Integrand size = 30, antiderivative size = 381 \[ \int \frac {1}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {231 e^3}{64 (b d-a e)^4 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{4 (b d-a e) (a+b x)^3 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {11 e}{24 (b d-a e)^2 (a+b x)^2 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {33 e^2}{32 (b d-a e)^3 (a+b x) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {385 e^4 (a+b x)}{64 (b d-a e)^5 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1155 b e^4 (a+b x)}{64 (b d-a e)^6 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1155 b^{3/2} e^4 (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 (b d-a e)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
231/64*e^3/(-a*e+b*d)^4/(e*x+d)^(3/2)/((b*x+a)^2)^(1/2)-1/4/(-a*e+b*d)/(b* x+a)^3/(e*x+d)^(3/2)/((b*x+a)^2)^(1/2)+11/24*e/(-a*e+b*d)^2/(b*x+a)^2/(e*x +d)^(3/2)/((b*x+a)^2)^(1/2)-33/32*e^2/(-a*e+b*d)^3/(b*x+a)/(e*x+d)^(3/2)/( (b*x+a)^2)^(1/2)+385/64*e^4*(b*x+a)/(-a*e+b*d)^5/(e*x+d)^(3/2)/((b*x+a)^2) ^(1/2)-1155/64*b^(3/2)*e^4*(b*x+a)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d )^(1/2))/(-a*e+b*d)^(13/2)/((b*x+a)^2)^(1/2)+1155/64*b*e^4*(b*x+a)/(-a*e+b *d)^6/(e*x+d)^(1/2)/((b*x+a)^2)^(1/2)
Time = 1.33 (sec) , antiderivative size = 301, normalized size of antiderivative = 0.79 \[ \int \frac {1}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {e^4 (a+b x)^5 \left (\frac {-128 a^5 e^5+128 a^4 b e^4 (16 d+11 e x)+a^3 b^2 e^3 \left (2295 d^2+12782 d e x+9207 e^2 x^2\right )+a^2 b^3 e^2 \left (-1030 d^3+3795 d^2 e x+22968 d e^2 x^2+16863 e^3 x^3\right )+a b^4 e \left (328 d^4-748 d^3 e x+2673 d^2 e^2 x^2+17094 d e^3 x^3+12705 e^4 x^4\right )+b^5 \left (-48 d^5+88 d^4 e x-198 d^3 e^2 x^2+693 d^2 e^3 x^3+4620 d e^4 x^4+3465 e^5 x^5\right )}{e^4 (b d-a e)^6 (a+b x)^4 (d+e x)^{3/2}}+\frac {3465 b^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{13/2}}\right )}{192 \left ((a+b x)^2\right )^{5/2}} \]
(e^4*(a + b*x)^5*((-128*a^5*e^5 + 128*a^4*b*e^4*(16*d + 11*e*x) + a^3*b^2* e^3*(2295*d^2 + 12782*d*e*x + 9207*e^2*x^2) + a^2*b^3*e^2*(-1030*d^3 + 379 5*d^2*e*x + 22968*d*e^2*x^2 + 16863*e^3*x^3) + a*b^4*e*(328*d^4 - 748*d^3* e*x + 2673*d^2*e^2*x^2 + 17094*d*e^3*x^3 + 12705*e^4*x^4) + b^5*(-48*d^5 + 88*d^4*e*x - 198*d^3*e^2*x^2 + 693*d^2*e^3*x^3 + 4620*d*e^4*x^4 + 3465*e^ 5*x^5))/(e^4*(b*d - a*e)^6*(a + b*x)^4*(d + e*x)^(3/2)) + (3465*b^(3/2)*Ar cTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(-(b*d) + a*e)^(13/2)))/ (192*((a + b*x)^2)^(5/2))
Time = 0.36 (sec) , antiderivative size = 313, normalized size of antiderivative = 0.82, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1102, 27, 52, 52, 52, 52, 61, 61, 73, 221}
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.
| \(\displaystyle \int \frac {1}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2} (d+e x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1102 |
| \(\displaystyle \frac {b^5 (a+b x) \int \frac {1}{b^5 (a+b x)^5 (d+e x)^{5/2}}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(a+b x) \int \frac {1}{(a+b x)^5 (d+e x)^{5/2}}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {(a+b x) \left (-\frac {11 e \int \frac {1}{(a+b x)^4 (d+e x)^{5/2}}dx}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{3/2} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {(a+b x) \left (-\frac {11 e \left (-\frac {3 e \int \frac {1}{(a+b x)^3 (d+e x)^{5/2}}dx}{2 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{3/2} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{3/2} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {(a+b x) \left (-\frac {11 e \left (-\frac {3 e \left (-\frac {7 e \int \frac {1}{(a+b x)^2 (d+e x)^{5/2}}dx}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{3/2} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{3/2} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {(a+b x) \left (-\frac {11 e \left (-\frac {3 e \left (-\frac {7 e \left (-\frac {5 e \int \frac {1}{(a+b x) (d+e x)^{5/2}}dx}{2 (b d-a e)}-\frac {1}{(a+b x) (d+e x)^{3/2} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{3/2} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{3/2} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {(a+b x) \left (-\frac {11 e \left (-\frac {3 e \left (-\frac {7 e \left (-\frac {5 e \left (\frac {b \int \frac {1}{(a+b x) (d+e x)^{3/2}}dx}{b d-a e}+\frac {2}{3 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) (d+e x)^{3/2} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{3/2} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{3/2} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {(a+b x) \left (-\frac {11 e \left (-\frac {3 e \left (-\frac {7 e \left (-\frac {5 e \left (\frac {b \left (\frac {b \int \frac {1}{(a+b x) \sqrt {d+e x}}dx}{b d-a e}+\frac {2}{\sqrt {d+e x} (b d-a e)}\right )}{b d-a e}+\frac {2}{3 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) (d+e x)^{3/2} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{3/2} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{3/2} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {(a+b x) \left (-\frac {11 e \left (-\frac {3 e \left (-\frac {7 e \left (-\frac {5 e \left (\frac {b \left (\frac {2 b \int \frac {1}{a+\frac {b (d+e x)}{e}-\frac {b d}{e}}d\sqrt {d+e x}}{e (b d-a e)}+\frac {2}{\sqrt {d+e x} (b d-a e)}\right )}{b d-a e}+\frac {2}{3 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) (d+e x)^{3/2} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{3/2} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{3/2} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {(a+b x) \left (-\frac {11 e \left (-\frac {3 e \left (-\frac {7 e \left (-\frac {5 e \left (\frac {b \left (\frac {2}{\sqrt {d+e x} (b d-a e)}-\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{3/2}}\right )}{b d-a e}+\frac {2}{3 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) (d+e x)^{3/2} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{3/2} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{3/2} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
((a + b*x)*(-1/4*1/((b*d - a*e)*(a + b*x)^4*(d + e*x)^(3/2)) - (11*e*(-1/3 *1/((b*d - a*e)*(a + b*x)^3*(d + e*x)^(3/2)) - (3*e*(-1/2*1/((b*d - a*e)*( a + b*x)^2*(d + e*x)^(3/2)) - (7*e*(-(1/((b*d - a*e)*(a + b*x)*(d + e*x)^( 3/2))) - (5*e*(2/(3*(b*d - a*e)*(d + e*x)^(3/2)) + (b*(2/((b*d - a*e)*Sqrt [d + e*x]) - (2*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/ (b*d - a*e)^(3/2)))/(b*d - a*e)))/(2*(b*d - a*e))))/(4*(b*d - a*e))))/(2*( b*d - a*e))))/(8*(b*d - a*e))))/Sqrt[a^2 + 2*a*b*x + b^2*x^2]
3.18.29.3.1 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] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
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*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(762\) vs. \(2(268)=536\).
Time = 2.56 (sec) , antiderivative size = 763, normalized size of antiderivative = 2.00
| method | result | size |
| default | \(\frac {\left (3465 \left (e x +d \right )^{\frac {3}{2}} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) b^{6} e^{4} x^{4}+13860 \left (e x +d \right )^{\frac {3}{2}} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a \,b^{5} e^{4} x^{3}+13860 \left (e x +d \right )^{\frac {3}{2}} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{3} b^{3} e^{4} x +16863 \sqrt {\left (a e -b d \right ) b}\, a^{2} b^{3} e^{5} x^{3}+2295 \sqrt {\left (a e -b d \right ) b}\, a^{3} b^{2} d^{2} e^{3}+17094 \sqrt {\left (a e -b d \right ) b}\, a \,b^{4} d \,e^{4} x^{3}+22968 \sqrt {\left (a e -b d \right ) b}\, a^{2} b^{3} d \,e^{4} x^{2}+2673 \sqrt {\left (a e -b d \right ) b}\, a \,b^{4} d^{2} e^{3} x^{2}+12782 \sqrt {\left (a e -b d \right ) b}\, a^{3} b^{2} d \,e^{4} x +3795 \sqrt {\left (a e -b d \right ) b}\, a^{2} b^{3} d^{2} e^{3} x -748 \sqrt {\left (a e -b d \right ) b}\, a \,b^{4} d^{3} e^{2} x +20790 \left (e x +d \right )^{\frac {3}{2}} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{2} b^{4} e^{4} x^{2}-48 \sqrt {\left (a e -b d \right ) b}\, b^{5} d^{5}+3465 \left (e x +d \right )^{\frac {3}{2}} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{4} b^{2} e^{4}+3465 \sqrt {\left (a e -b d \right ) b}\, b^{5} e^{5} x^{5}+4620 \sqrt {\left (a e -b d \right ) b}\, b^{5} d \,e^{4} x^{4}+693 \sqrt {\left (a e -b d \right ) b}\, b^{5} d^{2} e^{3} x^{3}-198 \sqrt {\left (a e -b d \right ) b}\, b^{5} d^{3} e^{2} x^{2}+88 \sqrt {\left (a e -b d \right ) b}\, b^{5} d^{4} e x +2048 \sqrt {\left (a e -b d \right ) b}\, a^{4} b d \,e^{4}-1030 \sqrt {\left (a e -b d \right ) b}\, a^{2} b^{3} d^{3} e^{2}+328 \sqrt {\left (a e -b d \right ) b}\, a \,b^{4} d^{4} e -128 \sqrt {\left (a e -b d \right ) b}\, a^{5} e^{5}+12705 \sqrt {\left (a e -b d \right ) b}\, a \,b^{4} e^{5} x^{4}+9207 \sqrt {\left (a e -b d \right ) b}\, a^{3} b^{2} e^{5} x^{2}+1408 \sqrt {\left (a e -b d \right ) b}\, a^{4} b \,e^{5} x \right ) \left (b x +a \right )}{192 \sqrt {\left (a e -b d \right ) b}\, \left (e x +d \right )^{\frac {3}{2}} \left (a e -b d \right )^{6} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\) | \(763\) |
1/192*(3465*(e*x+d)^(3/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*b^6* e^4*x^4+13860*(e*x+d)^(3/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*a* b^5*e^4*x^3+13860*(e*x+d)^(3/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2) )*a^3*b^3*e^4*x+16863*((a*e-b*d)*b)^(1/2)*a^2*b^3*e^5*x^3+2295*((a*e-b*d)* b)^(1/2)*a^3*b^2*d^2*e^3+17094*((a*e-b*d)*b)^(1/2)*a*b^4*d*e^4*x^3+22968*( (a*e-b*d)*b)^(1/2)*a^2*b^3*d*e^4*x^2+2673*((a*e-b*d)*b)^(1/2)*a*b^4*d^2*e^ 3*x^2+12782*((a*e-b*d)*b)^(1/2)*a^3*b^2*d*e^4*x+3795*((a*e-b*d)*b)^(1/2)*a ^2*b^3*d^2*e^3*x-748*((a*e-b*d)*b)^(1/2)*a*b^4*d^3*e^2*x+20790*(e*x+d)^(3/ 2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*a^2*b^4*e^4*x^2-48*((a*e-b* d)*b)^(1/2)*b^5*d^5+3465*(e*x+d)^(3/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b )^(1/2))*a^4*b^2*e^4+3465*((a*e-b*d)*b)^(1/2)*b^5*e^5*x^5+4620*((a*e-b*d)* b)^(1/2)*b^5*d*e^4*x^4+693*((a*e-b*d)*b)^(1/2)*b^5*d^2*e^3*x^3-198*((a*e-b *d)*b)^(1/2)*b^5*d^3*e^2*x^2+88*((a*e-b*d)*b)^(1/2)*b^5*d^4*e*x+2048*((a*e -b*d)*b)^(1/2)*a^4*b*d*e^4-1030*((a*e-b*d)*b)^(1/2)*a^2*b^3*d^3*e^2+328*(( a*e-b*d)*b)^(1/2)*a*b^4*d^4*e-128*((a*e-b*d)*b)^(1/2)*a^5*e^5+12705*((a*e- b*d)*b)^(1/2)*a*b^4*e^5*x^4+9207*((a*e-b*d)*b)^(1/2)*a^3*b^2*e^5*x^2+1408* ((a*e-b*d)*b)^(1/2)*a^4*b*e^5*x)*(b*x+a)/((a*e-b*d)*b)^(1/2)/(e*x+d)^(3/2) /(a*e-b*d)^6/((b*x+a)^2)^(5/2)
Leaf count of result is larger than twice the leaf count of optimal. 1242 vs. \(2 (268) = 536\).
Time = 0.78 (sec) , antiderivative size = 2494, normalized size of antiderivative = 6.55 \[ \int \frac {1}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Too large to display} \]
[1/384*(3465*(b^5*e^6*x^6 + a^4*b*d^2*e^4 + 2*(b^5*d*e^5 + 2*a*b^4*e^6)*x^ 5 + (b^5*d^2*e^4 + 8*a*b^4*d*e^5 + 6*a^2*b^3*e^6)*x^4 + 4*(a*b^4*d^2*e^4 + 3*a^2*b^3*d*e^5 + a^3*b^2*e^6)*x^3 + (6*a^2*b^3*d^2*e^4 + 8*a^3*b^2*d*e^5 + a^4*b*e^6)*x^2 + 2*(2*a^3*b^2*d^2*e^4 + a^4*b*d*e^5)*x)*sqrt(b/(b*d - a *e))*log((b*e*x + 2*b*d - a*e - 2*(b*d - a*e)*sqrt(e*x + d)*sqrt(b/(b*d - a*e)))/(b*x + a)) + 2*(3465*b^5*e^5*x^5 - 48*b^5*d^5 + 328*a*b^4*d^4*e - 1 030*a^2*b^3*d^3*e^2 + 2295*a^3*b^2*d^2*e^3 + 2048*a^4*b*d*e^4 - 128*a^5*e^ 5 + 1155*(4*b^5*d*e^4 + 11*a*b^4*e^5)*x^4 + 231*(3*b^5*d^2*e^3 + 74*a*b^4* d*e^4 + 73*a^2*b^3*e^5)*x^3 - 99*(2*b^5*d^3*e^2 - 27*a*b^4*d^2*e^3 - 232*a ^2*b^3*d*e^4 - 93*a^3*b^2*e^5)*x^2 + 11*(8*b^5*d^4*e - 68*a*b^4*d^3*e^2 + 345*a^2*b^3*d^2*e^3 + 1162*a^3*b^2*d*e^4 + 128*a^4*b*e^5)*x)*sqrt(e*x + d) )/(a^4*b^6*d^8 - 6*a^5*b^5*d^7*e + 15*a^6*b^4*d^6*e^2 - 20*a^7*b^3*d^5*e^3 + 15*a^8*b^2*d^4*e^4 - 6*a^9*b*d^3*e^5 + a^10*d^2*e^6 + (b^10*d^6*e^2 - 6 *a*b^9*d^5*e^3 + 15*a^2*b^8*d^4*e^4 - 20*a^3*b^7*d^3*e^5 + 15*a^4*b^6*d^2* e^6 - 6*a^5*b^5*d*e^7 + a^6*b^4*e^8)*x^6 + 2*(b^10*d^7*e - 4*a*b^9*d^6*e^2 + 3*a^2*b^8*d^5*e^3 + 10*a^3*b^7*d^4*e^4 - 25*a^4*b^6*d^3*e^5 + 24*a^5*b^ 5*d^2*e^6 - 11*a^6*b^4*d*e^7 + 2*a^7*b^3*e^8)*x^5 + (b^10*d^8 + 2*a*b^9*d^ 7*e - 27*a^2*b^8*d^6*e^2 + 64*a^3*b^7*d^5*e^3 - 55*a^4*b^6*d^4*e^4 - 6*a^5 *b^5*d^3*e^5 + 43*a^6*b^4*d^2*e^6 - 28*a^7*b^3*d*e^7 + 6*a^8*b^2*e^8)*x^4 + 4*(a*b^9*d^8 - 3*a^2*b^8*d^7*e - 2*a^3*b^7*d^6*e^2 + 19*a^4*b^6*d^5*e...
\[ \int \frac {1}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {1}{\left (d + e x\right )^{\frac {5}{2}} \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {1}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 635 vs. \(2 (268) = 536\).
Time = 0.31 (sec) , antiderivative size = 635, normalized size of antiderivative = 1.67 \[ \int \frac {1}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {1155 \, b^{2} e^{4} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{64 \, {\left (b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) - 6 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) - 20 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - 6 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + a^{6} e^{6} \mathrm {sgn}\left (b x + a\right )\right )} \sqrt {-b^{2} d + a b e}} + \frac {2 \, {\left (15 \, {\left (e x + d\right )} b e^{4} + b d e^{4} - a e^{5}\right )}}{3 \, {\left (b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) - 6 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) - 20 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - 6 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + a^{6} e^{6} \mathrm {sgn}\left (b x + a\right )\right )} {\left (e x + d\right )}^{\frac {3}{2}}} + \frac {1545 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{5} e^{4} - 5153 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{5} d e^{4} + 5855 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{5} d^{2} e^{4} - 2295 \, \sqrt {e x + d} b^{5} d^{3} e^{4} + 5153 \, {\left (e x + d\right )}^{\frac {5}{2}} a b^{4} e^{5} - 11710 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{4} d e^{5} + 6885 \, \sqrt {e x + d} a b^{4} d^{2} e^{5} + 5855 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b^{3} e^{6} - 6885 \, \sqrt {e x + d} a^{2} b^{3} d e^{6} + 2295 \, \sqrt {e x + d} a^{3} b^{2} e^{7}}{192 \, {\left (b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) - 6 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) - 20 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - 6 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + a^{6} e^{6} \mathrm {sgn}\left (b x + a\right )\right )} {\left ({\left (e x + d\right )} b - b d + a e\right )}^{4}} \]
1155/64*b^2*e^4*arctan(sqrt(e*x + d)*b/sqrt(-b^2*d + a*b*e))/((b^6*d^6*sgn (b*x + a) - 6*a*b^5*d^5*e*sgn(b*x + a) + 15*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) + 15*a^4*b^2*d^2*e^4*sgn(b*x + a) - 6*a^5 *b*d*e^5*sgn(b*x + a) + a^6*e^6*sgn(b*x + a))*sqrt(-b^2*d + a*b*e)) + 2/3* (15*(e*x + d)*b*e^4 + b*d*e^4 - a*e^5)/((b^6*d^6*sgn(b*x + a) - 6*a*b^5*d^ 5*e*sgn(b*x + a) + 15*a^2*b^4*d^4*e^2*sgn(b*x + a) - 20*a^3*b^3*d^3*e^3*sg n(b*x + a) + 15*a^4*b^2*d^2*e^4*sgn(b*x + a) - 6*a^5*b*d*e^5*sgn(b*x + a) + a^6*e^6*sgn(b*x + a))*(e*x + d)^(3/2)) + 1/192*(1545*(e*x + d)^(7/2)*b^5 *e^4 - 5153*(e*x + d)^(5/2)*b^5*d*e^4 + 5855*(e*x + d)^(3/2)*b^5*d^2*e^4 - 2295*sqrt(e*x + d)*b^5*d^3*e^4 + 5153*(e*x + d)^(5/2)*a*b^4*e^5 - 11710*( e*x + d)^(3/2)*a*b^4*d*e^5 + 6885*sqrt(e*x + d)*a*b^4*d^2*e^5 + 5855*(e*x + d)^(3/2)*a^2*b^3*e^6 - 6885*sqrt(e*x + d)*a^2*b^3*d*e^6 + 2295*sqrt(e*x + d)*a^3*b^2*e^7)/((b^6*d^6*sgn(b*x + a) - 6*a*b^5*d^5*e*sgn(b*x + a) + 15 *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) + 15*a^4*b ^2*d^2*e^4*sgn(b*x + a) - 6*a^5*b*d*e^5*sgn(b*x + a) + a^6*e^6*sgn(b*x + a ))*((e*x + d)*b - b*d + a*e)^4)
Timed out. \[ \int \frac {1}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {1}{{\left (d+e\,x\right )}^{5/2}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]