3.25.53 \(\int \frac {1}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^2} \, dx\)

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

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Rubi [A]  time = 0.12, 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 = {1970, 1341, 640, 621, 206} \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 206

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

Rule 621

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 640

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 1341

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 1970

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*x^(2*n))/d^(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 {\operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b \sqrt {x}+\frac {c x}{d}}} \, dx,x,\frac {d}{x}\right )}{d}\\ &=-\frac {2 \operatorname {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 \operatorname {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) \operatorname {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 [F]  time = 0.20, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [B]  time = 0.94, size = 212, normalized size = 2.28 \begin {gather*} -\frac {b d \sqrt {\frac {c}{d}} \log \left (-8 c d \sqrt {\frac {c}{d}} \sqrt {\frac {d}{x}} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}+4 a c d-b^2 d^2+4 b c d \sqrt {\frac {d}{x}}+\frac {8 c^2 d}{x}\right )}{2 c^2}+\frac {b \sqrt {d} \tanh ^{-1}\left (\frac {2 \sqrt {c} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{b \sqrt {d}}-\frac {2 \sqrt {c} \sqrt {\frac {c}{d}} \sqrt {\frac {d}{x}}}{b \sqrt {d}}\right )}{c^{3/2}}-\frac {2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[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[(2*Sqrt[c]*Sqrt[a + b*Sqrt[d/x] + c/x])/(b*Sqrt[d]) -
(2*Sqrt[c]*Sqrt[c/d]*Sqrt[d/x])/(b*Sqrt[d])])/c^(3/2) - (b*Sqrt[c/d]*d*Log[4*a*c*d - b^2*d^2 + 4*b*c*d*Sqrt[d/
x] - 8*c*Sqrt[c/d]*d*Sqrt[a + b*Sqrt[d/x] + c/x]*Sqrt[d/x] + (8*c^2*d)/x])/(2*c^2)

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fricas [F(-1)]  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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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \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]

Exception raised: NotImplementedError >> Unable to parse Giac output: Warning, checking for positivity of a ro
ot depending of parameters might return wrong sign: -dWarning, checking for positivity of a root depending of
parameters might return wrong sign: d-1/d/abs(d)*d^2*2*(1/d/c*sqrt(a*d^2+b*d^2*sqrt(d/x)+c*d*d/x)+2*b*d/4/c/sq
rt(c*d)*ln(abs(2*sqrt(c*d)*(sqrt(a*d^2+b*d^2*sqrt(d/x)+c*d*d/x)-sqrt(c*d)*sqrt(d/x))-d^2*b)))

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maple [A]  time = 0.14, size = 120, normalized size = 1.29 \begin {gather*} -\frac {\sqrt {\frac {a x +\sqrt {\frac {d}{x}}\, b x +c}{x}}\, \left (-\sqrt {\frac {d}{x}}\, b c x \ln \left (\frac {\sqrt {\frac {d}{x}}\, b x +2 c +2 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \sqrt {c}}{\sqrt {x}}\right )+2 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, c^{\frac {3}{2}}\right )}{\sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, c^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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