Integrand size = 28, antiderivative size = 200 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{10}} \, dx=\frac {(a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{9 (b d-a e) (d+e x)^9}+\frac {b (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{24 (b d-a e)^2 (d+e x)^8}+\frac {b^2 (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{84 (b d-a e)^3 (d+e x)^7}+\frac {b^3 (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{504 (b d-a e)^4 (d+e x)^6} \] Output:
1/9*(b*x+a)^5*((b*x+a)^2)^(1/2)/(-a*e+b*d)/(e*x+d)^9+1/24*b*(b*x+a)^5*((b* x+a)^2)^(1/2)/(-a*e+b*d)^2/(e*x+d)^8+1/84*b^2*(b*x+a)^5*((b*x+a)^2)^(1/2)/ (-a*e+b*d)^3/(e*x+d)^7+1/504*b^3*(b*x+a)^5*((b*x+a)^2)^(1/2)/(-a*e+b*d)^4/ (e*x+d)^6
Time = 1.07 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.12 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{10}} \, dx=-\frac {\sqrt {(a+b x)^2} \left (56 a^5 e^5+35 a^4 b e^4 (d+9 e x)+20 a^3 b^2 e^3 \left (d^2+9 d e x+36 e^2 x^2\right )+10 a^2 b^3 e^2 \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )+4 a b^4 e \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )+b^5 \left (d^5+9 d^4 e x+36 d^3 e^2 x^2+84 d^2 e^3 x^3+126 d e^4 x^4+126 e^5 x^5\right )\right )}{504 e^6 (a+b x) (d+e x)^9} \] Input:
Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^10,x]
Output:
-1/504*(Sqrt[(a + b*x)^2]*(56*a^5*e^5 + 35*a^4*b*e^4*(d + 9*e*x) + 20*a^3* b^2*e^3*(d^2 + 9*d*e*x + 36*e^2*x^2) + 10*a^2*b^3*e^2*(d^3 + 9*d^2*e*x + 3 6*d*e^2*x^2 + 84*e^3*x^3) + 4*a*b^4*e*(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) + b^5*(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)))/(e^6*(a + b*x)*(d + e*x)^9)
Time = 0.43 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.87, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {1102, 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.
| \(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{10}} \, dx\) |
| \(\Big \downarrow \) 1102 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {b^5 (a+b x)^5}{(d+e x)^{10}}dx}{b^5 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x)^5}{(d+e x)^{10}}dx}{a+b x}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {b \int \frac {(a+b x)^5}{(d+e x)^9}dx}{3 (b d-a e)}+\frac {(a+b x)^6}{9 (d+e x)^9 (b d-a e)}\right )}{a+b x}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {b \left (\frac {b \int \frac {(a+b x)^5}{(d+e x)^8}dx}{4 (b d-a e)}+\frac {(a+b x)^6}{8 (d+e x)^8 (b d-a e)}\right )}{3 (b d-a e)}+\frac {(a+b x)^6}{9 (d+e x)^9 (b d-a e)}\right )}{a+b x}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {b \left (\frac {b \left (\frac {b \int \frac {(a+b x)^5}{(d+e x)^7}dx}{7 (b d-a e)}+\frac {(a+b x)^6}{7 (d+e x)^7 (b d-a e)}\right )}{4 (b d-a e)}+\frac {(a+b x)^6}{8 (d+e x)^8 (b d-a e)}\right )}{3 (b d-a e)}+\frac {(a+b x)^6}{9 (d+e x)^9 (b d-a e)}\right )}{a+b x}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {(a+b x)^6}{9 (d+e x)^9 (b d-a e)}+\frac {b \left (\frac {(a+b x)^6}{8 (d+e x)^8 (b d-a e)}+\frac {b \left (\frac {b (a+b x)^6}{42 (d+e x)^6 (b d-a e)^2}+\frac {(a+b x)^6}{7 (d+e x)^7 (b d-a e)}\right )}{4 (b d-a e)}\right )}{3 (b d-a e)}\right )}{a+b x}\) |
