\(\int \frac {\sqrt {d^2-e^2 x^2}}{(d+e x)^4} \, dx\) [193]

Optimal result

Integrand size = 24, antiderivative size = 67 \[ \int \frac {\sqrt {d^2-e^2 x^2}}{(d+e x)^4} \, dx=-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{5 d e (d+e x)^4}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{15 d^2 e (d+e x)^3} \]

[Out]

-1/5*(-e^2*x^2+d^2)^(3/2)/d/e/(e*x+d)^4-1/15*(-e^2*x^2+d^2)^(3/2)/d^2/e/(e*x+d)^3

Rubi [A] (verified)

Time = 0.02 (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} \[ \int \frac {\sqrt {d^2-e^2 x^2}}{(d+e x)^4} \, dx=-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{15 d^2 e (d+e x)^3}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{5 d e (d+e x)^4} \]

[In]

Int[Sqrt[d^2 - e^2*x^2]/(d + e*x)^4,x]

[Out]

-1/5*(d^2 - e^2*x^2)^(3/2)/(d*e*(d + e*x)^4) - (d^2 - e^2*x^2)^(3/2)/(15*d^2*e*(d + e*x)^3)

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 {\left (d^2-e^2 x^2\right )^{3/2}}{5 d e (d+e x)^4}+\frac {\int \frac {\sqrt {d^2-e^2 x^2}}{(d+e x)^3} \, dx}{5 d} \\ & = -\frac {\left (d^2-e^2 x^2\right )^{3/2}}{5 d e (d+e x)^4}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{15 d^2 e (d+e x)^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {d^2-e^2 x^2}}{(d+e x)^4} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-4 d^2+3 d e x+e^2 x^2\right )}{15 d^2 e (d+e x)^3} \]

[In]

Integrate[Sqrt[d^2 - e^2*x^2]/(d + e*x)^4,x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-4*d^2 + 3*d*e*x + e^2*x^2))/(15*d^2*e*(d + e*x)^3)

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.64

method	result	size
gosper	\(-\frac {\left (-e x +d \right ) \left (e x +4 d \right ) \sqrt {-e^{2} x^{2}+d^{2}}}{15 \left (e x +d \right )^{3} d^{2} e}\)	\(43\)
trager	\(-\frac {\left (-e^{2} x^{2}-3 d e x +4 d^{2}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{15 d^{2} \left (e x +d \right )^{3} e}\)	\(49\)
default	\(\frac {-\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{5 d e \left (x +\frac {d}{e}\right )^{4}}-\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{15 d^{2} \left (x +\frac {d}{e}\right )^{3}}}{e^{4}}\)	\(93\)

[In]

int((-e^2*x^2+d^2)^(1/2)/(e*x+d)^4,x,method=_RETURNVERBOSE)

[Out]

-1/15*(-e*x+d)*(e*x+4*d)*(-e^2*x^2+d^2)^(1/2)/(e*x+d)^3/d^2/e

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.55 \[ \int \frac {\sqrt {d^2-e^2 x^2}}{(d+e x)^4} \, dx=-\frac {4 \, e^{3} x^{3} + 12 \, d e^{2} x^{2} + 12 \, d^{2} e x + 4 \, d^{3} - {\left (e^{2} x^{2} + 3 \, d e x - 4 \, d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d^{2} e^{4} x^{3} + 3 \, d^{3} e^{3} x^{2} + 3 \, d^{4} e^{2} x + d^{5} e\right )}} \]

[In]

integrate((-e^2*x^2+d^2)^(1/2)/(e*x+d)^4,x, algorithm="fricas")

[Out]

-1/15*(4*e^3*x^3 + 12*d*e^2*x^2 + 12*d^2*e*x + 4*d^3 - (e^2*x^2 + 3*d*e*x - 4*d^2)*sqrt(-e^2*x^2 + d^2))/(d^2*
e^4*x^3 + 3*d^3*e^3*x^2 + 3*d^4*e^2*x + d^5*e)

Sympy [F]

\[ \int \frac {\sqrt {d^2-e^2 x^2}}{(d+e x)^4} \, dx=\int \frac {\sqrt {- \left (- d + e x\right ) \left (d + e x\right )}}{\left (d + e x\right )^{4}}\, dx \]

[In]

integrate((-e**2*x**2+d**2)**(1/2)/(e*x+d)**4,x)

[Out]

Integral(sqrt(-(-d + e*x)*(d + e*x))/(d + e*x)**4, x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 123 vs. \(2 (59) = 118\).

Time = 0.20 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.84 \[ \int \frac {\sqrt {d^2-e^2 x^2}}{(d+e x)^4} \, dx=-\frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}}}{5 \, {\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} + \frac {\sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d e^{3} x^{2} + 2 \, d^{2} e^{2} x + d^{3} e\right )}} + \frac {\sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d^{2} e^{2} x + d^{3} e\right )}} \]

[In]

integrate((-e^2*x^2+d^2)^(1/2)/(e*x+d)^4,x, algorithm="maxima")

[Out]

-2/5*sqrt(-e^2*x^2 + d^2)/(e^4*x^3 + 3*d*e^3*x^2 + 3*d^2*e^2*x + d^3*e) + 1/15*sqrt(-e^2*x^2 + d^2)/(d*e^3*x^2
+ 2*d^2*e^2*x + d^3*e) + 1/15*sqrt(-e^2*x^2 + d^2)/(d^2*e^2*x + d^3*e)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 165 vs. \(2 (59) = 118\).

Time = 0.32 (sec) , antiderivative size = 165, normalized size of antiderivative = 2.46 \[ \int \frac {\sqrt {d^2-e^2 x^2}}{(d+e x)^4} \, dx=\frac {2 \, {\left (\frac {5 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}}{e^{2} x} + \frac {25 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2}}{e^{4} x^{2}} + \frac {15 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3}}{e^{6} x^{3}} + \frac {15 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4}}{e^{8} x^{4}} + 4\right )}}{15 \, d^{2} {\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} + 1\right )}^{5} {\left | e \right |}} \]

[In]

integrate((-e^2*x^2+d^2)^(1/2)/(e*x+d)^4,x, algorithm="giac")

[Out]

2/15*(5*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*x) + 25*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2/(e^4*x^2) + 15*
(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3/(e^6*x^3) + 15*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4/(e^8*x^4) + 4)/(d^2
*((d*e + sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*x) + 1)^5*abs(e))

Mupad [B] (verification not implemented)

Time = 11.68 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {d^2-e^2 x^2}}{(d+e x)^4} \, dx=\frac {\sqrt {d^2-e^2\,x^2}\,\left (-4\,d^2+3\,d\,e\,x+e^2\,x^2\right )}{15\,d^2\,e\,{\left (d+e\,x\right )}^3} \]

[In]

int((d^2 - e^2*x^2)^(1/2)/(d + e*x)^4,x)

[Out]

((d^2 - e^2*x^2)^(1/2)*(e^2*x^2 - 4*d^2 + 3*d*e*x))/(15*d^2*e*(d + e*x)^3)