Integrand size = 27, antiderivative size = 102 \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^4 (d+e x)^2} \, dx=\frac {e (d-e x) \sqrt {d^2-e^2 x^2}}{x^2}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{3 x^3}-e^3 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-e^3 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]
-1/3*(-e^2*x^2+d^2)^(3/2)/x^3-e^3*arctan(e*x/(-e^2*x^2+d^2)^(1/2))-e^3*arc tanh((-e^2*x^2+d^2)^(1/2)/d)+e*(-e*x+d)*(-e^2*x^2+d^2)^(1/2)/x^2
Time = 0.29 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.35 \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^4 (d+e x)^2} \, dx=\frac {\left (-d^2+3 d e x-2 e^2 x^2\right ) \sqrt {d^2-e^2 x^2}}{3 x^3}+2 e^3 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )-\frac {\sqrt {d^2} e^3 \log (x)}{d}+\frac {\sqrt {d^2} e^3 \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right )}{d} \]
((-d^2 + 3*d*e*x - 2*e^2*x^2)*Sqrt[d^2 - e^2*x^2])/(3*x^3) + 2*e^3*ArcTan[ (e*x)/(Sqrt[d^2] - Sqrt[d^2 - e^2*x^2])] - (Sqrt[d^2]*e^3*Log[x])/d + (Sqr t[d^2]*e^3*Log[Sqrt[d^2] - Sqrt[d^2 - e^2*x^2]])/d
Time = 0.30 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.04, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {570, 540, 27, 537, 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.
\(\displaystyle \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^4 (d+e x)^2} \, dx\) |
\(\Big \downarrow \) 570 |
\(\displaystyle \int \frac {(d-e x)^2 \sqrt {d^2-e^2 x^2}}{x^4}dx\) |
\(\Big \downarrow \) 540 |
\(\displaystyle -\frac {\int \frac {3 d^2 e (2 d-e x) \sqrt {d^2-e^2 x^2}}{x^3}dx}{3 d^2}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{3 x^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -e \int \frac {(2 d-e x) \sqrt {d^2-e^2 x^2}}{x^3}dx-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{3 x^3}\) |
\(\Big \downarrow \) 537 |
\(\displaystyle -e \left (\frac {1}{2} e^2 \int -\frac {2 (d-e x)}{x \sqrt {d^2-e^2 x^2}}dx-\frac {(d-e x) \sqrt {d^2-e^2 x^2}}{x^2}\right )-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{3 x^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -e \left (e^2 \left (-\int \frac {d-e x}{x \sqrt {d^2-e^2 x^2}}dx\right )-\frac {(d-e x) \sqrt {d^2-e^2 x^2}}{x^2}\right )-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{3 x^3}\) |
\(\Big \downarrow \) 538 |
\(\displaystyle -e \left (-\left (e^2 \left (d \int \frac {1}{x \sqrt {d^2-e^2 x^2}}dx-e \int \frac {1}{\sqrt {d^2-e^2 x^2}}dx\right )\right )-\frac {(d-e x) \sqrt {d^2-e^2 x^2}}{x^2}\right )-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{3 x^3}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle -e \left (-\left (e^2 \left (d \int \frac {1}{x \sqrt {d^2-e^2 x^2}}dx-e \int \frac {1}{\frac {e^2 x^2}{d^2-e^2 x^2}+1}d\frac {x}{\sqrt {d^2-e^2 x^2}}\right )\right )-\frac {(d-e x) \sqrt {d^2-e^2 x^2}}{x^2}\right )-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{3 x^3}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle -e \left (-\left (e^2 \left (d \int \frac {1}{x \sqrt {d^2-e^2 x^2}}dx-\arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )\right )\right )-\frac {(d-e x) \sqrt {d^2-e^2 x^2}}{x^2}\right )-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{3 x^3}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle -e \left (-\left (e^2 \left (\frac {1}{2} d \int \frac {1}{x^2 \sqrt {d^2-e^2 x^2}}dx^2-\arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )\right )\right )-\frac {(d-e x) \sqrt {d^2-e^2 x^2}}{x^2}\right )-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{3 x^3}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -e \left (-\left (e^2 \left (-\frac {d \int \frac {1}{\frac {d^2}{e^2}-\frac {x^4}{e^2}}d\sqrt {d^2-e^2 x^2}}{e^2}-\arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )\right )\right )-\frac {(d-e x) \sqrt {d^2-e^2 x^2}}{x^2}\right )-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{3 x^3}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -e \left (-\left (e^2 \left (-\arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )\right )\right )-\frac {(d-e x) \sqrt {d^2-e^2 x^2}}{x^2}\right )-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{3 x^3}\) |
-1/3*(d^2 - e^2*x^2)^(3/2)/x^3 - e*(-(((d - e*x)*Sqrt[d^2 - e^2*x^2])/x^2) - e^2*(-ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]] - ArcTanh[Sqrt[d^2 - e^2*x^2]/d ]))
3.2.66.3.1 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[(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[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c^(2*n)/a^n Int[(e*x)^m*((a + b*x^2)^(n + p)/(c - d*x)^ n), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b*c^2 + a*d^2, 0] && I LtQ[n, -1] && !(IGtQ[m, 0] && ILtQ[m + n, 0] && !GtQ[p, 1])
Time = 0.46 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.11
method | result | size |
risch | \(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (2 e^{2} x^{2}-3 d e x +d^{2}\right )}{3 x^{3}}-\frac {e^{4} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{\sqrt {e^{2}}}-\frac {e^{3} d \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\) | \(113\) |
default | \(\text {Expression too large to display}\) | \(983\) |
-1/3*(-e^2*x^2+d^2)^(1/2)*(2*e^2*x^2-3*d*e*x+d^2)/x^3-e^4/(e^2)^(1/2)*arct an((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))-e^3*d/(d^2)^(1/2)*ln((2*d^2+2*(d^2) ^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)
Time = 0.29 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.04 \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^4 (d+e x)^2} \, dx=\frac {6 \, e^{3} x^{3} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + 3 \, e^{3} x^{3} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - {\left (2 \, e^{2} x^{2} - 3 \, d e x + d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{3 \, x^{3}} \]
1/3*(6*e^3*x^3*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + 3*e^3*x^3*log(- (d - sqrt(-e^2*x^2 + d^2))/x) - (2*e^2*x^2 - 3*d*e*x + d^2)*sqrt(-e^2*x^2 + d^2))/x^3
Result contains complex when optimal does not.
Time = 3.93 (sec) , antiderivative size = 338, normalized size of antiderivative = 3.31 \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^4 (d+e x)^2} \, dx=d^{2} \left (\begin {cases} - \frac {e \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{3 x^{2}} + \frac {e^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{3 d^{2}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i e \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{3 x^{2}} + \frac {i e^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{3 d^{2}} & \text {otherwise} \end {cases}\right ) - 2 d e \left (\begin {cases} - \frac {e \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{2 x} + \frac {e^{2} \operatorname {acosh}{\left (\frac {d}{e x} \right )}}{2 d} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac {i d^{2}}{2 e x^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e}{2 x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e^{2} \operatorname {asin}{\left (\frac {d}{e x} \right )}}{2 d} & \text {otherwise} \end {cases}\right ) + e^{2} \left (\begin {cases} \frac {i d}{x \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + i e \operatorname {acosh}{\left (\frac {e x}{d} \right )} - \frac {i e^{2} x}{d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\- \frac {d}{x \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - e \operatorname {asin}{\left (\frac {e x}{d} \right )} + \frac {e^{2} x}{d \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) \]
d**2*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) ) - 2*d*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)) + 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))
Time = 0.28 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.43 \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^4 (d+e x)^2} \, dx=-\frac {e^{4} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{\sqrt {e^{2}}} - e^{3} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right ) + \frac {\sqrt {-e^{2} x^{2} + d^{2}} e^{3}}{d} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} e^{2}}{x} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e}{d x^{2}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}}}{3 \, x^{3}} \]
-e^4*arcsin(e^2*x/(d*sqrt(e^2)))/sqrt(e^2) - e^3*log(2*d^2/abs(x) + 2*sqrt (-e^2*x^2 + d^2)*d/abs(x)) + sqrt(-e^2*x^2 + d^2)*e^3/d - sqrt(-e^2*x^2 + d^2)*e^2/x + (-e^2*x^2 + d^2)^(3/2)*e/(d*x^2) - 1/3*(-e^2*x^2 + d^2)^(3/2) /x^3
Exception generated. \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^4 (d+e x)^2} \, dx=\text {Exception raised: NotImplementedError} \]
Exception raised: NotImplementedError >> unable to parse Giac output: abs( sageVARe)*(1/3*(12*sageVARe^2*sqrt(2*sageVARd*sageVARe*(sageVARe*sageVARx+ sageVARd)^-1/sageVARe-1)*(2*sageVARd*sageVARe*(sageVARe*sageVARx+sageVARd) ^-1/sageVARe-1)^2*s
Timed out. \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^4 (d+e x)^2} \, dx=\int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}}{x^4\,{\left (d+e\,x\right )}^2} \,d x \]