Integrand size = 33, antiderivative size = 206 \[ \int \frac {a+b x}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {315 e^4}{64 (b d-a e)^5 \sqrt {d+e x}}-\frac {1}{4 (b d-a e) (a+b x)^4 \sqrt {d+e x}}+\frac {3 e}{8 (b d-a e)^2 (a+b x)^3 \sqrt {d+e x}}-\frac {21 e^2}{32 (b d-a e)^3 (a+b x)^2 \sqrt {d+e x}}+\frac {105 e^3}{64 (b d-a e)^4 (a+b x) \sqrt {d+e x}}-\frac {315 \sqrt {b} e^4 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 (b d-a e)^{11/2}} \]
-315/64*e^4*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))*b^(1/2)/(-a*e+ b*d)^(11/2)+315/64*e^4/(-a*e+b*d)^5/(e*x+d)^(1/2)-1/4/(-a*e+b*d)/(b*x+a)^4 /(e*x+d)^(1/2)+3/8*e/(-a*e+b*d)^2/(b*x+a)^3/(e*x+d)^(1/2)-21/32*e^2/(-a*e+ b*d)^3/(b*x+a)^2/(e*x+d)^(1/2)+105/64*e^3/(-a*e+b*d)^4/(b*x+a)/(e*x+d)^(1/ 2)
Time = 0.38 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.06 \[ \int \frac {a+b x}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {1}{64} \left (\frac {128 a^4 e^4+a^3 b e^3 (325 d+837 e x)+3 a^2 b^2 e^2 \left (-70 d^2+185 d e x+511 e^2 x^2\right )+a b^3 e \left (88 d^3-156 d^2 e x+399 d e^2 x^2+1155 e^3 x^3\right )+b^4 \left (-16 d^4+24 d^3 e x-42 d^2 e^2 x^2+105 d e^3 x^3+315 e^4 x^4\right )}{(b d-a e)^5 (a+b x)^4 \sqrt {d+e x}}-\frac {315 \sqrt {b} e^4 \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{11/2}}\right ) \]
((128*a^4*e^4 + a^3*b*e^3*(325*d + 837*e*x) + 3*a^2*b^2*e^2*(-70*d^2 + 185 *d*e*x + 511*e^2*x^2) + a*b^3*e*(88*d^3 - 156*d^2*e*x + 399*d*e^2*x^2 + 11 55*e^3*x^3) + b^4*(-16*d^4 + 24*d^3*e*x - 42*d^2*e^2*x^2 + 105*d*e^3*x^3 + 315*e^4*x^4))/((b*d - a*e)^5*(a + b*x)^4*Sqrt[d + e*x]) - (315*Sqrt[b]*e^ 4*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(-(b*d) + a*e)^(11/2 ))/64
Time = 0.33 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.22, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1184, 27, 52, 52, 52, 52, 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.
| \(\displaystyle \int \frac {a+b x}{\left (a^2+2 a b x+b^2 x^2\right )^3 (d+e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle b^6 \int \frac {1}{b^6 (a+b x)^5 (d+e x)^{3/2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {1}{(a+b x)^5 (d+e x)^{3/2}}dx\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -\frac {9 e \int \frac {1}{(a+b x)^4 (d+e x)^{3/2}}dx}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 \sqrt {d+e x} (b d-a e)}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -\frac {9 e \left (-\frac {7 e \int \frac {1}{(a+b x)^3 (d+e x)^{3/2}}dx}{6 (b d-a e)}-\frac {1}{3 (a+b x)^3 \sqrt {d+e x} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 \sqrt {d+e x} (b d-a e)}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -\frac {9 e \left (-\frac {7 e \left (-\frac {5 e \int \frac {1}{(a+b x)^2 (d+e x)^{3/2}}dx}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 \sqrt {d+e x} (b d-a e)}\right )}{6 (b d-a e)}-\frac {1}{3 (a+b x)^3 \sqrt {d+e x} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 \sqrt {d+e x} (b d-a e)}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -\frac {9 e \left (-\frac {7 e \left (-\frac {5 e \left (-\frac {3 e \int \frac {1}{(a+b x) (d+e x)^{3/2}}dx}{2 (b d-a e)}-\frac {1}{(a+b x) \sqrt {d+e x} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 \sqrt {d+e x} (b d-a e)}\right )}{6 (b d-a e)}-\frac {1}{3 (a+b x)^3 \sqrt {d+e x} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 \sqrt {d+e x} (b d-a e)}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle -\frac {9 e \left (-\frac {7 e \left (-\frac {5 e \left (-\frac {3 e \left (\frac {b \int \frac {1}{(a+b x) \sqrt {d+e x}}dx}{b d-a e}+\frac {2}{\sqrt {d+e x} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) \sqrt {d+e x} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 \sqrt {d+e x} (b d-a e)}\right )}{6 (b d-a e)}-\frac {1}{3 (a+b x)^3 \sqrt {d+e x} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 \sqrt {d+e x} (b d-a e)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {9 e \left (-\frac {7 e \left (-\frac {5 e \left (-\frac {3 e \left (\frac {2 b \int \frac {1}{a+\frac {b (d+e x)}{e}-\frac {b d}{e}}d\sqrt {d+e x}}{e (b d-a e)}+\frac {2}{\sqrt {d+e x} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) \sqrt {d+e x} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 \sqrt {d+e x} (b d-a e)}\right )}{6 (b d-a e)}-\frac {1}{3 (a+b x)^3 \sqrt {d+e x} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 \sqrt {d+e x} (b d-a e)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {9 e \left (-\frac {7 e \left (-\frac {5 e \left (-\frac {3 e \left (\frac {2}{\sqrt {d+e x} (b d-a e)}-\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{3/2}}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) \sqrt {d+e x} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 \sqrt {d+e x} (b d-a e)}\right )}{6 (b d-a e)}-\frac {1}{3 (a+b x)^3 \sqrt {d+e x} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 \sqrt {d+e x} (b d-a e)}\) |
-1/4*1/((b*d - a*e)*(a + b*x)^4*Sqrt[d + e*x]) - (9*e*(-1/3*1/((b*d - a*e) *(a + b*x)^3*Sqrt[d + e*x]) - (7*e*(-1/2*1/((b*d - a*e)*(a + b*x)^2*Sqrt[d + e*x]) - (5*e*(-(1/((b*d - a*e)*(a + b*x)*Sqrt[d + e*x])) - (3*e*(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)))/(2*(b*d - a*e))))/(4*(b*d - a*e))))/(6*(b *d - a*e))))/(8*(b*d - a*e))
3.21.90.3.1 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 = 206, normalized size of antiderivative = 1.00
| method | result | size |
| derivativedivides | \(2 e^{4} \left (-\frac {1}{\left (a e -b d \right )^{5} \sqrt {e x +d}}-\frac {b \left (\frac {\frac {187 \left (e x +d \right )^{\frac {7}{2}} b^{3}}{128}+\frac {643 \left (a e -b d \right ) \left (e x +d \right )^{\frac {5}{2}} b^{2}}{128}+\left (\frac {765}{128} a^{2} b \,e^{2}-\frac {765}{64} a \,b^{2} d e +\frac {765}{128} b^{3} d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (\frac {325}{128} a^{3} e^{3}-\frac {975}{128} a^{2} b d \,e^{2}+\frac {975}{128} a \,b^{2} d^{2} e -\frac {325}{128} b^{3} d^{3}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{4}}+\frac {315 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{128 \sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right )^{5}}\right )\) | \(206\) |
| default | \(2 e^{4} \left (-\frac {1}{\left (a e -b d \right )^{5} \sqrt {e x +d}}-\frac {b \left (\frac {\frac {187 \left (e x +d \right )^{\frac {7}{2}} b^{3}}{128}+\frac {643 \left (a e -b d \right ) \left (e x +d \right )^{\frac {5}{2}} b^{2}}{128}+\left (\frac {765}{128} a^{2} b \,e^{2}-\frac {765}{64} a \,b^{2} d e +\frac {765}{128} b^{3} d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (\frac {325}{128} a^{3} e^{3}-\frac {975}{128} a^{2} b d \,e^{2}+\frac {975}{128} a \,b^{2} d^{2} e -\frac {325}{128} b^{3} d^{3}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{4}}+\frac {315 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{128 \sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right )^{5}}\right )\) | \(206\) |
| pseudoelliptic | \(-\frac {2 \left (\frac {315 b \,e^{4} \sqrt {e x +d}\, \left (b x +a \right )^{4} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{128}+\sqrt {\left (a e -b d \right ) b}\, \left (\left (\frac {315}{128} e^{4} x^{4}+\frac {105}{128} d \,e^{3} x^{3}-\frac {21}{64} d^{2} e^{2} x^{2}+\frac {3}{16} d^{3} e x -\frac {1}{8} d^{4}\right ) b^{4}+\frac {11 e \left (\frac {105}{8} e^{3} x^{3}+\frac {399}{88} d \,e^{2} x^{2}-\frac {39}{22} d^{2} e x +d^{3}\right ) a \,b^{3}}{16}-\frac {105 e^{2} \left (-\frac {73}{10} e^{2} x^{2}-\frac {37}{14} d e x +d^{2}\right ) a^{2} b^{2}}{64}+\frac {325 e^{3} a^{3} \left (\frac {837 e x}{325}+d \right ) b}{128}+e^{4} a^{4}\right )\right )}{\sqrt {\left (a e -b d \right ) b}\, \sqrt {e x +d}\, \left (b x +a \right )^{4} \left (a e -b d \right )^{5}}\) | \(226\) |
2*e^4*(-1/(a*e-b*d)^5/(e*x+d)^(1/2)-1/(a*e-b*d)^5*b*((187/128*(e*x+d)^(7/2 )*b^3+643/128*(a*e-b*d)*(e*x+d)^(5/2)*b^2+(765/128*a^2*b*e^2-765/64*a*b^2* d*e+765/128*b^3*d^2)*(e*x+d)^(3/2)+(325/128*a^3*e^3-975/128*a^2*b*d*e^2+97 5/128*a*b^2*d^2*e-325/128*b^3*d^3)*(e*x+d)^(1/2))/(b*(e*x+d)+a*e-b*d)^4+31 5/128/((a*e-b*d)*b)^(1/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 862 vs. \(2 (174) = 348\).
