Integrand size = 27, antiderivative size = 169 \[ \int \frac {(d+e x)^2}{x^6 \sqrt {d^2-e^2 x^2}} \, dx=-\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}-\frac {e \sqrt {d^2-e^2 x^2}}{2 d x^4}-\frac {3 e^2 \sqrt {d^2-e^2 x^2}}{5 d^2 x^3}-\frac {3 e^3 \sqrt {d^2-e^2 x^2}}{4 d^3 x^2}-\frac {6 e^4 \sqrt {d^2-e^2 x^2}}{5 d^4 x}-\frac {3 e^5 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{4 d^4} \]
-3/4*e^5*arctanh((-e^2*x^2+d^2)^(1/2)/d)/d^4-1/5*(-e^2*x^2+d^2)^(1/2)/x^5- 1/2*e*(-e^2*x^2+d^2)^(1/2)/d/x^4-3/5*e^2*(-e^2*x^2+d^2)^(1/2)/d^2/x^3-3/4* e^3*(-e^2*x^2+d^2)^(1/2)/d^3/x^2-6/5*e^4*(-e^2*x^2+d^2)^(1/2)/d^4/x
Time = 0.25 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.78 \[ \int \frac {(d+e x)^2}{x^6 \sqrt {d^2-e^2 x^2}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-4 d^4-10 d^3 e x-12 d^2 e^2 x^2-15 d e^3 x^3-24 e^4 x^4\right )}{20 d^4 x^5}-\frac {3 \sqrt {d^2} e^5 \log (x)}{4 d^5}+\frac {3 \sqrt {d^2} e^5 \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right )}{4 d^5} \]
(Sqrt[d^2 - e^2*x^2]*(-4*d^4 - 10*d^3*e*x - 12*d^2*e^2*x^2 - 15*d*e^3*x^3 - 24*e^4*x^4))/(20*d^4*x^5) - (3*Sqrt[d^2]*e^5*Log[x])/(4*d^5) + (3*Sqrt[d ^2]*e^5*Log[Sqrt[d^2] - Sqrt[d^2 - e^2*x^2]])/(4*d^5)
Time = 0.35 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.07, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.519, Rules used = {540, 25, 27, 539, 27, 539, 27, 539, 25, 27, 534, 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.
| \(\displaystyle \int \frac {(d+e x)^2}{x^6 \sqrt {d^2-e^2 x^2}} \, dx\) |
| \(\Big \downarrow \) 540 |
| \(\displaystyle -\frac {\int -\frac {d^2 e (10 d+9 e x)}{x^5 \sqrt {d^2-e^2 x^2}}dx}{5 d^2}-\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {d^2 e (10 d+9 e x)}{x^5 \sqrt {d^2-e^2 x^2}}dx}{5 d^2}-\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} e \int \frac {10 d+9 e x}{x^5 \sqrt {d^2-e^2 x^2}}dx-\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}\) |
| \(\Big \downarrow \) 539 |
| \(\displaystyle \frac {1}{5} e \left (-\frac {\int -\frac {6 d e (6 d+5 e x)}{x^4 \sqrt {d^2-e^2 x^2}}dx}{4 d^2}-\frac {5 \sqrt {d^2-e^2 x^2}}{2 d x^4}\right )-\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} e \left (\frac {3 e \int \frac {6 d+5 e x}{x^4 \sqrt {d^2-e^2 x^2}}dx}{2 d}-\frac {5 \sqrt {d^2-e^2 x^2}}{2 d x^4}\right )-\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}\) |
| \(\Big \downarrow \) 539 |
| \(\displaystyle \frac {1}{5} e \left (\frac {3 e \left (-\frac {\int -\frac {3 d e (5 d+4 e x)}{x^3 \sqrt {d^2-e^2 x^2}}dx}{3 d^2}-\frac {2 \sqrt {d^2-e^2 x^2}}{d x^3}\right )}{2 d}-\frac {5 \sqrt {d^2-e^2 x^2}}{2 d x^4}\right )-\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} e \left (\frac {3 e \left (\frac {e \int \frac {5 d+4 e x}{x^3 \sqrt {d^2-e^2 x^2}}dx}{d}-\frac {2 \sqrt {d^2-e^2 x^2}}{d x^3}\right )}{2 d}-\frac {5 \sqrt {d^2-e^2 x^2}}{2 d x^4}\right )-\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}\) |
| \(\Big \downarrow \) 539 |
| \(\displaystyle \frac {1}{5} e \left (\frac {3 e \left (\frac {e \left (-\frac {\int -\frac {d e (8 d+5 e x)}{x^2 \sqrt {d^2-e^2 x^2}}dx}{2 d^2}-\frac {5 \sqrt {d^2-e^2 x^2}}{2 d x^2}\right )}{d}-\frac {2 \sqrt {d^2-e^2 x^2}}{d x^3}\right )}{2 d}-\frac {5 \sqrt {d^2-e^2 x^2}}{2 d x^4}\right )-\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{5} e \left (\frac {3 e \left (\frac {e \left (\frac {\int \frac {d e (8 d+5 e x)}{x^2 \sqrt {d^2-e^2 x^2}}dx}{2 d^2}-\frac {5 \sqrt {d^2-e^2 x^2}}{2 d x^2}\right )}{d}-\frac {2 \sqrt {d^2-e^2 x^2}}{d x^3}\right )}{2 d}-\frac {5 \sqrt {d^2-e^2 x^2}}{2 d x^4}\right )-\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} e \left (\frac {3 e \left (\frac {e \left (\frac {e \int \frac {8 d+5 e x}{x^2 \sqrt {d^2-e^2 x^2}}dx}{2 d}-\frac {5 \sqrt {d^2-e^2 x^2}}{2 d x^2}\right )}{d}-\frac {2 \sqrt {d^2-e^2 x^2}}{d x^3}\right )}{2 d}-\frac {5 \sqrt {d^2-e^2 x^2}}{2 d x^4}\right )-\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}\) |
| \(\Big \downarrow \) 534 |
| \(\displaystyle \frac {1}{5} e \left (\frac {3 e \left (\frac {e \left (\frac {e \left (5 e \int \frac {1}{x \sqrt {d^2-e^2 x^2}}dx-\frac {8 \sqrt {d^2-e^2 x^2}}{d x}\right )}{2 d}-\frac {5 \sqrt {d^2-e^2 x^2}}{2 d x^2}\right )}{d}-\frac {2 \sqrt {d^2-e^2 x^2}}{d x^3}\right )}{2 d}-\frac {5 \sqrt {d^2-e^2 x^2}}{2 d x^4}\right )-\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {1}{5} e \left (\frac {3 e \left (\frac {e \left (\frac {e \left (\frac {5}{2} e \int \frac {1}{x^2 \sqrt {d^2-e^2 x^2}}dx^2-\frac {8 \sqrt {d^2-e^2 x^2}}{d x}\right )}{2 d}-\frac {5 \sqrt {d^2-e^2 x^2}}{2 d x^2}\right )}{d}-\frac {2 \sqrt {d^2-e^2 x^2}}{d x^3}\right )}{2 d}-\frac {5 \sqrt {d^2-e^2 x^2}}{2 d x^4}\right )-\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{5} e \left (\frac {3 e \left (\frac {e \left (\frac {e \left (-\frac {5 \int \frac {1}{\frac {d^2}{e^2}-\frac {x^4}{e^2}}d\sqrt {d^2-e^2 x^2}}{e}-\frac {8 \sqrt {d^2-e^2 x^2}}{d x}\right )}{2 d}-\frac {5 \sqrt {d^2-e^2 x^2}}{2 d x^2}\right )}{d}-\frac {2 \sqrt {d^2-e^2 x^2}}{d x^3}\right )}{2 d}-\frac {5 \sqrt {d^2-e^2 x^2}}{2 d x^4}\right )-\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{5} e \left (\frac {3 e \left (\frac {e \left (\frac {e \left (-\frac {5 e \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d}-\frac {8 \sqrt {d^2-e^2 x^2}}{d x}\right )}{2 d}-\frac {5 \sqrt {d^2-e^2 x^2}}{2 d x^2}\right )}{d}-\frac {2 \sqrt {d^2-e^2 x^2}}{d x^3}\right )}{2 d}-\frac {5 \sqrt {d^2-e^2 x^2}}{2 d x^4}\right )-\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}\) |
-1/5*Sqrt[d^2 - e^2*x^2]/x^5 + (e*((-5*Sqrt[d^2 - e^2*x^2])/(2*d*x^4) + (3 *e*((-2*Sqrt[d^2 - e^2*x^2])/(d*x^3) + (e*((-5*Sqrt[d^2 - e^2*x^2])/(2*d*x ^2) + (e*((-8*Sqrt[d^2 - e^2*x^2])/(d*x) - (5*e*ArcTanh[Sqrt[d^2 - e^2*x^2 ]/d])/d))/(2*d)))/d))/(2*d)))/5
3.1.43.3.1 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[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
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[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-c)*x^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1))), x] + Simp[d Int[ x^(m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x] && ILtQ[m, 0] && GtQ[p, -1] && EqQ[m + 2*p + 3, 0]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*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*(a*d*(m + 1) - b*c*(m + 2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d, p}, x] && ILtQ[m, -1] && GtQ[p, -1] && IntegerQ[2*p]
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]
Time = 0.40 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.65
| method | result | size |
| risch | \(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (24 e^{4} x^{4}+15 d \,e^{3} x^{3}+12 d^{2} e^{2} x^{2}+10 d^{3} e x +4 d^{4}\right )}{20 x^{5} d^{4}}-\frac {3 e^{5} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{4 d^{3} \sqrt {d^{2}}}\) | \(110\) |
