Integrand size = 33, antiderivative size = 233 \[ \int \frac {a+b x}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {385 e^4}{64 (b d-a e)^5 (d+e x)^{3/2}}-\frac {1}{4 (b d-a e) (a+b x)^4 (d+e x)^{3/2}}+\frac {11 e}{24 (b d-a e)^2 (a+b x)^3 (d+e x)^{3/2}}-\frac {33 e^2}{32 (b d-a e)^3 (a+b x)^2 (d+e x)^{3/2}}+\frac {231 e^3}{64 (b d-a e)^4 (a+b x) (d+e x)^{3/2}}+\frac {1155 b e^4}{64 (b d-a e)^6 \sqrt {d+e x}}-\frac {1155 b^{3/2} e^4 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 (b d-a e)^{13/2}} \]
385/64*e^4/(-a*e+b*d)^5/(e*x+d)^(3/2)-1/4/(-a*e+b*d)/(b*x+a)^4/(e*x+d)^(3/ 2)+11/24*e/(-a*e+b*d)^2/(b*x+a)^3/(e*x+d)^(3/2)-33/32*e^2/(-a*e+b*d)^3/(b* x+a)^2/(e*x+d)^(3/2)+231/64*e^3/(-a*e+b*d)^4/(b*x+a)/(e*x+d)^(3/2)-1155/64 *b^(3/2)*e^4*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))/(-a*e+b*d)^(1 3/2)+1155/64*b*e^4/(-a*e+b*d)^6/(e*x+d)^(1/2)
Time = 0.40 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.20 \[ \int \frac {a+b x}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {1}{192} \left (\frac {-128 a^5 e^5+128 a^4 b e^4 (16 d+11 e x)+a^3 b^2 e^3 \left (2295 d^2+12782 d e x+9207 e^2 x^2\right )+a^2 b^3 e^2 \left (-1030 d^3+3795 d^2 e x+22968 d e^2 x^2+16863 e^3 x^3\right )+a b^4 e \left (328 d^4-748 d^3 e x+2673 d^2 e^2 x^2+17094 d e^3 x^3+12705 e^4 x^4\right )+b^5 \left (-48 d^5+88 d^4 e x-198 d^3 e^2 x^2+693 d^2 e^3 x^3+4620 d e^4 x^4+3465 e^5 x^5\right )}{(b d-a e)^6 (a+b x)^4 (d+e x)^{3/2}}+\frac {3465 b^{3/2} e^4 \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{13/2}}\right ) \]
((-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 + 1 2782*d*e*x + 9207*e^2*x^2) + a^2*b^3*e^2*(-1030*d^3 + 3795*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))/((b*d - a *e)^6*(a + b*x)^4*(d + e*x)^(3/2)) + (3465*b^(3/2)*e^4*ArcTan[(Sqrt[b]*Sqr t[d + e*x])/Sqrt[-(b*d) + a*e]])/(-(b*d) + a*e)^(13/2))/192
Time = 0.35 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.23, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.303, Rules used = {1184, 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.
