Integrand size = 27, antiderivative size = 209 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^5} \, dx=-\frac {45}{8} d^2 e^4 (d-e x) \sqrt {d^2-e^2 x^2}+\frac {15 d e^3 (2 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{8 x}-\frac {3 e^2 (3 d+2 e x) \left (d^2-e^2 x^2\right )^{5/2}}{8 x^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}+\frac {45}{8} d^4 e^4 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {45}{8} d^4 e^4 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]
15/8*d*e^3*(-e*x+2*d)*(-e^2*x^2+d^2)^(3/2)/x-3/8*e^2*(2*e*x+3*d)*(-e^2*x^2 +d^2)^(5/2)/x^2-1/4*d*(-e^2*x^2+d^2)^(7/2)/x^4-e*(-e^2*x^2+d^2)^(7/2)/x^3+ 45/8*d^4*e^4*arctan(e*x/(-e^2*x^2+d^2)^(1/2))+45/8*d^4*e^4*arctanh((-e^2*x ^2+d^2)^(1/2)/d)-45/8*d^2*e^4*(-e*x+d)*(-e^2*x^2+d^2)^(1/2)
Time = 0.62 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.95 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^5} \, dx=\frac {1}{8} \left (\frac {\sqrt {d^2-e^2 x^2} \left (-2 d^7-8 d^6 e x-3 d^5 e^2 x^2+48 d^4 e^3 x^3-48 d^3 e^4 x^4+3 d^2 e^5 x^5+8 d e^6 x^6+2 e^7 x^7\right )}{x^4}-90 d^4 e^4 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )+45 d^3 \sqrt {d^2} e^4 \log (x)-45 d^3 \sqrt {d^2} e^4 \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right )\right ) \]
((Sqrt[d^2 - e^2*x^2]*(-2*d^7 - 8*d^6*e*x - 3*d^5*e^2*x^2 + 48*d^4*e^3*x^3 - 48*d^3*e^4*x^4 + 3*d^2*e^5*x^5 + 8*d*e^6*x^6 + 2*e^7*x^7))/x^4 - 90*d^4 *e^4*ArcTan[(e*x)/(Sqrt[d^2] - Sqrt[d^2 - e^2*x^2])] + 45*d^3*Sqrt[d^2]*e^ 4*Log[x] - 45*d^3*Sqrt[d^2]*e^4*Log[Sqrt[d^2] - Sqrt[d^2 - e^2*x^2]])/8
Time = 0.51 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.593, Rules used = {540, 25, 2338, 27, 537, 25, 535, 27, 535, 27, 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.
| \(\displaystyle \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^5} \, dx\) |
| \(\Big \downarrow \) 540 |
| \(\displaystyle -\frac {\int -\frac {\left (d^2-e^2 x^2\right )^{5/2} \left (12 e d^4+9 e^2 x d^3+4 e^3 x^2 d^2\right )}{x^4}dx}{4 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\left (d^2-e^2 x^2\right )^{5/2} \left (12 e d^4+9 e^2 x d^3+4 e^3 x^2 d^2\right )}{x^4}dx}{4 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle \frac {-\frac {\int -\frac {9 d^4 e^2 (3 d-4 e x) \left (d^2-e^2 x^2\right )^{5/2}}{x^3}dx}{3 d^2}-\frac {4 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}}{4 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 d^2 e^2 \int \frac {(3 d-4 e x) \left (d^2-e^2 x^2\right )^{5/2}}{x^3}dx-\frac {4 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}}{4 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}\) |
| \(\Big \downarrow \) 537 |
