Integrand size = 27, antiderivative size = 206 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^8} \, dx=-\frac {3 e^6 (16 d-5 e x) \sqrt {d^2-e^2 x^2}}{16 x}+\frac {e^4 (16 d+5 e x) \left (d^2-e^2 x^2\right )^{3/2}}{16 x^3}-\frac {e^2 (24 d+5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{40 x^5}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^6}-3 d e^7 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {15}{16} d e^7 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]
1/16*e^4*(5*e*x+16*d)*(-e^2*x^2+d^2)^(3/2)/x^3-1/40*e^2*(5*e*x+24*d)*(-e^2 *x^2+d^2)^(5/2)/x^5-1/7*d*(-e^2*x^2+d^2)^(7/2)/x^7-1/2*e*(-e^2*x^2+d^2)^(7 /2)/x^6-3*d*e^7*arctan(e*x/(-e^2*x^2+d^2)^(1/2))-15/16*d*e^7*arctanh((-e^2 *x^2+d^2)^(1/2)/d)-3/16*e^6*(-5*e*x+16*d)*(-e^2*x^2+d^2)^(1/2)/x
Time = 0.50 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.94 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^8} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-80 d^7-280 d^6 e x-96 d^5 e^2 x^2+770 d^4 e^3 x^3+992 d^3 e^4 x^4-525 d^2 e^5 x^5-2496 d e^6 x^6+560 e^7 x^7\right )}{560 x^7}+6 d e^7 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )-\frac {15}{16} \sqrt {d^2} e^7 \log (x)+\frac {15}{16} \sqrt {d^2} e^7 \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right ) \]
(Sqrt[d^2 - e^2*x^2]*(-80*d^7 - 280*d^6*e*x - 96*d^5*e^2*x^2 + 770*d^4*e^3 *x^3 + 992*d^3*e^4*x^4 - 525*d^2*e^5*x^5 - 2496*d*e^6*x^6 + 560*e^7*x^7))/ (560*x^7) + 6*d*e^7*ArcTan[(e*x)/(Sqrt[d^2] - Sqrt[d^2 - e^2*x^2])] - (15* Sqrt[d^2]*e^7*Log[x])/16 + (15*Sqrt[d^2]*e^7*Log[Sqrt[d^2] - Sqrt[d^2 - e^ 2*x^2]])/16
Time = 0.51 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.09, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {540, 27, 2338, 27, 537, 25, 537, 27, 536, 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.
| \(\displaystyle \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^8} \, dx\) |
| \(\Big \downarrow \) 540 |
| \(\displaystyle -\frac {\int -\frac {7 \left (d^2-e^2 x^2\right )^{5/2} \left (3 e d^4+3 e^2 x d^3+e^3 x^2 d^2\right )}{x^7}dx}{7 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\left (d^2-e^2 x^2\right )^{5/2} \left (3 e d^4+3 e^2 x d^3+e^3 x^2 d^2\right )}{x^7}dx}{d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle \frac {-\frac {\int -\frac {3 d^4 e^2 (6 d+e x) \left (d^2-e^2 x^2\right )^{5/2}}{x^6}dx}{6 d^2}-\frac {d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^6}}{d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{2} d^2 e^2 \int \frac {(6 d+e x) \left (d^2-e^2 x^2\right )^{5/2}}{x^6}dx-\frac {d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^6}}{d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}\) |
| \(\Big \downarrow \) 537 |
| \(\displaystyle \frac {\frac {1}{2} d^2 e^2 \left (\frac {1}{4} e^2 \int -\frac {(24 d+5 e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^4}dx-\frac {(24 d+5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^6}}{d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{2} d^2 e^2 \left (-\frac {1}{4} e^2 \int \frac {(24 d+5 e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^4}dx-\frac {(24 d+5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^6}}{d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}\) |
| \(\Big \downarrow \) 537 |