| \(\displaystyle \int \frac {a+b x}{\left (a^2+2 a b x+b^2 x^2\right )^3 (d+e x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle b^6 \int \frac {1}{b^6 (a+b x)^5 (d+e x)^{5/2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {1}{(a+b x)^5 (d+e x)^{5/2}}dx\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -\frac {11 e \int \frac {1}{(a+b x)^4 (d+e x)^{5/2}}dx}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{3/2} (b d-a e)}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -\frac {11 e \left (-\frac {3 e \int \frac {1}{(a+b x)^3 (d+e x)^{5/2}}dx}{2 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{3/2} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{3/2} (b d-a e)}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -\frac {11 e \left (-\frac {3 e \left (-\frac {7 e \int \frac {1}{(a+b x)^2 (d+e x)^{5/2}}dx}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{3/2} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{3/2} (b d-a e)}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -\frac {11 e \left (-\frac {3 e \left (-\frac {7 e \left (-\frac {5 e \int \frac {1}{(a+b x) (d+e x)^{5/2}}dx}{2 (b d-a e)}-\frac {1}{(a+b x) (d+e x)^{3/2} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{3/2} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{3/2} (b d-a e)}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle -\frac {11 e \left (-\frac {3 e \left (-\frac {7 e \left (-\frac {5 e \left (\frac {b \int \frac {1}{(a+b x) (d+e x)^{3/2}}dx}{b d-a e}+\frac {2}{3 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) (d+e x)^{3/2} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{3/2} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{3/2} (b d-a e)}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle -\frac {11 e \left (-\frac {3 e \left (-\frac {7 e \left (-\frac {5 e \left (\frac {b \left (\frac {b \int \frac {1}{(a+b x) \sqrt {d+e x}}dx}{b d-a e}+\frac {2}{\sqrt {d+e x} (b d-a e)}\right )}{b d-a e}+\frac {2}{3 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) (d+e x)^{3/2} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{3/2} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{3/2} (b d-a e)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {11 e \left (-\frac {3 e \left (-\frac {7 e \left (-\frac {5 e \left (\frac {b \left (\frac {2 b \int \frac {1}{a+\frac {b (d+e x)}{e}-\frac {b d}{e}}d\sqrt {d+e x}}{e (b d-a e)}+\frac {2}{\sqrt {d+e x} (b d-a e)}\right )}{b d-a e}+\frac {2}{3 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) (d+e x)^{3/2} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{3/2} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{3/2} (b d-a e)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {11 e \left (-\frac {3 e \left (-\frac {7 e \left (-\frac {5 e \left (\frac {b \left (\frac {2}{\sqrt {d+e x} (b d-a e)}-\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{3/2}}\right )}{b d-a e}+\frac {2}{3 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) (d+e x)^{3/2} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{3/2} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{3/2} (b d-a e)}\) |
-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))
3.21.91.3.1 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_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^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}, x] && E qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Time = 0.40 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.97
| method | result | size |
