\(\int \frac {(d+e x)^2}{\sqrt {d^2-e^2 x^2}} \, dx\) [829]

Optimal result

Integrand size = 24, antiderivative size = 74 \[ \int \frac {(d+e x)^2}{\sqrt {d^2-e^2 x^2}} \, dx=-\frac {2 d \sqrt {d^2-e^2 x^2}}{e}-\frac {1}{2} x \sqrt {d^2-e^2 x^2}+\frac {3 d^2 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e} \]

[Out]

3/2*d^2*arctan(e*x/(-e^2*x^2+d^2)^(1/2))/e-2*d*(-e^2*x^2+d^2)^(1/2)/e-1/2*x*(-e^2*x^2+d^2)^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.12, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {685, 655, 223, 209} \[ \int \frac {(d+e x)^2}{\sqrt {d^2-e^2 x^2}} \, dx=\frac {3 d^2 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e}-\frac {3 d \sqrt {d^2-e^2 x^2}}{2 e}-\frac {(d+e x) \sqrt {d^2-e^2 x^2}}{2 e} \]

[In]

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

[Out]

(-3*d*Sqrt[d^2 - e^2*x^2])/(2*e) - ((d + e*x)*Sqrt[d^2 - e^2*x^2])/(2*e) + (3*d^2*ArcTan[(e*x)/Sqrt[d^2 - e^2*
x^2]])/(2*e)

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 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 655

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[e*((a + c*x^2)^(p + 1)/(2*c*(p + 1))),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

Rule 685

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*((a + c*x^2)^(p
+ 1)/(c*(m + 2*p + 1))), x] + Dist[2*c*d*((m + p)/(c*(m + 2*p + 1))), Int[(d + e*x)^(m - 1)*(a + c*x^2)^p, x]
, x] /; FreeQ[{a, c, d, e, p}, x] && EqQ[c*d^2 + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p
]

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

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.93 \[ \int \frac {(d+e x)^2}{\sqrt {d^2-e^2 x^2}} \, dx=-\frac {(4 d+e x) \sqrt {d^2-e^2 x^2}+6 d^2 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )}{2 e} \]

[In]

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

[Out]

-1/2*((4*d + e*x)*Sqrt[d^2 - e^2*x^2] + 6*d^2*ArcTan[(e*x)/(Sqrt[d^2] - Sqrt[d^2 - e^2*x^2])])/e

Maple [A] (verified)

Time = 2.28 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.81

method	result	size
risch	\(-\frac {\left (e x +4 d \right ) \sqrt {-x^{2} e^{2}+d^{2}}}{2 e}+\frac {3 d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-x^{2} e^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\)	\(60\)
default	\(\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-x^{2} e^{2}+d^{2}}}\right )}{\sqrt {e^{2}}}+e^{2} \left (-\frac {x \sqrt {-x^{2} e^{2}+d^{2}}}{2 e^{2}}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-x^{2} e^{2}+d^{2}}}\right )}{2 e^{2} \sqrt {e^{2}}}\right )-\frac {2 d \sqrt {-x^{2} e^{2}+d^{2}}}{e}\)	\(113\)

[In]

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

[Out]

-1/2*(e*x+4*d)/e*(-e^2*x^2+d^2)^(1/2)+3/2*d^2/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.81 \[ \int \frac {(d+e x)^2}{\sqrt {d^2-e^2 x^2}} \, dx=-\frac {6 \, d^{2} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + \sqrt {-e^{2} x^{2} + d^{2}} {\left (e x + 4 \, d\right )}}{2 \, e} \]

[In]

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

[Out]

-1/2*(6*d^2*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + sqrt(-e^2*x^2 + d^2)*(e*x + 4*d))/e

Sympy [A] (verification not implemented)

Time = 0.49 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.51 \[ \int \frac {(d+e x)^2}{\sqrt {d^2-e^2 x^2}} \, dx=\begin {cases} \frac {3 d^{2} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{2} + \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {2 d}{e} - \frac {x}{2}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {\begin {cases} d^{2} x & \text {for}\: e = 0 \\\frac {\left (d + e x\right )^{3}}{3 e} & \text {otherwise} \end {cases}}{\sqrt {d^{2}}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((3*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)/sqrt(-e**2*x**2), True))/2 + sqrt(d**2 - e**2*x**2)*(-2*d/e - x/2), Ne(e**2, 0)), (Piecewise((d**2*x
, Eq(e, 0)), ((d + e*x)**3/(3*e), True))/sqrt(d**2), True))

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.84 \[ \int \frac {(d+e x)^2}{\sqrt {d^2-e^2 x^2}} \, dx=\frac {3 \, d^{2} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{2 \, \sqrt {e^{2}}} - \frac {1}{2} \, \sqrt {-e^{2} x^{2} + d^{2}} x - \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{e} \]

[In]

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

[Out]

3/2*d^2*arcsin(e^2*x/(d*sqrt(e^2)))/sqrt(e^2) - 1/2*sqrt(-e^2*x^2 + d^2)*x - 2*sqrt(-e^2*x^2 + d^2)*d/e

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.61 \[ \int \frac {(d+e x)^2}{\sqrt {d^2-e^2 x^2}} \, dx=\frac {3 \, d^{2} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{2 \, {\left | e \right |}} - \frac {1}{2} \, \sqrt {-e^{2} x^{2} + d^{2}} {\left (x + \frac {4 \, d}{e}\right )} \]

[In]

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

[Out]

3/2*d^2*arcsin(e*x/d)*sgn(d)*sgn(e)/abs(e) - 1/2*sqrt(-e^2*x^2 + d^2)*(x + 4*d/e)

Mupad [F(-1)]

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

[In]

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

[Out]

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