Integrand size = 25, antiderivative size = 113 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x} \, dx=\frac {1}{8} d^2 (8 d+3 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{12} (4 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac {3}{8} d^4 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-d^4 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]
1/12*(3*e*x+4*d)*(-e^2*x^2+d^2)^(3/2)+3/8*d^4*arctan(e*x/(-e^2*x^2+d^2)^(1 /2))-d^4*arctanh((-e^2*x^2+d^2)^(1/2)/d)+1/8*d^2*(3*e*x+8*d)*(-e^2*x^2+d^2 )^(1/2)
Time = 0.38 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.26 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x} \, dx=\frac {1}{24} \sqrt {d^2-e^2 x^2} \left (32 d^3+15 d^2 e x-8 d e^2 x^2-6 e^3 x^3\right )-\frac {3}{4} d^4 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )-d^3 \sqrt {d^2} \log (x)+d^3 \sqrt {d^2} \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right ) \]
(Sqrt[d^2 - e^2*x^2]*(32*d^3 + 15*d^2*e*x - 8*d*e^2*x^2 - 6*e^3*x^3))/24 - (3*d^4*ArcTan[(e*x)/(Sqrt[d^2] - Sqrt[d^2 - e^2*x^2])])/4 - d^3*Sqrt[d^2] *Log[x] + d^3*Sqrt[d^2]*Log[Sqrt[d^2] - Sqrt[d^2 - e^2*x^2]]
Time = 0.27 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.04, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {535, 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.
| \(\displaystyle \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x} \, dx\) |
| \(\Big \downarrow \) 535 |
| \(\displaystyle \frac {1}{4} d^2 \int \frac {(4 d+3 e x) \sqrt {d^2-e^2 x^2}}{x}dx+\frac {1}{12} (4 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}\) |
| \(\Big \downarrow \) 535 |
| \(\displaystyle \frac {1}{4} d^2 \left (\frac {1}{2} d^2 \int \frac {8 d+3 e x}{x \sqrt {d^2-e^2 x^2}}dx+\frac {1}{2} (8 d+3 e x) \sqrt {d^2-e^2 x^2}\right )+\frac {1}{12} (4 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}\) |
| \(\Big \downarrow \) 538 |
| \(\displaystyle \frac {1}{4} d^2 \left (\frac {1}{2} d^2 \left (3 e \int \frac {1}{\sqrt {d^2-e^2 x^2}}dx+8 d \int \frac {1}{x \sqrt {d^2-e^2 x^2}}dx\right )+\frac {1}{2} (8 d+3 e x) \sqrt {d^2-e^2 x^2}\right )+\frac {1}{12} (4 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {1}{4} d^2 \left (\frac {1}{2} d^2 \left (8 d \int \frac {1}{x \sqrt {d^2-e^2 x^2}}dx+3 e \int \frac {1}{\frac {e^2 x^2}{d^2-e^2 x^2}+1}d\frac {x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {1}{2} (8 d+3 e x) \sqrt {d^2-e^2 x^2}\right )+\frac {1}{12} (4 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {1}{4} d^2 \left (\frac {1}{2} d^2 \left (8 d \int \frac {1}{x \sqrt {d^2-e^2 x^2}}dx+3 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )\right )+\frac {1}{2} (8 d+3 e x) \sqrt {d^2-e^2 x^2}\right )+\frac {1}{12} (4 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {1}{4} d^2 \left (\frac {1}{2} d^2 \left (4 d \int \frac {1}{x^2 \sqrt {d^2-e^2 x^2}}dx^2+3 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )\right )+\frac {1}{2} (8 d+3 e x) \sqrt {d^2-e^2 x^2}\right )+\frac {1}{12} (4 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{4} d^2 \left (\frac {1}{2} d^2 \left (3 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {8 d \int \frac {1}{\frac {d^2}{e^2}-\frac {x^4}{e^2}}d\sqrt {d^2-e^2 x^2}}{e^2}\right )+\frac {1}{2} (8 d+3 e x) \sqrt {d^2-e^2 x^2}\right )+\frac {1}{12} (4 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{4} d^2 \left (\frac {1}{2} d^2 \left (3 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-8 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )\right )+\frac {1}{2} (8 d+3 e x) \sqrt {d^2-e^2 x^2}\right )+\frac {1}{12} (4 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}\) |
((4*d + 3*e*x)*(d^2 - e^2*x^2)^(3/2))/12 + (d^2*(((8*d + 3*e*x)*Sqrt[d^2 - e^2*x^2])/2 + (d^2*(3*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]] - 8*ArcTanh[Sqrt[ d^2 - e^2*x^2]/d]))/2))/4
3.1.7.3.1 Defintions of rubi rules used
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_))/((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]
Time = 0.34 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.39
| method | result | size |
| default | \(e \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )+d \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )\) | \(157\) |
e*(1/4*x*(-e^2*x^2+d^2)^(3/2)+3/4*d^2*(1/2*x*(-e^2*x^2+d^2)^(1/2)+1/2*d^2/ (e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))))+d*(1/3*(-e^2*x^2+ d^2)^(3/2)+d^2*((-e^2*x^2+d^2)^(1/2)-d^2/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/ 2)*(-e^2*x^2+d^2)^(1/2))/x)))
Time = 0.32 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.95 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x} \, dx=-\frac {3}{4} \, d^{4} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + d^{4} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - \frac {1}{24} \, {\left (6 \, e^{3} x^{3} + 8 \, d e^{2} x^{2} - 15 \, d^{2} e x - 32 \, d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}} \]
-3/4*d^4*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + d^4*log(-(d - sqrt(-e ^2*x^2 + d^2))/x) - 1/24*(6*e^3*x^3 + 8*d*e^2*x^2 - 15*d^2*e*x - 32*d^3)*s qrt(-e^2*x^2 + d^2)
Result contains complex when optimal does not.
