Integrand size = 28, antiderivative size = 239 \[ \int \frac {1}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {693 e^5}{128 (b d-a e)^6 \sqrt {d+e x}}-\frac {1}{5 (b d-a e) (a+b x)^5 \sqrt {d+e x}}+\frac {11 e}{40 (b d-a e)^2 (a+b x)^4 \sqrt {d+e x}}-\frac {33 e^2}{80 (b d-a e)^3 (a+b x)^3 \sqrt {d+e x}}+\frac {231 e^3}{320 (b d-a e)^4 (a+b x)^2 \sqrt {d+e x}}-\frac {231 e^4}{128 (b d-a e)^5 (a+b x) \sqrt {d+e x}}+\frac {693 \sqrt {b} e^5 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 (b d-a e)^{13/2}} \]
693/128*e^5*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))*b^(1/2)/(-a*e+ b*d)^(13/2)-693/128*e^5/(-a*e+b*d)^6/(e*x+d)^(1/2)-1/5/(-a*e+b*d)/(b*x+a)^ 5/(e*x+d)^(1/2)+11/40*e/(-a*e+b*d)^2/(b*x+a)^4/(e*x+d)^(1/2)-33/80*e^2/(-a *e+b*d)^3/(b*x+a)^3/(e*x+d)^(1/2)+231/320*e^3/(-a*e+b*d)^4/(b*x+a)^2/(e*x+ d)^(1/2)-231/128*e^4/(-a*e+b*d)^5/(b*x+a)/(e*x+d)^(1/2)
Time = 1.48 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.19 \[ \int \frac {1}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {1}{640} \left (\frac {-1280 a^5 e^5-5 a^4 b e^4 (843 d+2123 e x)-10 a^3 b^2 e^3 \left (-359 d^2+968 d e x+2607 e^2 x^2\right )-2 a^2 b^3 e^2 \left (1124 d^3-2013 d^2 e x+5247 d e^2 x^2+14784 e^3 x^3\right )-2 a b^4 e \left (-408 d^4+616 d^3 e x-1089 d^2 e^2 x^2+2772 d e^3 x^3+8085 e^4 x^4\right )-b^5 \left (128 d^5-176 d^4 e x+264 d^3 e^2 x^2-462 d^2 e^3 x^3+1155 d e^4 x^4+3465 e^5 x^5\right )}{(b d-a e)^6 (a+b x)^5 \sqrt {d+e x}}-\frac {3465 \sqrt {b} e^5 \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{13/2}}\right ) \]
((-1280*a^5*e^5 - 5*a^4*b*e^4*(843*d + 2123*e*x) - 10*a^3*b^2*e^3*(-359*d^ 2 + 968*d*e*x + 2607*e^2*x^2) - 2*a^2*b^3*e^2*(1124*d^3 - 2013*d^2*e*x + 5 247*d*e^2*x^2 + 14784*e^3*x^3) - 2*a*b^4*e*(-408*d^4 + 616*d^3*e*x - 1089* d^2*e^2*x^2 + 2772*d*e^3*x^3 + 8085*e^4*x^4) - b^5*(128*d^5 - 176*d^4*e*x + 264*d^3*e^2*x^2 - 462*d^2*e^3*x^3 + 1155*d*e^4*x^4 + 3465*e^5*x^5))/((b* d - a*e)^6*(a + b*x)^5*Sqrt[d + e*x]) - (3465*Sqrt[b]*e^5*ArcTan[(Sqrt[b]* Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(-(b*d) + a*e)^(13/2))/640
Time = 0.36 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.24, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {1098, 27, 52, 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.
| \(\displaystyle \int \frac {1}{\left (a^2+2 a b x+b^2 x^2\right )^3 (d+e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1098 |
| \(\displaystyle b^6 \int \frac {1}{b^6 (a+b x)^6 (d+e x)^{3/2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {1}{(a+b x)^6 (d+e x)^{3/2}}dx\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -\frac {11 e \int \frac {1}{(a+b x)^5 (d+e x)^{3/2}}dx}{10 (b d-a e)}-\frac {1}{5 (a+b x)^5 \sqrt {d+e x} (b d-a e)}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -\frac {11 e \left (-\frac {9 e \int \frac {1}{(a+b x)^4 (d+e x)^{3/2}}dx}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 \sqrt {d+e x} (b d-a e)}\right )}{10 (b d-a e)}-\frac {1}{5 (a+b x)^5 \sqrt {d+e x} (b d-a e)}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -\frac {11 e \left (-\frac {9 e \left (-\frac {7 e \int \frac {1}{(a+b x)^3 (d+e x)^{3/2}}dx}{6 (b d-a e)}-\frac {1}{3 (a+b x)^3 \sqrt {d+e x} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 \sqrt {d+e x} (b d-a e)}\right )}{10 (b d-a e)}-\frac {1}{5 (a+b x)^5 \sqrt {d+e x} (b d-a e)}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -\frac {11 e \left (-\frac {9 e \left (-\frac {7 e \left (-\frac {5 e \int \frac {1}{(a+b x)^2 (d+e x)^{3/2}}dx}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 \sqrt {d+e x} (b d-a e)}\right )}{6 (b d-a e)}-\frac {1}{3 (a+b x)^3 \sqrt {d+e x} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 \sqrt {d+e x} (b d-a e)}\right )}{10 (b d-a e)}-\frac {1}{5 (a+b x)^5 \sqrt {d+e x} (b d-a e)}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -\frac {11 e \left (-\frac {9 e \left (-\frac {7 e \left (-\frac {5 e \left (-\frac {3 e \int \frac {1}{(a+b x) (d+e x)^{3/2}}dx}{2 (b d-a e)}-\frac {1}{(a+b x) \sqrt {d+e x} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 \sqrt {d+e x} (b d-a e)}\right )}{6 (b d-a e)}-\frac {1}{3 (a+b x)^3 \sqrt {d+e x} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 \sqrt {d+e x} (b d-a e)}\right )}{10 (b d-a e)}-\frac {1}{5 (a+b x)^5 \sqrt {d+e x} (b d-a e)}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle -\frac {11 e \left (-\frac {9 e \left (-\frac {7 e \left (-\frac {5 e \left (-\frac {3 e \left (\frac {b \int \frac {1}{(a+b x) \sqrt {d+e x}}dx}{b d-a e}+\frac {2}{\sqrt {d+e x} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) \sqrt {d+e x} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 \sqrt {d+e x} (b d-a e)}\right )}{6 (b d-a e)}-\frac {1}{3 (a+b x)^3 \sqrt {d+e x} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 \sqrt {d+e x} (b d-a e)}\right )}{10 (b d-a e)}-\frac {1}{5 (a+b x)^5 \sqrt {d+e x} (b d-a e)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {11 e \left (-\frac {9 e \left (-\frac {7 e \left (-\frac {5 e \left (-\frac {3 e \left (\frac {2 b \int \frac {1}{a+\frac {b (d+e x)}{e}-\frac {b d}{e}}d\sqrt {d+e x}}{e (b d-a e)}+\frac {2}{\sqrt {d+e x} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) \sqrt {d+e x} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 \sqrt {d+e x} (b d-a e)}\right )}{6 (b d-a e)}-\frac {1}{3 (a+b x)^3 \sqrt {d+e x} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 \sqrt {d+e x} (b d-a e)}\right )}{10 (b d-a e)}-\frac {1}{5 (a+b x)^5 \sqrt {d+e x} (b d-a e)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {11 e \left (-\frac {9 e \left (-\frac {7 e \left (-\frac {5 e \left (-\frac {3 e \left (\frac {2}{\sqrt {d+e x} (b d-a e)}-\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{3/2}}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) \sqrt {d+e x} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 \sqrt {d+e x} (b d-a e)}\right )}{6 (b d-a e)}-\frac {1}{3 (a+b x)^3 \sqrt {d+e x} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 \sqrt {d+e x} (b d-a e)}\right )}{10 (b d-a e)}-\frac {1}{5 (a+b x)^5 \sqrt {d+e x} (b d-a e)}\) |
-1/5*1/((b*d - a*e)*(a + b*x)^5*Sqrt[d + e*x]) - (11*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))))/(10*(b*d - a*e))
3.17.73.3.1 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_ Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[ {a, b, c, d, e, m}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Time = 2.50 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.13
| method | result | size |
| derivativedivides | \(2 e^{5} \left (-\frac {1}{\left (a e -b d \right )^{6} \sqrt {e x +d}}-\frac {b \left (\frac {\frac {437 b^{4} \left (e x +d \right )^{\frac {9}{2}}}{256}+\frac {977 \left (a e -b d \right ) b^{3} \left (e x +d \right )^{\frac {7}{2}}}{128}+\left (\frac {131}{10} a^{2} b^{2} e^{2}-\frac {131}{5} a \,b^{3} d e +\frac {131}{10} b^{4} d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}+\left (\frac {1327}{128} a^{3} b \,e^{3}-\frac {3981}{128} a^{2} b^{2} d \,e^{2}+\frac {3981}{128} a \,b^{3} d^{2} e -\frac {1327}{128} b^{4} d^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (\frac {843}{256} e^{4} a^{4}-\frac {843}{64} b \,e^{3} d \,a^{3}+\frac {2529}{128} b^{2} e^{2} d^{2} a^{2}-\frac {843}{64} a \,b^{3} d^{3} e +\frac {843}{256} b^{4} d^{4}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {693 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{256 \sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right )^{6}}\right )\) | \(270\) |
