3.2.93 \(\int \frac {\sqrt {d^2-e^2 x^2}}{(d+e x)^4} \, dx\) [193]

Optimal. Leaf size=67 \[ -\frac {\left (d^2-e^2 x^2\right )^{3/2}}{5 d e (d+e x)^4}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{15 d^2 e (d+e x)^3} \]

[Out]

-1/5*(-e^2*x^2+d^2)^(3/2)/d/e/(e*x+d)^4-1/15*(-e^2*x^2+d^2)^(3/2)/d^2/e/(e*x+d)^3

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Rubi [A]
time = 0.01, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {673, 665} \begin {gather*} -\frac {\left (d^2-e^2 x^2\right )^{3/2}}{15 d^2 e (d+e x)^3}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{5 d e (d+e x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/5*(d^2 - e^2*x^2)^(3/2)/(d*e*(d + e*x)^4) - (d^2 - e^2*x^2)^(3/2)/(15*d^2*e*(d + e*x)^3)

Rule 665

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]

Rule 673

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*(m + p + 1))), x] + Dist[Simplify[m + 2*p + 2]/(2*d*(m + p + 1)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^
p, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[Simplify[m + 2*p +
2], 0]

Rubi steps

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

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Mathematica [A]
time = 0.37, size = 51, normalized size = 0.76 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-4 d^2+3 d e x+e^2 x^2\right )}{15 d^2 e (d+e x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.06, size = 93, normalized size = 1.39

method	result	size
gosper	\(-\frac {\left (-e x +d \right ) \left (e x +4 d \right ) \sqrt {-e^{2} x^{2}+d^{2}}}{15 \left (e x +d \right )^{3} d^{2} e}\)	\(43\)
trager	\(-\frac {\left (-e^{2} x^{2}-3 d e x +4 d^{2}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{15 d^{2} \left (e x +d \right )^{3} e}\)	\(49\)
default	\(\frac {-\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{5 d e \left (x +\frac {d}{e}\right )^{4}}-\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{15 d^{2} \left (x +\frac {d}{e}\right )^{3}}}{e^{4}}\)	\(93\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/e^4*(-1/5/d/e/(x+d/e)^4*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(3/2)-1/15/d^2/(x+d/e)^3*(-(x+d/e)^2*e^2+2*d*e*(x+d/e
))^(3/2))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (57) = 114\).
time = 0.36, size = 117, normalized size = 1.75 \begin {gather*} -\frac {2 \, \sqrt {-x^{2} e^{2} + d^{2}}}{5 \, {\left (x^{3} e^{4} + 3 \, d x^{2} e^{3} + 3 \, d^{2} x e^{2} + d^{3} e\right )}} + \frac {\sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (d x^{2} e^{3} + 2 \, d^{2} x e^{2} + d^{3} e\right )}} + \frac {\sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (d^{2} x e^{2} + d^{3} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5*sqrt(-x^2*e^2 + d^2)/(x^3*e^4 + 3*d*x^2*e^3 + 3*d^2*x*e^2 + d^3*e) + 1/15*sqrt(-x^2*e^2 + d^2)/(d*x^2*e^3
+ 2*d^2*x*e^2 + d^3*e) + 1/15*sqrt(-x^2*e^2 + d^2)/(d^2*x*e^2 + d^3*e)

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Fricas [A]
time = 1.78, size = 100, normalized size = 1.49 \begin {gather*} -\frac {4 \, x^{3} e^{3} + 12 \, d x^{2} e^{2} + 12 \, d^{2} x e + 4 \, d^{3} - {\left (x^{2} e^{2} + 3 \, d x e - 4 \, d^{2}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (d^{2} x^{3} e^{4} + 3 \, d^{3} x^{2} e^{3} + 3 \, d^{4} x e^{2} + d^{5} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(4*x^3*e^3 + 12*d*x^2*e^2 + 12*d^2*x*e + 4*d^3 - (x^2*e^2 + 3*d*x*e - 4*d^2)*sqrt(-x^2*e^2 + d^2))/(d^2*
x^3*e^4 + 3*d^3*x^2*e^3 + 3*d^4*x*e^2 + d^5*e)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- \left (- d + e x\right ) \left (d + e x\right )}}{\left (d + e x\right )^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(-(-d + e*x)*(d + e*x))/(d + e*x)**4, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 158 vs. \(2 (57) = 114\).
time = 1.55, size = 158, normalized size = 2.36 \begin {gather*} \frac {2 \, {\left (\frac {5 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + \frac {25 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{\left (-4\right )}}{x^{2}} + \frac {15 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{\left (-6\right )}}{x^{3}} + \frac {15 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{\left (-8\right )}}{x^{4}} + 4\right )} e^{\left (-1\right )}}{15 \, d^{2} {\left (\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1\right )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(5*(d*e + sqrt(-x^2*e^2 + d^2)*e)*e^(-2)/x + 25*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*e^(-4)/x^2 + 15*(d*e + s
qrt(-x^2*e^2 + d^2)*e)^3*e^(-6)/x^3 + 15*(d*e + sqrt(-x^2*e^2 + d^2)*e)^4*e^(-8)/x^4 + 4)*e^(-1)/(d^2*((d*e +
sqrt(-x^2*e^2 + d^2)*e)*e^(-2)/x + 1)^5)

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Mupad [B]
time = 2.78, size = 47, normalized size = 0.70 \begin {gather*} \frac {\sqrt {d^2-e^2\,x^2}\,\left (-4\,d^2+3\,d\,e\,x+e^2\,x^2\right )}{15\,d^2\,e\,{\left (d+e\,x\right )}^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((d^2 - e^2*x^2)^(1/2)*(e^2*x^2 - 4*d^2 + 3*d*e*x))/(15*d^2*e*(d + e*x)^3)

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