Integrand size = 33, antiderivative size = 262 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{10}} \, dx=-\frac {(B d-A e) (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{9 e (b d-a e) (d+e x)^9}+\frac {(2 b B d+A b e-3 a B e) (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{24 e (b d-a e)^2 (d+e x)^8}+\frac {b (2 b B d+A b e-3 a B e) (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{84 e (b d-a e)^3 (d+e x)^7}+\frac {b^2 (2 b B d+A b e-3 a B e) (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{504 e (b d-a e)^4 (d+e x)^6} \]
-1/9*(-A*e+B*d)*(b*x+a)^5*((b*x+a)^2)^(1/2)/e/(-a*e+b*d)/(e*x+d)^9+1/24*(A *b*e-3*B*a*e+2*B*b*d)*(b*x+a)^5*((b*x+a)^2)^(1/2)/e/(-a*e+b*d)^2/(e*x+d)^8 +1/84*b*(A*b*e-3*B*a*e+2*B*b*d)*(b*x+a)^5*((b*x+a)^2)^(1/2)/e/(-a*e+b*d)^3 /(e*x+d)^7+1/504*b^2*(A*b*e-3*B*a*e+2*B*b*d)*(b*x+a)^5*((b*x+a)^2)^(1/2)/e /(-a*e+b*d)^4/(e*x+d)^6
Time = 1.15 (sec) , antiderivative size = 468, normalized size of antiderivative = 1.79 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{10}} \, dx=-\frac {\sqrt {(a+b x)^2} \left (7 a^5 e^5 (8 A e+B (d+9 e x))+5 a^4 b e^4 \left (7 A e (d+9 e x)+2 B \left (d^2+9 d e x+36 e^2 x^2\right )\right )+10 a^3 b^2 e^3 \left (2 A e \left (d^2+9 d e x+36 e^2 x^2\right )+B \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )\right )+2 a^2 b^3 e^2 \left (5 A e \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )+4 B \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )\right )+a b^4 e \left (4 A e \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )+5 B \left (d^5+9 d^4 e x+36 d^3 e^2 x^2+84 d^2 e^3 x^3+126 d e^4 x^4+126 e^5 x^5\right )\right )+b^5 \left (A e \left (d^5+9 d^4 e x+36 d^3 e^2 x^2+84 d^2 e^3 x^3+126 d e^4 x^4+126 e^5 x^5\right )+2 B \left (d^6+9 d^5 e x+36 d^4 e^2 x^2+84 d^3 e^3 x^3+126 d^2 e^4 x^4+126 d e^5 x^5+84 e^6 x^6\right )\right )\right )}{504 e^7 (a+b x) (d+e x)^9} \]
-1/504*(Sqrt[(a + b*x)^2]*(7*a^5*e^5*(8*A*e + B*(d + 9*e*x)) + 5*a^4*b*e^4 *(7*A*e*(d + 9*e*x) + 2*B*(d^2 + 9*d*e*x + 36*e^2*x^2)) + 10*a^3*b^2*e^3*( 2*A*e*(d^2 + 9*d*e*x + 36*e^2*x^2) + B*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 8 4*e^3*x^3)) + 2*a^2*b^3*e^2*(5*A*e*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^ 3*x^3) + 4*B*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 + 84*d*e^3*x^3 + 126*e^4*x^ 4)) + a*b^4*e*(4*A*e*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 + 84*d*e^3*x^3 + 12 6*e^4*x^4) + 5*B*(d^5 + 9*d^4*e*x + 36*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 126* d*e^4*x^4 + 126*e^5*x^5)) + b^5*(A*e*(d^5 + 9*d^4*e*x + 36*d^3*e^2*x^2 + 8 4*d^2*e^3*x^3 + 126*d*e^4*x^4 + 126*e^5*x^5) + 2*B*(d^6 + 9*d^5*e*x + 36*d ^4*e^2*x^2 + 84*d^3*e^3*x^3 + 126*d^2*e^4*x^4 + 126*d*e^5*x^5 + 84*e^6*x^6 ))))/(e^7*(a + b*x)*(d + e*x)^9)
Time = 0.31 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.77, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1187, 27, 87, 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.