Input:
Int[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^10,x]
Output:
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*((a + b*x)^6/(9*(b*d - a*e)*(d + e*x)^9) + (b*((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*(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_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*F racPart[p])) Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0]
Time = 8.11 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.31
| method | result | size |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {b^{5} x^{5}}{4 e}-\frac {b^{4} \left (4 a e +b d \right ) x^{4}}{4 e^{2}}-\frac {b^{3} \left (10 e^{2} a^{2}+4 a b d e +b^{2} d^{2}\right ) x^{3}}{6 e^{3}}-\frac {b^{2} \left (20 e^{3} a^{3}+10 a^{2} b d \,e^{2}+4 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x^{2}}{14 e^{4}}-\frac {b \left (35 a^{4} e^{4}+20 a^{3} b d \,e^{3}+10 a^{2} b^{2} d^{2} e^{2}+4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) x}{56 e^{5}}-\frac {56 e^{5} a^{5}+35 a^{4} b d \,e^{4}+20 a^{3} b^{2} d^{2} e^{3}+10 a^{2} b^{3} d^{3} e^{2}+4 a \,b^{4} d^{4} e +b^{5} d^{5}}{504 e^{6}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{9}}\) | \(262\) |
| gosper | \(-\frac {\left (126 x^{5} e^{5} b^{5}+504 x^{4} a \,b^{4} e^{5}+126 x^{4} b^{5} d \,e^{4}+840 x^{3} a^{2} b^{3} e^{5}+336 x^{3} a \,b^{4} d \,e^{4}+84 x^{3} b^{5} d^{2} e^{3}+720 x^{2} a^{3} b^{2} e^{5}+360 x^{2} a^{2} b^{3} d \,e^{4}+144 x^{2} a \,b^{4} d^{2} e^{3}+36 x^{2} b^{5} d^{3} e^{2}+315 a^{4} b \,e^{5} x +180 a^{3} b^{2} d \,e^{4} x +90 x \,a^{2} b^{3} d^{2} e^{3}+36 x a \,b^{4} d^{3} e^{2}+9 b^{5} d^{4} e x +56 e^{5} a^{5}+35 a^{4} b d \,e^{4}+20 a^{3} b^{2} d^{2} e^{3}+10 a^{2} b^{3} d^{3} e^{2}+4 a \,b^{4} d^{4} e +b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{504 e^{6} \left (e x +d \right )^{9} \left (b x +a \right )^{5}}\) | \(288\) |
| default | \(-\frac {\left (126 x^{5} e^{5} b^{5}+504 x^{4} a \,b^{4} e^{5}+126 x^{4} b^{5} d \,e^{4}+840 x^{3} a^{2} b^{3} e^{5}+336 x^{3} a \,b^{4} d \,e^{4}+84 x^{3} b^{5} d^{2} e^{3}+720 x^{2} a^{3} b^{2} e^{5}+360 x^{2} a^{2} b^{3} d \,e^{4}+144 x^{2} a \,b^{4} d^{2} e^{3}+36 x^{2} b^{5} d^{3} e^{2}+315 a^{4} b \,e^{5} x +180 a^{3} b^{2} d \,e^{4} x +90 x \,a^{2} b^{3} d^{2} e^{3}+36 x a \,b^{4} d^{3} e^{2}+9 b^{5} d^{4} e x +56 e^{5} a^{5}+35 a^{4} b d \,e^{4}+20 a^{3} b^{2} d^{2} e^{3}+10 a^{2} b^{3} d^{3} e^{2}+4 a \,b^{4} d^{4} e +b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{504 e^{6} \left (e x +d \right )^{9} \left (b x +a \right )^{5}}\) | \(288\) |
| orering | \(-\frac {\left (126 x^{5} e^{5} b^{5}+504 x^{4} a \,b^{4} e^{5}+126 x^{4} b^{5} d \,e^{4}+840 x^{3} a^{2} b^{3} e^{5}+336 x^{3} a \,b^{4} d \,e^{4}+84 x^{3} b^{5} d^{2} e^{3}+720 x^{2} a^{3} b^{2} e^{5}+360 x^{2} a^{2} b^{3} d \,e^{4}+144 x^{2} a \,b^{4} d^{2} e^{3}+36 x^{2} b^{5} d^{3} e^{2}+315 a^{4} b \,e^{5} x +180 a^{3} b^{2} d \,e^{4} x +90 x \,a^{2} b^{3} d^{2} e^{3}+36 x a \,b^{4} d^{3} e^{2}+9 b^{5} d^{4} e x +56 e^{5} a^{5}+35 a^{4} b d \,e^{4}+20 a^{3} b^{2} d^{2} e^{3}+10 a^{2} b^{3} d^{3} e^{2}+4 a \,b^{4} d^{4} e +b^{5} d^{5}\right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{\frac {5}{2}}}{504 e^{6} \left (b x +a \right )^{5} \left (e x +d \right )^{9}}\) | \(297\) |
