3.2.86 \(\int \frac {1}{x^2 (d+e x)^3 \sqrt {d^2-e^2 x^2}} \, dx\) [186]

Optimal. Leaf size=146 \[ -\frac {4 e (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}-\frac {e (5 d-7 e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {e (15 d-19 e x)}{5 d^5 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{d^5 x}+\frac {3 e \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^5} \]

[Out]

-4/5*e*(-e*x+d)/d/(-e^2*x^2+d^2)^(5/2)-1/5*e*(-7*e*x+5*d)/d^3/(-e^2*x^2+d^2)^(3/2)+3*e*arctanh((-e^2*x^2+d^2)^
(1/2)/d)/d^5-1/5*e*(-19*e*x+15*d)/d^5/(-e^2*x^2+d^2)^(1/2)-(-e^2*x^2+d^2)^(1/2)/d^5/x

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Rubi [A]
time = 0.19, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {866, 1819, 821, 272, 65, 214} \begin {gather*} -\frac {4 e (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}-\frac {e (15 d-19 e x)}{5 d^5 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{d^5 x}+\frac {3 e \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^5}-\frac {e (5 d-7 e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*e*(d - e*x))/(5*d*(d^2 - e^2*x^2)^(5/2)) - (e*(5*d - 7*e*x))/(5*d^3*(d^2 - e^2*x^2)^(3/2)) - (e*(15*d - 19
*e*x))/(5*d^5*Sqrt[d^2 - e^2*x^2]) - Sqrt[d^2 - e^2*x^2]/(d^5*x) + (3*e*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/d^5

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 272

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

Rule 821

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g
))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e
^2), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0
] && EqQ[Simplify[m + 2*p + 3], 0]

Rule 866

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

Rule 1819

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(c*x)^m*Pq,
a + b*x^2, x], f = Coeff[PolynomialRemainder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[
(c*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b*f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Dist[1/(2*a*
(p + 1)), Int[(c*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x],
x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {1}{x^2 (d+e x)^3 \sqrt {d^2-e^2 x^2}} \, dx &=\int \frac {(d-e x)^3}{x^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx\\ &=-\frac {4 e (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {-5 d^3+15 d^2 e x-16 d e^2 x^2}{x^2 \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2}\\ &=-\frac {4 e (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}-\frac {e (5 d-7 e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {15 d^3-45 d^2 e x+42 d e^2 x^2}{x^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^4}\\ &=-\frac {4 e (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}-\frac {e (5 d-7 e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {e (15 d-19 e x)}{5 d^5 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {-15 d^3+45 d^2 e x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{15 d^6}\\ &=-\frac {4 e (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}-\frac {e (5 d-7 e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {e (15 d-19 e x)}{5 d^5 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{d^5 x}-\frac {(3 e) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{d^4}\\ &=-\frac {4 e (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}-\frac {e (5 d-7 e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {e (15 d-19 e x)}{5 d^5 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{d^5 x}-\frac {(3 e) \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{2 d^4}\\ &=-\frac {4 e (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}-\frac {e (5 d-7 e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {e (15 d-19 e x)}{5 d^5 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{d^5 x}+\frac {3 \text {Subst}\left (\int \frac {1}{\frac {d^2}{e^2}-\frac {x^2}{e^2}} \, dx,x,\sqrt {d^2-e^2 x^2}\right )}{d^4 e}\\ &=-\frac {4 e (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}-\frac {e (5 d-7 e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {e (15 d-19 e x)}{5 d^5 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{d^5 x}+\frac {3 e \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^5}\\ \end {align*}

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Mathematica [A]
time = 0.43, size = 103, normalized size = 0.71 \begin {gather*} -\frac {\frac {\sqrt {d^2-e^2 x^2} \left (5 d^3+39 d^2 e x+57 d e^2 x^2+24 e^3 x^3\right )}{x (d+e x)^3}+30 e \tanh ^{-1}\left (\frac {\sqrt {-e^2} x-\sqrt {d^2-e^2 x^2}}{d}\right )}{5 d^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/5*((Sqrt[d^2 - e^2*x^2]*(5*d^3 + 39*d^2*e*x + 57*d*e^2*x^2 + 24*e^3*x^3))/(x*(d + e*x)^3) + 30*e*ArcTanh[(S
qrt[-e^2]*x - Sqrt[d^2 - e^2*x^2])/d])/d^5