| \(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2} (A+B x)}{(d+e x)^{10}} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {b^5 (a+b x)^5 (A+B x)}{(d+e x)^{10}}dx}{b^5 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x)^5 (A+B x)}{(d+e x)^{10}}dx}{a+b x}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {(-3 a B e+A b e+2 b B d) \int \frac {(a+b x)^5}{(d+e x)^9}dx}{3 e (b d-a e)}-\frac {(a+b x)^6 (B d-A e)}{9 e (d+e x)^9 (b d-a e)}\right )}{a+b x}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {(-3 a B e+A b e+2 b B d) \left (\frac {b \int \frac {(a+b x)^5}{(d+e x)^8}dx}{4 (b d-a e)}+\frac {(a+b x)^6}{8 (d+e x)^8 (b d-a e)}\right )}{3 e (b d-a e)}-\frac {(a+b x)^6 (B d-A e)}{9 e (d+e x)^9 (b d-a e)}\right )}{a+b x}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {(-3 a B e+A b e+2 b B d) \left (\frac {b \left (\frac {b \int \frac {(a+b x)^5}{(d+e x)^7}dx}{7 (b d-a e)}+\frac {(a+b x)^6}{7 (d+e x)^7 (b d-a e)}\right )}{4 (b d-a e)}+\frac {(a+b x)^6}{8 (d+e x)^8 (b d-a e)}\right )}{3 e (b d-a e)}-\frac {(a+b x)^6 (B d-A e)}{9 e (d+e x)^9 (b d-a e)}\right )}{a+b x}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {\left (\frac {(a+b x)^6}{8 (d+e x)^8 (b d-a e)}+\frac {b \left (\frac {b (a+b x)^6}{42 (d+e x)^6 (b d-a e)^2}+\frac {(a+b x)^6}{7 (d+e x)^7 (b d-a e)}\right )}{4 (b d-a e)}\right ) (-3 a B e+A b e+2 b B d)}{3 e (b d-a e)}-\frac {(a+b x)^6 (B d-A e)}{9 e (d+e x)^9 (b d-a e)}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-1/9*((B*d - A*e)*(a + b*x)^6)/(e*(b*d - a *e)*(d + e*x)^9) + ((2*b*B*d + A*b*e - 3*a*B*e)*((a + b*x)^6/(8*(b*d - a*e )*(d + e*x)^8) + (b*((a + b*x)^6/(7*(b*d - a*e)*(d + e*x)^7) + (b*(a + b*x )^6)/(42*(b*d - a*e)^2*(d + e*x)^6)))/(4*(b*d - a*e))))/(3*e*(b*d - a*e))) )/(a + b*x)
3.18.54.3.1 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] :> 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[((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. \(595\) vs. \(2(210)=420\).
