3.2.31 \(\int \frac {x}{(d+e x) (d^2-e^2 x^2)^{3/2}} \, dx\) [131]

Optimal. Leaf size=58 \[ \frac {x}{3 d^2 e \sqrt {d^2-e^2 x^2}}+\frac {1}{3 e^2 (d+e x) \sqrt {d^2-e^2 x^2}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {807, 197} \begin {gather*} \frac {x}{3 d^2 e \sqrt {d^2-e^2 x^2}}+\frac {1}{3 e^2 (d+e x) \sqrt {d^2-e^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 807

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]

Rubi steps

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

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Mathematica [A]
time = 0.23, size = 56, normalized size = 0.97 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (d^2+d e x+e^2 x^2\right )}{3 d^2 e^2 (d-e x) (d+e x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(128\) vs. \(2(50)=100\).
time = 0.07, size = 129, normalized size = 2.22

method	result	size
gosper	\(\frac {\left (-e x +d \right ) \left (e^{2} x^{2}+d e x +d^{2}\right )}{3 d^{2} e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}\)	\(44\)
trager	\(\frac {\left (e^{2} x^{2}+d e x +d^{2}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{3 e^{2} d^{2} \left (e x +d \right )^{2} \left (-e x +d \right )}\)	\(53\)
default	\(\frac {x}{d^{2} e \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {d \left (-\frac {1}{3 d e \left (x +\frac {d}{e}\right ) \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{3 e \,d^{3} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{e^{2}}\)	\(129\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 61, normalized size = 1.05 \begin {gather*} \frac {x e^{\left (-1\right )}}{3 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{2}} + \frac {1}{3 \, {\left (\sqrt {-x^{2} e^{2} + d^{2}} x e^{3} + \sqrt {-x^{2} e^{2} + d^{2}} d e^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (47) = 94\).
time = 1.57, size = 95, normalized size = 1.64 \begin {gather*} \frac {x^{3} e^{3} + d x^{2} e^{2} - d^{2} x e - d^{3} - {\left (x^{2} e^{2} + d x e + d^{2}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{3 \, {\left (d^{2} x^{3} e^{5} + d^{3} x^{2} e^{4} - d^{4} x e^{3} - d^{5} e^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Mupad [B]
time = 2.71, size = 52, normalized size = 0.90 \begin {gather*} \frac {\sqrt {d^2-e^2\,x^2}\,\left (d^2+d\,e\,x+e^2\,x^2\right )}{3\,d^2\,e^2\,{\left (d+e\,x\right )}^2\,\left (d-e\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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