| \(\displaystyle \frac {\frac {1}{2} d^2 e^2 \left (-\frac {1}{4} e^2 \left (\frac {1}{2} e^2 \int -\frac {3 (16 d+5 e x) \sqrt {d^2-e^2 x^2}}{x^2}dx-\frac {(16 d+5 e x) \left (d^2-e^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {(24 d+5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^6}}{d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{2} d^2 e^2 \left (-\frac {1}{4} e^2 \left (-\frac {3}{2} e^2 \int \frac {(16 d+5 e x) \sqrt {d^2-e^2 x^2}}{x^2}dx-\frac {(16 d+5 e x) \left (d^2-e^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {(24 d+5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^6}}{d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}\) |
| \(\Big \downarrow \) 536 |
| \(\displaystyle \frac {\frac {1}{2} d^2 e^2 \left (-\frac {1}{4} e^2 \left (-\frac {3}{2} e^2 \left (\int \frac {5 d^2 e-16 d e^2 x}{x \sqrt {d^2-e^2 x^2}}dx-\frac {(16 d-5 e x) \sqrt {d^2-e^2 x^2}}{x}\right )-\frac {(16 d+5 e x) \left (d^2-e^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {(24 d+5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^6}}{d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}\) |
| \(\Big \downarrow \) 538 |
| \(\displaystyle \frac {\frac {1}{2} d^2 e^2 \left (-\frac {1}{4} e^2 \left (-\frac {3}{2} e^2 \left (5 d^2 e \int \frac {1}{x \sqrt {d^2-e^2 x^2}}dx-16 d e^2 \int \frac {1}{\sqrt {d^2-e^2 x^2}}dx-\frac {(16 d-5 e x) \sqrt {d^2-e^2 x^2}}{x}\right )-\frac {(16 d+5 e x) \left (d^2-e^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {(24 d+5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^6}}{d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\frac {1}{2} d^2 e^2 \left (-\frac {1}{4} e^2 \left (-\frac {3}{2} e^2 \left (5 d^2 e \int \frac {1}{x \sqrt {d^2-e^2 x^2}}dx-16 d e^2 \int \frac {1}{\frac {e^2 x^2}{d^2-e^2 x^2}+1}d\frac {x}{\sqrt {d^2-e^2 x^2}}-\frac {(16 d-5 e x) \sqrt {d^2-e^2 x^2}}{x}\right )-\frac {(16 d+5 e x) \left (d^2-e^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {(24 d+5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^6}}{d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\frac {1}{2} d^2 e^2 \left (-\frac {1}{4} e^2 \left (-\frac {3}{2} e^2 \left (5 d^2 e \int \frac {1}{x \sqrt {d^2-e^2 x^2}}dx-16 d e \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {(16 d-5 e x) \sqrt {d^2-e^2 x^2}}{x}\right )-\frac {(16 d+5 e x) \left (d^2-e^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {(24 d+5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^6}}{d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {\frac {1}{2} d^2 e^2 \left (-\frac {1}{4} e^2 \left (-\frac {3}{2} e^2 \left (\frac {5}{2} d^2 e \int \frac {1}{x^2 \sqrt {d^2-e^2 x^2}}dx^2-16 d e \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {(16 d-5 e x) \sqrt {d^2-e^2 x^2}}{x}\right )-\frac {(16 d+5 e x) \left (d^2-e^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {(24 d+5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^6}}{d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {1}{2} d^2 e^2 \left (-\frac {1}{4} e^2 \left (-\frac {3}{2} e^2 \left (-\frac {5 d^2 \int \frac {1}{\frac {d^2}{e^2}-\frac {x^4}{e^2}}d\sqrt {d^2-e^2 x^2}}{e}-16 d e \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {(16 d-5 e x) \sqrt {d^2-e^2 x^2}}{x}\right )-\frac {(16 d+5 e x) \left (d^2-e^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {(24 d+5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^6}}{d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {1}{2} d^2 e^2 \left (-\frac {1}{4} e^2 \left (-\frac {3}{2} e^2 \left (-16 d e \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-5 d e \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )-\frac {\sqrt {d^2-e^2 x^2} (16 d-5 e x)}{x}\right )-\frac {(16 d+5 e x) \left (d^2-e^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {(24 d+5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^6}}{d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}\) |
