\(\int \frac {(d+e x)^2}{(d^2-e^2 x^2)^{5/2}} \, dx\) [840]

Optimal result

Integrand size = 24, antiderivative size = 53 \[ \int \frac {(d+e x)^2}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {2 (d+e x)}{3 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x}{3 d^2 \sqrt {d^2-e^2 x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 53, 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 = {667, 197} \[ \int \frac {(d+e x)^2}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {x}{3 d^2 \sqrt {d^2-e^2 x^2}}+\frac {2 (d+e x)}{3 e \left (d^2-e^2 x^2\right )^{3/2}} \]

[In]

Int[(d + e*x)^2/(d^2 - e^2*x^2)^(5/2),x]

[Out]

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

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 667

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

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

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.79 \[ \int \frac {(d+e x)^2}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {(2 d-e x) \sqrt {d^2-e^2 x^2}}{3 d^2 e (d-e x)^2} \]

[In]

Integrate[(d + e*x)^2/(d^2 - e^2*x^2)^(5/2),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 = 2.32 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.74

method	result	size
trager	\(\frac {\left (-e x +2 d \right ) \sqrt {-x^{2} e^{2}+d^{2}}}{3 d^{2} \left (-e x +d \right )^{2} e}\)	\(39\)
gosper	\(\frac {\left (e x +d \right )^{3} \left (-e x +d \right ) \left (-e x +2 d \right )}{3 d^{2} e \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}\)	\(44\)
default	\(d^{2} \left (\frac {x}{3 d^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{4} \sqrt {-x^{2} e^{2}+d^{2}}}\right )+e^{2} \left (\frac {x}{2 e^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {d^{2} \left (\frac {x}{3 d^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{4} \sqrt {-x^{2} e^{2}+d^{2}}}\right )}{2 e^{2}}\right )+\frac {2 d}{3 e \left (-x^{2} e^{2}+d^{2}\right )^{\frac {3}{2}}}\)	\(141\)

[In]

int((e*x+d)^2/(-e^2*x^2+d^2)^(5/2),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 = 0.27 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.34 \[ \int \frac {(d+e x)^2}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {2 \, e^{2} x^{2} - 4 \, d e x + 2 \, d^{2} - \sqrt {-e^{2} x^{2} + d^{2}} {\left (e x - 2 \, d\right )}}{3 \, {\left (d^{2} e^{3} x^{2} - 2 \, d^{3} e^{2} x + d^{4} e\right )}} \]

[In]

integrate((e*x+d)^2/(-e^2*x^2+d^2)^(5/2),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]

\[ \int \frac {(d+e x)^2}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=\int \frac {\left (d + e x\right )^{2}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate((e*x+d)**2/(-e**2*x**2+d**2)**(5/2),x)

[Out]

Integral((d + e*x)**2/(-(-d + e*x)*(d + e*x))**(5/2), x)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.09 \[ \int \frac {(d+e x)^2}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {2 \, x}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}}} + \frac {2 \, d}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e} + \frac {x}{3 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2}} \]

[In]

integrate((e*x+d)^2/(-e^2*x^2+d^2)^(5/2),x, algorithm="maxima")

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (45) = 90\).

Time = 0.30 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.94 \[ \int \frac {(d+e x)^2}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=-\frac {2 \, {\left (\frac {3 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}}{e^{2} x} - \frac {3 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2}}{e^{4} x^{2}} - 2\right )}}{3 \, d^{2} {\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} - 1\right )}^{3} {\left | e \right |}} \]

[In]

integrate((e*x+d)^2/(-e^2*x^2+d^2)^(5/2),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.80 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.72 \[ \int \frac {(d+e x)^2}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {\sqrt {d^2-e^2\,x^2}\,\left (2\,d-e\,x\right )}{3\,d^2\,e\,{\left (d-e\,x\right )}^2} \]

[In]

int((d + e*x)^2/(d^2 - e^2*x^2)^(5/2),x)

[Out]

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