| \(\displaystyle \frac {3 d^2 e^2 \left (\frac {5}{2} e^2 \int -\frac {(3 d-8 e x) \left (d^2-e^2 x^2\right )^{3/2}}{x}dx-\frac {(3 d-8 e x) \left (d^2-e^2 x^2\right )^{5/2}}{2 x^2}\right )-\frac {4 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}}{4 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3 d^2 e^2 \left (-\frac {5}{2} e^2 \int \frac {(3 d-8 e x) \left (d^2-e^2 x^2\right )^{3/2}}{x}dx-\frac {(3 d-8 e x) \left (d^2-e^2 x^2\right )^{5/2}}{2 x^2}\right )-\frac {4 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}}{4 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}\) |
| \(\Big \downarrow \) 535 |
| \(\displaystyle \frac {3 d^2 e^2 \left (-\frac {5}{2} e^2 \left (\frac {1}{4} d^2 \int \frac {12 (d-2 e x) \sqrt {d^2-e^2 x^2}}{x}dx+(d-2 e x) \left (d^2-e^2 x^2\right )^{3/2}\right )-\frac {(3 d-8 e x) \left (d^2-e^2 x^2\right )^{5/2}}{2 x^2}\right )-\frac {4 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}}{4 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 d^2 e^2 \left (-\frac {5}{2} e^2 \left (3 d^2 \int \frac {(d-2 e x) \sqrt {d^2-e^2 x^2}}{x}dx+(d-2 e x) \left (d^2-e^2 x^2\right )^{3/2}\right )-\frac {(3 d-8 e x) \left (d^2-e^2 x^2\right )^{5/2}}{2 x^2}\right )-\frac {4 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}}{4 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}\) |
| \(\Big \downarrow \) 535 |
| \(\displaystyle \frac {3 d^2 e^2 \left (-\frac {5}{2} e^2 \left (3 d^2 \left (\frac {1}{2} d^2 \int \frac {2 (d-e x)}{x \sqrt {d^2-e^2 x^2}}dx+(d-e x) \sqrt {d^2-e^2 x^2}\right )+(d-2 e x) \left (d^2-e^2 x^2\right )^{3/2}\right )-\frac {(3 d-8 e x) \left (d^2-e^2 x^2\right )^{5/2}}{2 x^2}\right )-\frac {4 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}}{4 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 d^2 e^2 \left (-\frac {5}{2} e^2 \left (3 d^2 \left (d^2 \int \frac {d-e x}{x \sqrt {d^2-e^2 x^2}}dx+(d-e x) \sqrt {d^2-e^2 x^2}\right )+(d-2 e x) \left (d^2-e^2 x^2\right )^{3/2}\right )-\frac {(3 d-8 e x) \left (d^2-e^2 x^2\right )^{5/2}}{2 x^2}\right )-\frac {4 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}}{4 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}\) |
| \(\Big \downarrow \) 538 |
| \(\displaystyle \frac {3 d^2 e^2 \left (-\frac {5}{2} e^2 \left (3 d^2 \left (d^2 \left (d \int \frac {1}{x \sqrt {d^2-e^2 x^2}}dx-e \int \frac {1}{\sqrt {d^2-e^2 x^2}}dx\right )+(d-e x) \sqrt {d^2-e^2 x^2}\right )+(d-2 e x) \left (d^2-e^2 x^2\right )^{3/2}\right )-\frac {(3 d-8 e x) \left (d^2-e^2 x^2\right )^{5/2}}{2 x^2}\right )-\frac {4 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}}{4 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {3 d^2 e^2 \left (-\frac {5}{2} e^2 \left (3 d^2 \left (d^2 \left (d \int \frac {1}{x \sqrt {d^2-e^2 x^2}}dx-e \int \frac {1}{\frac {e^2 x^2}{d^2-e^2 x^2}+1}d\frac {x}{\sqrt {d^2-e^2 x^2}}\right )+(d-e x) \sqrt {d^2-e^2 x^2}\right )+(d-2 e x) \left (d^2-e^2 x^2\right )^{3/2}\right )-\frac {(3 d-8 e x) \left (d^2-e^2 x^2\right )^{5/2}}{2 x^2}\right )-\frac {4 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}}{4 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {3 d^2 e^2 \left (-\frac {5}{2} e^2 \left (3 d^2 \left (d^2 \left (d \int \frac {1}{x \sqrt {d^2-e^2 x^2}}dx-\arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )\right )+(d-e x) \sqrt {d^2-e^2 x^2}\right )+(d-2 e x) \left (d^2-e^2 x^2\right )^{3/2}\right )-\frac {(3 d-8 e x) \left (d^2-e^2 x^2\right )^{5/2}}{2 x^2}\right )-\frac {4 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}}{4 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {3 d^2 e^2 \left (-\frac {5}{2} e^2 \left (3 d^2 \left (d^2 \left (\frac {1}{2} d \int \frac {1}{x^2 \sqrt {d^2-e^2 x^2}}dx^2-\arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )\right )+(d-e x) \sqrt {d^2-e^2 x^2}\right )+(d-2 e x) \left (d^2-e^2 x^2\right )^{3/2}\right )-\frac {(3 d-8 e x) \left (d^2-e^2 x^2\right )^{5/2}}{2 x^2}\right )-\frac {4 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}}{4 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {3 d^2 e^2 \left (-\frac {5}{2} e^2 \left (3 d^2 \left (d^2 \left (-\frac {d \int \frac {1}{\frac {d^2}{e^2}-\frac {x^4}{e^2}}d\sqrt {d^2-e^2 x^2}}{e^2}-\arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )\right )+(d-e x) \sqrt {d^2-e^2 x^2}\right )+(d-2 e x) \left (d^2-e^2 x^2\right )^{3/2}\right )-\frac {(3 d-8 e x) \left (d^2-e^2 x^2\right )^{5/2}}{2 x^2}\right )-\frac {4 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}}{4 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {3 d^2 e^2 \left (-\frac {5}{2} e^2 \left (3 d^2 \left (d^2 \left (-\arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )\right )+(d-e x) \sqrt {d^2-e^2 x^2}\right )+(d-2 e x) \left (d^2-e^2 x^2\right )^{3/2}\right )-\frac {(3 d-8 e x) \left (d^2-e^2 x^2\right )^{5/2}}{2 x^2}\right )-\frac {4 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}}{4 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}\) |
-1/4*(d*(d^2 - e^2*x^2)^(7/2))/x^4 + ((-4*d^2*e*(d^2 - e^2*x^2)^(7/2))/x^3 + 3*d^2*e^2*(-1/2*((3*d - 8*e*x)*(d^2 - e^2*x^2)^(5/2))/x^2 - (5*e^2*((d - 2*e*x)*(d^2 - e^2*x^2)^(3/2) + 3*d^2*((d - e*x)*Sqrt[d^2 - e^2*x^2] + d^ 2*(-ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]] - ArcTanh[Sqrt[d^2 - e^2*x^2]/d])))) /2))/(4*d^2)
3.1.75.3.1 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[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x^(m + 1)*(c*(m + 2) + d*(m + 1)*x)*((a + b*x^2)^p/((m + 1)*(m + 2))), x] - Simp[2*b*(p/((m + 1)*(m + 2))) Int[x^(m + 2)*(c*(m + 2) + d*(m + 1) *x)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, -2] && GtQ[p, 0] && !ILtQ[m + 2*p + 3, 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_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 = 0.43 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.07
| method | result | size |
