\(\int \frac {1}{(d+e x)^2 \sqrt {d^2-e^2 x^2}} \, dx\) [832]

Optimal result

Integrand size = 24, antiderivative size = 67 \[ \int \frac {1}{(d+e x)^2 \sqrt {d^2-e^2 x^2}} \, dx=-\frac {\sqrt {d^2-e^2 x^2}}{3 d e (d+e x)^2}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^2 e (d+e x)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {673, 665} \[ \int \frac {1}{(d+e x)^2 \sqrt {d^2-e^2 x^2}} \, dx=-\frac {\sqrt {d^2-e^2 x^2}}{3 d^2 e (d+e x)}-\frac {\sqrt {d^2-e^2 x^2}}{3 d e (d+e x)^2} \]

[In]

Int[1/((d + e*x)^2*Sqrt[d^2 - e^2*x^2]),x]

[Out]

-1/3*Sqrt[d^2 - e^2*x^2]/(d*e*(d + e*x)^2) - Sqrt[d^2 - e^2*x^2]/(3*d^2*e*(d + e*x))

Rule 665

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

Rule 673

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

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

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.61 \[ \int \frac {1}{(d+e x)^2 \sqrt {d^2-e^2 x^2}} \, dx=\frac {(-2 d-e x) \sqrt {d^2-e^2 x^2}}{3 d^2 e (d+e x)^2} \]

[In]

Integrate[1/((d + e*x)^2*Sqrt[d^2 - e^2*x^2]),x]

[Out]

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

Maple [A] (verified)

Time = 2.38 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.55

method	result	size
trager	\(-\frac {\left (e x +2 d \right ) \sqrt {-x^{2} e^{2}+d^{2}}}{3 d^{2} \left (e x +d \right )^{2} e}\)	\(37\)
gosper	\(-\frac {\left (-e x +d \right ) \left (e x +2 d \right )}{3 \left (e x +d \right ) d^{2} e \sqrt {-x^{2} e^{2}+d^{2}}}\)	\(43\)
default	\(\frac {-\frac {\sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}{3 d e \left (x +\frac {d}{e}\right )^{2}}-\frac {\sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}{3 d^{2} \left (x +\frac {d}{e}\right )}}{e^{2}}\)	\(93\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.04 \[ \int \frac {1}{(d+e x)^2 \sqrt {d^2-e^2 x^2}} \, dx=-\frac {2 \, e^{2} x^{2} + 4 \, d e x + 2 \, d^{2} + \sqrt {-e^{2} x^{2} + d^{2}} {\left (e x + 2 \, d\right )}}{3 \, {\left (d^{2} e^{3} x^{2} + 2 \, d^{3} e^{2} x + d^{4} e\right )}} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {1}{(d+e x)^2 \sqrt {d^2-e^2 x^2}} \, dx=\int \frac {1}{\sqrt {- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(d+e x)^2 \sqrt {d^2-e^2 x^2}} \, dx=-\frac {\sqrt {-e^{2} x^{2} + d^{2}}}{3 \, {\left (d e^{3} x^{2} + 2 \, d^{2} e^{2} x + d^{3} e\right )}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}}}{3 \, {\left (d^{2} e^{2} x + d^{3} e\right )}} \]

[In]

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

[Out]

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

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.29 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(d+e x)^2 \sqrt {d^2-e^2 x^2}} \, dx=\frac {i \, \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{3 \, d^{2} {\left | e \right |}} - \frac {{\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {3}{2}} + 3 \, \sqrt {\frac {2 \, d}{e x + d} - 1}}{6 \, d^{2} {\left | e \right |} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )} \]

[In]

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

[Out]

1/3*I*sgn(1/(e*x + d))*sgn(e)/(d^2*abs(e)) - 1/6*((2*d/(e*x + d) - 1)^(3/2) + 3*sqrt(2*d/(e*x + d) - 1))/(d^2*
abs(e)*sgn(1/(e*x + d))*sgn(e))

Mupad [B] (verification not implemented)

Time = 9.54 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.54 \[ \int \frac {1}{(d+e x)^2 \sqrt {d^2-e^2 x^2}} \, dx=-\frac {\sqrt {d^2-e^2\,x^2}\,\left (2\,d+e\,x\right )}{3\,d^2\,e\,{\left (d+e\,x\right )}^2} \]

[In]

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

[Out]

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