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

Optimal. Leaf size=58 \[ \frac {2 x}{3 d^3 \sqrt {d^2-e^2 x^2}}-\frac {1}{3 d e (d+e x) \sqrt {d^2-e^2 x^2}} \]

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Rubi [A]  time = 0.01, antiderivative size = 58, 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 = {659, 191} \begin {gather*} \frac {2 x}{3 d^3 \sqrt {d^2-e^2 x^2}}-\frac {1}{3 d e (d+e x) \sqrt {d^2-e^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

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]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 58, normalized size = 1.00 \begin {gather*} -\frac {\left (d^2-2 d e x-2 e^2 x^2\right ) \sqrt {d^2-e^2 x^2}}{3 d^3 e (d-e x) (d+e x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.38, size = 60, normalized size = 1.03 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-d^2+2 d e x+2 e^2 x^2\right )}{3 d^3 e (d-e x) (d+e x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.40, size = 102, normalized size = 1.76 \begin {gather*} -\frac {e^{3} x^{3} + d e^{2} x^{2} - d^{2} e x - d^{3} + {\left (2 \, e^{2} x^{2} + 2 \, d e x - d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{3 \, {\left (d^{3} e^{4} x^{3} + d^{4} e^{3} x^{2} - d^{5} e^{2} x - d^{6} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(e^3*x^3 + d*e^2*x^2 - d^2*e*x - d^3 + (2*e^2*x^2 + 2*d*e*x - d^2)*sqrt(-e^2*x^2 + d^2))/(d^3*e^4*x^3 + d
^4*e^3*x^2 - d^5*e^2*x - d^6*e)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {undef} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

undef

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maple [A]  time = 0.01, size = 46, normalized size = 0.79 \begin {gather*} -\frac {\left (-e x +d \right ) \left (-2 e^{2} x^{2}-2 d e x +d^{2}\right )}{3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} d^{3} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.45, size = 65, normalized size = 1.12 \begin {gather*} -\frac {1}{3 \, {\left (\sqrt {-e^{2} x^{2} + d^{2}} d e^{2} x + \sqrt {-e^{2} x^{2} + d^{2}} d^{2} e\right )}} + \frac {2 \, x}{3 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 2.71, size = 56, normalized size = 0.97 \begin {gather*} \frac {\sqrt {d^2-e^2\,x^2}\,\left (-d^2+2\,d\,e\,x+2\,e^2\,x^2\right )}{3\,d^3\,e\,{\left (d+e\,x\right )}^2\,\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)),x)

[Out]

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

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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 )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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