-1/7*(d*(d^2 - e^2*x^2)^(7/2))/x^7 + (-1/2*(d^2*e*(d^2 - e^2*x^2)^(7/2))/x ^6 + (d^2*e^2*(-1/20*((24*d + 5*e*x)*(d^2 - e^2*x^2)^(5/2))/x^5 - (e^2*(-1 /2*((16*d + 5*e*x)*(d^2 - e^2*x^2)^(3/2))/x^3 - (3*e^2*(-(((16*d - 5*e*x)* Sqrt[d^2 - e^2*x^2])/x) - 16*d*e*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]] - 5*d*e *ArcTanh[Sqrt[d^2 - e^2*x^2]/d]))/2))/4))/2)/d^2
3.1.78.3.1 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_)^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[(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 = 0.52 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.84
| method | result | size |
| risch | \(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (-560 e^{7} x^{7}+2496 d \,e^{6} x^{6}+525 d^{2} e^{5} x^{5}-992 d^{3} e^{4} x^{4}-770 d^{4} e^{3} x^{3}+96 d^{5} e^{2} x^{2}+280 d^{6} e x +80 d^{7}\right )}{560 x^{7}}-\frac {3 d \,e^{8} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{\sqrt {e^{2}}}-\frac {15 d^{2} e^{7} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{16 \sqrt {d^{2}}}\) | \(173\) |
| default | \(e^{3} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{4 d^{2} x^{4}}-\frac {3 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{2 d^{2} x^{2}}-\frac {5 e^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5}+d^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )\right )}{2 d^{2}}\right )}{4 d^{2}}\right )-\frac {d \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{7 x^{7}}+3 d^{2} e \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{6 d^{2} x^{6}}-\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{4 d^{2} x^{4}}-\frac {3 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{2 d^{2} x^{2}}-\frac {5 e^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5}+d^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )\right )}{2 d^{2}}\right )}{4 d^{2}}\right )}{6 d^{2}}\right )+3 d \,e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{5 d^{2} x^{5}}-\frac {2 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{3 d^{2} x^{3}}-\frac {4 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{d^{2} x}-\frac {6 e^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{d^{2}}\right )}{3 d^{2}}\right )}{5 d^{2}}\right )\) | \(580\) |
-1/560*(-e^2*x^2+d^2)^(1/2)*(-560*e^7*x^7+2496*d*e^6*x^6+525*d^2*e^5*x^5-9 92*d^3*e^4*x^4-770*d^4*e^3*x^3+96*d^5*e^2*x^2+280*d^6*e*x+80*d^7)/x^7-3*d* e^8/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))-15/16*d^2*e^7/( d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)
Time = 0.27 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.84 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^8} \, dx=\frac {3360 \, d e^{7} x^{7} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + 525 \, d e^{7} x^{7} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + 560 \, d e^{7} x^{7} + {\left (560 \, e^{7} x^{7} - 2496 \, d e^{6} x^{6} - 525 \, d^{2} e^{5} x^{5} + 992 \, d^{3} e^{4} x^{4} + 770 \, d^{4} e^{3} x^{3} - 96 \, d^{5} e^{2} x^{2} - 280 \, d^{6} e x - 80 \, d^{7}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{560 \, x^{7}} \]
1/560*(3360*d*e^7*x^7*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + 525*d*e^ 7*x^7*log(-(d - sqrt(-e^2*x^2 + d^2))/x) + 560*d*e^7*x^7 + (560*e^7*x^7 - 2496*d*e^6*x^6 - 525*d^2*e^5*x^5 + 992*d^3*e^4*x^4 + 770*d^4*e^3*x^3 - 96* d^5*e^2*x^2 - 280*d^6*e*x - 80*d^7)*sqrt(-e^2*x^2 + d^2))/x^7
Result contains complex when optimal does not.
Time = 11.75 (sec) , antiderivative size = 1513, normalized size of antiderivative = 7.34 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^8} \, dx=\text {Too large to display} \]
d**7*Piecewise((-e*sqrt(d**2/(e**2*x**2) - 1)/(7*x**6) + e**3*sqrt(d**2/(e **2*x**2) - 1)/(35*d**2*x**4) + 4*e**5*sqrt(d**2/(e**2*x**2) - 1)/(105*d** 4*x**2) + 8*e**7*sqrt(d**2/(e**2*x**2) - 1)/(105*d**6), Abs(d**2/(e**2*x** 2)) > 1), (-I*e*sqrt(-d**2/(e**2*x**2) + 1)/(7*x**6) + I*e**3*sqrt(-d**2/( e**2*x**2) + 1)/(35*d**2*x**4) + 4*I*e**5*sqrt(-d**2/(e**2*x**2) + 1)/(105 *d**4*x**2) + 8*I*e**7*sqrt(-d**2/(e**2*x**2) + 1)/(105*d**6), True)) + 3* d**6*e*Piecewise((-d**2/(6*e*x**7*sqrt(d**2/(e**2*x**2) - 1)) + 5*e/(24*x* *5*sqrt(d**2/(e**2*x**2) - 1)) + e**3/(48*d**2*x**3*sqrt(d**2/(e**2*x**2) - 1)) - e**5/(16*d**4*x*sqrt(d**2/(e**2*x**2) - 1)) + e**6*acosh(d/(e*x))/ (16*d**5), Abs(d**2/(e**2*x**2)) > 1), (I*d**2/(6*e*x**7*sqrt(-d**2/(e**2* x**2) + 1)) - 5*I*e/(24*x**5*sqrt(-d**2/(e**2*x**2) + 1)) - I*e**3/(48*d** 2*x**3*sqrt(-d**2/(e**2*x**2) + 1)) + I*e**5/(16*d**4*x*sqrt(-d**2/(e**2*x **2) + 1)) - I*e**6*asin(d/(e*x))/(16*d**5), True)) + d**5*e**2*Piecewise( (3*I*d**3*sqrt(-1 + e**2*x**2/d**2)/(-15*d**2*x**5 + 15*e**2*x**7) - 4*I*d *e**2*x**2*sqrt(-1 + e**2*x**2/d**2)/(-15*d**2*x**5 + 15*e**2*x**7) + 2*I* e**6*x**6*sqrt(-1 + e**2*x**2/d**2)/(-15*d**5*x**5 + 15*d**3*e**2*x**7) - I*e**4*x**4*sqrt(-1 + e**2*x**2/d**2)/(-15*d**3*x**5 + 15*d*e**2*x**7), Ab s(e**2*x**2/d**2) > 1), (3*d**3*sqrt(1 - e**2*x**2/d**2)/(-15*d**2*x**5 + 15*e**2*x**7) - 4*d*e**2*x**2*sqrt(1 - e**2*x**2/d**2)/(-15*d**2*x**5 + 15 *e**2*x**7) + 2*e**6*x**6*sqrt(1 - e**2*x**2/d**2)/(-15*d**5*x**5 + 15*...