| derivativedivides | \(2 e^{4} \left (\frac {b^{2} \left (\frac {\frac {515 \left (e x +d \right )^{\frac {7}{2}} b^{3}}{128}+\frac {5153 \left (a e -b d \right ) \left (e x +d \right )^{\frac {5}{2}} b^{2}}{384}+\left (\frac {5855}{384} a^{2} b \,e^{2}-\frac {5855}{192} a \,b^{2} d e +\frac {5855}{384} b^{3} d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (\frac {765}{128} a^{3} e^{3}-\frac {2295}{128} a^{2} b d \,e^{2}+\frac {2295}{128} a \,b^{2} d^{2} e -\frac {765}{128} b^{3} d^{3}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{4}}+\frac {1155 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{128 \sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right )^{6}}-\frac {1}{3 \left (a e -b d \right )^{5} \left (e x +d \right )^{\frac {3}{2}}}+\frac {5 b}{\left (a e -b d \right )^{6} \sqrt {e x +d}}\right )\) | \(227\) |
| default | \(2 e^{4} \left (\frac {b^{2} \left (\frac {\frac {515 \left (e x +d \right )^{\frac {7}{2}} b^{3}}{128}+\frac {5153 \left (a e -b d \right ) \left (e x +d \right )^{\frac {5}{2}} b^{2}}{384}+\left (\frac {5855}{384} a^{2} b \,e^{2}-\frac {5855}{192} a \,b^{2} d e +\frac {5855}{384} b^{3} d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (\frac {765}{128} a^{3} e^{3}-\frac {2295}{128} a^{2} b d \,e^{2}+\frac {2295}{128} a \,b^{2} d^{2} e -\frac {765}{128} b^{3} d^{3}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{4}}+\frac {1155 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{128 \sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right )^{6}}-\frac {1}{3 \left (a e -b d \right )^{5} \left (e x +d \right )^{\frac {3}{2}}}+\frac {5 b}{\left (a e -b d \right )^{6} \sqrt {e x +d}}\right )\) | \(227\) |
| pseudoelliptic | \(-\frac {2 \left (-\frac {3465 b^{2} e^{4} \left (e x +d \right )^{\frac {3}{2}} \left (b x +a \right )^{4} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{128}+\sqrt {\left (a e -b d \right ) b}\, \left (\left (-\frac {1155}{32} d \,e^{4} x^{4}-\frac {693}{128} d^{2} e^{3} x^{3}+\frac {99}{64} d^{3} e^{2} x^{2}-\frac {11}{16} d^{4} e x +\frac {3}{8} d^{5}-\frac {3465}{128} e^{5} x^{5}\right ) b^{5}-\frac {41 e \left (\frac {12705}{328} e^{4} x^{4}+\frac {8547}{164} d \,e^{3} x^{3}+\frac {2673}{328} d^{2} e^{2} x^{2}-\frac {187}{82} d^{3} e x +d^{4}\right ) a \,b^{4}}{16}+\frac {515 e^{2} \left (-\frac {16863}{1030} e^{3} x^{3}-\frac {11484}{515} d \,e^{2} x^{2}-\frac {759}{206} d^{2} e x +d^{3}\right ) a^{2} b^{3}}{64}-\frac {2295 e^{3} \left (\frac {341}{85} e^{2} x^{2}+\frac {12782}{2295} d e x +d^{2}\right ) a^{3} b^{2}}{128}-16 e^{4} a^{4} \left (\frac {11 e x}{16}+d \right ) b +e^{5} a^{5}\right )\right )}{3 \sqrt {\left (a e -b d \right ) b}\, \left (e x +d \right )^{\frac {3}{2}} \left (b x +a \right )^{4} \left (a e -b d \right )^{6}}\) | \(289\) |
2*e^4*(1/(a*e-b*d)^6*b^2*((515/128*(e*x+d)^(7/2)*b^3+5153/384*(a*e-b*d)*(e *x+d)^(5/2)*b^2+(5855/384*a^2*b*e^2-5855/192*a*b^2*d*e+5855/384*b^3*d^2)*( e*x+d)^(3/2)+(765/128*a^3*e^3-2295/128*a^2*b*d*e^2+2295/128*a*b^2*d^2*e-76 5/128*b^3*d^3)*(e*x+d)^(1/2))/(b*(e*x+d)+a*e-b*d)^4+1155/128/((a*e-b*d)*b) ^(1/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)))-1/3/(a*e-b*d)^5/(e*x+d )^(3/2)+5/(a*e-b*d)^6*b/(e*x+d)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 1242 vs. \(2 (197) = 394\).
Time = 0.61 (sec) , antiderivative size = 2494, normalized size of antiderivative = 10.70 \[ \int \frac {a+b x}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Too large to display} \]
[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...
Timed out. \[ \int \frac {a+b x}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {a+b x}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
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. 509 vs. \(2 (197) = 394\).
