Integrand size = 27, antiderivative size = 190 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x} \, dx=\frac {1}{128} d^6 (128 d+125 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{192} d^4 (64 d+125 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{240} d^2 (48 d+125 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac {3}{7} d \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{8} e x \left (d^2-e^2 x^2\right )^{7/2}+\frac {125}{128} d^8 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-d^8 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]
1/192*d^4*(125*e*x+64*d)*(-e^2*x^2+d^2)^(3/2)+1/240*d^2*(125*e*x+48*d)*(-e ^2*x^2+d^2)^(5/2)-3/7*d*(-e^2*x^2+d^2)^(7/2)-1/8*e*x*(-e^2*x^2+d^2)^(7/2)+ 125/128*d^8*arctan(e*x/(-e^2*x^2+d^2)^(1/2))-d^8*arctanh((-e^2*x^2+d^2)^(1 /2)/d)+1/128*d^6*(125*e*x+128*d)*(-e^2*x^2+d^2)^(1/2)
Time = 0.60 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.98 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (14848 d^7+27195 d^6 e x+7424 d^5 e^2 x^2-17710 d^4 e^3 x^3-14592 d^3 e^4 x^4+1960 d^2 e^5 x^5+5760 d e^6 x^6+1680 e^7 x^7\right )}{13440}-\frac {125}{64} d^8 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )-d^7 \sqrt {d^2} \log (x)+d^7 \sqrt {d^2} \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right ) \]
(Sqrt[d^2 - e^2*x^2]*(14848*d^7 + 27195*d^6*e*x + 7424*d^5*e^2*x^2 - 17710 *d^4*e^3*x^3 - 14592*d^3*e^4*x^4 + 1960*d^2*e^5*x^5 + 5760*d*e^6*x^6 + 168 0*e^7*x^7))/13440 - (125*d^8*ArcTan[(e*x)/(Sqrt[d^2] - Sqrt[d^2 - e^2*x^2] )])/64 - d^7*Sqrt[d^2]*Log[x] + d^7*Sqrt[d^2]*Log[Sqrt[d^2] - Sqrt[d^2 - e ^2*x^2]]
Time = 0.50 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.14, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.519, Rules used = {541, 25, 2340, 27, 535, 535, 27, 535, 538, 224, 216, 243, 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.
| \(\displaystyle \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x} \, dx\) |
| \(\Big \downarrow \) 541 |
| \(\displaystyle -\frac {\int -\frac {\left (d^2-e^2 x^2\right )^{5/2} \left (24 d x^2 e^4+25 d^2 x e^3+8 d^3 e^2\right )}{x}dx}{8 e^2}-\frac {1}{8} e x \left (d^2-e^2 x^2\right )^{7/2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\left (d^2-e^2 x^2\right )^{5/2} \left (24 d x^2 e^4+25 d^2 x e^3+8 d^3 e^2\right )}{x}dx}{8 e^2}-\frac {1}{8} e x \left (d^2-e^2 x^2\right )^{7/2}\) |
| \(\Big \downarrow \) 2340 |
| \(\displaystyle \frac {-\frac {\int -\frac {7 d^2 e^4 (8 d+25 e x) \left (d^2-e^2 x^2\right )^{5/2}}{x}dx}{7 e^2}-\frac {24}{7} d e^2 \left (d^2-e^2 x^2\right )^{7/2}}{8 e^2}-\frac {1}{8} e x \left (d^2-e^2 x^2\right )^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d^2 e^2 \int \frac {(8 d+25 e x) \left (d^2-e^2 x^2\right )^{5/2}}{x}dx-\frac {24}{7} d e^2 \left (d^2-e^2 x^2\right )^{7/2}}{8 e^2}-\frac {1}{8} e x \left (d^2-e^2 x^2\right )^{7/2}\) |
| \(\Big \downarrow \) 535 |
| \(\displaystyle \frac {d^2 e^2 \left (\frac {1}{6} d^2 \int \frac {(48 d+125 e x) \left (d^2-e^2 x^2\right )^{3/2}}{x}dx+\frac {1}{30} (48 d+125 e x) \left (d^2-e^2 x^2\right )^{5/2}\right )-\frac {24}{7} d e^2 \left (d^2-e^2 x^2\right )^{7/2}}{8 e^2}-\frac {1}{8} e x \left (d^2-e^2 x^2\right )^{7/2}\) |
| \(\Big \downarrow \) 535 |
| \(\displaystyle \frac {d^2 e^2 \left (\frac {1}{6} d^2 \left (\frac {1}{4} d^2 \int \frac {3 (64 d+125 e x) \sqrt {d^2-e^2 x^2}}{x}dx+\frac {1}{4} (64 d+125 e x) \left (d^2-e^2 x^2\right )^{3/2}\right )+\frac {1}{30} (48 d+125 e x) \left (d^2-e^2 x^2\right )^{5/2}\right )-\frac {24}{7} d e^2 \left (d^2-e^2 x^2\right )^{7/2}}{8 e^2}-\frac {1}{8} e x \left (d^2-e^2 x^2\right )^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d^2 e^2 \left (\frac {1}{6} d^2 \left (\frac {3}{4} d^2 \int \frac {(64 d+125 e x) \sqrt {d^2-e^2 x^2}}{x}dx+\frac {1}{4} (64 d+125 e x) \left (d^2-e^2 x^2\right )^{3/2}\right )+\frac {1}{30} (48 d+125 e x) \left (d^2-e^2 x^2\right )^{5/2}\right )-\frac {24}{7} d e^2 \left (d^2-e^2 x^2\right )^{7/2}}{8 e^2}-\frac {1}{8} e x \left (d^2-e^2 x^2\right )^{7/2}\) |
| \(\Big \downarrow \) 535 |
| \(\displaystyle \frac {d^2 e^2 \left (\frac {1}{6} d^2 \left (\frac {3}{4} d^2 \left (\frac {1}{2} d^2 \int \frac {128 d+125 e x}{x \sqrt {d^2-e^2 x^2}}dx+\frac {1}{2} (128 d+125 e x) \sqrt {d^2-e^2 x^2}\right )+\frac {1}{4} (64 d+125 e x) \left (d^2-e^2 x^2\right )^{3/2}\right )+\frac {1}{30} (48 d+125 e x) \left (d^2-e^2 x^2\right )^{5/2}\right )-\frac {24}{7} d e^2 \left (d^2-e^2 x^2\right )^{7/2}}{8 e^2}-\frac {1}{8} e x \left (d^2-e^2 x^2\right )^{7/2}\) |
| \(\Big \downarrow \) 538 |
| \(\displaystyle \frac {d^2 e^2 \left (\frac {1}{6} d^2 \left (\frac {3}{4} d^2 \left (\frac {1}{2} d^2 \left (125 e \int \frac {1}{\sqrt {d^2-e^2 x^2}}dx+128 d \int \frac {1}{x \sqrt {d^2-e^2 x^2}}dx\right )+\frac {1}{2} (128 d+125 e x) \sqrt {d^2-e^2 x^2}\right )+\frac {1}{4} (64 d+125 e x) \left (d^2-e^2 x^2\right )^{3/2}\right )+\frac {1}{30} (48 d+125 e x) \left (d^2-e^2 x^2\right )^{5/2}\right )-\frac {24}{7} d e^2 \left (d^2-e^2 x^2\right )^{7/2}}{8 e^2}-\frac {1}{8} e x \left (d^2-e^2 x^2\right )^{7/2}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {d^2 e^2 \left (\frac {1}{6} d^2 \left (\frac {3}{4} d^2 \left (\frac {1}{2} d^2 \left (128 d \int \frac {1}{x \sqrt {d^2-e^2 x^2}}dx+125 e \int \frac {1}{\frac {e^2 x^2}{d^2-e^2 x^2}+1}d\frac {x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {1}{2} (128 d+125 e x) \sqrt {d^2-e^2 x^2}\right )+\frac {1}{4} (64 d+125 e x) \left (d^2-e^2 x^2\right )^{3/2}\right )+\frac {1}{30} (48 d+125 e x) \left (d^2-e^2 x^2\right )^{5/2}\right )-\frac {24}{7} d e^2 \left (d^2-e^2 x^2\right )^{7/2}}{8 e^2}-\frac {1}{8} e x \left (d^2-e^2 x^2\right )^{7/2}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {d^2 e^2 \left (\frac {1}{6} d^2 \left (\frac {3}{4} d^2 \left (\frac {1}{2} d^2 \left (128 d \int \frac {1}{x \sqrt {d^2-e^2 x^2}}dx+125 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )\right )+\frac {1}{2} (128 d+125 e x) \sqrt {d^2-e^2 x^2}\right )+\frac {1}{4} (64 d+125 e x) \left (d^2-e^2 x^2\right )^{3/2}\right )+\frac {1}{30} (48 d+125 e x) \left (d^2-e^2 x^2\right )^{5/2}\right )-\frac {24}{7} d e^2 \left (d^2-e^2 x^2\right )^{7/2}}{8 e^2}-\frac {1}{8} e x \left (d^2-e^2 x^2\right )^{7/2}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {d^2 e^2 \left (\frac {1}{6} d^2 \left (\frac {3}{4} d^2 \left (\frac {1}{2} d^2 \left (64 d \int \frac {1}{x^2 \sqrt {d^2-e^2 x^2}}dx^2+125 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )\right )+\frac {1}{2} (128 d+125 e x) \sqrt {d^2-e^2 x^2}\right )+\frac {1}{4} (64 d+125 e x) \left (d^2-e^2 x^2\right )^{3/2}\right )+\frac {1}{30} (48 d+125 e x) \left (d^2-e^2 x^2\right )^{5/2}\right )-\frac {24}{7} d e^2 \left (d^2-e^2 x^2\right )^{7/2}}{8 e^2}-\frac {1}{8} e x \left (d^2-e^2 x^2\right )^{7/2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {d^2 e^2 \left (\frac {1}{6} d^2 \left (\frac {3}{4} d^2 \left (\frac {1}{2} d^2 \left (125 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {128 d \int \frac {1}{\frac {d^2}{e^2}-\frac {x^4}{e^2}}d\sqrt {d^2-e^2 x^2}}{e^2}\right )+\frac {1}{2} (128 d+125 e x) \sqrt {d^2-e^2 x^2}\right )+\frac {1}{4} (64 d+125 e x) \left (d^2-e^2 x^2\right )^{3/2}\right )+\frac {1}{30} (48 d+125 e x) \left (d^2-e^2 x^2\right )^{5/2}\right )-\frac {24}{7} d e^2 \left (d^2-e^2 x^2\right )^{7/2}}{8 e^2}-\frac {1}{8} e x \left (d^2-e^2 x^2\right )^{7/2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {d^2 e^2 \left (\frac {1}{6} d^2 \left (\frac {3}{4} d^2 \left (\frac {1}{2} d^2 \left (125 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-128 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )\right )+\frac {1}{2} (128 d+125 e x) \sqrt {d^2-e^2 x^2}\right )+\frac {1}{4} (64 d+125 e x) \left (d^2-e^2 x^2\right )^{3/2}\right )+\frac {1}{30} (48 d+125 e x) \left (d^2-e^2 x^2\right )^{5/2}\right )-\frac {24}{7} d e^2 \left (d^2-e^2 x^2\right )^{7/2}}{8 e^2}-\frac {1}{8} e x \left (d^2-e^2 x^2\right )^{7/2}\) |
-1/8*(e*x*(d^2 - e^2*x^2)^(7/2)) + ((-24*d*e^2*(d^2 - e^2*x^2)^(7/2))/7 + d^2*e^2*(((48*d + 125*e*x)*(d^2 - e^2*x^2)^(5/2))/30 + (d^2*(((64*d + 125* e*x)*(d^2 - e^2*x^2)^(3/2))/4 + (3*d^2*(((128*d + 125*e*x)*Sqrt[d^2 - e^2* x^2])/2 + (d^2*(125*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]] - 128*ArcTanh[Sqrt[d ^2 - e^2*x^2]/d]))/2))/4))/6))/(8*e^2)
3.1.71.3.1 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] :> 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[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_))/(x_), x_Symbol] :> Sim p[(c*(2*p + 1) + 2*d*p*x)*((a + b*x^2)^p/(2*p*(2*p + 1))), x] + Simp[a/(2*p + 1) Int[(c*(2*p + 1) + 2*d*p*x)*((a + b*x^2)^(p - 1)/x), x], x] /; Free Q[{a, b, c, d}, x] && GtQ[p, 0] && IntegerQ[2*p]
Int[((c_) + (d_.)*(x_))/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp [c Int[1/(x*Sqrt[a + b*x^2]), x], x] + Simp[d Int[1/Sqrt[a + b*x^2], x] , x] /; FreeQ[{a, b, c, d}, x]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> Simp[d^n*x^(m + n - 1)*((a + b*x^2)^(p + 1)/(b*(m + n + 2*p + 1))), x ] + Simp[1/(b*(m + n + 2*p + 1)) Int[x^m*(a + b*x^2)^p*ExpandToSum[b*(m + n + 2*p + 1)*(c + d*x)^n - b*d^n*(m + n + 2*p + 1)*x^n - a*d^n*(m + n - 1) *x^(n - 2), x], x], x] /; FreeQ[{a, b, c, d, m, p}, x] && IGtQ[n, 1] && IGt Q[m, -2] && GtQ[p, -1] && IntegerQ[2*p]
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ {q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(c*x)^(m + q - 1 )*((a + b*x^2)^(p + 1)/(b*c^(q - 1)*(m + q + 2*p + 1))), x] + Simp[1/(b*(m + q + 2*p + 1)) Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1) *Pq - b*f*(m + q + 2*p + 1)*x^q - a*f*(m + q - 1)*x^(q - 2), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ [Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])
Leaf count of result is larger than twice the leaf count of optimal. \(352\) vs. \(2(164)=328\).
