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

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Rubi [A]  time = 0.03, antiderivative size = 60, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.05, size = 60, normalized size = 1.00 \[ \frac {\sqrt {d^2-e^2 x^2} \left (2 d^2+2 d e x-e^2 x^2\right )}{3 d e^3 (d-e x) (d+e x)^2} \]

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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fricas [A]  time = 0.67, size = 103, normalized size = 1.72 \[ \frac {2 \, e^{3} x^{3} + 2 \, d e^{2} x^{2} - 2 \, d^{2} e x - 2 \, d^{3} + {\left (e^{2} x^{2} - 2 \, d e x - 2 \, d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{3 \, {\left (d e^{6} x^{3} + d^{2} e^{5} x^{2} - d^{3} e^{4} x - d^{4} e^{3}\right )}} \]

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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giac [A]  time = 0.24, size = 1, normalized size = 0.02 \[ +\infty \]

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

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maple [A]  time = 0.01, size = 48, normalized size = 0.80 \[ \frac {\left (-e x +d \right ) \left (-e^{2} x^{2}+2 d e x +2 d^{2}\right )}{3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} d \,e^{3}} \]

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 [A]  time = 0.46, size = 86, normalized size = 1.43 \[ -\frac {d}{3 \, {\left (\sqrt {-e^{2} x^{2} + d^{2}} e^{4} x + \sqrt {-e^{2} x^{2} + d^{2}} d e^{3}\right )}} - \frac {x}{3 \, \sqrt {-e^{2} x^{2} + d^{2}} d e^{2}} + \frac {1}{\sqrt {-e^{2} x^{2} + d^{2}} e^{3}} \]

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]

-1/3*d/(sqrt(-e^2*x^2 + d^2)*e^4*x + sqrt(-e^2*x^2 + d^2)*d*e^3) - 1/3*x/(sqrt(-e^2*x^2 + d^2)*d*e^2) + 1/(sqr
t(-e^2*x^2 + d^2)*e^3)

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mupad [B]  time = 2.71, size = 56, normalized size = 0.93 \[ \frac {\sqrt {d^2-e^2\,x^2}\,\left (2\,d^2+2\,d\,e\,x-e^2\,x^2\right )}{3\,d\,e^3\,{\left (d+e\,x\right )}^2\,\left (d-e\,x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}} \left (d + e x\right )}\, dx \]

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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