| risch | \(-\frac {d^{4} \sqrt {-e^{2} x^{2}+d^{2}}\, \left (-48 e^{3} x^{3}+3 d \,e^{2} x^{2}+8 d^{2} e x +2 d^{3}\right )}{8 x^{4}}+\frac {e^{7} x^{3} \sqrt {-e^{2} x^{2}+d^{2}}}{4}+\frac {3 e^{5} d^{2} x \sqrt {-e^{2} x^{2}+d^{2}}}{8}+\frac {45 e^{5} d^{4} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{8 \sqrt {e^{2}}}+\frac {45 e^{4} d^{5} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{8 \sqrt {d^{2}}}+e^{6} d \,x^{2} \sqrt {-e^{2} x^{2}+d^{2}}-6 e^{4} d^{3} \sqrt {-e^{2} x^{2}+d^{2}}\) | \(223\) |
| default | \(d^{3} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{4 d^{2} x^{4}}-\frac {3 e^{2} \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 )}{4 d^{2}}\right )+e^{3} \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 \,e^{2} \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^{2} e \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 )\) | \(598\) |
-1/8*d^4*(-e^2*x^2+d^2)^(1/2)*(-48*e^3*x^3+3*d*e^2*x^2+8*d^2*e*x+2*d^3)/x^ 4+1/4*e^7*x^3*(-e^2*x^2+d^2)^(1/2)+3/8*e^5*d^2*x*(-e^2*x^2+d^2)^(1/2)+45/8 *e^5*d^4/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))+45/8*e^4*d ^5/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)+e^6*d*x^2* (-e^2*x^2+d^2)^(1/2)-6*e^4*d^3*(-e^2*x^2+d^2)^(1/2)
Time = 0.28 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.86 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^5} \, dx=-\frac {90 \, d^{4} e^{4} x^{4} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + 45 \, d^{4} e^{4} x^{4} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + 48 \, d^{4} e^{4} x^{4} - {\left (2 \, e^{7} x^{7} + 8 \, d e^{6} x^{6} + 3 \, d^{2} e^{5} x^{5} - 48 \, d^{3} e^{4} x^{4} + 48 \, d^{4} e^{3} x^{3} - 3 \, d^{5} e^{2} x^{2} - 8 \, d^{6} e x - 2 \, d^{7}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{8 \, x^{4}} \]
-1/8*(90*d^4*e^4*x^4*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + 45*d^4*e^ 4*x^4*log(-(d - sqrt(-e^2*x^2 + d^2))/x) + 48*d^4*e^4*x^4 - (2*e^7*x^7 + 8 *d*e^6*x^6 + 3*d^2*e^5*x^5 - 48*d^3*e^4*x^4 + 48*d^4*e^3*x^3 - 3*d^5*e^2*x ^2 - 8*d^6*e*x - 2*d^7)*sqrt(-e^2*x^2 + d^2))/x^4
Result contains complex when optimal does not.
Time = 5.95 (sec) , antiderivative size = 959, normalized size of antiderivative = 4.59 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^5} \, dx=\text {Too large to display} \]
d**7*Piecewise((-d**2/(4*e*x**5*sqrt(d**2/(e**2*x**2) - 1)) + 3*e/(8*x**3* sqrt(d**2/(e**2*x**2) - 1)) - e**3/(8*d**2*x*sqrt(d**2/(e**2*x**2) - 1)) + e**4*acosh(d/(e*x))/(8*d**3), Abs(d**2/(e**2*x**2)) > 1), (I*d**2/(4*e*x* *5*sqrt(-d**2/(e**2*x**2) + 1)) - 3*I*e/(8*x**3*sqrt(-d**2/(e**2*x**2) + 1 )) + I*e**3/(8*d**2*x*sqrt(-d**2/(e**2*x**2) + 1)) - I*e**4*asin(d/(e*x))/ (8*d**3), True)) + 3*d**6*e*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)) + d**5*e**2*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)) - 5*d**4*e**3*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**3*e** 4*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*sq rt(-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)) + d**2*e**5*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...
