Integrand size = 27, antiderivative size = 210 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^4} \, dx=-\frac {1}{8} d^3 e^3 (52 d+25 e x) \sqrt {d^2-e^2 x^2}-\frac {1}{12} d e^3 (26 d+25 e x) \left (d^2-e^2 x^2\right )^{3/2}-\frac {e^2 (50 d+39 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 x}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{3 x^3}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac {25}{8} d^5 e^3 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {13}{2} d^5 e^3 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]
-1/12*d*e^3*(25*e*x+26*d)*(-e^2*x^2+d^2)^(3/2)-1/30*e^2*(39*e*x+50*d)*(-e^ 2*x^2+d^2)^(5/2)/x-1/3*d*(-e^2*x^2+d^2)^(7/2)/x^3-3/2*e*(-e^2*x^2+d^2)^(7/ 2)/x^2-25/8*d^5*e^3*arctan(e*x/(-e^2*x^2+d^2)^(1/2))+13/2*d^5*e^3*arctanh( (-e^2*x^2+d^2)^(1/2)/d)-1/8*d^3*e^3*(25*e*x+52*d)*(-e^2*x^2+d^2)^(1/2)
Time = 0.58 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.90 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^4} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-40 d^7-180 d^6 e x-80 d^5 e^2 x^2-656 d^4 e^3 x^3-345 d^3 e^4 x^4+32 d^2 e^5 x^5+90 d e^6 x^6+24 e^7 x^7\right )}{120 x^3}-13 d^5 e^3 \text {arctanh}\left (\frac {\sqrt {-e^2} x}{d}-\frac {\sqrt {d^2-e^2 x^2}}{d}\right )+\frac {25}{8} d^5 \left (-e^2\right )^{3/2} \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right ) \]
(Sqrt[d^2 - e^2*x^2]*(-40*d^7 - 180*d^6*e*x - 80*d^5*e^2*x^2 - 656*d^4*e^3 *x^3 - 345*d^3*e^4*x^4 + 32*d^2*e^5*x^5 + 90*d*e^6*x^6 + 24*e^7*x^7))/(120 *x^3) - 13*d^5*e^3*ArcTanh[(Sqrt[-e^2]*x)/d - Sqrt[d^2 - e^2*x^2]/d] + (25 *d^5*(-e^2)^(3/2)*Log[-(Sqrt[-e^2]*x) + Sqrt[d^2 - e^2*x^2]])/8
Time = 0.50 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.06, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {540, 25, 2338, 25, 27, 536, 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.
| \(\displaystyle \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^4} \, dx\) |
| \(\Big \downarrow \) 540 |
| \(\displaystyle -\frac {\int -\frac {\left (d^2-e^2 x^2\right )^{5/2} \left (9 e d^4+5 e^2 x d^3+3 e^3 x^2 d^2\right )}{x^3}dx}{3 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{3 x^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\left (d^2-e^2 x^2\right )^{5/2} \left (9 e d^4+5 e^2 x d^3+3 e^3 x^2 d^2\right )}{x^3}dx}{3 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{3 x^3}\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle \frac {-\frac {\int -\frac {d^4 e^2 (10 d-39 e x) \left (d^2-e^2 x^2\right )^{5/2}}{x^2}dx}{2 d^2}-\frac {9 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}}{3 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{3 x^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {d^4 e^2 (10 d-39 e x) \left (d^2-e^2 x^2\right )^{5/2}}{x^2}dx}{2 d^2}-\frac {9 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}}{3 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{3 x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{2} d^2 e^2 \int \frac {(10 d-39 e x) \left (d^2-e^2 x^2\right )^{5/2}}{x^2}dx-\frac {9 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}}{3 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{3 x^3}\) |
| \(\Big \downarrow \) 536 |
| \(\displaystyle \frac {\frac {1}{2} d^2 e^2 \left (\int \frac {\left (-39 e d^2-50 e^2 x d\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x}dx-\frac {(50 d+39 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5 x}\right )-\frac {9 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}}{3 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{3 x^3}\) |