Time = 5.23 (sec) , antiderivative size = 596, normalized size of antiderivative = 2.27
| method | result | size |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {B \,b^{5} x^{6}}{3 e}-\frac {b^{4} \left (A b e +5 B a e +2 B b d \right ) x^{5}}{4 e^{2}}-\frac {b^{3} \left (4 A a b \,e^{2}+A \,b^{2} d e +8 a^{2} B \,e^{2}+5 B a b d e +2 B \,b^{2} d^{2}\right ) x^{4}}{4 e^{3}}-\frac {b^{2} \left (10 A \,a^{2} b \,e^{3}+4 A a \,b^{2} d \,e^{2}+A \,b^{3} d^{2} e +10 B \,e^{3} a^{3}+8 B \,a^{2} b d \,e^{2}+5 B a \,b^{2} d^{2} e +2 B \,b^{3} d^{3}\right ) x^{3}}{6 e^{4}}-\frac {b \left (20 A \,a^{3} b \,e^{4}+10 A \,a^{2} b^{2} d \,e^{3}+4 A a \,b^{3} d^{2} e^{2}+A \,b^{4} d^{3} e +10 B \,a^{4} e^{4}+10 B \,a^{3} b d \,e^{3}+8 B \,a^{2} b^{2} d^{2} e^{2}+5 B a \,b^{3} d^{3} e +2 b^{4} B \,d^{4}\right ) x^{2}}{14 e^{5}}-\frac {\left (35 A \,a^{4} b \,e^{5}+20 A \,a^{3} b^{2} d \,e^{4}+10 A \,a^{2} b^{3} d^{2} e^{3}+4 A a \,b^{4} d^{3} e^{2}+A \,b^{5} d^{4} e +7 B \,a^{5} e^{5}+10 B \,a^{4} b d \,e^{4}+10 B \,a^{3} b^{2} d^{2} e^{3}+8 B \,a^{2} b^{3} d^{3} e^{2}+5 B a \,b^{4} d^{4} e +2 B \,b^{5} d^{5}\right ) x}{56 e^{6}}-\frac {56 A \,a^{5} e^{6}+35 A \,a^{4} b d \,e^{5}+20 A \,a^{3} b^{2} d^{2} e^{4}+10 A \,a^{2} b^{3} d^{3} e^{3}+4 A a \,b^{4} d^{4} e^{2}+A \,b^{5} d^{5} e +7 B \,a^{5} d \,e^{5}+10 B \,a^{4} b \,d^{2} e^{4}+10 B \,a^{3} b^{2} d^{3} e^{3}+8 B \,a^{2} b^{3} d^{4} e^{2}+5 B a \,b^{4} d^{5} e +2 B \,b^{5} d^{6}}{504 e^{7}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{9}}\) | \(596\) |
| gosper | \(-\frac {\left (168 B \,b^{5} e^{6} x^{6}+126 A \,b^{5} e^{6} x^{5}+630 B a \,b^{4} e^{6} x^{5}+252 B \,b^{5} d \,e^{5} x^{5}+504 A a \,b^{4} e^{6} x^{4}+126 A \,b^{5} d \,e^{5} x^{4}+1008 B \,a^{2} b^{3} e^{6} x^{4}+630 B a \,b^{4} d \,e^{5} x^{4}+252 B \,b^{5} d^{2} e^{4} x^{4}+840 A \,a^{2} b^{3} e^{6} x^{3}+336 A a \,b^{4} d \,e^{5} x^{3}+84 A \,b^{5} d^{2} e^{4} x^{3}+840 B \,a^{3} b^{2} e^{6} x^{3}+672 B \,a^{2} b^{3} d \,e^{5} x^{3}+420 B a \,b^{4} d^{2} e^{4} x^{3}+168 B \,b^{5} d^{3} e^{3} x^{3}+720 A \,a^{3} b^{2} e^{6} x^{2}+360 A \,a^{2} b^{3} d \,e^{5} x^{2}+144 A a \,b^{4} d^{2} e^{4} x^{2}+36 A \,b^{5} d^{3} e^{3} x^{2}+360 B \,a^{4} b \,e^{6} x^{2}+360 B \,a^{3} b^{2} d \,e^{5} x^{2}+288 B \,a^{2} b^{3} d^{2} e^{4} x^{2}+180 B a \,b^{4} d^{3} e^{3} x^{2}+72 B \,b^{5} d^{4} e^{2} x^{2}+315 A \,a^{4} b \,e^{6} x +180 A \,a^{3} b^{2} d \,e^{5} x +90 A \,a^{2} b^{3} d^{2} e^{4} x +36 A a \,b^{4} d^{3} e^{3} x +9 A \,b^{5} d^{4} e^{2} x +63 B \,a^{5} e^{6} x +90 B \,a^{4} b d \,e^{5} x +90 B \,a^{3} b^{2} d^{2} e^{4} x +72 B \,a^{2} b^{3} d^{3} e^{3} x +45 B a \,b^{4} d^{4} e^{2} x +18 B \,b^{5} d^{5} e x +56 A \,a^{5} e^{6}+35 A \,a^{4} b d \,e^{5}+20 A \,a^{3} b^{2} d^{2} e^{4}+10 A \,a^{2} b^{3} d^{3} e^{3}+4 A a \,b^{4} d^{4} e^{2}+A \,b^{5} d^{5} e +7 B \,a^{5} d \,e^{5}+10 B \,a^{4} b \,d^{2} e^{4}+10 B \,a^{3} b^{2} d^{3} e^{3}+8 B \,a^{2} b^{3} d^{4} e^{2}+5 B a \,b^{4} d^{5} e +2 B \,b^{5} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{504 e^{7} \left (e x +d \right )^{9} \left (b x +a \right )^{5}}\) | \(688\) |
| default | \(-\frac {\left (168 B \,b^{5} e^{6} x^{6}+126 A \,b^{5} e^{6} x^{5}+630 B a \,b^{4} e^{6} x^{5}+252 B \,b^{5} d \,e^{5} x^{5}+504 A a \,b^{4} e^{6} x^{4}+126 A \,b^{5} d \,e^{5} x^{4}+1008 B \,a^{2} b^{3} e^{6} x^{4}+630 B a \,b^{4} d \,e^{5} x^{4}+252 B \,b^{5} d^{2} e^{4} x^{4}+840 A \,a^{2} b^{3} e^{6} x^{3}+336 A a \,b^{4} d \,e^{5} x^{3}+84 A \,b^{5} d^{2} e^{4} x^{3}+840 B \,a^{3} b^{2} e^{6} x^{3}+672 B \,a^{2} b^{3} d \,e^{5} x^{3}+420 B a \,b^{4} d^{2} e^{4} x^{3}+168 B \,b^{5} d^{3} e^{3} x^{3}+720 A \,a^{3} b^{2} e^{6} x^{2}+360 A \,a^{2} b^{3} d \,e^{5} x^{2}+144 A a \,b^{4} d^{2} e^{4} x^{2}+36 A \,b^{5} d^{3} e^{3} x^{2}+360 B \,a^{4} b \,e^{6} x^{2}+360 B \,a^{3} b^{2} d \,e^{5} x^{2}+288 B \,a^{2} b^{3} d^{2} e^{4} x^{2}+180 B a \,b^{4} d^{3} e^{3} x^{2}+72 B \,b^{5} d^{4} e^{2} x^{2}+315 A \,a^{4} b \,e^{6} x +180 A \,a^{3} b^{2} d \,e^{5} x +90 A \,a^{2} b^{3} d^{2} e^{4} x +36 A a \,b^{4} d^{3} e^{3} x +9 A \,b^{5} d^{4} e^{2} x +63 B \,a^{5} e^{6} x +90 B \,a^{4} b d \,e^{5} x +90 B \,a^{3} b^{2} d^{2} e^{4} x +72 B \,a^{2} b^{3} d^{3} e^{3} x +45 B a \,b^{4} d^{4} e^{2} x +18 B \,b^{5} d^{5} e x +56 A \,a^{5} e^{6}+35 A \,a^{4} b d \,e^{5}+20 A \,a^{3} b^{2} d^{2} e^{4}+10 A \,a^{2} b^{3} d^{3} e^{3}+4 A a \,b^{4} d^{4} e^{2}+A \,b^{5} d^{5} e +7 B \,a^{5} d \,e^{5}+10 B \,a^{4} b \,d^{2} e^{4}+10 B \,a^{3} b^{2} d^{3} e^{3}+8 B \,a^{2} b^{3} d^{4} e^{2}+5 B a \,b^{4} d^{5} e +2 B \,b^{5} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{504 e^{7} \left (e x +d \right )^{9} \left (b x +a \right )^{5}}\) | \(688\) |
((b*x+a)^2)^(1/2)/(b*x+a)*(-1/3*B*b^5/e*x^6-1/4*b^4/e^2*(A*b*e+5*B*a*e+2*B *b*d)*x^5-1/4*b^3/e^3*(4*A*a*b*e^2+A*b^2*d*e+8*B*a^2*e^2+5*B*a*b*d*e+2*B*b ^2*d^2)*x^4-1/6*b^2/e^4*(10*A*a^2*b*e^3+4*A*a*b^2*d*e^2+A*b^3*d^2*e+10*B*a ^3*e^3+8*B*a^2*b*d*e^2+5*B*a*b^2*d^2*e+2*B*b^3*d^3)*x^3-1/14*b/e^5*(20*A*a ^3*b*e^4+10*A*a^2*b^2*d*e^3+4*A*a*b^3*d^2*e^2+A*b^4*d^3*e+10*B*a^4*e^4+10* B*a^3*b*d*e^3+8*B*a^2*b^2*d^2*e^2+5*B*a*b^3*d^3*e+2*B*b^4*d^4)*x^2-1/56/e^ 6*(35*A*a^4*b*e^5+20*A*a^3*b^2*d*e^4+10*A*a^2*b^3*d^2*e^3+4*A*a*b^4*d^3*e^ 2+A*b^5*d^4*e+7*B*a^5*e^5+10*B*a^4*b*d*e^4+10*B*a^3*b^2*d^2*e^3+8*B*a^2*b^ 3*d^3*e^2+5*B*a*b^4*d^4*e+2*B*b^5*d^5)*x-1/504/e^7*(56*A*a^5*e^6+35*A*a^4* b*d*e^5+20*A*a^3*b^2*d^2*e^4+10*A*a^2*b^3*d^3*e^3+4*A*a*b^4*d^4*e^2+A*b^5* d^5*e+7*B*a^5*d*e^5+10*B*a^4*b*d^2*e^4+10*B*a^3*b^2*d^3*e^3+8*B*a^2*b^3*d^ 4*e^2+5*B*a*b^4*d^5*e+2*B*b^5*d^6))/(e*x+d)^9