| default | \(2 e^{5} \left (-\frac {1}{\left (a e -b d \right )^{6} \sqrt {e x +d}}-\frac {b \left (\frac {\frac {437 b^{4} \left (e x +d \right )^{\frac {9}{2}}}{256}+\frac {977 \left (a e -b d \right ) b^{3} \left (e x +d \right )^{\frac {7}{2}}}{128}+\left (\frac {131}{10} a^{2} b^{2} e^{2}-\frac {131}{5} a \,b^{3} d e +\frac {131}{10} b^{4} d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}+\left (\frac {1327}{128} a^{3} b \,e^{3}-\frac {3981}{128} a^{2} b^{2} d \,e^{2}+\frac {3981}{128} a \,b^{3} d^{2} e -\frac {1327}{128} b^{4} d^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (\frac {843}{256} e^{4} a^{4}-\frac {843}{64} b \,e^{3} d \,a^{3}+\frac {2529}{128} b^{2} e^{2} d^{2} a^{2}-\frac {843}{64} a \,b^{3} d^{3} e +\frac {843}{256} b^{4} d^{4}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {693 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{256 \sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right )^{6}}\right )\) | \(270\) |
| pseudoelliptic | \(-\frac {693 \left (b \,e^{5} \sqrt {e x +d}\, \left (b x +a \right )^{5} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )+\frac {256 \left (\left (\frac {693}{256} e^{5} x^{5}+\frac {231}{256} x^{4} d \,e^{4}-\frac {231}{640} d^{2} e^{3} x^{3}+\frac {33}{160} d^{3} e^{2} x^{2}-\frac {11}{80} d^{4} e x +\frac {1}{10} d^{5}\right ) b^{5}-\frac {51 \left (-\frac {2695}{136} e^{4} x^{4}-\frac {231}{34} d \,e^{3} x^{3}+\frac {363}{136} d^{2} e^{2} x^{2}-\frac {77}{51} d^{3} e x +d^{4}\right ) e a \,b^{4}}{80}+\frac {281 e^{2} \left (\frac {3696}{281} e^{3} x^{3}+\frac {5247}{1124} d \,e^{2} x^{2}-\frac {2013}{1124} d^{2} e x +d^{3}\right ) a^{2} b^{3}}{160}-\frac {359 \left (-\frac {2607}{359} x^{2} e^{2}-\frac {968}{359} d e x +d^{2}\right ) e^{3} a^{3} b^{2}}{128}+\frac {843 \left (\frac {2123 e x}{843}+d \right ) e^{4} a^{4} b}{256}+a^{5} e^{5}\right ) \sqrt {\left (a e -b d \right ) b}}{693}\right )}{128 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, \left (b x +a \right )^{5} \left (a e -b d \right )^{6}}\) | \(287\) |
2*e^5*(-1/(a*e-b*d)^6/(e*x+d)^(1/2)-1/(a*e-b*d)^6*b*((437/256*b^4*(e*x+d)^ (9/2)+977/128*(a*e-b*d)*b^3*(e*x+d)^(7/2)+(131/10*a^2*b^2*e^2-131/5*a*b^3* d*e+131/10*b^4*d^2)*(e*x+d)^(5/2)+(1327/128*a^3*b*e^3-3981/128*a^2*b^2*d*e ^2+3981/128*a*b^3*d^2*e-1327/128*b^4*d^3)*(e*x+d)^(3/2)+(843/256*e^4*a^4-8 43/64*b*e^3*d*a^3+2529/128*b^2*e^2*d^2*a^2-843/64*a*b^3*d^3*e+843/256*b^4* d^4)*(e*x+d)^(1/2))/(b*(e*x+d)+a*e-b*d)^5+693/256/((a*e-b*d)*b)^(1/2)*arct an(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. 1152 vs. \(2 (203) = 406\).
Time = 1.02 (sec) , antiderivative size = 2314, normalized size of antiderivative = 9.68 \[ \int \frac {1}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Too large to display} \]
[1/1280*(3465*(b^5*e^6*x^6 + a^5*d*e^5 + (b^5*d*e^5 + 5*a*b^4*e^6)*x^5 + 5 *(a*b^4*d*e^5 + 2*a^2*b^3*e^6)*x^4 + 10*(a^2*b^3*d*e^5 + a^3*b^2*e^6)*x^3 + 5*(2*a^3*b^2*d*e^5 + a^4*b*e^6)*x^2 + (5*a^4*b*d*e^5 + a^5*e^6)*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 + 128*b^5*d^5 - 816*a*b^ 4*d^4*e + 2248*a^2*b^3*d^3*e^2 - 3590*a^3*b^2*d^2*e^3 + 4215*a^4*b*d*e^4 + 1280*a^5*e^5 + 1155*(b^5*d*e^4 + 14*a*b^4*e^5)*x^4 - 462*(b^5*d^2*e^3 - 1 2*a*b^4*d*e^4 - 64*a^2*b^3*e^5)*x^3 + 66*(4*b^5*d^3*e^2 - 33*a*b^4*d^2*e^3 + 159*a^2*b^3*d*e^4 + 395*a^3*b^2*e^5)*x^2 - 11*(16*b^5*d^4*e - 112*a*b^4 *d^3*e^2 + 366*a^2*b^3*d^2*e^3 - 880*a^3*b^2*d*e^4 - 965*a^4*b*e^5)*x)*sqr t(e*x + d))/(a^5*b^6*d^7 - 6*a^6*b^5*d^6*e + 15*a^7*b^4*d^5*e^2 - 20*a^8*b ^3*d^4*e^3 + 15*a^9*b^2*d^3*e^4 - 6*a^10*b*d^2*e^5 + a^11*d*e^6 + (b^11*d^ 6*e - 6*a*b^10*d^5*e^2 + 15*a^2*b^9*d^4*e^3 - 20*a^3*b^8*d^3*e^4 + 15*a^4* b^7*d^2*e^5 - 6*a^5*b^6*d*e^6 + a^6*b^5*e^7)*x^6 + (b^11*d^7 - a*b^10*d^6* e - 15*a^2*b^9*d^5*e^2 + 55*a^3*b^8*d^4*e^3 - 85*a^4*b^7*d^3*e^4 + 69*a^5* b^6*d^2*e^5 - 29*a^6*b^5*d*e^6 + 5*a^7*b^4*e^7)*x^5 + 5*(a*b^10*d^7 - 4*a^ 2*b^9*d^6*e + 3*a^3*b^8*d^5*e^2 + 10*a^4*b^7*d^4*e^3 - 25*a^5*b^6*d^3*e^4 + 24*a^6*b^5*d^2*e^5 - 11*a^7*b^4*d*e^6 + 2*a^8*b^3*e^7)*x^4 + 10*(a^2*b^9 *d^7 - 5*a^3*b^8*d^6*e + 9*a^4*b^7*d^5*e^2 - 5*a^5*b^6*d^4*e^3 - 5*a^6*b^5 *d^3*e^4 + 9*a^7*b^4*d^2*e^5 - 5*a^8*b^3*d*e^6 + a^9*b^2*e^7)*x^3 + 5*(...
Timed out. \[ \int \frac {1}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {1}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
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. 579 vs. \(2 (203) = 406\).
Time = 0.30 (sec) , antiderivative size = 579, normalized size of antiderivative = 2.42 \[ \int \frac {1}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {693 \, b e^{5} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{128 \, {\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} \sqrt {-b^{2} d + a b e}} - \frac {2 \, e^{5}}{{\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} \sqrt {e x + d}} - \frac {2185 \, {\left (e x + d\right )}^{\frac {9}{2}} b^{5} e^{5} - 9770 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{5} d e^{5} + 16768 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{5} d^{2} e^{5} - 13270 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{5} d^{3} e^{5} + 4215 \, \sqrt {e x + d} b^{5} d^{4} e^{5} + 9770 \, {\left (e x + d\right )}^{\frac {7}{2}} a b^{4} e^{6} - 33536 \, {\left (e x + d\right )}^{\frac {5}{2}} a b^{4} d e^{6} + 39810 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{4} d^{2} e^{6} - 16860 \, \sqrt {e x + d} a b^{4} d^{3} e^{6} + 16768 \, {\left (e x + d\right )}^{\frac {5}{2}} a^{2} b^{3} e^{7} - 39810 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b^{3} d e^{7} + 25290 \, \sqrt {e x + d} a^{2} b^{3} d^{2} e^{7} + 13270 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{3} b^{2} e^{8} - 16860 \, \sqrt {e x + d} a^{3} b^{2} d e^{8} + 4215 \, \sqrt {e x + d} a^{4} b e^{9}}{640 \, {\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} {\left ({\left (e x + d\right )} b - b d + a e\right )}^{5}} \]
