\(\int \frac {(d+e x)^2}{x \sqrt {d^2-e^2 x^2}} \, dx\) [38]

Optimal result

Integrand size = 27, antiderivative size = 66 \[ \int \frac {(d+e x)^2}{x \sqrt {d^2-e^2 x^2}} \, dx=-\sqrt {d^2-e^2 x^2}+2 d \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-d \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]

[Out]

2*d*arctan(e*x/(-e^2*x^2+d^2)^(1/2))-d*arctanh((-e^2*x^2+d^2)^(1/2)/d)-(-e^2*x^2+d^2)^(1/2)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {1823, 858, 223, 209, 272, 65, 214} \[ \int \frac {(d+e x)^2}{x \sqrt {d^2-e^2 x^2}} \, dx=2 d \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-d \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )-\sqrt {d^2-e^2 x^2} \]

[In]

Int[(d + e*x)^2/(x*Sqrt[d^2 - e^2*x^2]),x]

[Out]

-Sqrt[d^2 - e^2*x^2] + 2*d*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]] - d*ArcTanh[Sqrt[d^2 - e^2*x^2]/d]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[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] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 214

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]

Rule 223

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]

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 858

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rule 1823

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x,
Expon[Pq, x]]}, Simp[f*(c*x)^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*c^(q - 1)*(m + q + 2*p + 1))), x] + Dist[1/(
b*(m + q + 2*p + 1)), Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*x^q
- a*f*(m + q - 1)*x^(q - 2), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x]
&& PolyQ[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])

Rubi steps \begin{align*} \text {integral}& = -\sqrt {d^2-e^2 x^2}-\frac {\int \frac {-d^2 e^2-2 d e^3 x}{x \sqrt {d^2-e^2 x^2}} \, dx}{e^2} \\ & = -\sqrt {d^2-e^2 x^2}+d^2 \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx+(2 d e) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx \\ & = -\sqrt {d^2-e^2 x^2}+\frac {1}{2} d^2 \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )+(2 d e) \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right ) \\ & = -\sqrt {d^2-e^2 x^2}+2 d \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {d^2 \text {Subst}\left (\int \frac {1}{\frac {d^2}{e^2}-\frac {x^2}{e^2}} \, dx,x,\sqrt {d^2-e^2 x^2}\right )}{e^2} \\ & = -\sqrt {d^2-e^2 x^2}+2 d \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-d \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.52 \[ \int \frac {(d+e x)^2}{x \sqrt {d^2-e^2 x^2}} \, dx=-\sqrt {d^2-e^2 x^2}-4 d \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )-\sqrt {d^2} \log (x)+\sqrt {d^2} \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right ) \]

[In]

Integrate[(d + e*x)^2/(x*Sqrt[d^2 - e^2*x^2]),x]

[Out]

-Sqrt[d^2 - e^2*x^2] - 4*d*ArcTan[(e*x)/(Sqrt[d^2] - Sqrt[d^2 - e^2*x^2])] - Sqrt[d^2]*Log[x] + Sqrt[d^2]*Log[
Sqrt[d^2] - Sqrt[d^2 - e^2*x^2]]

Maple [A] (verified)

Time = 0.35 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.38

method	result	size
default	\(-\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}+\frac {2 d e \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{\sqrt {e^{2}}}\)	\(91\)

[In]

int((e*x+d)^2/x/(-e^2*x^2+d^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-(-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)+2*d*e/(e^2)^(1/2)*arcta
n((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.11 \[ \int \frac {(d+e x)^2}{x \sqrt {d^2-e^2 x^2}} \, dx=-4 \, d \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + d \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - \sqrt {-e^{2} x^{2} + d^{2}} \]

[In]

integrate((e*x+d)^2/x/(-e^2*x^2+d^2)^(1/2),x, algorithm="fricas")

[Out]

-4*d*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + d*log(-(d - sqrt(-e^2*x^2 + d^2))/x) - sqrt(-e^2*x^2 + d^2)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.29 (sec) , antiderivative size = 144, normalized size of antiderivative = 2.18 \[ \int \frac {(d+e x)^2}{x \sqrt {d^2-e^2 x^2}} \, dx=d^{2} \left (\begin {cases} - \frac {\operatorname {acosh}{\left (\frac {d}{e x} \right )}}{d} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac {i \operatorname {asin}{\left (\frac {d}{e x} \right )}}{d} & \text {otherwise} \end {cases}\right ) + 2 d e \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \wedge e^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {for}\: e^{2} \neq 0 \\\frac {x}{\sqrt {d^{2}}} & \text {otherwise} \end {cases}\right ) + e^{2} \left (\begin {cases} - \frac {\sqrt {d^{2} - e^{2} x^{2}}}{e^{2}} & \text {for}\: e^{2} \neq 0 \\\frac {x^{2}}{2 \sqrt {d^{2}}} & \text {otherwise} \end {cases}\right ) \]

[In]

integrate((e*x+d)**2/x/(-e**2*x**2+d**2)**(1/2),x)

[Out]

d**2*Piecewise((-acosh(d/(e*x))/d, Abs(d**2/(e**2*x**2)) > 1), (I*asin(d/(e*x))/d, True)) + 2*d*e*Piecewise((l
og(-2*e**2*x + 2*sqrt(-e**2)*sqrt(d**2 - e**2*x**2))/sqrt(-e**2), Ne(d**2, 0) & Ne(e**2, 0)), (x*log(x)/sqrt(-
e**2*x**2), Ne(e**2, 0)), (x/sqrt(d**2), True)) + e**2*Piecewise((-sqrt(d**2 - e**2*x**2)/e**2, Ne(e**2, 0)),
(x**2/(2*sqrt(d**2)), True))

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.14 \[ \int \frac {(d+e x)^2}{x \sqrt {d^2-e^2 x^2}} \, dx=\frac {2 \, d e \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{\sqrt {e^{2}}} - d \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right ) - \sqrt {-e^{2} x^{2} + d^{2}} \]

[In]

integrate((e*x+d)^2/x/(-e^2*x^2+d^2)^(1/2),x, algorithm="maxima")

[Out]

2*d*e*arcsin(e^2*x/(d*sqrt(e^2)))/sqrt(e^2) - d*log(2*d^2/abs(x) + 2*sqrt(-e^2*x^2 + d^2)*d/abs(x)) - sqrt(-e^
2*x^2 + d^2)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.18 \[ \int \frac {(d+e x)^2}{x \sqrt {d^2-e^2 x^2}} \, dx=\frac {2 \, d e \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{{\left | e \right |}} - \frac {d e \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |} \right |}}{2 \, e^{2} {\left | x \right |}}\right )}{{\left | e \right |}} - \sqrt {-e^{2} x^{2} + d^{2}} \]

[In]

integrate((e*x+d)^2/x/(-e^2*x^2+d^2)^(1/2),x, algorithm="giac")

[Out]

2*d*e*arcsin(e*x/d)*sgn(d)*sgn(e)/abs(e) - d*e*log(1/2*abs(-2*d*e - 2*sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*abs(x)
))/abs(e) - sqrt(-e^2*x^2 + d^2)

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^2}{x \sqrt {d^2-e^2 x^2}} \, dx=\int \frac {{\left (d+e\,x\right )}^2}{x\,\sqrt {d^2-e^2\,x^2}} \,d x \]

[In]

int((d + e*x)^2/(x*(d^2 - e^2*x^2)^(1/2)),x)

[Out]

int((d + e*x)^2/(x*(d^2 - e^2*x^2)^(1/2)), x)