| \(\Big \downarrow \) 535 |
| \(\displaystyle \frac {\frac {1}{2} d^2 e^2 \left (\frac {1}{4} d^2 \int -\frac {6 d e (26 d+25 e x) \sqrt {d^2-e^2 x^2}}{x}dx-\frac {(50 d+39 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5 x}-\frac {1}{2} d e (26 d+25 e x) \left (d^2-e^2 x^2\right )^{3/2}\right )-\frac {9 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}}{3 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{3 x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{2} d^2 e^2 \left (-\frac {3}{2} d^3 e \int \frac {(26 d+25 e x) \sqrt {d^2-e^2 x^2}}{x}dx-\frac {1}{2} d e (26 d+25 e x) \left (d^2-e^2 x^2\right )^{3/2}-\frac {(50 d+39 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5 x}\right )-\frac {9 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}}{3 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{3 x^3}\) |
| \(\Big \downarrow \) 535 |
| \(\displaystyle \frac {\frac {1}{2} d^2 e^2 \left (-\frac {3}{2} d^3 e \left (\frac {1}{2} d^2 \int \frac {52 d+25 e x}{x \sqrt {d^2-e^2 x^2}}dx+\frac {1}{2} (52 d+25 e x) \sqrt {d^2-e^2 x^2}\right )-\frac {1}{2} d e (26 d+25 e x) \left (d^2-e^2 x^2\right )^{3/2}-\frac {(50 d+39 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5 x}\right )-\frac {9 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}}{3 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{3 x^3}\) |
| \(\Big \downarrow \) 538 |
| \(\displaystyle \frac {\frac {1}{2} d^2 e^2 \left (-\frac {3}{2} d^3 e \left (\frac {1}{2} d^2 \left (25 e \int \frac {1}{\sqrt {d^2-e^2 x^2}}dx+52 d \int \frac {1}{x \sqrt {d^2-e^2 x^2}}dx\right )+\frac {1}{2} (52 d+25 e x) \sqrt {d^2-e^2 x^2}\right )-\frac {1}{2} d e (26 d+25 e x) \left (d^2-e^2 x^2\right )^{3/2}-\frac {(50 d+39 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5 x}\right )-\frac {9 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}}{3 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{3 x^3}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\frac {1}{2} d^2 e^2 \left (-\frac {3}{2} d^3 e \left (\frac {1}{2} d^2 \left (52 d \int \frac {1}{x \sqrt {d^2-e^2 x^2}}dx+25 e \int \frac {1}{\frac {e^2 x^2}{d^2-e^2 x^2}+1}d\frac {x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {1}{2} (52 d+25 e x) \sqrt {d^2-e^2 x^2}\right )-\frac {1}{2} d e (26 d+25 e x) \left (d^2-e^2 x^2\right )^{3/2}-\frac {(50 d+39 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5 x}\right )-\frac {9 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}}{3 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{3 x^3}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\frac {1}{2} d^2 e^2 \left (-\frac {3}{2} d^3 e \left (\frac {1}{2} d^2 \left (52 d \int \frac {1}{x \sqrt {d^2-e^2 x^2}}dx+25 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )\right )+\frac {1}{2} (52 d+25 e x) \sqrt {d^2-e^2 x^2}\right )-\frac {1}{2} d e (26 d+25 e x) \left (d^2-e^2 x^2\right )^{3/2}-\frac {(50 d+39 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5 x}\right )-\frac {9 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}}{3 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{3 x^3}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {\frac {1}{2} d^2 e^2 \left (-\frac {3}{2} d^3 e \left (\frac {1}{2} d^2 \left (26 d \int \frac {1}{x^2 \sqrt {d^2-e^2 x^2}}dx^2+25 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )\right )+\frac {1}{2} (52 d+25 e x) \sqrt {d^2-e^2 x^2}\right )-\frac {1}{2} d e (26 d+25 e x) \left (d^2-e^2 x^2\right )^{3/2}-\frac {(50 d+39 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5 x}\right )-\frac {9 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}}{3 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{3 x^3}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {1}{2} d^2 e^2 \left (-\frac {3}{2} d^3 e \left (\frac {1}{2} d^2 \left (25 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {52 d \int \frac {1}{\frac {d^2}{e^2}-\frac {x^4}{e^2}}d\sqrt {d^2-e^2 x^2}}{e^2}\right )+\frac {1}{2} (52 d+25 e x) \sqrt {d^2-e^2 x^2}\right )-\frac {1}{2} d e (26 d+25 e x) \left (d^2-e^2 x^2\right )^{3/2}-\frac {(50 d+39 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5 x}\right )-\frac {9 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}}{3 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{3 x^3}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {1}{2} d^2 e^2 \left (-\frac {3}{2} d^3 e \left (\frac {1}{2} d^2 \left (25 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-52 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )\right )+\frac {1}{2} (52 d+25 e x) \sqrt {d^2-e^2 x^2}\right )-\frac {1}{2} d e (26 d+25 e x) \left (d^2-e^2 x^2\right )^{3/2}-\frac {(50 d+39 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5 x}\right )-\frac {9 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}}{3 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{3 x^3}\) |
-1/3*(d*(d^2 - e^2*x^2)^(7/2))/x^3 + ((-9*d^2*e*(d^2 - e^2*x^2)^(7/2))/(2* x^2) + (d^2*e^2*(-1/2*(d*e*(26*d + 25*e*x)*(d^2 - e^2*x^2)^(3/2)) - ((50*d + 39*e*x)*(d^2 - e^2*x^2)^(5/2))/(5*x) - (3*d^3*e*(((52*d + 25*e*x)*Sqrt[ d^2 - e^2*x^2])/2 + (d^2*(25*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]] - 52*ArcTan h[Sqrt[d^2 - e^2*x^2]/d]))/2))/2))/2)/(3*d^2)
3.1.74.3.1 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_))*((a_) + (b_.)*(x_)^2)^(p_))/(x_)^2, x_Symbol] :> S imp[(-(2*c*p - d*x))*((a + b*x^2)^p/(2*p*x)), x] + Int[(a*d + 2*b*c*p*x)*(( a + b*x^2)^(p - 1)/x), x] /; FreeQ[{a, b, c, d}, x] && GtQ[p, 0] && Integer Q[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_Symbol ] :> With[{Qx = PolynomialQuotient[(c + d*x)^n, x, x], R = PolynomialRemain der[(c + d*x)^n, x, x]}, Simp[R*x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))) , x] + Simp[1/(a*(m + 1)) Int[x^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*(m + 1)*Qx - b*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, p}, x] && IG tQ[n, 1] && ILtQ[m, -1] && GtQ[p, -1] && IntegerQ[2*p]
Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{ Q = PolynomialQuotient[Pq, c*x, x], R = PolynomialRemainder[Pq, c*x, x]}, S imp[R*(c*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] + Simp[1/(a*c*( m + 1)) Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*( m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && Lt Q[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])
Time = 0.40 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.13
| method | result | size |
| risch | \(-\frac {d^{5} \sqrt {-e^{2} x^{2}+d^{2}}\, \left (4 e^{2} x^{2}+9 d e x +2 d^{2}\right )}{6 x^{3}}+\frac {e^{7} x^{4} \sqrt {-e^{2} x^{2}+d^{2}}}{5}+\frac {4 e^{5} d^{2} x^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{15}-\frac {82 e^{3} d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}{15}-\frac {25 e^{4} d^{5} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{8 \sqrt {e^{2}}}+\frac {13 e^{3} d^{6} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 \sqrt {d^{2}}}+\frac {3 e^{6} d \,x^{3} \sqrt {-e^{2} x^{2}+d^{2}}}{4}-\frac {23 e^{4} d^{3} x \sqrt {-e^{2} x^{2}+d^{2}}}{8}\) | \(238\) |
