3.2.75 \(\int \frac {1}{(d+e x)^2 (d^2-e^2 x^2)^{3/2}} \, dx\) [175]

Optimal. Leaf size=91 \[ \frac {2 x}{5 d^4 \sqrt {d^2-e^2 x^2}}-\frac {1}{5 d e (d+e x)^2 \sqrt {d^2-e^2 x^2}}-\frac {1}{5 d^2 e (d+e x) \sqrt {d^2-e^2 x^2}} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {673, 197} \begin {gather*} -\frac {1}{5 d^2 e (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {1}{5 d e (d+e x)^2 \sqrt {d^2-e^2 x^2}}+\frac {2 x}{5 d^4 \sqrt {d^2-e^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x)/(5*d^4*Sqrt[d^2 - e^2*x^2]) - 1/(5*d*e*(d + e*x)^2*Sqrt[d^2 - e^2*x^2]) - 1/(5*d^2*e*(d + e*x)*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 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*} \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx &=-\frac {1}{5 d e (d+e x)^2 \sqrt {d^2-e^2 x^2}}+\frac {3 \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{5 d}\\ &=-\frac {1}{5 d e (d+e x)^2 \sqrt {d^2-e^2 x^2}}-\frac {1}{5 d^2 e (d+e x) \sqrt {d^2-e^2 x^2}}+\frac {2 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{5 d^2}\\ &=\frac {2 x}{5 d^4 \sqrt {d^2-e^2 x^2}}-\frac {1}{5 d e (d+e x)^2 \sqrt {d^2-e^2 x^2}}-\frac {1}{5 d^2 e (d+e x) \sqrt {d^2-e^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.33, size = 70, normalized size = 0.77 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-2 d^3+d^2 e x+4 d e^2 x^2+2 e^3 x^3\right )}{5 d^4 e (d-e x) (d+e x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(-2*d^3 + d^2*e*x + 4*d*e^2*x^2 + 2*e^3*x^3))/(5*d^4*e*(d - e*x)*(d + e*x)^3)

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Maple [A]
time = 0.06, size = 156, normalized size = 1.71

method	result	size
gosper	\(-\frac {\left (-e x +d \right ) \left (-2 e^{3} x^{3}-4 d \,e^{2} x^{2}-d^{2} e x +2 d^{3}\right )}{5 \left (e x +d \right ) d^{4} e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}\)	\(66\)
trager	\(-\frac {\left (-2 e^{3} x^{3}-4 d \,e^{2} x^{2}-d^{2} e x +2 d^{3}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{5 d^{4} \left (e x +d \right )^{3} e \left (-e x +d \right )}\)	\(68\)
default	\(\frac {-\frac {1}{5 d e \left (x +\frac {d}{e}\right )^{2} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}+\frac {3 e \left (-\frac {1}{3 d e \left (x +\frac {d}{e}\right ) \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{3 e \,d^{3} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{5 d}}{e^{2}}\)	\(156\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^2/(-e^2*x^2+d^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 129, normalized size = 1.42 \begin {gather*} -\frac {1}{5 \, {\left (\sqrt {-x^{2} e^{2} + d^{2}} d x^{2} e^{3} + 2 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{2} x e^{2} + \sqrt {-x^{2} e^{2} + d^{2}} d^{3} e\right )}} - \frac {1}{5 \, {\left (\sqrt {-x^{2} e^{2} + d^{2}} d^{2} x e^{2} + \sqrt {-x^{2} e^{2} + d^{2}} d^{3} e\right )}} + \frac {2 \, x}{5 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^2/(-e^2*x^2+d^2)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 1.68, size = 110, normalized size = 1.21 \begin {gather*} -\frac {2 \, x^{4} e^{4} + 4 \, d x^{3} e^{3} - 4 \, d^{3} x e - 2 \, d^{4} + {\left (2 \, x^{3} e^{3} + 4 \, d x^{2} e^{2} + d^{2} x e - 2 \, d^{3}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{5 \, {\left (d^{4} x^{4} e^{5} + 2 \, d^{5} x^{3} e^{4} - 2 \, d^{7} x e^{2} - d^{8} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^2/(-e^2*x^2+d^2)^(3/2),x, algorithm="fricas")

[Out]

-1/5*(2*x^4*e^4 + 4*d*x^3*e^3 - 4*d^3*x*e - 2*d^4 + (2*x^3*e^3 + 4*d*x^2*e^2 + d^2*x*e - 2*d^3)*sqrt(-x^2*e^2
+ d^2))/(d^4*x^4*e^5 + 2*d^5*x^3*e^4 - 2*d^7*x*e^2 - d^8*e)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [C] Result contains complex when optimal does not.
time = 1.10, size = 173, normalized size = 1.90 \begin {gather*} \frac {1}{40} \, {\left ({\left (\frac {5 \, e^{\left (-3\right )}}{d^{4} \sqrt {\frac {2 \, d}{x e + d} - 1} \mathrm {sgn}\left (\frac {1}{x e + d}\right )} - \frac {{\left (d^{16} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {5}{2}} e^{12} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{4} + 5 \, d^{16} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {3}{2}} e^{12} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{4} + 15 \, d^{16} \sqrt {\frac {2 \, d}{x e + d} - 1} e^{12} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{4}\right )} e^{\left (-15\right )}}{d^{20} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{5}}\right )} e^{3} + \frac {16 i \, \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{d^{4}}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^2/(-e^2*x^2+d^2)^(3/2),x, algorithm="giac")

[Out]

1/40*((5*e^(-3)/(d^4*sqrt(2*d/(x*e + d) - 1)*sgn(1/(x*e + d))) - (d^16*(2*d/(x*e + d) - 1)^(5/2)*e^12*sgn(1/(x
*e + d))^4 + 5*d^16*(2*d/(x*e + d) - 1)^(3/2)*e^12*sgn(1/(x*e + d))^4 + 15*d^16*sqrt(2*d/(x*e + d) - 1)*e^12*s
gn(1/(x*e + d))^4)*e^(-15)/(d^20*sgn(1/(x*e + d))^5))*e^3 + 16*I*sgn(1/(x*e + d))/d^4)*e^(-1)

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Mupad [B]
time = 2.85, size = 66, normalized size = 0.73 \begin {gather*} \frac {\sqrt {d^2-e^2\,x^2}\,\left (-2\,d^3+d^2\,e\,x+4\,d\,e^2\,x^2+2\,e^3\,x^3\right )}{5\,d^4\,e\,{\left (d+e\,x\right )}^3\,\left (d-e\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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