3.2.18 \(\int \frac {\sqrt {b x^2+c x^4}}{x^6} \, dx\)

Optimal. Leaf size=84 \[ \frac {c^2 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{8 b^{3/2}}-\frac {\sqrt {b x^2+c x^4}}{4 x^5}-\frac {c \sqrt {b x^2+c x^4}}{8 b x^3} \]

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Rubi [A]  time = 0.10, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2020, 2025, 2008, 206} \begin {gather*} \frac {c^2 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{8 b^{3/2}}-\frac {c \sqrt {b x^2+c x^4}}{8 b x^3}-\frac {\sqrt {b x^2+c x^4}}{4 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[b*x^2 + c*x^4]/x^6,x]

[Out]

-Sqrt[b*x^2 + c*x^4]/(4*x^5) - (c*Sqrt[b*x^2 + c*x^4])/(8*b*x^3) + (c^2*ArcTanh[(Sqrt[b]*x)/Sqrt[b*x^2 + c*x^4
]])/(8*b^(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 2008

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

Rule 2020

Int[((c_.)*(x_))^(m_)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a*x^j + b*
x^n)^p)/(c*(m + j*p + 1)), x] - Dist[(b*p*(n - j))/(c^n*(m + j*p + 1)), Int[(c*x)^(m + n)*(a*x^j + b*x^n)^(p -
1), x], x] /; FreeQ[{a, b, c}, x] &&  !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && GtQ[p
, 0] && LtQ[m + j*p + 1, 0]

Rule 2025

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(c^(j - 1)*(c*x)^(m - j +
1)*(a*x^j + b*x^n)^(p + 1))/(a*(m + j*p + 1)), x] - Dist[(b*(m + n*p + n - j + 1))/(a*c^(n - j)*(m + j*p + 1)
), Int[(c*x)^(m + n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IntegerQ[p] && LtQ[0, j,
n] && (IntegersQ[j, n] || GtQ[c, 0]) && LtQ[m + j*p + 1, 0]

Rubi steps

\begin {align*} \int \frac {\sqrt {b x^2+c x^4}}{x^6} \, dx &=-\frac {\sqrt {b x^2+c x^4}}{4 x^5}+\frac {1}{4} c \int \frac {1}{x^2 \sqrt {b x^2+c x^4}} \, dx\\ &=-\frac {\sqrt {b x^2+c x^4}}{4 x^5}-\frac {c \sqrt {b x^2+c x^4}}{8 b x^3}-\frac {c^2 \int \frac {1}{\sqrt {b x^2+c x^4}} \, dx}{8 b}\\ &=-\frac {\sqrt {b x^2+c x^4}}{4 x^5}-\frac {c \sqrt {b x^2+c x^4}}{8 b x^3}+\frac {c^2 \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {b x^2+c x^4}}\right )}{8 b}\\ &=-\frac {\sqrt {b x^2+c x^4}}{4 x^5}-\frac {c \sqrt {b x^2+c x^4}}{8 b x^3}+\frac {c^2 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{8 b^{3/2}}\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 46, normalized size = 0.55 \begin {gather*} -\frac {c^2 \left (x^2 \left (b+c x^2\right )\right )^{3/2} \, _2F_1\left (\frac {3}{2},3;\frac {5}{2};\frac {c x^2}{b}+1\right )}{3 b^3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[b*x^2 + c*x^4]/x^6,x]

[Out]

-1/3*(c^2*(x^2*(b + c*x^2))^(3/2)*Hypergeometric2F1[3/2, 3, 5/2, 1 + (c*x^2)/b])/(b^3*x^3)

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IntegrateAlgebraic [A]  time = 0.13, size = 71, normalized size = 0.85 \begin {gather*} \frac {c^2 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{8 b^{3/2}}+\frac {\left (-2 b-c x^2\right ) \sqrt {b x^2+c x^4}}{8 b x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[b*x^2 + c*x^4]/x^6,x]

[Out]

((-2*b - c*x^2)*Sqrt[b*x^2 + c*x^4])/(8*b*x^5) + (c^2*ArcTanh[(Sqrt[b]*x)/Sqrt[b*x^2 + c*x^4]])/(8*b^(3/2))