| default | \(e^{3} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5}+d^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )\right )+d^{3} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{3 d^{2} x^{3}}-\frac {4 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{d^{2} x}-\frac {6 e^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{d^{2}}\right )}{3 d^{2}}\right )+3 d^{2} e \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{2 d^{2} x^{2}}-\frac {5 e^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5}+d^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )\right )}{2 d^{2}}\right )+3 d \,e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{d^{2} x}-\frac {6 e^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{d^{2}}\right )\) | \(536\) |
-1/6*d^5*(-e^2*x^2+d^2)^(1/2)*(4*e^2*x^2+9*d*e*x+2*d^2)/x^3+1/5*e^7*x^4*(- e^2*x^2+d^2)^(1/2)+4/15*e^5*d^2*x^2*(-e^2*x^2+d^2)^(1/2)-82/15*e^3*d^4*(-e ^2*x^2+d^2)^(1/2)-25/8*e^4*d^5/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+ d^2)^(1/2))+13/2*e^3*d^6/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2 )^(1/2))/x)+3/4*e^6*d*x^3*(-e^2*x^2+d^2)^(1/2)-23/8*e^4*d^3*x*(-e^2*x^2+d^ 2)^(1/2)
Time = 0.27 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.85 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^4} \, dx=\frac {750 \, d^{5} e^{3} x^{3} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - 780 \, d^{5} e^{3} x^{3} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - 656 \, d^{5} e^{3} x^{3} + {\left (24 \, e^{7} x^{7} + 90 \, d e^{6} x^{6} + 32 \, d^{2} e^{5} x^{5} - 345 \, d^{3} e^{4} x^{4} - 656 \, d^{4} e^{3} x^{3} - 80 \, d^{5} e^{2} x^{2} - 180 \, d^{6} e x - 40 \, d^{7}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{120 \, x^{3}} \]
1/120*(750*d^5*e^3*x^3*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - 780*d^5 *e^3*x^3*log(-(d - sqrt(-e^2*x^2 + d^2))/x) - 656*d^5*e^3*x^3 + (24*e^7*x^ 7 + 90*d*e^6*x^6 + 32*d^2*e^5*x^5 - 345*d^3*e^4*x^4 - 656*d^4*e^3*x^3 - 80 *d^5*e^2*x^2 - 180*d^6*e*x - 40*d^7)*sqrt(-e^2*x^2 + d^2))/x^3
Result contains complex when optimal does not.
Time = 4.18 (sec) , antiderivative size = 843, normalized size of antiderivative = 4.01 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^4} \, dx=\text {Too large to display} \]
d**7*Piecewise((-e*sqrt(d**2/(e**2*x**2) - 1)/(3*x**2) + e**3*sqrt(d**2/(e **2*x**2) - 1)/(3*d**2), Abs(d**2/(e**2*x**2)) > 1), (-I*e*sqrt(-d**2/(e** 2*x**2) + 1)/(3*x**2) + I*e**3*sqrt(-d**2/(e**2*x**2) + 1)/(3*d**2), True) ) + 3*d**6*e*Piecewise((-e*sqrt(d**2/(e**2*x**2) - 1)/(2*x) + e**2*acosh(d /(e*x))/(2*d), Abs(d**2/(e**2*x**2)) > 1), (I*d**2/(2*e*x**3*sqrt(-d**2/(e **2*x**2) + 1)) - I*e/(2*x*sqrt(-d**2/(e**2*x**2) + 1)) - I*e**2*asin(d/(e *x))/(2*d), True)) + d**5*e**2*Piecewise((I*d/(x*sqrt(-1 + e**2*x**2/d**2) ) + I*e*acosh(e*x/d) - I*e**2*x/(d*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x* *2/d**2) > 1), (-d/(x*sqrt(1 - e**2*x**2/d**2)) - e*asin(e*x/d) + e**2*x/( d*sqrt(1 - e**2*x**2/d**2)), True)) - 5*d**4*e**3*Piecewise((d**2/(e*x*sqr t(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)) - 5*d**3* e**4*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**2 *e**5*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)) + 3*d*e**6*Piecewis e((d**4*Piecewise((log(-2*e**2*x + 2*sqrt(-e**2)*sqrt(d**2 - e**2*x**2))/s qrt(-e**2), Ne(d**2, 0)), (x*log(x)/sqrt(-e**2*x**2), True))/(8*e**2) -...
Time = 0.28 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.13 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^4} \, dx=-\frac {25 \, d^{5} e^{4} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{8 \, \sqrt {e^{2}}} + \frac {13}{2} \, d^{5} e^{3} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right ) - \frac {25}{8} \, \sqrt {-e^{2} x^{2} + d^{2}} d^{3} e^{4} x - \frac {13}{2} \, \sqrt {-e^{2} x^{2} + d^{2}} d^{4} e^{3} - \frac {25}{12} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d e^{4} x - \frac {13}{6} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2} e^{3} - \frac {13}{10} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{3} - \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d e^{2}}{3 \, x} - \frac {3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e}{2 \, x^{2}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d}{3 \, x^{3}} \]
-25/8*d^5*e^4*arcsin(e^2*x/(d*sqrt(e^2)))/sqrt(e^2) + 13/2*d^5*e^3*log(2*d ^2/abs(x) + 2*sqrt(-e^2*x^2 + d^2)*d/abs(x)) - 25/8*sqrt(-e^2*x^2 + d^2)*d ^3*e^4*x - 13/2*sqrt(-e^2*x^2 + d^2)*d^4*e^3 - 25/12*(-e^2*x^2 + d^2)^(3/2 )*d*e^4*x - 13/6*(-e^2*x^2 + d^2)^(3/2)*d^2*e^3 - 13/10*(-e^2*x^2 + d^2)^( 5/2)*e^3 - 5/3*(-e^2*x^2 + d^2)^(5/2)*d*e^2/x - 3/2*(-e^2*x^2 + d^2)^(7/2) *e/x^2 - 1/3*(-e^2*x^2 + d^2)^(7/2)*d/x^3
Time = 0.29 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.65 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^4} \, dx=-\frac {25 \, d^{5} e^{4} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{8 \, {\left | e \right |}} + \frac {13 \, d^{5} e^{4} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |} \right |}}{2 \, e^{2} {\left | x \right |}}\right )}{2 \, {\left | e \right |}} + \frac {{\left (d^{5} e^{4} + \frac {9 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{5} e^{2}}{x} + \frac {9 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{5}}{x^{2}}\right )} e^{6} x^{3}}{24 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} {\left | e \right |}} - \frac {1}{120} \, {\left (656 \, d^{4} e^{3} + {\left (345 \, d^{3} e^{4} - 2 \, {\left (16 \, d^{2} e^{5} + 3 \, {\left (4 \, e^{7} x + 15 \, d e^{6}\right )} x\right )} x\right )} x\right )} \sqrt {-e^{2} x^{2} + d^{2}} - \frac {\frac {9 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{5} e^{4}}{x} + \frac {9 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{5} e^{2}}{x^{2}} + \frac {{\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d^{5}}{x^{3}}}{24 \, e^{2} {\left | e \right |}} \]
-25/8*d^5*e^4*arcsin(e*x/d)*sgn(d)*sgn(e)/abs(e) + 13/2*d^5*e^4*log(1/2*ab s(-2*d*e - 2*sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*abs(x)))/abs(e) + 1/24*(d^5 *e^4 + 9*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*d^5*e^2/x + 9*(d*e + sqrt(-e^ 2*x^2 + d^2)*abs(e))^2*d^5/x^2)*e^6*x^3/((d*e + sqrt(-e^2*x^2 + d^2)*abs(e ))^3*abs(e)) - 1/120*(656*d^4*e^3 + (345*d^3*e^4 - 2*(16*d^2*e^5 + 3*(4*e^ 7*x + 15*d*e^6)*x)*x)*x)*sqrt(-e^2*x^2 + d^2) - 1/24*(9*(d*e + sqrt(-e^2*x ^2 + d^2)*abs(e))*d^5*e^4/x + 9*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d^5* e^2/x^2 + (d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*d^5/x^3)/(e^2*abs(e))
Timed out. \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^4} \, dx=\int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^3}{x^4} \,d x \]