Time = 0.28 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.25 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^5} \, dx=\frac {45 \, d^{4} e^{5} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{8 \, \sqrt {e^{2}}} + \frac {45}{8} \, d^{4} e^{4} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right ) + \frac {45}{8} \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2} e^{5} x - \frac {45}{8} \, \sqrt {-e^{2} x^{2} + d^{2}} d^{3} e^{4} + \frac {15}{4} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{5} x - \frac {15}{8} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d e^{4} - \frac {9 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}}{8 \, d} + \frac {3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{3}}{x} - \frac {9 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{2}}{8 \, d x^{2}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e}{x^{3}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d}{4 \, x^{4}} \]
45/8*d^4*e^5*arcsin(e^2*x/(d*sqrt(e^2)))/sqrt(e^2) + 45/8*d^4*e^4*log(2*d^ 2/abs(x) + 2*sqrt(-e^2*x^2 + d^2)*d/abs(x)) + 45/8*sqrt(-e^2*x^2 + d^2)*d^ 2*e^5*x - 45/8*sqrt(-e^2*x^2 + d^2)*d^3*e^4 + 15/4*(-e^2*x^2 + d^2)^(3/2)* e^5*x - 15/8*(-e^2*x^2 + d^2)^(3/2)*d*e^4 - 9/8*(-e^2*x^2 + d^2)^(5/2)*e^4 /d + 3*(-e^2*x^2 + d^2)^(5/2)*e^3/x - 9/8*(-e^2*x^2 + d^2)^(7/2)*e^2/(d*x^ 2) - (-e^2*x^2 + d^2)^(7/2)*e/x^3 - 1/4*(-e^2*x^2 + d^2)^(7/2)*d/x^4
Leaf count of result is larger than twice the leaf count of optimal. 409 vs. \(2 (183) = 366\).
Time = 0.30 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.96 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^5} \, dx=\frac {45 \, d^{4} e^{5} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{8 \, {\left | e \right |}} + \frac {45 \, d^{4} e^{5} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |} \right |}}{2 \, e^{2} {\left | x \right |}}\right )}{8 \, {\left | e \right |}} + \frac {{\left (d^{4} e^{5} + \frac {8 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{4} e^{3}}{x} + \frac {8 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{4} e}{x^{2}} - \frac {184 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d^{4}}{e x^{3}}\right )} e^{8} x^{4}}{64 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} {\left | e \right |}} - \frac {1}{8} \, {\left (48 \, d^{3} e^{4} - {\left (3 \, d^{2} e^{5} + 2 \, {\left (e^{7} x + 4 \, d e^{6}\right )} x\right )} x\right )} \sqrt {-e^{2} x^{2} + d^{2}} + \frac {\frac {184 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{4} e^{5} {\left | e \right |}}{x} - \frac {8 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{4} e^{3} {\left | e \right |}}{x^{2}} - \frac {8 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d^{4} e {\left | e \right |}}{x^{3}} - \frac {{\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} d^{4} {\left | e \right |}}{e x^{4}}}{64 \, e^{4}} \]
45/8*d^4*e^5*arcsin(e*x/d)*sgn(d)*sgn(e)/abs(e) + 45/8*d^4*e^5*log(1/2*abs (-2*d*e - 2*sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*abs(x)))/abs(e) + 1/64*(d^4* e^5 + 8*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*d^4*e^3/x + 8*(d*e + sqrt(-e^2 *x^2 + d^2)*abs(e))^2*d^4*e/x^2 - 184*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^ 3*d^4/(e*x^3))*e^8*x^4/((d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4*abs(e)) - 1/ 8*(48*d^3*e^4 - (3*d^2*e^5 + 2*(e^7*x + 4*d*e^6)*x)*x)*sqrt(-e^2*x^2 + d^2 ) + 1/64*(184*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*d^4*e^5*abs(e)/x - 8*(d* e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d^4*e^3*abs(e)/x^2 - 8*(d*e + sqrt(-e^2 *x^2 + d^2)*abs(e))^3*d^4*e*abs(e)/x^3 - (d*e + sqrt(-e^2*x^2 + d^2)*abs(e ))^4*d^4*abs(e)/(e*x^4))/e^4
Timed out. \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^5} \, dx=\int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^3}{x^5} \,d x \]