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

Optimal. Leaf size=91 \[ \frac{4 x}{15 d^3 e \sqrt{d^2-e^2 x^2}}-\frac{2}{15 d e^2 (d+e x) \sqrt{d^2-e^2 x^2}}+\frac{1}{5 e^2 (d+e x)^2 \sqrt{d^2-e^2 x^2}} \]

[Out]

(4*x)/(15*d^3*e*Sqrt[d^2 - e^2*x^2]) + 1/(5*e^2*(d + e*x)^2*Sqrt[d^2 - e^2*x^2]) - 2/(15*d*e^2*(d + e*x)*Sqrt[
d^2 - e^2*x^2])

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Rubi [A]  time = 0.0364116, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {793, 659, 191} \[ \frac{4 x}{15 d^3 e \sqrt{d^2-e^2 x^2}}-\frac{2}{15 d e^2 (d+e x) \sqrt{d^2-e^2 x^2}}+\frac{1}{5 e^2 (d+e x)^2 \sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*x)/(15*d^3*e*Sqrt[d^2 - e^2*x^2]) + 1/(5*e^2*(d + e*x)^2*Sqrt[d^2 - e^2*x^2]) - 2/(15*d*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 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{x}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx &=\frac{1}{5 e^2 (d+e x)^2 \sqrt{d^2-e^2 x^2}}+\frac{2 \int \frac{1}{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{5 e}\\ &=\frac{1}{5 e^2 (d+e x)^2 \sqrt{d^2-e^2 x^2}}-\frac{2}{15 d e^2 (d+e x) \sqrt{d^2-e^2 x^2}}+\frac{4 \int \frac{1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d e}\\ &=\frac{4 x}{15 d^3 e \sqrt{d^2-e^2 x^2}}+\frac{1}{5 e^2 (d+e x)^2 \sqrt{d^2-e^2 x^2}}-\frac{2}{15 d e^2 (d+e x) \sqrt{d^2-e^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0627545, size = 69, normalized size = 0.76 \[ \frac{\sqrt{d^2-e^2 x^2} \left (2 d^2 e x+d^3+8 d e^2 x^2+4 e^3 x^3\right )}{15 d^3 e^2 (d-e x) (d+e x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(d^3 + 2*d^2*e*x + 8*d*e^2*x^2 + 4*e^3*x^3))/(15*d^3*e^2*(d - e*x)*(d + e*x)^3)

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Maple [A]  time = 0.049, size = 64, normalized size = 0.7 \begin{align*}{\frac{ \left ( -ex+d \right ) \left ( 4\,{e}^{3}{x}^{3}+8\,d{e}^{2}{x}^{2}+2\,{d}^{2}ex+{d}^{3} \right ) }{ \left ( 15\,ex+15\,d \right ){d}^{3}{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)^2/(-e^2*x^2+d^2)^(3/2),x)

[Out]

1/15*(-e*x+d)*(4*e^3*x^3+8*d*e^2*x^2+2*d^2*e*x+d^3)/(e*x+d)/d^3/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)^2/(-e^2*x^2+d^2)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.59638, size = 228, normalized size = 2.51 \begin{align*} \frac{e^{4} x^{4} + 2 \, d e^{3} x^{3} - 2 \, d^{3} e x - d^{4} -{\left (4 \, e^{3} x^{3} + 8 \, d e^{2} x^{2} + 2 \, d^{2} e x + d^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (d^{3} e^{6} x^{4} + 2 \, d^{4} e^{5} x^{3} - 2 \, d^{6} e^{3} x - d^{7} e^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*(e^4*x^4 + 2*d*e^3*x^3 - 2*d^3*e*x - d^4 - (4*e^3*x^3 + 8*d*e^2*x^2 + 2*d^2*e*x + d^3)*sqrt(-e^2*x^2 + d^
2))/(d^3*e^6*x^4 + 2*d^4*e^5*x^3 - 2*d^6*e^3*x - d^7*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 )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x