-693/128*b*e^5*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*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)*sqrt(e*x + d)) - 1/640*(2185*(e*x + d)^(9/2)*b^5 *e^5 - 9770*(e*x + d)^(7/2)*b^5*d*e^5 + 16768*(e*x + d)^(5/2)*b^5*d^2*e^5 - 13270*(e*x + d)^(3/2)*b^5*d^3*e^5 + 4215*sqrt(e*x + d)*b^5*d^4*e^5 + 977 0*(e*x + d)^(7/2)*a*b^4*e^6 - 33536*(e*x + d)^(5/2)*a*b^4*d*e^6 + 39810*(e *x + d)^(3/2)*a*b^4*d^2*e^6 - 16860*sqrt(e*x + d)*a*b^4*d^3*e^6 + 16768*(e *x + d)^(5/2)*a^2*b^3*e^7 - 39810*(e*x + d)^(3/2)*a^2*b^3*d*e^7 + 25290*sq rt(e*x + d)*a^2*b^3*d^2*e^7 + 13270*(e*x + d)^(3/2)*a^3*b^2*e^8 - 16860*sq rt(e*x + d)*a^3*b^2*d*e^8 + 4215*sqrt(e*x + d)*a^4*b*e^9)/((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)^5)
Time = 10.10 (sec) , antiderivative size = 515, normalized size of antiderivative = 2.15 \[ \int \frac {1}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {\frac {2\,e^5}{a\,e-b\,d}+\frac {2607\,b^2\,e^5\,{\left (d+e\,x\right )}^2}{64\,{\left (a\,e-b\,d\right )}^3}+\frac {231\,b^3\,e^5\,{\left (d+e\,x\right )}^3}{5\,{\left (a\,e-b\,d\right )}^4}+\frac {1617\,b^4\,e^5\,{\left (d+e\,x\right )}^4}{64\,{\left (a\,e-b\,d\right )}^5}+\frac {693\,b^5\,e^5\,{\left (d+e\,x\right )}^5}{128\,{\left (a\,e-b\,d\right )}^6}+\frac {2123\,b\,e^5\,\left (d+e\,x\right )}{128\,{\left (a\,e-b\,d\right )}^2}}{\sqrt {d+e\,x}\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )-{\left (d+e\,x\right )}^{5/2}\,\left (-10\,a^3\,b^2\,e^3+30\,a^2\,b^3\,d\,e^2-30\,a\,b^4\,d^2\,e+10\,b^5\,d^3\right )+{\left (d+e\,x\right )}^{3/2}\,\left (5\,a^4\,b\,e^4-20\,a^3\,b^2\,d\,e^3+30\,a^2\,b^3\,d^2\,e^2-20\,a\,b^4\,d^3\,e+5\,b^5\,d^4\right )+b^5\,{\left (d+e\,x\right )}^{11/2}-\left (5\,b^5\,d-5\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^{9/2}+{\left (d+e\,x\right )}^{7/2}\,\left (10\,a^2\,b^3\,e^2-20\,a\,b^4\,d\,e+10\,b^5\,d^2\right )}-\frac {693\,\sqrt {b}\,e^5\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}\,\left (a^6\,e^6-6\,a^5\,b\,d\,e^5+15\,a^4\,b^2\,d^2\,e^4-20\,a^3\,b^3\,d^3\,e^3+15\,a^2\,b^4\,d^4\,e^2-6\,a\,b^5\,d^5\,e+b^6\,d^6\right )}{{\left (a\,e-b\,d\right )}^{13/2}}\right )}{128\,{\left (a\,e-b\,d\right )}^{13/2}} \]
- ((2*e^5)/(a*e - b*d) + (2607*b^2*e^5*(d + e*x)^2)/(64*(a*e - b*d)^3) + ( 231*b^3*e^5*(d + e*x)^3)/(5*(a*e - b*d)^4) + (1617*b^4*e^5*(d + e*x)^4)/(6 4*(a*e - b*d)^5) + (693*b^5*e^5*(d + e*x)^5)/(128*(a*e - b*d)^6) + (2123*b *e^5*(d + e*x))/(128*(a*e - b*d)^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) - (d + e*x)^(5/2)*(10*b^5*d^3 - 10*a^3*b^2*e^3 + 30*a^2*b^3*d*e^2 - 30*a*b ^4*d^2*e) + (d + e*x)^(3/2)*(5*b^5*d^4 + 5*a^4*b*e^4 - 20*a^3*b^2*d*e^3 + 30*a^2*b^3*d^2*e^2 - 20*a*b^4*d^3*e) + b^5*(d + e*x)^(11/2) - (5*b^5*d - 5 *a*b^4*e)*(d + e*x)^(9/2) + (d + e*x)^(7/2)*(10*b^5*d^2 + 10*a^2*b^3*e^2 - 20*a*b^4*d*e)) - (693*b^(1/2)*e^5*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)))/(128*(a*e - b*d)^(13 /2))