Time = 0.46 (sec) , antiderivative size = 1734, normalized size of antiderivative = 8.42 \[ \int \frac {a+b x}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Too large to display} \]
[-1/128*(315*(b^4*e^5*x^5 + a^4*d*e^4 + (b^4*d*e^4 + 4*a*b^3*e^5)*x^4 + 2* (2*a*b^3*d*e^4 + 3*a^2*b^2*e^5)*x^3 + 2*(3*a^2*b^2*d*e^4 + 2*a^3*b*e^5)*x^ 2 + (4*a^3*b*d*e^4 + a^4*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*(315 *b^4*e^4*x^4 - 16*b^4*d^4 + 88*a*b^3*d^3*e - 210*a^2*b^2*d^2*e^2 + 325*a^3 *b*d*e^3 + 128*a^4*e^4 + 105*(b^4*d*e^3 + 11*a*b^3*e^4)*x^3 - 21*(2*b^4*d^ 2*e^2 - 19*a*b^3*d*e^3 - 73*a^2*b^2*e^4)*x^2 + 3*(8*b^4*d^3*e - 52*a*b^3*d ^2*e^2 + 185*a^2*b^2*d*e^3 + 279*a^3*b*e^4)*x)*sqrt(e*x + d))/(a^4*b^5*d^6 - 5*a^5*b^4*d^5*e + 10*a^6*b^3*d^4*e^2 - 10*a^7*b^2*d^3*e^3 + 5*a^8*b*d^2 *e^4 - a^9*d*e^5 + (b^9*d^5*e - 5*a*b^8*d^4*e^2 + 10*a^2*b^7*d^3*e^3 - 10* a^3*b^6*d^2*e^4 + 5*a^4*b^5*d*e^5 - a^5*b^4*e^6)*x^5 + (b^9*d^6 - a*b^8*d^ 5*e - 10*a^2*b^7*d^4*e^2 + 30*a^3*b^6*d^3*e^3 - 35*a^4*b^5*d^2*e^4 + 19*a^ 5*b^4*d*e^5 - 4*a^6*b^3*e^6)*x^4 + 2*(2*a*b^8*d^6 - 7*a^2*b^7*d^5*e + 5*a^ 3*b^6*d^4*e^2 + 10*a^4*b^5*d^3*e^3 - 20*a^5*b^4*d^2*e^4 + 13*a^6*b^3*d*e^5 - 3*a^7*b^2*e^6)*x^3 + 2*(3*a^2*b^7*d^6 - 13*a^3*b^6*d^5*e + 20*a^4*b^5*d ^4*e^2 - 10*a^5*b^4*d^3*e^3 - 5*a^6*b^3*d^2*e^4 + 7*a^7*b^2*d*e^5 - 2*a^8* b*e^6)*x^2 + (4*a^3*b^6*d^6 - 19*a^4*b^5*d^5*e + 35*a^5*b^4*d^4*e^2 - 30*a ^6*b^3*d^3*e^3 + 10*a^7*b^2*d^2*e^4 + a^8*b*d*e^5 - a^9*e^6)*x), -1/64*(31 5*(b^4*e^5*x^5 + a^4*d*e^4 + (b^4*d*e^4 + 4*a*b^3*e^5)*x^4 + 2*(2*a*b^3*d* e^4 + 3*a^2*b^2*e^5)*x^3 + 2*(3*a^2*b^2*d*e^4 + 2*a^3*b*e^5)*x^2 + (4*a...
Timed out. \[ \int \frac {a+b x}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {a+b x}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
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. 445 vs. \(2 (174) = 348\).
Time = 0.28 (sec) , antiderivative size = 445, normalized size of antiderivative = 2.16 \[ \int \frac {a+b x}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {315 \, b e^{4} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{64 \, {\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} \sqrt {-b^{2} d + a b e}} + \frac {2 \, e^{4}}{{\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} \sqrt {e x + d}} + \frac {187 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{4} e^{4} - 643 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{4} d e^{4} + 765 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{4} d^{2} e^{4} - 325 \, \sqrt {e x + d} b^{4} d^{3} e^{4} + 643 \, {\left (e x + d\right )}^{\frac {5}{2}} a b^{3} e^{5} - 1530 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{3} d e^{5} + 975 \, \sqrt {e x + d} a b^{3} d^{2} e^{5} + 765 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b^{2} e^{6} - 975 \, \sqrt {e x + d} a^{2} b^{2} d e^{6} + 325 \, \sqrt {e x + d} a^{3} b e^{7}}{64 \, {\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} {\left ({\left (e x + d\right )} b - b d + a e\right )}^{4}} \]
315/64*b*e^4*arctan(sqrt(e*x + d)*b/sqrt(-b^2*d + a*b*e))/((b^5*d^5 - 5*a* b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5* e^5)*sqrt(-b^2*d + a*b*e)) + 2*e^4/((b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3* d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5)*sqrt(e*x + d)) + 1 /64*(187*(e*x + d)^(7/2)*b^4*e^4 - 643*(e*x + d)^(5/2)*b^4*d*e^4 + 765*(e* x + d)^(3/2)*b^4*d^2*e^4 - 325*sqrt(e*x + d)*b^4*d^3*e^4 + 643*(e*x + d)^( 5/2)*a*b^3*e^5 - 1530*(e*x + d)^(3/2)*a*b^3*d*e^5 + 975*sqrt(e*x + d)*a*b^ 3*d^2*e^5 + 765*(e*x + d)^(3/2)*a^2*b^2*e^6 - 975*sqrt(e*x + d)*a^2*b^2*d* e^6 + 325*sqrt(e*x + d)*a^3*b*e^7)/((b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3* d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5)*((e*x + d)*b - b*d + a*e)^4)
Time = 11.30 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.93 \[ \int \frac {a+b x}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {\frac {2\,e^4}{a\,e-b\,d}+\frac {1533\,b^2\,e^4\,{\left (d+e\,x\right )}^2}{64\,{\left (a\,e-b\,d\right )}^3}+\frac {1155\,b^3\,e^4\,{\left (d+e\,x\right )}^3}{64\,{\left (a\,e-b\,d\right )}^4}+\frac {315\,b^4\,e^4\,{\left (d+e\,x\right )}^4}{64\,{\left (a\,e-b\,d\right )}^5}+\frac {837\,b\,e^4\,\left (d+e\,x\right )}{64\,{\left (a\,e-b\,d\right )}^2}}{b^4\,{\left (d+e\,x\right )}^{9/2}-\left (4\,b^4\,d-4\,a\,b^3\,e\right )\,{\left (d+e\,x\right )}^{7/2}+\sqrt {d+e\,x}\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )+{\left (d+e\,x\right )}^{5/2}\,\left (6\,a^2\,b^2\,e^2-12\,a\,b^3\,d\,e+6\,b^4\,d^2\right )-{\left (d+e\,x\right )}^{3/2}\,\left (-4\,a^3\,b\,e^3+12\,a^2\,b^2\,d\,e^2-12\,a\,b^3\,d^2\,e+4\,b^4\,d^3\right )}-\frac {315\,\sqrt {b}\,e^4\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )}{{\left (a\,e-b\,d\right )}^{11/2}}\right )}{64\,{\left (a\,e-b\,d\right )}^{11/2}} \]
- ((2*e^4)/(a*e - b*d) + (1533*b^2*e^4*(d + e*x)^2)/(64*(a*e - b*d)^3) + ( 1155*b^3*e^4*(d + e*x)^3)/(64*(a*e - b*d)^4) + (315*b^4*e^4*(d + e*x)^4)/( 64*(a*e - b*d)^5) + (837*b*e^4*(d + e*x))/(64*(a*e - b*d)^2))/(b^4*(d + e* x)^(9/2) - (4*b^4*d - 4*a*b^3*e)*(d + e*x)^(7/2) + (d + e*x)^(1/2)*(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)^(5/2)*(6*b^4*d^2 + 6*a^2*b^2*e^2 - 12*a*b^3*d*e) - (d + e*x)^(3/2)*(4*b ^4*d^3 - 4*a^3*b*e^3 + 12*a^2*b^2*d*e^2 - 12*a*b^3*d^2*e)) - (315*b^(1/2)* e^4*atan((b^(1/2)*(d + e*x)^(1/2)*(a^5*e^5 - b^5*d^5 - 10*a^2*b^3*d^3*e^2 + 10*a^3*b^2*d^2*e^3 + 5*a*b^4*d^4*e - 5*a^4*b*d*e^4))/(a*e - b*d)^(11/2)) )/(64*(a*e - b*d)^(11/2))