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

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

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Rubi [A]  time = 0.02, 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 = {659, 651} \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]

-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 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]

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)^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 = 0.04, size = 40, normalized size = 0.60 \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]

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

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IntegrateAlgebraic [A]  time = 0.38, 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]

IntegrateAlgebraic[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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fricas [A]  time = 0.39, size = 70, normalized size = 1.04 \begin {gather*} -\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 {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*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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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(t_nostep)]index.cc index_m i_lex_is_greater Error: Bad Argument Value

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maple [A]  time = 0.05, size = 43, normalized size = 0.64 \begin {gather*} -\frac {\left (-e x +d \right ) \left (e x +2 d \right )}{3 \left (e x +d \right ) \sqrt {-e^{2} x^{2}+d^{2}}\, d^{2} e} \end {gather*}

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 [A]  time = 3.01, size = 74, normalized size = 1.10 \begin {gather*} -\frac {\sqrt {-e^{2} x^{2} + d^{2}}}{3 \, {\left (d e^{3} x^{2} + 2 \, d^{2} e^{2} x + d^{3} e\right )}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}}}{3 \, {\left (d^{2} e^{2} x + 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(-e^2*x^2 + d^2)/(d*e^3*x^2 + 2*d^2*e^2*x + d^3*e) - 1/3*sqrt(-e^2*x^2 + d^2)/(d^2*e^2*x + d^3*e)

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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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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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