Input:
int((b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^10,x,method=_RETURNVERBOSE)
Output:
((b*x+a)^2)^(1/2)/(b*x+a)*(-1/4*b^5/e*x^5-1/4/e^2*b^4*(4*a*e+b*d)*x^4-1/6* b^3/e^3*(10*a^2*e^2+4*a*b*d*e+b^2*d^2)*x^3-1/14*b^2/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^2-1/56*b/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-1/504/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))/(e*x+ d)^9
Leaf count of result is larger than twice the leaf count of optimal. 348 vs. \(2 (148) = 296\).
Time = 0.08 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.74 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{10}} \, dx=-\frac {126 \, b^{5} e^{5} x^{5} + b^{5} d^{5} + 4 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} + 20 \, a^{3} b^{2} d^{2} e^{3} + 35 \, a^{4} b d e^{4} + 56 \, a^{5} e^{5} + 126 \, {\left (b^{5} d e^{4} + 4 \, a b^{4} e^{5}\right )} x^{4} + 84 \, {\left (b^{5} d^{2} e^{3} + 4 \, a b^{4} d e^{4} + 10 \, a^{2} b^{3} e^{5}\right )} x^{3} + 36 \, {\left (b^{5} d^{3} e^{2} + 4 \, a b^{4} d^{2} e^{3} + 10 \, a^{2} b^{3} d e^{4} + 20 \, a^{3} b^{2} e^{5}\right )} x^{2} + 9 \, {\left (b^{5} d^{4} e + 4 \, a b^{4} d^{3} e^{2} + 10 \, a^{2} b^{3} d^{2} e^{3} + 20 \, a^{3} b^{2} d e^{4} + 35 \, a^{4} b e^{5}\right )} x}{504 \, {\left (e^{15} x^{9} + 9 \, d e^{14} x^{8} + 36 \, d^{2} e^{13} x^{7} + 84 \, d^{3} e^{12} x^{6} + 126 \, d^{4} e^{11} x^{5} + 126 \, d^{5} e^{10} x^{4} + 84 \, d^{6} e^{9} x^{3} + 36 \, d^{7} e^{8} x^{2} + 9 \, d^{8} e^{7} x + d^{9} e^{6}\right )}} \] Input:
integrate((b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^10,x, algorithm="fricas")
Output:
-1/504*(126*b^5*e^5*x^5 + b^5*d^5 + 4*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 + 2 0*a^3*b^2*d^2*e^3 + 35*a^4*b*d*e^4 + 56*a^5*e^5 + 126*(b^5*d*e^4 + 4*a*b^4 *e^5)*x^4 + 84*(b^5*d^2*e^3 + 4*a*b^4*d*e^4 + 10*a^2*b^3*e^5)*x^3 + 36*(b^ 5*d^3*e^2 + 4*a*b^4*d^2*e^3 + 10*a^2*b^3*d*e^4 + 20*a^3*b^2*e^5)*x^2 + 9*( b^5*d^4*e + 4*a*b^4*d^3*e^2 + 10*a^2*b^3*d^2*e^3 + 20*a^3*b^2*d*e^4 + 35*a ^4*b*e^5)*x)/(e^15*x^9 + 9*d*e^14*x^8 + 36*d^2*e^13*x^7 + 84*d^3*e^12*x^6 + 126*d^4*e^11*x^5 + 126*d^5*e^10*x^4 + 84*d^6*e^9*x^3 + 36*d^7*e^8*x^2 + 9*d^8*e^7*x + d^9*e^6)
\[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{10}} \, dx=\int \frac {\left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{10}}\, dx \] Input:
integrate((b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**10,x)
Output:
Integral(((a + b*x)**2)**(5/2)/(d + e*x)**10, x)
Exception generated. \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{10}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^10,x, algorithm="maxima")
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. 465 vs. \(2 (148) = 296\).
Time = 0.35 (sec) , antiderivative size = 465, normalized size of antiderivative = 2.32 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{10}} \, dx=\frac {b^{9} \mathrm {sgn}\left (b x + a\right )}{504 \, {\left (b^{4} d^{4} e^{6} - 4 \, a b^{3} d^{3} e^{7} + 6 \, a^{2} b^{2} d^{2} e^{8} - 4 \, a^{3} b d e^{9} + a^{4} e^{10}\right )}} - \frac {126 \, b^{5} e^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + 126 \, b^{5} d e^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 504 \, a b^{4} e^{5} x^{4} \mathrm {sgn}\left (b x + a\right ) + 84 \, b^{5} d^{2} e^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 336 \, a b^{4} d e^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + 840 \, a^{2} b^{3} e^{5} x^{3} \mathrm {sgn}\left (b x + a\right ) + 36 \, b^{5} d^{3} e^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 144 \, a b^{4} d^{2} e^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + 360 \, a^{2} b^{3} d e^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + 720 \, a^{3} b^{2} e^{5} x^{2} \mathrm {sgn}\left (b x + a\right ) + 9 \, b^{5} d^{4} e x \mathrm {sgn}\left (b x + a\right ) + 36 \, a b^{4} d^{3} e^{2} x \mathrm {sgn}\left (b x + a\right ) + 90 \, a^{2} b^{3} d^{2} e^{3} x \mathrm {sgn}\left (b x + a\right ) + 180 \, a^{3} b^{2} d e^{4} x \mathrm {sgn}\left (b x + a\right ) + 315 \, a^{4} b e^{5} x \mathrm {sgn}\left (b x + a\right ) + b^{5} d^{5} \mathrm {sgn}\left (b x + a\right ) + 4 \, a b^{4} d^{4} e \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{3} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + 20 \, a^{3} b^{2} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{4} b d e^{4} \mathrm {sgn}\left (b x + a\right ) + 56 \, a^{5} e^{5} \mathrm {sgn}\left (b x + a\right )}{504 \, {\left (e x + d\right )}^{9} e^{6}} \] Input:
integrate((b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^10,x, algorithm="giac")
Output:
1/504*b^9*sgn(b*x + a)/(b^4*d^4*e^6 - 4*a*b^3*d^3*e^7 + 6*a^2*b^2*d^2*e^8 - 4*a^3*b*d*e^9 + a^4*e^10) - 1/504*(126*b^5*e^5*x^5*sgn(b*x + a) + 126*b^ 5*d*e^4*x^4*sgn(b*x + a) + 504*a*b^4*e^5*x^4*sgn(b*x + a) + 84*b^5*d^2*e^3 *x^3*sgn(b*x + a) + 336*a*b^4*d*e^4*x^3*sgn(b*x + a) + 840*a^2*b^3*e^5*x^3 *sgn(b*x + a) + 36*b^5*d^3*e^2*x^2*sgn(b*x + a) + 144*a*b^4*d^2*e^3*x^2*sg n(b*x + a) + 360*a^2*b^3*d*e^4*x^2*sgn(b*x + a) + 720*a^3*b^2*e^5*x^2*sgn( b*x + a) + 9*b^5*d^4*e*x*sgn(b*x + a) + 36*a*b^4*d^3*e^2*x*sgn(b*x + a) + 90*a^2*b^3*d^2*e^3*x*sgn(b*x + a) + 180*a^3*b^2*d*e^4*x*sgn(b*x + a) + 315 *a^4*b*e^5*x*sgn(b*x + a) + b^5*d^5*sgn(b*x + a) + 4*a*b^4*d^4*e*sgn(b*x + a) + 10*a^2*b^3*d^3*e^2*sgn(b*x + a) + 20*a^3*b^2*d^2*e^3*sgn(b*x + a) + 35*a^4*b*d*e^4*sgn(b*x + a) + 56*a^5*e^5*sgn(b*x + a))/((e*x + d)^9*e^6)
Time = 5.36 (sec) , antiderivative size = 687, normalized size of antiderivative = 3.44 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{10}} \, dx=\frac {\left (\frac {4\,b^5\,d-5\,a\,b^4\,e}{5\,e^6}+\frac {b^5\,d}{5\,e^6}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^5}-\frac {\left (\frac {5\,a^4\,b\,e^4-10\,a^3\,b^2\,d\,e^3+10\,a^2\,b^3\,d^2\,e^2-5\,a\,b^4\,d^3\,e+b^5\,d^4}{8\,e^6}+\frac {d\,\left (\frac {-10\,a^3\,b^2\,e^4+10\,a^2\,b^3\,d\,e^3-5\,a\,b^4\,d^2\,e^2+b^5\,d^3\,e}{8\,e^6}+\frac {d\,\left (\frac {d\,\left (\frac {b^5\,d}{8\,e^3}-\frac {b^4\,\left (5\,a\,e-b\,d\right )}{8\,e^3}\right )}{e}+\frac {b^3\,\left (10\,a^2\,e^2-5\,a\,b\,d\,e+b^2\,d^2\right )}{8\,e^4}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^8}-\frac {\left (\frac {10\,a^2\,b^3\,e^2-15\,a\,b^4\,d\,e+6\,b^5\,d^2}{6\,e^6}+\frac {d\,\left (\frac {b^5\,d}{6\,e^5}-\frac {b^4\,\left (5\,a\,e-3\,b\,d\right )}{6\,e^5}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^6}-\frac {\left (\frac {a^5}{9\,e}-\frac {d\,\left (\frac {5\,a^4\,b}{9\,e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {5\,a\,b^4}{9\,e}-\frac {b^5\,d}{9\,e^2}\right )}{e}-\frac {10\,a^2\,b^3}{9\,e}\right )}{e}+\frac {10\,a^3\,b^2}{9\,e}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^9}+\frac {\left (\frac {-10\,a^3\,b^2\,e^3+20\,a^2\,b^3\,d\,e^2-15\,a\,b^4\,d^2\,e+4\,b^5\,d^3}{7\,e^6}+\frac {d\,\left (\frac {d\,\left (\frac {b^5\,d}{7\,e^4}-\frac {b^4\,\left (5\,a\,e-2\,b\,d\right )}{7\,e^4}\right )}{e}+\frac {b^3\,\left (10\,a^2\,e^2-10\,a\,b\,d\,e+3\,b^2\,d^2\right )}{7\,e^5}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^7}-\frac {b^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{4\,e^6\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^4} \] Input:
int((a^2 + b^2*x^2 + 2*a*b*x)^(5/2)/(d + e*x)^10,x)
Output:
(((4*b^5*d - 5*a*b^4*e)/(5*e^6) + (b^5*d)/(5*e^6))*(a^2 + b^2*x^2 + 2*a*b* x)^(1/2))/((a + b*x)*(d + e*x)^5) - (((b^5*d^4 + 5*a^4*b*e^4 - 10*a^3*b^2* d*e^3 + 10*a^2*b^3*d^2*e^2 - 5*a*b^4*d^3*e)/(8*e^6) + (d*((b^5*d^3*e - 10* a^3*b^2*e^4 - 5*a*b^4*d^2*e^2 + 10*a^2*b^3*d*e^3)/(8*e^6) + (d*((d*((b^5*d )/(8*e^3) - (b^4*(5*a*e - b*d))/(8*e^3)))/e + (b^3*(10*a^2*e^2 + b^2*d^2 - 5*a*b*d*e))/(8*e^4)))/e))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)* (d + e*x)^8) - (((6*b^5*d^2 + 10*a^2*b^3*e^2 - 15*a*b^4*d*e)/(6*e^6) + (d* ((b^5*d)/(6*e^5) - (b^4*(5*a*e - 3*b*d))/(6*e^5)))/e)*(a^2 + b^2*x^2 + 2*a *b*x)^(1/2))/((a + b*x)*(d + e*x)^6) - ((a^5/(9*e) - (d*((5*a^4*b)/(9*e) - (d*((d*((d*((5*a*b^4)/(9*e) - (b^5*d)/(9*e^2)))/e - (10*a^2*b^3)/(9*e)))/ e + (10*a^3*b^2)/(9*e)))/e))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x )*(d + e*x)^9) + (((4*b^5*d^3 - 10*a^3*b^2*e^3 + 20*a^2*b^3*d*e^2 - 15*a*b ^4*d^2*e)/(7*e^6) + (d*((d*((b^5*d)/(7*e^4) - (b^4*(5*a*e - 2*b*d))/(7*e^4 )))/e + (b^3*(10*a^2*e^2 + 3*b^2*d^2 - 10*a*b*d*e))/(7*e^5)))/e)*(a^2 + b^ 2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^7) - (b^5*(a^2 + b^2*x^2 + 2* a*b*x)^(1/2))/(4*e^6*(a + b*x)*(d + e*x)^4)
Time = 0.26 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.80 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{10}} \, dx=\frac {-126 b^{5} e^{5} x^{5}-504 a \,b^{4} e^{5} x^{4}-126 b^{5} d \,e^{4} x^{4}-840 a^{2} b^{3} e^{5} x^{3}-336 a \,b^{4} d \,e^{4} x^{3}-84 b^{5} d^{2} e^{3} x^{3}-720 a^{3} b^{2} e^{5} x^{2}-360 a^{2} b^{3} d \,e^{4} x^{2}-144 a \,b^{4} d^{2} e^{3} x^{2}-36 b^{5} d^{3} e^{2} x^{2}-315 a^{4} b \,e^{5} x -180 a^{3} b^{2} d \,e^{4} x -90 a^{2} b^{3} d^{2} e^{3} x -36 a \,b^{4} d^{3} e^{2} x -9 b^{5} d^{4} e x -56 a^{5} e^{5}-35 a^{4} b d \,e^{4}-20 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}-4 a \,b^{4} d^{4} e -b^{5} d^{5}}{504 e^{6} \left (e^{9} x^{9}+9 d \,e^{8} x^{8}+36 d^{2} e^{7} x^{7}+84 d^{3} e^{6} x^{6}+126 d^{4} e^{5} x^{5}+126 d^{5} e^{4} x^{4}+84 d^{6} e^{3} x^{3}+36 d^{7} e^{2} x^{2}+9 d^{8} e x +d^{9}\right )} \] Input:
int((b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^10,x)
Output:
( - 56*a**5*e**5 - 35*a**4*b*d*e**4 - 315*a**4*b*e**5*x - 20*a**3*b**2*d** 2*e**3 - 180*a**3*b**2*d*e**4*x - 720*a**3*b**2*e**5*x**2 - 10*a**2*b**3*d **3*e**2 - 90*a**2*b**3*d**2*e**3*x - 360*a**2*b**3*d*e**4*x**2 - 840*a**2 *b**3*e**5*x**3 - 4*a*b**4*d**4*e - 36*a*b**4*d**3*e**2*x - 144*a*b**4*d** 2*e**3*x**2 - 336*a*b**4*d*e**4*x**3 - 504*a*b**4*e**5*x**4 - b**5*d**5 - 9*b**5*d**4*e*x - 36*b**5*d**3*e**2*x**2 - 84*b**5*d**2*e**3*x**3 - 126*b* *5*d*e**4*x**4 - 126*b**5*e**5*x**5)/(504*e**6*(d**9 + 9*d**8*e*x + 36*d** 7*e**2*x**2 + 84*d**6*e**3*x**3 + 126*d**5*e**4*x**4 + 126*d**4*e**5*x**5 + 84*d**3*e**6*x**6 + 36*d**2*e**7*x**7 + 9*d*e**8*x**8 + e**9*x**9))