3.175 \(\int \frac{1}{(d+e x)^2 (d^2-e^2 x^2)^{3/2}} \, dx\)

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

[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])

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Rubi [A]  time = 0.0311912, antiderivative size = 91, normalized size of antiderivative = 1., 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 = {659, 191} \[ \frac{2 x}{5 d^4 \sqrt{d^2-e^2 x^2}}-\frac{1}{5 d^2 e (d+e x) \sqrt{d^2-e^2 x^2}}-\frac{1}{5 d e (d+e x)^2 \sqrt{d^2-e^2 x^2}} \]

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 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 191

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]

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*}

Mathematica [A]  time = 0.0397725, size = 70, normalized size = 0.77 \[ \frac{\sqrt{d^2-e^2 x^2} \left (d^2 e x-2 d^3+4 d e^2 x^2+2 e^3 x^3\right )}{5 d^4 e (d-e x) (d+e x)^3} \]

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 = 0.05, size = 66, normalized size = 0.7 \begin{align*} -{\frac{ \left ( -ex+d \right ) \left ( -2\,{e}^{3}{x}^{3}-4\,{e}^{2}{x}^{2}d-x{d}^{2}e+2\,{d}^{3} \right ) }{ \left ( 5\,ex+5\,d \right ){d}^{4}e} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{3}{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)^(3/2),x)

[Out]

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

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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)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.54651, size = 231, normalized size = 2.54 \begin{align*} -\frac{2 \, e^{4} x^{4} + 4 \, d e^{3} x^{3} - 4 \, d^{3} e x - 2 \, d^{4} +{\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 \,{\left (d^{4} e^{5} x^{4} + 2 \, d^{5} e^{4} x^{3} - 2 \, d^{7} e^{2} x - d^{8} 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)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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{align*}

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 [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \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)^(3/2),x, algorithm="giac")

[Out]

Exception raised: RuntimeError