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

Optimal. Leaf size=67 \[ -\frac {\sqrt {d^2-e^2 x^2}}{3 d e (d+e x)^2}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^2 e (d+e x)} \]

[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)

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Rubi [A]
time = 0.01, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {673, 665} \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[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*} \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 {\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*}

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Mathematica [A]
time = 0.22, size = 41, normalized size = 0.61 \begin {gather*} \frac {(-2 d-e x) \sqrt {d^2-e^2 x^2}}{3 d^2 e (d+e x)^2} \end {gather*}

Antiderivative was successfully verified.

[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)

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Maple [A]
time = 0.47, size = 93, normalized size = 1.39

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.49, size = 71, normalized size = 1.06 \begin {gather*} -\frac {\sqrt {-x^{2} e^{2} + d^{2}}}{3 \, {\left (d x^{2} e^{3} + 2 \, d^{2} x e^{2} + d^{3} e\right )}} - \frac {\sqrt {-x^{2} e^{2} + d^{2}}}{3 \, {\left (d^{2} x e^{2} + d^{3} 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)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 3.20, size = 69, normalized size = 1.03 \begin {gather*} -\frac {2 \, x^{2} e^{2} + 4 \, d x e + 2 \, d^{2} + \sqrt {-x^{2} e^{2} + d^{2}} {\left (x e + 2 \, d\right )}}{3 \, {\left (d^{2} x^{2} e^{3} + 2 \, d^{3} x e^{2} + d^{4} 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)^(1/2),x, algorithm="fricas")

[Out]

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

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

[Out]

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

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Giac [C] Result contains complex when optimal does not.
time = 1.68, size = 68, normalized size = 1.01 \begin {gather*} \frac {i \, e^{\left (-1\right )} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{3 \, d^{2}} - \frac {{\left ({\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {3}{2}} + 3 \, \sqrt {\frac {2 \, d}{x e + d} - 1}\right )} e^{\left (-1\right )}}{6 \, d^{2} \mathrm {sgn}\left (\frac {1}{x e + d}\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)^(1/2),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.45, size = 36, normalized size = 0.54 \begin {gather*} -\frac {\sqrt {d^2-e^2\,x^2}\,\left (2\,d+e\,x\right )}{3\,d^2\,e\,{\left (d+e\,x\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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