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

Optimal. Leaf size=97 \[ -\frac{\sqrt{d^2-e^2 x^2}}{5 d^2 e^2 (d+e x)}-\frac{\sqrt{d^2-e^2 x^2}}{5 d e^2 (d+e x)^2}+\frac{\sqrt{d^2-e^2 x^2}}{5 e^2 (d+e x)^3} \]

[Out]

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

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Rubi [A]  time = 0.0446837, antiderivative size = 97, 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, 651} \[ -\frac{\sqrt{d^2-e^2 x^2}}{5 d^2 e^2 (d+e x)}-\frac{\sqrt{d^2-e^2 x^2}}{5 d e^2 (d+e x)^2}+\frac{\sqrt{d^2-e^2 x^2}}{5 e^2 (d+e x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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]

Rubi steps

\begin{align*} \int \frac{x}{(d+e x)^3 \sqrt{d^2-e^2 x^2}} \, dx &=\frac{\sqrt{d^2-e^2 x^2}}{5 e^2 (d+e x)^3}+\frac{3 \int \frac{1}{(d+e x)^2 \sqrt{d^2-e^2 x^2}} \, dx}{5 e}\\ &=\frac{\sqrt{d^2-e^2 x^2}}{5 e^2 (d+e x)^3}-\frac{\sqrt{d^2-e^2 x^2}}{5 d e^2 (d+e x)^2}+\frac{\int \frac{1}{(d+e x) \sqrt{d^2-e^2 x^2}} \, dx}{5 d e}\\ &=\frac{\sqrt{d^2-e^2 x^2}}{5 e^2 (d+e x)^3}-\frac{\sqrt{d^2-e^2 x^2}}{5 d e^2 (d+e x)^2}-\frac{\sqrt{d^2-e^2 x^2}}{5 d^2 e^2 (d+e x)}\\ \end{align*}

Mathematica [A]  time = 0.0546501, size = 49, normalized size = 0.51 \[ -\frac{\sqrt{d^2-e^2 x^2} \left (d^2+3 d e x+e^2 x^2\right )}{5 d^2 e^2 (d+e x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.048, size = 52, normalized size = 0.5 \begin{align*} -{\frac{ \left ( -ex+d \right ) \left ({x}^{2}{e}^{2}+3\,dex+{d}^{2} \right ) }{5\,{e}^{2}{d}^{2} \left ( ex+d \right ) ^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.62409, size = 204, normalized size = 2.1 \begin{align*} -\frac{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3} +{\left (e^{2} x^{2} + 3 \, d e x + d^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{5 \,{\left (d^{2} e^{5} x^{3} + 3 \, d^{3} e^{4} x^{2} + 3 \, 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)^3/(-e^2*x^2+d^2)^(1/2),x, algorithm="fricas")

[Out]

-1/5*(e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3 + (e^2*x^2 + 3*d*e*x + d^2)*sqrt(-e^2*x^2 + d^2))/(d^2*e^5*x^3 +
3*d^3*e^4*x^2 + 3*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}{\sqrt{- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError