3.1.18 \(\int \frac {x^2 (d+e x)}{(d^2-e^2 x^2)^{5/2}} \, dx\) [18]

Optimal. Leaf size=58 \[ \frac {x^2 (d+e x)}{3 d e \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{3 e^3 \sqrt {d^2-e^2 x^2}} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {810, 12, 267} \begin {gather*} \frac {x^2 (d+e x)}{3 d e \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{3 e^3 \sqrt {d^2-e^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^2*(d + e*x))/(d^2 - e^2*x^2)^(5/2),x]

[Out]

(x^2*(d + e*x))/(3*d*e*(d^2 - e^2*x^2)^(3/2)) - 2/(3*e^3*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 267

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 810

Int[(x_)^2*((f_) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x^2*(a*g - c*f*x)*((a + c*x^2)^(p
+ 1)/(2*a*c*(p + 1))), x] - Dist[1/(2*a*c*(p + 1)), Int[x*Simp[2*a*g - c*f*(2*p + 5)*x, x]*(a + c*x^2)^(p + 1
), x], x] /; FreeQ[{a, c, f, g}, x] && EqQ[a*g^2 + f^2*c, 0] && LtQ[p, -2]

Rubi steps

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

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Mathematica [A]
time = 0.23, size = 59, normalized size = 1.02 \begin {gather*} \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)^2 (d+e x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^2*(d + e*x))/(d^2 - e^2*x^2)^(5/2),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)^2*(d + e*x))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(119\) vs. \(2(50)=100\).
time = 0.06, size = 120, normalized size = 2.07

method	result	size
gosper	\(-\frac {\left (-e x +d \right ) \left (e x +d \right )^{2} \left (-e^{2} x^{2}-2 d e x +2 d^{2}\right )}{3 d \,e^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\)	\(55\)
trager	\(-\frac {\left (-e^{2} x^{2}-2 d e x +2 d^{2}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{3 e^{3} d \left (-e x +d \right )^{2} \left (e x +d \right )}\)	\(57\)
default	\(e \left (\frac {x^{2}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {2 d^{2}}{3 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}\right )+d \left (\frac {x}{2 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {d^{2} \left (\frac {x}{3 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 e^{2}}\right )\)	\(120\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(e*x+d)/(-e^2*x^2+d^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 80, normalized size = 1.38 \begin {gather*} \frac {x^{2} e^{\left (-1\right )}}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}}} + \frac {d x e^{\left (-2\right )}}{3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}}} - \frac {2 \, d^{2} e^{\left (-3\right )}}{3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}}} - \frac {x e^{\left (-2\right )}}{3 \, \sqrt {-x^{2} e^{2} + d^{2}} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(e*x+d)/(-e^2*x^2+d^2)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (47) = 94\).
time = 1.96, size = 98, normalized size = 1.69 \begin {gather*} -\frac {2 \, x^{3} e^{3} - 2 \, d x^{2} e^{2} - 2 \, d^{2} x e + 2 \, d^{3} - {\left (x^{2} e^{2} + 2 \, d x e - 2 \, d^{2}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{3 \, {\left (d x^{3} e^{6} - d^{2} x^{2} e^{5} - d^{3} x e^{4} + d^{4} e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(e*x+d)/(-e^2*x^2+d^2)^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [C] Result contains complex when optimal does not.
time = 3.75, size = 231, normalized size = 3.98 \begin {gather*} d \left (\begin {cases} \frac {i x^{3}}{- 3 d^{5} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 3 d^{3} e^{2} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\- \frac {x^{3}}{- 3 d^{5} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} + 3 d^{3} e^{2} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) + e \left (\begin {cases} \frac {2 d^{2}}{- 3 d^{2} e^{4} \sqrt {d^{2} - e^{2} x^{2}} + 3 e^{6} x^{2} \sqrt {d^{2} - e^{2} x^{2}}} - \frac {3 e^{2} x^{2}}{- 3 d^{2} e^{4} \sqrt {d^{2} - e^{2} x^{2}} + 3 e^{6} x^{2} \sqrt {d^{2} - e^{2} x^{2}}} & \text {for}\: e \neq 0 \\\frac {x^{4}}{4 \left (d^{2}\right )^{\frac {5}{2}}} & \text {otherwise} \end {cases}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(e*x+d)/(-e**2*x**2+d**2)**(5/2),x)

[Out]

d*Piecewise((I*x**3/(-3*d**5*sqrt(-1 + e**2*x**2/d**2) + 3*d**3*e**2*x**2*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2
*x**2/d**2) > 1), (-x**3/(-3*d**5*sqrt(1 - e**2*x**2/d**2) + 3*d**3*e**2*x**2*sqrt(1 - e**2*x**2/d**2)), True)
) + e*Piecewise((2*d**2/(-3*d**2*e**4*sqrt(d**2 - e**2*x**2) + 3*e**6*x**2*sqrt(d**2 - e**2*x**2)) - 3*e**2*x*
*2/(-3*d**2*e**4*sqrt(d**2 - e**2*x**2) + 3*e**6*x**2*sqrt(d**2 - e**2*x**2)), Ne(e, 0)), (x**4/(4*(d**2)**(5/
2)), True))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(e*x+d)/(-e^2*x^2+d^2)^(5/2),x, algorithm="giac")

[Out]

integrate((x*e + d)*x^2/(-x^2*e^2 + d^2)^(5/2), x)

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Mupad [B]
time = 2.59, size = 55, normalized size = 0.95 \begin {gather*} \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 )\,{\left (d-e\,x\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*(d + e*x))/(d^2 - e^2*x^2)^(5/2),x)

[Out]

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

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