3.31.65 \(\int \frac {1}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^2} \, dx\) [3065]

Optimal. Leaf size=93 \[ -\frac {2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{c}+\frac {b \sqrt {d} \tanh ^{-1}\left (\frac {b d+2 c \sqrt {\frac {d}{x}}}{2 \sqrt {c} \sqrt {d} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{c^{3/2}} \]

[Out]

b*arctanh(1/2*(b*d+2*c*(d/x)^(1/2))/c^(1/2)/d^(1/2)/(a+c/x+b*(d/x)^(1/2))^(1/2))*d^(1/2)/c^(3/2)-2*(a+c/x+b*(d
/x)^(1/2))^(1/2)/c

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Rubi [A]
time = 0.09, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {1994, 1355, 654, 635, 212} \begin {gather*} \frac {b \sqrt {d} \tanh ^{-1}\left (\frac {b d+2 c \sqrt {\frac {d}{x}}}{2 \sqrt {c} \sqrt {d} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{c^{3/2}}-\frac {2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[a + b*Sqrt[d/x] + c/x]*x^2),x]

[Out]

(-2*Sqrt[a + b*Sqrt[d/x] + c/x])/c + (b*Sqrt[d]*ArcTanh[(b*d + 2*c*Sqrt[d/x])/(2*Sqrt[c]*Sqrt[d]*Sqrt[a + b*Sq
rt[d/x] + c/x])])/c^(3/2)

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 635

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 654

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*((a + b*x + c*x^2)^(p +
1)/(2*c*(p + 1))), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 1355

Int[((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[n]}, Dist[k, Subst[I
nt[x^(k - 1)*(a + b*x^(k*n) + c*x^(2*k*n))^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && EqQ[n2, 2*n] &
& FractionQ[n]

Rule 1994

Int[(x_)^(m_.)*((a_) + (b_.)*((d_.)/(x_))^(n_) + (c_.)*(x_)^(n2_.))^(p_), x_Symbol] :> Dist[-d^(m + 1), Subst[
Int[(a + b*x^n + (c/d^(2*n))*x^(2*n))^p/x^(m + 2), x], x, d/x], x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[n2,
-2*n] && IntegerQ[2*n] && IntegerQ[m]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^2} \, dx &=-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {a+b \sqrt {x}+\frac {c x}{d}}} \, dx,x,\frac {d}{x}\right )}{d}\\ &=-\frac {2 \text {Subst}\left (\int \frac {x}{\sqrt {a+b x+\frac {c x^2}{d}}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{d}\\ &=-\frac {2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{c}+\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+\frac {c x^2}{d}}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{c}\\ &=-\frac {2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{c}+\frac {(2 b) \text {Subst}\left (\int \frac {1}{\frac {4 c}{d}-x^2} \, dx,x,\frac {b+\frac {2 c \sqrt {\frac {d}{x}}}{d}}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{c}\\ &=-\frac {2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{c}+\frac {b \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d} \left (b+\frac {2 c \sqrt {\frac {d}{x}}}{d}\right )}{2 \sqrt {c} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{c^{3/2}}\\ \end {align*}

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Mathematica [A]
time = 0.35, size = 147, normalized size = 1.58 \begin {gather*} -\frac {\sqrt {\frac {d \left (c+\left (a+b \sqrt {\frac {d}{x}}\right ) x\right )}{x}} \left (2 \sqrt {c} \sqrt {\frac {d \left (c+a x+b \sqrt {\frac {d}{x}} x\right )}{x}}+b d \log \left (c \left (b d+2 c \sqrt {\frac {d}{x}}-2 \sqrt {c} \sqrt {\frac {d \left (c+a x+b \sqrt {\frac {d}{x}} x\right )}{x}}\right )\right )\right )}{c^{3/2} d \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[a + b*Sqrt[d/x] + c/x]*x^2),x]

[Out]

-((Sqrt[(d*(c + (a + b*Sqrt[d/x])*x))/x]*(2*Sqrt[c]*Sqrt[(d*(c + a*x + b*Sqrt[d/x]*x))/x] + b*d*Log[c*(b*d + 2
*c*Sqrt[d/x] - 2*Sqrt[c]*Sqrt[(d*(c + a*x + b*Sqrt[d/x]*x))/x])]))/(c^(3/2)*d*Sqrt[a + b*Sqrt[d/x] + c/x]))

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Maple [A]
time = 0.07, size = 118, normalized size = 1.27

method	result	size
default	\(\frac {\sqrt {\frac {b \sqrt {\frac {d}{x}}\, x +a x +c}{x}}\, \left (b \sqrt {\frac {d}{x}}\, x \ln \left (\frac {2 c +b \sqrt {\frac {d}{x}}\, x +2 \sqrt {c}\, \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}}{\sqrt {x}}\right ) c -2 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, c^{\frac {3}{2}}\right )}{\sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, c^{\frac {5}{2}}}\)	\(118\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(a+c/x+b*(d/x)^(1/2))^(1/2),x,method=_RETURNVERBOSE)

[Out]

((b*(d/x)^(1/2)*x+a*x+c)/x)^(1/2)*(b*(d/x)^(1/2)*x*ln((2*c+b*(d/x)^(1/2)*x+2*c^(1/2)*(b*(d/x)^(1/2)*x+a*x+c)^(
1/2))/x^(1/2))*c-2*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*c^(3/2))/(b*(d/x)^(1/2)*x+a*x+c)^(1/2)/c^(5/2)

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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/(a+c/x+b*(d/x)^(1/2))^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(b*sqrt(d/x) + a + c/x)*x^2), x)

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(a+c/x+b*(d/x)^(1/2))^(1/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{2} \sqrt {a + b \sqrt {\frac {d}{x}} + \frac {c}{x}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(a+c/x+b*(d/x)**(1/2))**(1/2),x)

[Out]

Integral(1/(x**2*sqrt(a + b*sqrt(d/x) + c/x)), x)

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(a+c/x+b*(d/x)^(1/2))^(1/2),x, algorithm="giac")

[Out]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x^2\,\sqrt {a+\frac {c}{x}+b\,\sqrt {\frac {d}{x}}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(a + c/x + b*(d/x)^(1/2))^(1/2)),x)

[Out]

int(1/(x^2*(a + c/x + b*(d/x)^(1/2))^(1/2)), x)

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