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

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

[Out]

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

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Rubi [A]  time = 0.02218, antiderivative size = 67, normalized size of antiderivative = 1., 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 = {659, 651} \[ -\frac{\sqrt{d^2-e^2 x^2}}{3 d^2 e (d+e x)}-\frac{\sqrt{d^2-e^2 x^2}}{3 d e (d+e x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 659

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]

Rule 651

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]

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*}

Mathematica [A]  time = 0.0415059, size = 40, normalized size = 0.6 \[ -\frac{(2 d+e x) \sqrt{d^2-e^2 x^2}}{3 d^2 e (d+e x)^2} \]

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 = 0.044, size = 43, normalized size = 0.6 \begin{align*} -{\frac{ \left ( -ex+d \right ) \left ( ex+2\,d \right ) }{ \left ( 3\,ex+3\,d \right ){d}^{2}e}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \end{align*}

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)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [A]  time = 2.12626, size = 144, normalized size = 2.15 \begin{align*} -\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 )}} \end{align*}

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*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)

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

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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

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]

sage0*x