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(348\) vs. \(2(130)=260\).
time = 0.10, size = 349, normalized size = 2.39

method	result	size
risch	\(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{d^{5} x}-\frac {19 \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{5 d^{5} \left (x +\frac {d}{e}\right )}+\frac {3 e \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{d^{4} \sqrt {d^{2}}}-\frac {4 \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{5 e \,d^{4} \left (x +\frac {d}{e}\right )^{2}}-\frac {\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{5 d^{3} e^{2} \left (x +\frac {d}{e}\right )^{3}}\)	\(199\)
default	\(-\frac {3 \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{d^{5} \left (x +\frac {d}{e}\right )}+\frac {-\frac {\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{5 d e \left (x +\frac {d}{e}\right )^{3}}+\frac {2 e \left (-\frac {\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{3 d e \left (x +\frac {d}{e}\right )^{2}}-\frac {\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{3 d^{2} \left (x +\frac {d}{e}\right )}\right )}{5 d}}{e \,d^{2}}-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{d^{5} x}+\frac {-\frac {2 \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{3 d e \left (x +\frac {d}{e}\right )^{2}}-\frac {2 \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{3 d^{2} \left (x +\frac {d}{e}\right )}}{d^{3}}+\frac {3 e \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{d^{4} \sqrt {d^{2}}}\)	\(349\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(e*x+d)^3/(-e^2*x^2+d^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(e*x+d)^3/(-e^2*x^2+d^2)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 2.44, size = 173, normalized size = 1.18 \begin {gather*} -\frac {24 \, x^{4} e^{4} + 72 \, d x^{3} e^{3} + 72 \, d^{2} x^{2} e^{2} + 24 \, d^{3} x e + 15 \, {\left (x^{4} e^{4} + 3 \, d x^{3} e^{3} + 3 \, d^{2} x^{2} e^{2} + d^{3} x e\right )} \log \left (-\frac {d - \sqrt {-x^{2} e^{2} + d^{2}}}{x}\right ) + {\left (24 \, x^{3} e^{3} + 57 \, d x^{2} e^{2} + 39 \, d^{2} x e + 5 \, d^{3}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{5 \, {\left (d^{5} x^{4} e^{3} + 3 \, d^{6} x^{3} e^{2} + 3 \, d^{7} x^{2} e + d^{8} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(e*x+d)^3/(-e^2*x^2+d^2)^(1/2),x, algorithm="fricas")

[Out]

-1/5*(24*x^4*e^4 + 72*d*x^3*e^3 + 72*d^2*x^2*e^2 + 24*d^3*x*e + 15*(x^4*e^4 + 3*d*x^3*e^3 + 3*d^2*x^2*e^2 + d^
3*x*e)*log(-(d - sqrt(-x^2*e^2 + d^2))/x) + (24*x^3*e^3 + 57*d*x^2*e^2 + 39*d^2*x*e + 5*d^3)*sqrt(-x^2*e^2 + d
^2))/(d^5*x^4*e^3 + 3*d^6*x^3*e^2 + 3*d^7*x^2*e + d^8*x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{2} \sqrt {- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 287 vs. \(2 (133) = 266\).
time = 2.85, size = 287, normalized size = 1.97 \begin {gather*} \frac {3 \, e \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \, {\left | x \right |}}\right )}{d^{5}} - \frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-1\right )}}{2 \, d^{5} x} + \frac {x {\left (\frac {121 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-1\right )}}{x} + \frac {410 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{\left (-3\right )}}{x^{2}} + \frac {610 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{\left (-5\right )}}{x^{3}} + \frac {425 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{\left (-7\right )}}{x^{4}} + \frac {125 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} e^{\left (-9\right )}}{x^{5}} + 5 \, e\right )} e^{2}}{10 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{5} {\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(1/x^2/(e*x+d)^3/(-e^2*x^2+d^2)^(1/2),x, algorithm="giac")

[Out]

3*e*log(1/2*abs(-2*d*e - 2*sqrt(-x^2*e^2 + d^2)*e)*e^(-2)/abs(x))/d^5 - 1/2*(d*e + sqrt(-x^2*e^2 + d^2)*e)*e^(
-1)/(d^5*x) + 1/10*x*(121*(d*e + sqrt(-x^2*e^2 + d^2)*e)*e^(-1)/x + 410*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*e^(-3
)/x^2 + 610*(d*e + sqrt(-x^2*e^2 + d^2)*e)^3*e^(-5)/x^3 + 425*(d*e + sqrt(-x^2*e^2 + d^2)*e)^4*e^(-7)/x^4 + 12
5*(d*e + sqrt(-x^2*e^2 + d^2)*e)^5*e^(-9)/x^5 + 5*e)*e^2/((d*e + sqrt(-x^2*e^2 + d^2)*e)*d^5*((d*e + sqrt(-x^2
*e^2 + d^2)*e)*e^(-2)/x + 1)^5)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x^2\,\sqrt {d^2-e^2\,x^2}\,{\left (d+e\,x\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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