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fricas [A]  time = 1.36, size = 159, normalized size = 1.89 \begin {gather*} \left [\frac {\sqrt {b} c^{2} x^{5} \log \left (-\frac {c x^{3} + 2 \, b x + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {b}}{x^{3}}\right ) - 2 \, \sqrt {c x^{4} + b x^{2}} {\left (b c x^{2} + 2 \, b^{2}\right )}}{16 \, b^{2} x^{5}}, -\frac {\sqrt {-b} c^{2} x^{5} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {-b}}{c x^{3} + b x}\right ) + \sqrt {c x^{4} + b x^{2}} {\left (b c x^{2} + 2 \, b^{2}\right )}}{8 \, b^{2} x^{5}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^(1/2)/x^6,x, algorithm="fricas")

[Out]

[1/16*(sqrt(b)*c^2*x^5*log(-(c*x^3 + 2*b*x + 2*sqrt(c*x^4 + b*x^2)*sqrt(b))/x^3) - 2*sqrt(c*x^4 + b*x^2)*(b*c*
x^2 + 2*b^2))/(b^2*x^5), -1/8*(sqrt(-b)*c^2*x^5*arctan(sqrt(c*x^4 + b*x^2)*sqrt(-b)/(c*x^3 + b*x)) + sqrt(c*x^
4 + b*x^2)*(b*c*x^2 + 2*b^2))/(b^2*x^5)]

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giac [A]  time = 0.21, size = 78, normalized size = 0.93 \begin {gather*} -\frac {\frac {c^{3} \arctan \left (\frac {\sqrt {c x^{2} + b}}{\sqrt {-b}}\right ) \mathrm {sgn}\relax (x)}{\sqrt {-b} b} + \frac {{\left (c x^{2} + b\right )}^{\frac {3}{2}} c^{3} \mathrm {sgn}\relax (x) + \sqrt {c x^{2} + b} b c^{3} \mathrm {sgn}\relax (x)}{b c^{2} x^{4}}}{8 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^(1/2)/x^6,x, algorithm="giac")

[Out]

-1/8*(c^3*arctan(sqrt(c*x^2 + b)/sqrt(-b))*sgn(x)/(sqrt(-b)*b) + ((c*x^2 + b)^(3/2)*c^3*sgn(x) + sqrt(c*x^2 +
b)*b*c^3*sgn(x))/(b*c^2*x^4))/c

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maple [A]  time = 0.01, size = 106, normalized size = 1.26 \begin {gather*} \frac {\sqrt {c \,x^{4}+b \,x^{2}}\, \left (\sqrt {b}\, c^{2} x^{4} \ln \left (\frac {2 b +2 \sqrt {c \,x^{2}+b}\, \sqrt {b}}{x}\right )-\sqrt {c \,x^{2}+b}\, c^{2} x^{4}+\left (c \,x^{2}+b \right )^{\frac {3}{2}} c \,x^{2}-2 \left (c \,x^{2}+b \right )^{\frac {3}{2}} b \right )}{8 \sqrt {c \,x^{2}+b}\, b^{2} x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2)^(1/2)/x^6,x)

[Out]

1/8*(c*x^4+b*x^2)^(1/2)*(b^(1/2)*ln(2*(b+(c*x^2+b)^(1/2)*b^(1/2))/x)*x^4*c^2-(c*x^2+b)^(1/2)*x^4*c^2+(c*x^2+b)
^(3/2)*x^2*c-2*(c*x^2+b)^(3/2)*b)/x^5/(c*x^2+b)^(1/2)/b^2

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {c x^{4} + b x^{2}}}{x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^(1/2)/x^6,x, algorithm="maxima")

[Out]

integrate(sqrt(c*x^4 + b*x^2)/x^6, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {c\,x^4+b\,x^2}}{x^6} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2 + c*x^4)^(1/2)/x^6,x)

[Out]

int((b*x^2 + c*x^4)^(1/2)/x^6, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{2} \left (b + c x^{2}\right )}}{x^{6}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2)**(1/2)/x**6,x)

[Out]

Integral(sqrt(x**2*(b + c*x**2))/x**6, x)

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