Leaf count of result is larger than twice the leaf count of optimal. 645 vs. \(2 (210) = 420\).
Time = 0.42 (sec) , antiderivative size = 645, normalized size of antiderivative = 2.46 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{10}} \, dx=-\frac {168 \, B b^{5} e^{6} x^{6} + 2 \, B b^{5} d^{6} + 56 \, A a^{5} e^{6} + {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e + 4 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} + 10 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} + 10 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{4} + 7 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{5} + 126 \, {\left (2 \, B b^{5} d e^{5} + {\left (5 \, B a b^{4} + A b^{5}\right )} e^{6}\right )} x^{5} + 126 \, {\left (2 \, B b^{5} d^{2} e^{4} + {\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} + 4 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{4} + 84 \, {\left (2 \, B b^{5} d^{3} e^{3} + {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + 4 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5} + 10 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{6}\right )} x^{3} + 36 \, {\left (2 \, B b^{5} d^{4} e^{2} + {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + 4 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} + 10 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5} + 10 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{6}\right )} x^{2} + 9 \, {\left (2 \, B b^{5} d^{5} e + {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 4 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} + 10 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} + 10 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{5} + 7 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} e^{6}\right )} x}{504 \, {\left (e^{16} x^{9} + 9 \, d e^{15} x^{8} + 36 \, d^{2} e^{14} x^{7} + 84 \, d^{3} e^{13} x^{6} + 126 \, d^{4} e^{12} x^{5} + 126 \, d^{5} e^{11} x^{4} + 84 \, d^{6} e^{10} x^{3} + 36 \, d^{7} e^{9} x^{2} + 9 \, d^{8} e^{8} x + d^{9} e^{7}\right )}} \]
-1/504*(168*B*b^5*e^6*x^6 + 2*B*b^5*d^6 + 56*A*a^5*e^6 + (5*B*a*b^4 + A*b^ 5)*d^5*e + 4*(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 + 10*(B*a^4*b + 2*A*a^3*b^2)*d^2*e^4 + 7*(B*a^5 + 5*A*a^4*b)*d*e^5 + 126*(2*B*b^5*d*e^5 + (5*B*a*b^4 + A*b^5)*e^6)*x^5 + 126*(2*B*b^5*d^2*e^ 4 + (5*B*a*b^4 + A*b^5)*d*e^5 + 4*(2*B*a^2*b^3 + A*a*b^4)*e^6)*x^4 + 84*(2 *B*b^5*d^3*e^3 + (5*B*a*b^4 + A*b^5)*d^2*e^4 + 4*(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 + 36*(2*B*b^5*d^4*e^2 + (5*B*a* b^4 + A*b^5)*d^3*e^3 + 4*(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 + 10*(B*a^4*b + 2*A*a^3*b^2)*e^6)*x^2 + 9*(2*B*b^5*d^5*e + (5*B*a*b^4 + A*b^5)*d^4*e^2 + 4*(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 + 10*(B*a^4*b + 2*A*a^3*b^2)*d*e^5 + 7*(B*a^ 5 + 5*A*a^4*b)*e^6)*x)/(e^16*x^9 + 9*d*e^15*x^8 + 36*d^2*e^14*x^7 + 84*d^3 *e^13*x^6 + 126*d^4*e^12*x^5 + 126*d^5*e^11*x^4 + 84*d^6*e^10*x^3 + 36*d^7 *e^9*x^2 + 9*d^8*e^8*x + d^9*e^7)
Timed out. \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{10}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{10}} \, dx=\text {Exception raised: ValueError} \]
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. 1046 vs. \(2 (210) = 420\).
Time = 0.30 (sec) , antiderivative size = 1046, normalized size of antiderivative = 3.99 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{10}} \, dx=\text {Too large to display} \]
1/504*(2*B*b^9*d - 3*B*a*b^8*e + A*b^9*e)*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) - 1/504*(168* B*b^5*e^6*x^6*sgn(b*x + a) + 252*B*b^5*d*e^5*x^5*sgn(b*x + a) + 630*B*a*b^ 4*e^6*x^5*sgn(b*x + a) + 126*A*b^5*e^6*x^5*sgn(b*x + a) + 252*B*b^5*d^2*e^ 4*x^4*sgn(b*x + a) + 630*B*a*b^4*d*e^5*x^4*sgn(b*x + a) + 126*A*b^5*d*e^5* x^4*sgn(b*x + a) + 1008*B*a^2*b^3*e^6*x^4*sgn(b*x + a) + 504*A*a*b^4*e^6*x ^4*sgn(b*x + a) + 168*B*b^5*d^3*e^3*x^3*sgn(b*x + a) + 420*B*a*b^4*d^2*e^4 *x^3*sgn(b*x + a) + 84*A*b^5*d^2*e^4*x^3*sgn(b*x + a) + 672*B*a^2*b^3*d*e^ 5*x^3*sgn(b*x + a) + 336*A*a*b^4*d*e^5*x^3*sgn(b*x + a) + 840*B*a^3*b^2*e^ 6*x^3*sgn(b*x + a) + 840*A*a^2*b^3*e^6*x^3*sgn(b*x + a) + 72*B*b^5*d^4*e^2 *x^2*sgn(b*x + a) + 180*B*a*b^4*d^3*e^3*x^2*sgn(b*x + a) + 36*A*b^5*d^3*e^ 3*x^2*sgn(b*x + a) + 288*B*a^2*b^3*d^2*e^4*x^2*sgn(b*x + a) + 144*A*a*b^4* d^2*e^4*x^2*sgn(b*x + a) + 360*B*a^3*b^2*d*e^5*x^2*sgn(b*x + a) + 360*A*a^ 2*b^3*d*e^5*x^2*sgn(b*x + a) + 360*B*a^4*b*e^6*x^2*sgn(b*x + a) + 720*A*a^ 3*b^2*e^6*x^2*sgn(b*x + a) + 18*B*b^5*d^5*e*x*sgn(b*x + a) + 45*B*a*b^4*d^ 4*e^2*x*sgn(b*x + a) + 9*A*b^5*d^4*e^2*x*sgn(b*x + a) + 72*B*a^2*b^3*d^3*e ^3*x*sgn(b*x + a) + 36*A*a*b^4*d^3*e^3*x*sgn(b*x + a) + 90*B*a^3*b^2*d^2*e ^4*x*sgn(b*x + a) + 90*A*a^2*b^3*d^2*e^4*x*sgn(b*x + a) + 90*B*a^4*b*d*e^5 *x*sgn(b*x + a) + 180*A*a^3*b^2*d*e^5*x*sgn(b*x + a) + 63*B*a^5*e^6*x*sgn( b*x + a) + 315*A*a^4*b*e^6*x*sgn(b*x + a) + 2*B*b^5*d^6*sgn(b*x + a) + ...
