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

Optimal. Leaf size=60 \[ \frac{2}{3 e^3 \sqrt{d^2-e^2 x^2}}-\frac{x^2}{3 d e (d+e x) \sqrt{d^2-e^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0346268, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {855, 12, 261} \[ \frac{2}{3 e^3 \sqrt{d^2-e^2 x^2}}-\frac{x^2}{3 d e (d+e x) \sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 855

Int[(((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_))/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(d*(f + g*x)
^n*(a + c*x^2)^(p + 1))/(2*a*e*p*(d + e*x)), x] - Dist[1/(2*d*e*p), Int[(f + g*x)^(n - 1)*(a + c*x^2)^p*Simp[d
*g*n - e*f*(2*p + 1) - e*g*(n + 2*p + 1)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] &&
EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && IGtQ[n, 0] && ILtQ[n + 2*p, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

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

Mathematica [A]  time = 0.0568615, size = 60, normalized size = 1. \[ \frac{\sqrt{d^2-e^2 x^2} \left (2 d^2+2 d e x-e^2 x^2\right )}{3 d e^3 (d-e x) (d+e x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.048, size = 48, normalized size = 0.8 \begin{align*}{\frac{ \left ( -ex+d \right ) \left ( -{x}^{2}{e}^{2}+2\,dex+2\,{d}^{2} \right ) }{3\,d{e}^{3}} \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^2/(e*x+d)/(-e^2*x^2+d^2)^(3/2),x)

[Out]

1/3*(-e*x+d)*(-e^2*x^2+2*d*e*x+2*d^2)/d/e^3/(-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^2/(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 = 1.6291, size = 203, normalized size = 3.38 \begin{align*} \frac{2 \, e^{3} x^{3} + 2 \, d e^{2} x^{2} - 2 \, d^{2} e x - 2 \, d^{3} +{\left (e^{2} x^{2} - 2 \, d e x - 2 \, d^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{3 \,{\left (d e^{6} x^{3} + d^{2} e^{5} x^{2} - d^{3} e^{4} x - d^{4} e^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(2*e^3*x^3 + 2*d*e^2*x^2 - 2*d^2*e*x - 2*d^3 + (e^2*x^2 - 2*d*e*x - 2*d^2)*sqrt(-e^2*x^2 + d^2))/(d*e^6*x^
3 + d^2*e^5*x^2 - d^3*e^4*x - d^4*e^3)

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

[Out]

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

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Giac [A]  time = 1.25927, size = 1, normalized size = 0.02 \begin{align*} +\infty \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

+Infinity