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

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

[Out]

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

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Rubi [A]  time = 0.0203455, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {793, 191} \[ \frac{x}{3 d^2 e \sqrt{d^2-e^2 x^2}}+\frac{1}{3 e^2 (d+e x) \sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 793

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d*g - e*f)*(
d + e*x)^m*(a + c*x^2)^(p + 1))/(2*c*d*(m + p + 1)), x] + Dist[(m*(g*c*d + c*e*f) + 2*e*c*f*(p + 1))/(e*(2*c*d
)*(m + p + 1)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[c*d^2
+ a*e^2, 0] && ((LtQ[m, -1] &&  !IGtQ[m + p + 1, 0]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) &&
NeQ[m + p + 1, 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{x}{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx &=\frac{1}{3 e^2 (d+e x) \sqrt{d^2-e^2 x^2}}+\frac{\int \frac{1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{3 e}\\ &=\frac{x}{3 d^2 e \sqrt{d^2-e^2 x^2}}+\frac{1}{3 e^2 (d+e x) \sqrt{d^2-e^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0543328, size = 56, normalized size = 0.97 \[ \frac{\sqrt{d^2-e^2 x^2} \left (d^2+d e x+e^2 x^2\right )}{3 d^2 e^2 (d-e x) (d+e x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.048, size = 44, normalized size = 0.8 \begin{align*}{\frac{ \left ( -ex+d \right ) \left ({x}^{2}{e}^{2}+dex+{d}^{2} \right ) }{3\,{d}^{2}{e}^{2}} \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(x/(e*x+d)/(-e^2*x^2+d^2)^(3/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.56244, size = 189, normalized size = 3.26 \begin{align*} \frac{e^{3} x^{3} + d e^{2} x^{2} - d^{2} e x - d^{3} -{\left (e^{2} x^{2} + d e x + d^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{3 \,{\left (d^{2} e^{5} x^{3} + d^{3} e^{4} x^{2} - d^{4} e^{3} x - d^{5} e^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{3}{2}} \left (d + e x\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{undef} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

undef