Time = 11.24 (sec) , antiderivative size = 1489, normalized size of antiderivative = 5.68 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{10}} \, dx=\text {Too large to display} \]
- (((10*B*b^5*d^2 - 4*A*b^5*d*e + 5*A*a*b^4*e^2 + 10*B*a^2*b^3*e^2 - 20*B* a*b^4*d*e)/(5*e^7) - (d*((b^4*(A*b*e + 5*B*a*e - 4*B*b*d))/(5*e^6) - (B*b^ 5*d)/(5*e^6)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^5) - (((A*b^5*e - 5*B*b^5*d + 5*B*a*b^4*e)/(4*e^7) - (B*b^5*d)/(4*e^7))*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^4) - (((A*a^5)/(9*e) - ( d*((B*a^5 + 5*A*a^4*b)/(9*e) + (d*((d*((d*((d*((A*b^5 + 5*B*a*b^4)/(9*e) - (B*b^5*d)/(9*e^2)))/e - (5*a*b^3*(A*b + 2*B*a))/(9*e)))/e + (10*a^2*b^2*( A*b + B*a))/(9*e)))/e - (5*a^3*b*(2*A*b + B*a))/(9*e)))/e))/e)*(a^2 + b^2* x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^9) - (((6*A*b^5*d^2*e - 10*B*b^ 5*d^3 + 10*A*a^2*b^3*e^3 + 10*B*a^3*b^2*e^3 - 30*B*a^2*b^3*d*e^2 - 15*A*a* b^4*d*e^2 + 30*B*a*b^4*d^2*e)/(6*e^7) - (d*((5*A*a*b^4*e^3 - 3*A*b^5*d*e^2 + 6*B*b^5*d^2*e + 10*B*a^2*b^3*e^3 - 15*B*a*b^4*d*e^2)/(6*e^7) - (d*((b^4 *(A*b*e + 5*B*a*e - 3*B*b*d))/(6*e^5) - (B*b^5*d)/(6*e^5)))/e))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^6) - (((B*a^5*e^5 - B*b^5*d ^5 + 5*A*a^4*b*e^5 + A*b^5*d^4*e - 5*A*a*b^4*d^3*e^2 - 10*A*a^3*b^2*d*e^4 + 10*A*a^2*b^3*d^2*e^3 - 10*B*a^2*b^3*d^3*e^2 + 10*B*a^3*b^2*d^2*e^3 + 5*B *a*b^4*d^4*e - 5*B*a^4*b*d*e^4)/(8*e^7) - (d*((5*B*a^4*b*e^5 + B*b^5*d^4*e + 10*A*a^3*b^2*e^5 - A*b^5*d^3*e^2 + 5*A*a*b^4*d^2*e^3 - 10*A*a^2*b^3*d*e ^4 - 5*B*a*b^4*d^3*e^2 - 10*B*a^3*b^2*d*e^4 + 10*B*a^2*b^3*d^2*e^3)/(8*e^7 ) - (d*((10*A*a^2*b^3*e^5 + 10*B*a^3*b^2*e^5 + A*b^5*d^2*e^3 - B*b^5*d^...