Time = 5.95 (sec) , antiderivative size = 400, normalized size of antiderivative = 3.54 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x} \, dx=d^{3} \left (\begin {cases} \frac {d^{2}}{e x \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} - d \operatorname {acosh}{\left (\frac {d}{e x} \right )} - \frac {e x}{\sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i d^{2}}{e x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + i d \operatorname {asin}{\left (\frac {d}{e x} \right )} + \frac {i e x}{\sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} & \text {otherwise} \end {cases}\right ) + d^{2} e \left (\begin {cases} \frac {d^{2} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{2} + \frac {x \sqrt {d^{2} - e^{2} x^{2}}}{2} & \text {for}\: e^{2} \neq 0 \\x \sqrt {d^{2}} & \text {otherwise} \end {cases}\right ) - d e^{2} \left (\begin {cases} - \frac {d^{2} \sqrt {d^{2} - e^{2} x^{2}}}{3 e^{2}} + \frac {x^{2} \sqrt {d^{2} - e^{2} x^{2}}}{3} & \text {for}\: e^{2} \neq 0 \\\frac {x^{2} \sqrt {d^{2}}}{2} & \text {otherwise} \end {cases}\right ) - e^{3} \left (\begin {cases} \frac {d^{4} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{8 e^{2}} - \frac {d^{2} x \sqrt {d^{2} - e^{2} x^{2}}}{8 e^{2}} + \frac {x^{3} \sqrt {d^{2} - e^{2} x^{2}}}{4} & \text {for}\: e^{2} \neq 0 \\\frac {x^{3} \sqrt {d^{2}}}{3} & \text {otherwise} \end {cases}\right ) \]
d**3*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 *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)) + d**2*e*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)/s qrt(-e**2*x**2), True))/2 + x*sqrt(d**2 - e**2*x**2)/2, Ne(e**2, 0)), (x*s qrt(d**2), True)) - d*e**2*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) ) - e**3*Piecewise((d**4*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), Tru e))/(8*e**2) - d**2*x*sqrt(d**2 - e**2*x**2)/(8*e**2) + x**3*sqrt(d**2 - e **2*x**2)/4, Ne(e**2, 0)), (x**3*sqrt(d**2)/3, True))
Time = 0.29 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.21 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x} \, dx=\frac {3 \, d^{4} e \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{8 \, \sqrt {e^{2}}} - d^{4} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right ) + \frac {3}{8} \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2} e x + \sqrt {-e^{2} x^{2} + d^{2}} d^{3} + \frac {1}{4} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e x + \frac {1}{3} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d \]
3/8*d^4*e*arcsin(e^2*x/(d*sqrt(e^2)))/sqrt(e^2) - d^4*log(2*d^2/abs(x) + 2 *sqrt(-e^2*x^2 + d^2)*d/abs(x)) + 3/8*sqrt(-e^2*x^2 + d^2)*d^2*e*x + sqrt( -e^2*x^2 + d^2)*d^3 + 1/4*(-e^2*x^2 + d^2)^(3/2)*e*x + 1/3*(-e^2*x^2 + d^2 )^(3/2)*d
Time = 0.31 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x} \, dx=\frac {3 \, d^{4} e \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{8 \, {\left | e \right |}} - \frac {d^{4} e \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |} \right |}}{2 \, e^{2} {\left | x \right |}}\right )}{{\left | e \right |}} + \frac {1}{24} \, \sqrt {-e^{2} x^{2} + d^{2}} {\left (32 \, d^{3} + {\left (15 \, d^{2} e - 2 \, {\left (3 \, e^{3} x + 4 \, d e^{2}\right )} x\right )} x\right )} \]
3/8*d^4*e*arcsin(e*x/d)*sgn(d)*sgn(e)/abs(e) - d^4*e*log(1/2*abs(-2*d*e - 2*sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*abs(x)))/abs(e) + 1/24*sqrt(-e^2*x^2 + d^2)*(32*d^3 + (15*d^2*e - 2*(3*e^3*x + 4*d*e^2)*x)*x)
Time = 11.74 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.95 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x} \, dx=\frac {d\,{\left (d^2-e^2\,x^2\right )}^{3/2}}{3}-d^4\,\mathrm {atanh}\left (\frac {\sqrt {d^2-e^2\,x^2}}{d}\right )+d^3\,\sqrt {d^2-e^2\,x^2}+\frac {e\,x\,{\left (d^2-e^2\,x^2\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},\frac {1}{2};\ \frac {3}{2};\ \frac {e^2\,x^2}{d^2}\right )}{{\left (1-\frac {e^2\,x^2}{d^2}\right )}^{3/2}} \]