Time = 0.28 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.64 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^8} \, dx=-\frac {3 \, d e^{8} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{\sqrt {e^{2}}} - \frac {15}{16} \, d e^{7} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right ) - \frac {3 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{8} x}{d} + \frac {15}{16} \, \sqrt {-e^{2} x^{2} + d^{2}} e^{7} - \frac {2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{8} x}{d^{3}} + \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{7}}{16 \, d^{2}} + \frac {3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{7}}{16 \, d^{4}} - \frac {8 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{6}}{5 \, d^{3} x} + \frac {3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{5}}{16 \, d^{4} x^{2}} + \frac {2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{4}}{5 \, d^{3} x^{3}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{3}}{8 \, d^{2} x^{4}} - \frac {3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{2}}{5 \, d x^{5}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e}{2 \, x^{6}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d}{7 \, x^{7}} \]
-3*d*e^8*arcsin(e^2*x/(d*sqrt(e^2)))/sqrt(e^2) - 15/16*d*e^7*log(2*d^2/abs (x) + 2*sqrt(-e^2*x^2 + d^2)*d/abs(x)) - 3*sqrt(-e^2*x^2 + d^2)*e^8*x/d + 15/16*sqrt(-e^2*x^2 + d^2)*e^7 - 2*(-e^2*x^2 + d^2)^(3/2)*e^8*x/d^3 + 5/16 *(-e^2*x^2 + d^2)^(3/2)*e^7/d^2 + 3/16*(-e^2*x^2 + d^2)^(5/2)*e^7/d^4 - 8/ 5*(-e^2*x^2 + d^2)^(5/2)*e^6/(d^3*x) + 3/16*(-e^2*x^2 + d^2)^(7/2)*e^5/(d^ 4*x^2) + 2/5*(-e^2*x^2 + d^2)^(7/2)*e^4/(d^3*x^3) - 1/8*(-e^2*x^2 + d^2)^( 7/2)*e^3/(d^2*x^4) - 3/5*(-e^2*x^2 + d^2)^(7/2)*e^2/(d*x^5) - 1/2*(-e^2*x^ 2 + d^2)^(7/2)*e/x^6 - 1/7*(-e^2*x^2 + d^2)^(7/2)*d/x^7
Leaf count of result is larger than twice the leaf count of optimal. 542 vs. \(2 (180) = 360\).
Time = 0.31 (sec) , antiderivative size = 542, normalized size of antiderivative = 2.63 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^8} \, dx=\frac {{\left (5 \, d e^{8} + \frac {35 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d e^{6}}{x} + \frac {49 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d e^{4}}{x^{2}} - \frac {245 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d e^{2}}{x^{3}} - \frac {875 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} d}{x^{4}} + \frac {455 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{5} d}{e^{2} x^{5}} + \frac {9065 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{6} d}{e^{4} x^{6}}\right )} e^{14} x^{7}}{4480 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{7} {\left | e \right |}} - \frac {3 \, d e^{8} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{{\left | e \right |}} - \frac {15 \, d e^{8} \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 )}{16 \, {\left | e \right |}} + \sqrt {-e^{2} x^{2} + d^{2}} e^{7} - \frac {\frac {9065 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d e^{12}}{x} + \frac {455 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d e^{10}}{x^{2}} - \frac {875 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d e^{8}}{x^{3}} - \frac {245 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} d e^{6}}{x^{4}} + \frac {49 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{5} d e^{4}}{x^{5}} + \frac {35 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{6} d e^{2}}{x^{6}} + \frac {5 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{7} d}{x^{7}}}{4480 \, e^{6} {\left | e \right |}} \]
1/4480*(5*d*e^8 + 35*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*d*e^6/x + 49*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d*e^4/x^2 - 245*(d*e + sqrt(-e^2*x^2 + d ^2)*abs(e))^3*d*e^2/x^3 - 875*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4*d/x^4 + 455*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^5*d/(e^2*x^5) + 9065*(d*e + sqrt (-e^2*x^2 + d^2)*abs(e))^6*d/(e^4*x^6))*e^14*x^7/((d*e + sqrt(-e^2*x^2 + d ^2)*abs(e))^7*abs(e)) - 3*d*e^8*arcsin(e*x/d)*sgn(d)*sgn(e)/abs(e) - 15/16 *d*e^8*log(1/2*abs(-2*d*e - 2*sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*abs(x)))/a bs(e) + sqrt(-e^2*x^2 + d^2)*e^7 - 1/4480*(9065*(d*e + sqrt(-e^2*x^2 + d^2 )*abs(e))*d*e^12/x + 455*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d*e^10/x^2 - 875*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*d*e^8/x^3 - 245*(d*e + sqrt(-e ^2*x^2 + d^2)*abs(e))^4*d*e^6/x^4 + 49*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e)) ^5*d*e^4/x^5 + 35*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^6*d*e^2/x^6 + 5*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^7*d/x^7)/(e^6*abs(e))
Timed out. \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^8} \, dx=\int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^3}{x^8} \,d x \]