3.8.68 \(\int \frac {\sqrt {a+c x^4}}{x^5} \, dx\) [768]

Optimal. Leaf size=47 \[ -\frac {\sqrt {a+c x^4}}{4 x^4}-\frac {c \tanh ^{-1}\left (\frac {\sqrt {a+c x^4}}{\sqrt {a}}\right )}{4 \sqrt {a}} \]

[Out]

-1/4*c*arctanh((c*x^4+a)^(1/2)/a^(1/2))/a^(1/2)-1/4*(c*x^4+a)^(1/2)/x^4

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Rubi [A]
time = 0.02, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {272, 43, 65, 214} \begin {gather*} -\frac {\sqrt {a+c x^4}}{4 x^4}-\frac {c \tanh ^{-1}\left (\frac {\sqrt {a+c x^4}}{\sqrt {a}}\right )}{4 \sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + c*x^4]/x^5,x]

[Out]

-1/4*Sqrt[a + c*x^4]/x^4 - (c*ArcTanh[Sqrt[a + c*x^4]/Sqrt[a]])/(4*Sqrt[a])

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + 1))), x] - Dist[d*(n/(b*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

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]]

Rubi steps

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

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Mathematica [A]
time = 0.07, size = 47, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {a+c x^4}}{4 x^4}-\frac {c \tanh ^{-1}\left (\frac {\sqrt {a+c x^4}}{\sqrt {a}}\right )}{4 \sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + c*x^4]/x^5,x]

[Out]

-1/4*Sqrt[a + c*x^4]/x^4 - (c*ArcTanh[Sqrt[a + c*x^4]/Sqrt[a]])/(4*Sqrt[a])

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Maple [A]
time = 0.17, size = 63, normalized size = 1.34

method	result	size
risch	\(-\frac {\sqrt {x^{4} c +a}}{4 x^{4}}-\frac {c \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {x^{4} c +a}}{x^{2}}\right )}{4 \sqrt {a}}\)	\(45\)
default	\(-\frac {\left (x^{4} c +a \right )^{\frac {3}{2}}}{4 a \,x^{4}}-\frac {c \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {x^{4} c +a}}{x^{2}}\right )}{4 \sqrt {a}}+\frac {c \sqrt {x^{4} c +a}}{4 a}\)	\(63\)
elliptic	\(-\frac {\left (x^{4} c +a \right )^{\frac {3}{2}}}{4 a \,x^{4}}-\frac {c \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {x^{4} c +a}}{x^{2}}\right )}{4 \sqrt {a}}+\frac {c \sqrt {x^{4} c +a}}{4 a}\)	\(63\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+a)^(1/2)/x^5,x,method=_RETURNVERBOSE)

[Out]

-1/4/a/x^4*(c*x^4+a)^(3/2)-1/4/a^(1/2)*c*ln((2*a+2*a^(1/2)*(c*x^4+a)^(1/2))/x^2)+1/4/a*c*(c*x^4+a)^(1/2)

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Maxima [A]
time = 0.49, size = 53, normalized size = 1.13 \begin {gather*} \frac {c \log \left (\frac {\sqrt {c x^{4} + a} - \sqrt {a}}{\sqrt {c x^{4} + a} + \sqrt {a}}\right )}{8 \, \sqrt {a}} - \frac {\sqrt {c x^{4} + a}}{4 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)^(1/2)/x^5,x, algorithm="maxima")

[Out]

1/8*c*log((sqrt(c*x^4 + a) - sqrt(a))/(sqrt(c*x^4 + a) + sqrt(a)))/sqrt(a) - 1/4*sqrt(c*x^4 + a)/x^4

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Fricas [A]
time = 0.38, size = 108, normalized size = 2.30 \begin {gather*} \left [\frac {\sqrt {a} c x^{4} \log \left (\frac {c x^{4} - 2 \, \sqrt {c x^{4} + a} \sqrt {a} + 2 \, a}{x^{4}}\right ) - 2 \, \sqrt {c x^{4} + a} a}{8 \, a x^{4}}, \frac {\sqrt {-a} c x^{4} \arctan \left (\frac {\sqrt {c x^{4} + a} \sqrt {-a}}{a}\right ) - \sqrt {c x^{4} + a} a}{4 \, a x^{4}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)^(1/2)/x^5,x, algorithm="fricas")

[Out]

[1/8*(sqrt(a)*c*x^4*log((c*x^4 - 2*sqrt(c*x^4 + a)*sqrt(a) + 2*a)/x^4) - 2*sqrt(c*x^4 + a)*a)/(a*x^4), 1/4*(sq
rt(-a)*c*x^4*arctan(sqrt(c*x^4 + a)*sqrt(-a)/a) - sqrt(c*x^4 + a)*a)/(a*x^4)]

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Sympy [A]
time = 0.95, size = 46, normalized size = 0.98 \begin {gather*} - \frac {\sqrt {c} \sqrt {\frac {a}{c x^{4}} + 1}}{4 x^{2}} - \frac {c \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {c} x^{2}} \right )}}{4 \sqrt {a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+a)**(1/2)/x**5,x)

[Out]

-sqrt(c)*sqrt(a/(c*x**4) + 1)/(4*x**2) - c*asinh(sqrt(a)/(sqrt(c)*x**2))/(4*sqrt(a))

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Giac [A]
time = 0.52, size = 46, normalized size = 0.98 \begin {gather*} \frac {\frac {c^{2} \arctan \left (\frac {\sqrt {c x^{4} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {\sqrt {c x^{4} + a} c}{x^{4}}}{4 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)^(1/2)/x^5,x, algorithm="giac")

[Out]

1/4*(c^2*arctan(sqrt(c*x^4 + a)/sqrt(-a))/sqrt(-a) - sqrt(c*x^4 + a)*c/x^4)/c

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Mupad [B]
time = 1.28, size = 35, normalized size = 0.74 \begin {gather*} -\frac {\sqrt {c\,x^4+a}}{4\,x^4}-\frac {c\,\mathrm {atanh}\left (\frac {\sqrt {c\,x^4+a}}{\sqrt {a}}\right )}{4\,\sqrt {a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^4)^(1/2)/x^5,x)

[Out]

- (a + c*x^4)^(1/2)/(4*x^4) - (c*atanh((a + c*x^4)^(1/2)/a^(1/2)))/(4*a^(1/2))

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