Time = 0.29 (sec) , antiderivative size = 509, normalized size of antiderivative = 2.18 \[ \int \frac {a+b x}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {1155 \, b^{2} e^{4} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{64 \, {\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} \sqrt {-b^{2} d + a b e}} + \frac {2 \, {\left (15 \, {\left (e x + d\right )} b e^{4} + b d e^{4} - a e^{5}\right )}}{3 \, {\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} {\left (e x + d\right )}^{\frac {3}{2}}} + \frac {1545 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{5} e^{4} - 5153 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{5} d e^{4} + 5855 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{5} d^{2} e^{4} - 2295 \, \sqrt {e x + d} b^{5} d^{3} e^{4} + 5153 \, {\left (e x + d\right )}^{\frac {5}{2}} a b^{4} e^{5} - 11710 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{4} d e^{5} + 6885 \, \sqrt {e x + d} a b^{4} d^{2} e^{5} + 5855 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b^{3} e^{6} - 6885 \, \sqrt {e x + d} a^{2} b^{3} d e^{6} + 2295 \, \sqrt {e x + d} a^{3} b^{2} e^{7}}{192 \, {\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} {\left ({\left (e x + d\right )} b - b d + a e\right )}^{4}} \]
1155/64*b^2*e^4*arctan(sqrt(e*x + d)*b/sqrt(-b^2*d + a*b*e))/((b^6*d^6 - 6 *a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^ 4 - 6*a^5*b*d*e^5 + a^6*e^6)*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 - 6*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 20* a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e^5 + a^6*e^6)*(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 - 6*a* b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e^5 + a^6*e^6)*((e*x + d)*b - b*d + a*e)^4)
Time = 11.53 (sec) , antiderivative size = 436, normalized size of antiderivative = 1.87 \[ \int \frac {a+b x}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\frac {3069\,b^2\,e^4\,{\left (d+e\,x\right )}^2}{64\,{\left (a\,e-b\,d\right )}^3}-\frac {2\,e^4}{3\,\left (a\,e-b\,d\right )}+\frac {5621\,b^3\,e^4\,{\left (d+e\,x\right )}^3}{64\,{\left (a\,e-b\,d\right )}^4}+\frac {4235\,b^4\,e^4\,{\left (d+e\,x\right )}^4}{64\,{\left (a\,e-b\,d\right )}^5}+\frac {1155\,b^5\,e^4\,{\left (d+e\,x\right )}^5}{64\,{\left (a\,e-b\,d\right )}^6}+\frac {22\,b\,e^4\,\left (d+e\,x\right )}{3\,{\left (a\,e-b\,d\right )}^2}}{b^4\,{\left (d+e\,x\right )}^{11/2}-\left (4\,b^4\,d-4\,a\,b^3\,e\right )\,{\left (d+e\,x\right )}^{9/2}+{\left (d+e\,x\right )}^{3/2}\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )+{\left (d+e\,x\right )}^{7/2}\,\left (6\,a^2\,b^2\,e^2-12\,a\,b^3\,d\,e+6\,b^4\,d^2\right )-{\left (d+e\,x\right )}^{5/2}\,\left (-4\,a^3\,b\,e^3+12\,a^2\,b^2\,d\,e^2-12\,a\,b^3\,d^2\,e+4\,b^4\,d^3\right )}+\frac {1155\,b^{3/2}\,e^4\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}\,\left (a^6\,e^6-6\,a^5\,b\,d\,e^5+15\,a^4\,b^2\,d^2\,e^4-20\,a^3\,b^3\,d^3\,e^3+15\,a^2\,b^4\,d^4\,e^2-6\,a\,b^5\,d^5\,e+b^6\,d^6\right )}{{\left (a\,e-b\,d\right )}^{13/2}}\right )}{64\,{\left (a\,e-b\,d\right )}^{13/2}} \]
((3069*b^2*e^4*(d + e*x)^2)/(64*(a*e - b*d)^3) - (2*e^4)/(3*(a*e - b*d)) + (5621*b^3*e^4*(d + e*x)^3)/(64*(a*e - b*d)^4) + (4235*b^4*e^4*(d + e*x)^4 )/(64*(a*e - b*d)^5) + (1155*b^5*e^4*(d + e*x)^5)/(64*(a*e - b*d)^6) + (22 *b*e^4*(d + e*x))/(3*(a*e - b*d)^2))/(b^4*(d + e*x)^(11/2) - (4*b^4*d - 4* a*b^3*e)*(d + e*x)^(9/2) + (d + e*x)^(3/2)*(a^4*e^4 + b^4*d^4 + 6*a^2*b^2* d^2*e^2 - 4*a*b^3*d^3*e - 4*a^3*b*d*e^3) + (d + e*x)^(7/2)*(6*b^4*d^2 + 6* a^2*b^2*e^2 - 12*a*b^3*d*e) - (d + e*x)^(5/2)*(4*b^4*d^3 - 4*a^3*b*e^3 + 1 2*a^2*b^2*d*e^2 - 12*a*b^3*d^2*e)) + (1155*b^(3/2)*e^4*atan((b^(1/2)*(d + e*x)^(1/2)*(a^6*e^6 + b^6*d^6 + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a*b^5*d^5*e - 6*a^5*b*d*e^5))/(a*e - b*d)^(13/2)))/ (64*(a*e - b*d)^(13/2))