| default | \(d^{2} \left (-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{5 d^{2} x^{5}}+\frac {4 e^{2} \left (-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{3 d^{2} x^{3}}-\frac {2 e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{3 d^{4} x}\right )}{5 d^{2}}\right )+e^{2} \left (-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{3 d^{2} x^{3}}-\frac {2 e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{3 d^{4} x}\right )+2 d e \left (-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{4 d^{2} x^{4}}+\frac {3 e^{2} \left (-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{2 d^{2} x^{2}}-\frac {e^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 d^{2} \sqrt {d^{2}}}\right )}{4 d^{2}}\right )\) | \(240\) |
-1/20*(-e^2*x^2+d^2)^(1/2)*(24*e^4*x^4+15*d*e^3*x^3+12*d^2*e^2*x^2+10*d^3* e*x+4*d^4)/x^5/d^4-3/4/d^3*e^5/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x ^2+d^2)^(1/2))/x)
Time = 0.34 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.58 \[ \int \frac {(d+e x)^2}{x^6 \sqrt {d^2-e^2 x^2}} \, dx=\frac {15 \, e^{5} x^{5} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - {\left (24 \, e^{4} x^{4} + 15 \, d e^{3} x^{3} + 12 \, d^{2} e^{2} x^{2} + 10 \, d^{3} e x + 4 \, d^{4}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{20 \, d^{4} x^{5}} \]
1/20*(15*e^5*x^5*log(-(d - sqrt(-e^2*x^2 + d^2))/x) - (24*e^4*x^4 + 15*d*e ^3*x^3 + 12*d^2*e^2*x^2 + 10*d^3*e*x + 4*d^4)*sqrt(-e^2*x^2 + d^2))/(d^4*x ^5)
Result contains complex when optimal does not.
Time = 3.96 (sec) , antiderivative size = 510, normalized size of antiderivative = 3.02 \[ \int \frac {(d+e x)^2}{x^6 \sqrt {d^2-e^2 x^2}} \, dx=d^{2} \left (\begin {cases} - \frac {e \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{5 d^{2} x^{4}} - \frac {4 e^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{15 d^{4} x^{2}} - \frac {8 e^{5} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{15 d^{6}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i e \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{5 d^{2} x^{4}} - \frac {4 i e^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{15 d^{4} x^{2}} - \frac {8 i e^{5} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{15 d^{6}} & \text {otherwise} \end {cases}\right ) + 2 d e \left (\begin {cases} - \frac {1}{4 e x^{5} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} - \frac {e}{8 d^{2} x^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {3 e^{3}}{8 d^{4} x \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} - \frac {3 e^{4} \operatorname {acosh}{\left (\frac {d}{e x} \right )}}{8 d^{5}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac {i}{4 e x^{5} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + \frac {i e}{8 d^{2} x^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {3 i e^{3}}{8 d^{4} x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + \frac {3 i e^{4} \operatorname {asin}{\left (\frac {d}{e x} \right )}}{8 d^{5}} & \text {otherwise} \end {cases}\right ) + e^{2} \left (\begin {cases} - \frac {e \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{3 d^{2} x^{2}} - \frac {2 e^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{3 d^{4}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i e \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{3 d^{2} x^{2}} - \frac {2 i e^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{3 d^{4}} & \text {otherwise} \end {cases}\right ) \]
d**2*Piecewise((-e*sqrt(d**2/(e**2*x**2) - 1)/(5*d**2*x**4) - 4*e**3*sqrt( d**2/(e**2*x**2) - 1)/(15*d**4*x**2) - 8*e**5*sqrt(d**2/(e**2*x**2) - 1)/( 15*d**6), Abs(d**2/(e**2*x**2)) > 1), (-I*e*sqrt(-d**2/(e**2*x**2) + 1)/(5 *d**2*x**4) - 4*I*e**3*sqrt(-d**2/(e**2*x**2) + 1)/(15*d**4*x**2) - 8*I*e* *5*sqrt(-d**2/(e**2*x**2) + 1)/(15*d**6), True)) + 2*d*e*Piecewise((-1/(4* e*x**5*sqrt(d**2/(e**2*x**2) - 1)) - e/(8*d**2*x**3*sqrt(d**2/(e**2*x**2) - 1)) + 3*e**3/(8*d**4*x*sqrt(d**2/(e**2*x**2) - 1)) - 3*e**4*acosh(d/(e*x ))/(8*d**5), Abs(d**2/(e**2*x**2)) > 1), (I/(4*e*x**5*sqrt(-d**2/(e**2*x** 2) + 1)) + I*e/(8*d**2*x**3*sqrt(-d**2/(e**2*x**2) + 1)) - 3*I*e**3/(8*d** 4*x*sqrt(-d**2/(e**2*x**2) + 1)) + 3*I*e**4*asin(d/(e*x))/(8*d**5), True)) + e**2*Piecewise((-e*sqrt(d**2/(e**2*x**2) - 1)/(3*d**2*x**2) - 2*e**3*sq rt(d**2/(e**2*x**2) - 1)/(3*d**4), Abs(d**2/(e**2*x**2)) > 1), (-I*e*sqrt( -d**2/(e**2*x**2) + 1)/(3*d**2*x**2) - 2*I*e**3*sqrt(-d**2/(e**2*x**2) + 1 )/(3*d**4), True))
Time = 0.28 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.93 \[ \int \frac {(d+e x)^2}{x^6 \sqrt {d^2-e^2 x^2}} \, dx=-\frac {3 \, e^{5} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{4 \, d^{4}} - \frac {6 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{4}}{5 \, d^{4} x} - \frac {3 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{3}}{4 \, d^{3} x^{2}} - \frac {3 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{2}}{5 \, d^{2} x^{3}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} e}{2 \, d x^{4}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}}}{5 \, x^{5}} \]
-3/4*e^5*log(2*d^2/abs(x) + 2*sqrt(-e^2*x^2 + d^2)*d/abs(x))/d^4 - 6/5*sqr t(-e^2*x^2 + d^2)*e^4/(d^4*x) - 3/4*sqrt(-e^2*x^2 + d^2)*e^3/(d^3*x^2) - 3 /5*sqrt(-e^2*x^2 + d^2)*e^2/(d^2*x^3) - 1/2*sqrt(-e^2*x^2 + d^2)*e/(d*x^4) - 1/5*sqrt(-e^2*x^2 + d^2)/x^5
Leaf count of result is larger than twice the leaf count of optimal. 385 vs. \(2 (145) = 290\).
Time = 0.29 (sec) , antiderivative size = 385, normalized size of antiderivative = 2.28 \[ \int \frac {(d+e x)^2}{x^6 \sqrt {d^2-e^2 x^2}} \, dx=\frac {{\left (e^{6} + \frac {5 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} e^{4}}{x} + \frac {15 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} e^{2}}{x^{2}} + \frac {40 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3}}{x^{3}} + \frac {110 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4}}{e^{2} x^{4}}\right )} e^{10} x^{5}}{160 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{5} d^{4} {\left | e \right |}} - \frac {3 \, e^{6} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |} \right |}}{2 \, e^{2} {\left | x \right |}}\right )}{4 \, d^{4} {\left | e \right |}} - \frac {\frac {110 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{16} e^{8}}{x} + \frac {40 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{16} e^{6}}{x^{2}} + \frac {15 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d^{16} e^{4}}{x^{3}} + \frac {5 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} d^{16} e^{2}}{x^{4}} + \frac {{\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{5} d^{16}}{x^{5}}}{160 \, d^{20} e^{4} {\left | e \right |}} \]
1/160*(e^6 + 5*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*e^4/x + 15*(d*e + sqrt( -e^2*x^2 + d^2)*abs(e))^2*e^2/x^2 + 40*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e)) ^3/x^3 + 110*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4/(e^2*x^4))*e^10*x^5/((d *e + sqrt(-e^2*x^2 + d^2)*abs(e))^5*d^4*abs(e)) - 3/4*e^6*log(1/2*abs(-2*d *e - 2*sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*abs(x)))/(d^4*abs(e)) - 1/160*(11 0*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*d^16*e^8/x + 40*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d^16*e^6/x^2 + 15*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3* d^16*e^4/x^3 + 5*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4*d^16*e^2/x^4 + (d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^5*d^16/x^5)/(d^20*e^4*abs(e))
Timed out. \[ \int \frac {(d+e x)^2}{x^6 \sqrt {d^2-e^2 x^2}} \, dx=\int \frac {{\left (d+e\,x\right )}^2}{x^6\,\sqrt {d^2-e^2\,x^2}} \,d x \]