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

Optimal. Leaf size=67 \[ -\frac{\left (d^2-e^2 x^2\right )^{3/2}}{15 d^2 e (d+e x)^3}-\frac{\left (d^2-e^2 x^2\right )^{3/2}}{5 d e (d+e x)^4} \]

[Out]

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

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Rubi [A]  time = 0.0236743, 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} \[ -\frac{\left (d^2-e^2 x^2\right )^{3/2}}{15 d^2 e (d+e x)^3}-\frac{\left (d^2-e^2 x^2\right )^{3/2}}{5 d e (d+e x)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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{\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{\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]  time = 0.0297135, size = 51, normalized size = 0.76 \[ \frac{\sqrt{d^2-e^2 x^2} \left (-4 d^2+3 d e x+e^2 x^2\right )}{15 d^2 e (d+e x)^3} \]

Antiderivative was successfully verified.

[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)

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Maple [A]  time = 0.045, size = 43, normalized size = 0.6 \begin{align*} -{\frac{ \left ( ex+4\,d \right ) \left ( -ex+d \right ) }{15\, \left ( ex+d \right ) ^{3}{d}^{2}e}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.69081, size = 213, normalized size = 3.18 \begin{align*} -\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- \left (- d + e x\right ) \left (d + e x\right )}}{\left (d + e x\right )^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError