3.192 \(\int \frac{x \sqrt{d^2-e^2 x^2}}{(d+e x)^4} \, dx\)

Optimal. Leaf size=64 \[ \frac{\left (d^2-e^2 x^2\right )^{3/2}}{5 e^2 (d+e x)^4}-\frac{4 \left (d^2-e^2 x^2\right )^{3/2}}{15 d e^2 (d+e x)^3} \]

[Out]

(d^2 - e^2*x^2)^(3/2)/(5*e^2*(d + e*x)^4) - (4*(d^2 - e^2*x^2)^(3/2))/(15*d*e^2*(d + e*x)^3)

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Rubi [A]  time = 0.0274207, antiderivative size = 64, 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, 651} \[ \frac{\left (d^2-e^2 x^2\right )^{3/2}}{5 e^2 (d+e x)^4}-\frac{4 \left (d^2-e^2 x^2\right )^{3/2}}{15 d e^2 (d+e x)^3} \]

Antiderivative was successfully verified.

[In]

Int[(x*Sqrt[d^2 - e^2*x^2])/(d + e*x)^4,x]

[Out]

(d^2 - e^2*x^2)^(3/2)/(5*e^2*(d + e*x)^4) - (4*(d^2 - e^2*x^2)^(3/2))/(15*d*e^2*(d + e*x)^3)

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

Mathematica [A]  time = 0.0509858, size = 50, normalized size = 0.78 \[ -\frac{\left (d^2+3 d e x-4 e^2 x^2\right ) \sqrt{d^2-e^2 x^2}}{15 d e^2 (d+e x)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(x*Sqrt[d^2 - e^2*x^2])/(d + e*x)^4,x]

[Out]

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

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Maple [A]  time = 0.046, size = 42, normalized size = 0.7 \begin{align*} -{\frac{ \left ( 4\,ex+d \right ) \left ( -ex+d \right ) }{15\, \left ( ex+d \right ) ^{3}d{e}^{2}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(-e^2*x^2+d^2)^(1/2)/(e*x+d)^4,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.62987, size = 205, normalized size = 3.2 \begin{align*} -\frac{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3} -{\left (4 \, e^{2} x^{2} - 3 \, d e x - d^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (d e^{5} x^{3} + 3 \, d^{2} e^{4} x^{2} + 3 \, d^{3} e^{3} x + d^{4} e^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-e^2*x^2+d^2)^(1/2)/(e*x+d)^4,x, algorithm="fricas")

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-e**2*x**2+d**2)**(1/2)/(e*x+d)**4,x)

[Out]

Integral(x*sqrt(-(-d + e*x)*(d + e*x))/(d + e*x)**4, 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^2*x^2+d^2)^(1/2)/(e*x+d)^4,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError