )} x^{4} + 120 \, {\left (b^{6} d^{3} e^{3} + 4 \, a b^{5} d^{2} e^{4} + 10 \, a^{2} b^{4} d e^{5} + 20 \, a^{3} b^{3} e^{6}\right )} x^{3} + 45 \, {\left (b^{6} d^{4} e^{2} + 4 \, a b^{5} d^{3} e^{3} + 10 \, a^{2} b^{4} d^{2} e^{4} + 20 \, a^{3} b^{3} d e^{5} + 35 \, a^{4} b^{2} e^{6}\right )} x^{2} + 10 \, {\left (b^{6} d^{5} e + 4 \, a b^{5} d^{4} e^{2} + 10 \, a^{2} b^{4} d^{3} e^{3} + 20 \, a^{3} b^{3} d^{2} e^{4} + 35 \, a^{4} b^{2} d e^{5} + 56 \, a^{5} b e^{6}\right )} x}{840 \, {\left (e^{17} x^{10} + 10 \, d e^{16} x^{9} + 45 \, d^{2} e^{15} x^{8} + 120 \, d^{3} e^{14} x^{7} + 210 \, d^{4} e^{13} x^{6} + 252 \, d^{5} e^{12} x^{5} + 210 \, d^{6} e^{11} x^{4} + 120 \, d^{7} e^{10} x^{3} + 45 \, d^{8} e^{9} x^{2} + 10 \, d^{9} e^{8} x + d^{10} e^{7}\right )}} \] Input:
integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^11,x, algorithm="fri cas")
Output:
-1/840*(210*b^6*e^6*x^6 + b^6*d^6 + 4*a*b^5*d^5*e + 10*a^2*b^4*d^4*e^2 + 2 0*a^3*b^3*d^3*e^3 + 35*a^4*b^2*d^2*e^4 + 56*a^5*b*d*e^5 + 84*a^6*e^6 + 252 *(b^6*d*e^5 + 4*a*b^5*e^6)*x^5 + 210*(b^6*d^2*e^4 + 4*a*b^5*d*e^5 + 10*a^2 *b^4*e^6)*x^4 + 120*(b^6*d^3*e^3 + 4*a*b^5*d^2*e^4 + 10*a^2*b^4*d*e^5 + 20 *a^3*b^3*e^6)*x^3 + 45*(b^6*d^4*e^2 + 4*a*b^5*d^3*e^3 + 10*a^2*b^4*d^2*e^4 + 20*a^3*b^3*d*e^5 + 35*a^4*b^2*e^6)*x^2 + 10*(b^6*d^5*e + 4*a*b^5*d^4*e^ 2 + 10*a^2*b^4*d^3*e^3 + 20*a^3*b^3*d^2*e^4 + 35*a^4*b^2*d*e^5 + 56*a^5*b* e^6)*x)/(e^17*x^10 + 10*d*e^16*x^9 + 45*d^2*e^15*x^8 + 120*d^3*e^14*x^7 + 210*d^4*e^13*x^6 + 252*d^5*e^12*x^5 + 210*d^6*e^11*x^4 + 120*d^7*e^10*x^3 + 45*d^8*e^9*x^2 + 10*d^9*e^8*x + d^10*e^7)
Timed out. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11}} \, dx=\text {Timed out} \] Input:
integrate((b*x+a)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**11,x)
Output:
Timed out
Exception generated. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^11,x, algorithm="max ima")
Output:
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. 611 vs. \(2 (156) = 312\).
Time = 0.17 (sec) , antiderivative size = 611, normalized size of antiderivative = 3.55 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11}} \, dx =\text {Too large to display} \] Input:
integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^11,x, algorithm="gia c")
Output:
1/840*b^10*sgn(b*x + a)/(b^4*d^4*e^7 - 4*a*b^3*d^3*e^8 + 6*a^2*b^2*d^2*e^9 - 4*a^3*b*d*e^10 + a^4*e^11) - 1/840*(210*b^6*e^6*x^6*sgn(b*x + a) + 252* b^6*d*e^5*x^5*sgn(b*x + a) + 1008*a*b^5*e^6*x^5*sgn(b*x + a) + 210*b^6*d^2 *e^4*x^4*sgn(b*x + a) + 840*a*b^5*d*e^5*x^4*sgn(b*x + a) + 2100*a^2*b^4*e^ 6*x^4*sgn(b*x + a) + 120*b^6*d^3*e^3*x^3*sgn(b*x + a) + 480*a*b^5*d^2*e^4* x^3*sgn(b*x + a) + 1200*a^2*b^4*d*e^5*x^3*sgn(b*x + a) + 2400*a^3*b^3*e^6* x^3*sgn(b*x + a) + 45*b^6*d^4*e^2*x^2*sgn(b*x + a) + 180*a*b^5*d^3*e^3*x^2 *sgn(b*x + a) + 450*a^2*b^4*d^2*e^4*x^2*sgn(b*x + a) + 900*a^3*b^3*d*e^5*x ^2*sgn(b*x + a) + 1575*a^4*b^2*e^6*x^2*sgn(b*x + a) + 10*b^6*d^5*e*x*sgn(b *x + a) + 40*a*b^5*d^4*e^2*x*sgn(b*x + a) + 100*a^2*b^4*d^3*e^3*x*sgn(b*x + a) + 200*a^3*b^3*d^2*e^4*x*sgn(b*x + a) + 350*a^4*b^2*d*e^5*x*sgn(b*x + a) + 560*a^5*b*e^6*x*sgn(b*x + a) + b^6*d^6*sgn(b*x + a) + 4*a*b^5*d^5*e*s gn(b*x + a) + 10*a^2*b^4*d^4*e^2*sgn(b*x + a) + 20*a^3*b^3*d^3*e^3*sgn(b*x + a) + 35*a^4*b^2*d^2*e^4*sgn(b*x + a) + 56*a^5*b*d*e^5*sgn(b*x + a) + 84 *a^6*e^6*sgn(b*x + a))/((e*x + d)^10*e^7)
Time = 11.51 (sec) , antiderivative size = 1010, normalized size of antiderivative = 5.87 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11}} \, dx =\text {Too large to display} \] Input:
int(((a + b*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2))/(d + e*x)^11,x)
Output:
(((b^6*d^5 - 6*a^5*b*e^5 + 15*a^4*b^2*d*e^4 + 15*a^2*b^4*d^3*e^2 - 20*a^3* b^3*d^2*e^3 - 6*a*b^5*d^4*e)/(9*e^7) + (d*((b^6*d^4*e + 15*a^4*b^2*e^5 - 6 *a*b^5*d^3*e^2 - 20*a^3*b^3*d*e^4 + 15*a^2*b^4*d^2*e^3)/(9*e^7) - (d*((20* a^3*b^3*e^5 - b^6*d^3*e^2 + 6*a*b^5*d^2*e^3 - 15*a^2*b^4*d*e^4)/(9*e^7) - (d*((d*((b^6*d)/(9*e^3) - (b^5*(6*a*e - b*d))/(9*e^3)))/e + (b^4*(15*a^2*e ^2 + b^2*d^2 - 6*a*b*d*e))/(9*e^4)))/e))/e))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^ (1/2))/((a + b*x)*(d + e*x)^9) - (((10*b^6*d^2 + 15*a^2*b^4*e^2 - 24*a*b^5 *d*e)/(6*e^7) + (d*((b^6*d)/(6*e^6) - (b^5*(3*a*e - 2*b*d))/(3*e^6)))/e)*( a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^6) - ((a^6/(10*e) - ( d*((d*((d*((d*((d*((3*a*b^5)/(5*e) - (b^6*d)/(10*e^2)))/e - (3*a^2*b^4)/(2 *e)))/e + (2*a^3*b^3)/e))/e - (3*a^4*b^2)/(2*e)))/e + (3*a^5*b)/(5*e)))/e) *(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^10) - (((5*b^6*d^4 + 15*a^4*b^2*e^4 - 40*a^3*b^3*d*e^3 + 45*a^2*b^4*d^2*e^2 - 24*a*b^5*d^3*e) /(8*e^7) + (d*((4*b^6*d^3*e - 20*a^3*b^3*e^4 - 18*a*b^5*d^2*e^2 + 30*a^2*b ^4*d*e^3)/(8*e^7) + (d*((d*((b^6*d)/(8*e^4) - (b^5*(3*a*e - b*d))/(4*e^4)) )/e + (3*b^4*(5*a^2*e^2 + b^2*d^2 - 4*a*b*d*e))/(8*e^5)))/e))/e)*(a^2 + b^ 2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^8) + (((5*b^6*d - 6*a*b^5*e)/ (5*e^7) + (b^6*d)/(5*e^7))*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^5) + (((10*b^6*d^3 - 20*a^3*b^3*e^3 + 45*a^2*b^4*d*e^2 - 36*a*b^5*d ^2*e)/(7*e^7) + (d*((d*((b^6*d)/(7*e^5) - (3*b^5*(2*a*e - b*d))/(7*e^5)...
Time = 0.22 (sec) , antiderivative size = 475, normalized size of antiderivative = 2.76 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11}} \, dx=\frac {-210 b^{6} e^{6} x^{6}-1008 a \,b^{5} e^{6} x^{5}-252 b^{6} d \,e^{5} x^{5}-2100 a^{2} b^{4} e^{6} x^{4}-840 a \,b^{5} d \,e^{5} x^{4}-210 b^{6} d^{2} e^{4} x^{4}-2400 a^{3} b^{3} e^{6} x^{3}-1200 a^{2} b^{4} d \,e^{5} x^{3}-480 a \,b^{5} d^{2} e^{4} x^{3}-120 b^{6} d^{3} e^{3} x^{3}-1575 a^{4} b^{2} e^{6} x^{2}-900 a^{3} b^{3} d \,e^{5} x^{2}-450 a^{2} b^{4} d^{2} e^{4} x^{2}-180 a \,b^{5} d^{3} e^{3} x^{2}-45 b^{6} d^{4} e^{2} x^{2}-560 a^{5} b \,e^{6} x -350 a^{4} b^{2} d \,e^{5} x -200 a^{3} b^{3} d^{2} e^{4} x -100 a^{2} b^{4} d^{3} e^{3} x -40 a \,b^{5} d^{4} e^{2} x -10 b^{6} d^{5} e x -84 a^{6} e^{6}-56 a^{5} b d \,e^{5}-35 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}-10 a^{2} b^{4} d^{4} e^{2}-4 a \,b^{5} d^{5} e -b^{6} d^{6}}{840 e^{7} \left (e^{10} x^{10}+10 d \,e^{9} x^{9}+45 d^{2} e^{8} x^{8}+120 d^{3} e^{7} x^{7}+210 d^{4} e^{6} x^{6}+252 d^{5} e^{5} x^{5}+210 d^{6} e^{4} x^{4}+120 d^{7} e^{3} x^{3}+45 d^{8} e^{2} x^{2}+10 d^{9} e x +d^{10}\right )} \] Input:
int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^11,x)
Output:
( - 84*a**6*e**6 - 56*a**5*b*d*e**5 - 560*a**5*b*e**6*x - 35*a**4*b**2*d** 2*e**4 - 350*a**4*b**2*d*e**5*x - 1575*a**4*b**2*e**6*x**2 - 20*a**3*b**3* d**3*e**3 - 200*a**3*b**3*d**2*e**4*x - 900*a**3*b**3*d*e**5*x**2 - 2400*a **3*b**3*e**6*x**3 - 10*a**2*b**4*d**4*e**2 - 100*a**2*b**4*d**3*e**3*x - 450*a**2*b**4*d**2*e**4*x**2 - 1200*a**2*b**4*d*e**5*x**3 - 2100*a**2*b**4 *e**6*x**4 - 4*a*b**5*d**5*e - 40*a*b**5*d**4*e**2*x - 180*a*b**5*d**3*e** 3*x**2 - 480*a*b**5*d**2*e**4*x**3 - 840*a*b**5*d*e**5*x**4 - 1008*a*b**5* e**6*x**5 - b**6*d**6 - 10*b**6*d**5*e*x - 45*b**6*d**4*e**2*x**2 - 120*b* *6*d**3*e**3*x**3 - 210*b**6*d**2*e**4*x**4 - 252*b**6*d*e**5*x**5 - 210*b **6*e**6*x**6)/(840*e**7*(d**10 + 10*d**9*e*x + 45*d**8*e**2*x**2 + 120*d* *7*e**3*x**3 + 210*d**6*e**4*x**4 + 252*d**5*e**5*x**5 + 210*d**4*e**6*x** 6 + 120*d**3*e**7*x**7 + 45*d**2*e**8*x**8 + 10*d*e**9*x**9 + e**10*x**10) )