Time = 0.38 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.86
| method | result | size |
| default | \(e^{3} \left (-\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{8 e^{2}}+\frac {d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{8 e^{2}}\right )+d^{3} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5}+d^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )\right )+3 d^{2} e \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )-\frac {3 d \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{7}\) | \(353\) |
e^3*(-1/8*x*(-e^2*x^2+d^2)^(7/2)/e^2+1/8*d^2/e^2*(1/6*x*(-e^2*x^2+d^2)^(5/ 2)+5/6*d^2*(1/4*x*(-e^2*x^2+d^2)^(3/2)+3/4*d^2*(1/2*x*(-e^2*x^2+d^2)^(1/2) +1/2*d^2/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))))))+d^3*(1 /5*(-e^2*x^2+d^2)^(5/2)+d^2*(1/3*(-e^2*x^2+d^2)^(3/2)+d^2*((-e^2*x^2+d^2)^ (1/2)-d^2/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x))))+ 3*d^2*e*(1/6*x*(-e^2*x^2+d^2)^(5/2)+5/6*d^2*(1/4*x*(-e^2*x^2+d^2)^(3/2)+3/ 4*d^2*(1/2*x*(-e^2*x^2+d^2)^(1/2)+1/2*d^2/(e^2)^(1/2)*arctan((e^2)^(1/2)*x /(-e^2*x^2+d^2)^(1/2)))))-3/7*d*(-e^2*x^2+d^2)^(7/2)
Time = 0.28 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.79 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x} \, dx=-\frac {125}{64} \, d^{8} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + d^{8} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + \frac {1}{13440} \, {\left (1680 \, e^{7} x^{7} + 5760 \, d e^{6} x^{6} + 1960 \, d^{2} e^{5} x^{5} - 14592 \, d^{3} e^{4} x^{4} - 17710 \, d^{4} e^{3} x^{3} + 7424 \, d^{5} e^{2} x^{2} + 27195 \, d^{6} e x + 14848 \, d^{7}\right )} \sqrt {-e^{2} x^{2} + d^{2}} \]
-125/64*d^8*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + d^8*log(-(d - sqrt (-e^2*x^2 + d^2))/x) + 1/13440*(1680*e^7*x^7 + 5760*d*e^6*x^6 + 1960*d^2*e ^5*x^5 - 14592*d^3*e^4*x^4 - 17710*d^4*e^3*x^3 + 7424*d^5*e^2*x^2 + 27195* d^6*e*x + 14848*d^7)*sqrt(-e^2*x^2 + d^2)
Result contains complex when optimal does not.
Time = 8.89 (sec) , antiderivative size = 954, normalized size of antiderivative = 5.02 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x} \, dx=\text {Too large to display} \]
d**7*Piecewise((d**2/(e*x*sqrt(d**2/(e**2*x**2) - 1)) - d*acosh(d/(e*x)) - e*x/sqrt(d**2/(e**2*x**2) - 1), Abs(d**2/(e**2*x**2)) > 1), (-I*d**2/(e*x *sqrt(-d**2/(e**2*x**2) + 1)) + I*d*asin(d/(e*x)) + I*e*x/sqrt(-d**2/(e**2 *x**2) + 1), True)) + 3*d**6*e*Piecewise((d**2*Piecewise((log(-2*e**2*x + 2*sqrt(-e**2)*sqrt(d**2 - e**2*x**2))/sqrt(-e**2), Ne(d**2, 0)), (x*log(x) /sqrt(-e**2*x**2), True))/2 + x*sqrt(d**2 - e**2*x**2)/2, Ne(e**2, 0)), (x *sqrt(d**2), True)) + d**5*e**2*Piecewise((-d**2*sqrt(d**2 - e**2*x**2)/(3 *e**2) + x**2*sqrt(d**2 - e**2*x**2)/3, Ne(e**2, 0)), (x**2*sqrt(d**2)/2, True)) - 5*d**4*e**3*Piecewise((d**4*Piecewise((log(-2*e**2*x + 2*sqrt(-e* *2)*sqrt(d**2 - e**2*x**2))/sqrt(-e**2), Ne(d**2, 0)), (x*log(x)/sqrt(-e** 2*x**2), True))/(8*e**2) - d**2*x*sqrt(d**2 - e**2*x**2)/(8*e**2) + x**3*s qrt(d**2 - e**2*x**2)/4, Ne(e**2, 0)), (x**3*sqrt(d**2)/3, True)) - 5*d**3 *e**4*Piecewise((-2*d**4*sqrt(d**2 - e**2*x**2)/(15*e**4) - d**2*x**2*sqrt (d**2 - e**2*x**2)/(15*e**2) + x**4*sqrt(d**2 - e**2*x**2)/5, Ne(e**2, 0)) , (x**4*sqrt(d**2)/4, True)) + d**2*e**5*Piecewise((d**6*Piecewise((log(-2 *e**2*x + 2*sqrt(-e**2)*sqrt(d**2 - e**2*x**2))/sqrt(-e**2), Ne(d**2, 0)), (x*log(x)/sqrt(-e**2*x**2), True))/(16*e**4) - d**4*x*sqrt(d**2 - e**2*x* *2)/(16*e**4) - d**2*x**3*sqrt(d**2 - e**2*x**2)/(24*e**2) + x**5*sqrt(d** 2 - e**2*x**2)/6, Ne(e**2, 0)), (x**5*sqrt(d**2)/5, True)) + 3*d*e**6*Piec ewise((-8*d**6*sqrt(d**2 - e**2*x**2)/(105*e**6) - 4*d**4*x**2*sqrt(d**...
Time = 0.28 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.14 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x} \, dx=\frac {125 \, d^{8} e \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{128 \, \sqrt {e^{2}}} - d^{8} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right ) + \frac {125}{128} \, \sqrt {-e^{2} x^{2} + d^{2}} d^{6} e x + \sqrt {-e^{2} x^{2} + d^{2}} d^{7} + \frac {125}{192} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4} e x + \frac {1}{3} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5} + \frac {25}{48} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e x + \frac {1}{5} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} - \frac {1}{8} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e x - \frac {3}{7} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d \]
125/128*d^8*e*arcsin(e^2*x/(d*sqrt(e^2)))/sqrt(e^2) - d^8*log(2*d^2/abs(x) + 2*sqrt(-e^2*x^2 + d^2)*d/abs(x)) + 125/128*sqrt(-e^2*x^2 + d^2)*d^6*e*x + sqrt(-e^2*x^2 + d^2)*d^7 + 125/192*(-e^2*x^2 + d^2)^(3/2)*d^4*e*x + 1/3 *(-e^2*x^2 + d^2)^(3/2)*d^5 + 25/48*(-e^2*x^2 + d^2)^(5/2)*d^2*e*x + 1/5*( -e^2*x^2 + d^2)^(5/2)*d^3 - 1/8*(-e^2*x^2 + d^2)^(7/2)*e*x - 3/7*(-e^2*x^2 + d^2)^(7/2)*d
Time = 0.31 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.85 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x} \, dx=\frac {125 \, d^{8} e \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{128 \, {\left | e \right |}} - \frac {d^{8} e \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |} \right |}}{2 \, e^{2} {\left | x \right |}}\right )}{{\left | e \right |}} + \frac {1}{13440} \, {\left (14848 \, d^{7} + {\left (27195 \, d^{6} e + 2 \, {\left (3712 \, d^{5} e^{2} - {\left (8855 \, d^{4} e^{3} + 4 \, {\left (1824 \, d^{3} e^{4} - 5 \, {\left (49 \, d^{2} e^{5} + 6 \, {\left (7 \, e^{7} x + 24 \, d e^{6}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-e^{2} x^{2} + d^{2}} \]
125/128*d^8*e*arcsin(e*x/d)*sgn(d)*sgn(e)/abs(e) - d^8*e*log(1/2*abs(-2*d* e - 2*sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*abs(x)))/abs(e) + 1/13440*(14848*d ^7 + (27195*d^6*e + 2*(3712*d^5*e^2 - (8855*d^4*e^3 + 4*(1824*d^3*e^4 - 5* (49*d^2*e^5 + 6*(7*e^7*x + 24*d*e^6)*x)*x)*x)*x)*x)*x)*sqrt(-e^2*x^2 + d^2 )
Timed out. \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x} \, dx=\int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^3}{x} \,d x \]