3.394 \(\int \frac {1}{c x \sqrt {a+b x^n}} \, dx\)

Optimal. Leaf size=31 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{\sqrt {a} c n} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {12, 266, 63, 208} \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{\sqrt {a} c n} \]

Antiderivative was successfully verified.

[In]

Int[1/(c*x*Sqrt[a + b*x^n]),x]

[Out]

(-2*ArcTanh[Sqrt[a + b*x^n]/Sqrt[a]])/(Sqrt[a]*c*n)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 63

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 208

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

Rule 266

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 {1}{c x \sqrt {a+b x^n}} \, dx &=\frac {\int \frac {1}{x \sqrt {a+b x^n}} \, dx}{c}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^n\right )}{c n}\\ &=\frac {2 \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^n}\right )}{b c n}\\ &=-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{\sqrt {a} c n}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 31, normalized size = 1.00 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{\sqrt {a} c n} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(c*x*Sqrt[a + b*x^n]),x]

[Out]

(-2*ArcTanh[Sqrt[a + b*x^n]/Sqrt[a]])/(Sqrt[a]*c*n)

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fricas [A]  time = 0.42, size = 76, normalized size = 2.45 \[ \left [\frac {\log \left (\frac {b x^{n} - 2 \, \sqrt {b x^{n} + a} \sqrt {a} + 2 \, a}{x^{n}}\right )}{\sqrt {a} c n}, \frac {2 \, \sqrt {-a} \arctan \left (\frac {\sqrt {b x^{n} + a} \sqrt {-a}}{a}\right )}{a c n}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[log((b*x^n - 2*sqrt(b*x^n + a)*sqrt(a) + 2*a)/x^n)/(sqrt(a)*c*n), 2*sqrt(-a)*arctan(sqrt(b*x^n + a)*sqrt(-a)/
a)/(a*c*n)]

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b x^{n} + a} c x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(b*x^n + a)*c*x), x)

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maple [A]  time = 0.05, size = 26, normalized size = 0.84 \[ -\frac {2 \arctanh \left (\frac {\sqrt {b \,x^{n}+a}}{\sqrt {a}}\right )}{\sqrt {a}\, c n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.96, size = 42, normalized size = 1.35 \[ \frac {\log \left (\frac {\sqrt {b x^{n} + a} - \sqrt {a}}{\sqrt {b x^{n} + a} + \sqrt {a}}\right )}{\sqrt {a} c n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log((sqrt(b*x^n + a) - sqrt(a))/(sqrt(b*x^n + a) + sqrt(a)))/(sqrt(a)*c*n)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{c\,x\,\sqrt {a+b\,x^n}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*x*(a + b*x^n)^(1/2)),x)

[Out]

int(1/(c*x*(a + b*x^n)^(1/2)), x)

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sympy [A]  time = 1.98, size = 27, normalized size = 0.87 \[ - \frac {2 \operatorname {asinh}{\left (\frac {\sqrt {a} x^{- \frac {n}{2}}}{\sqrt {b}} \right )}}{\sqrt {a} c n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*asinh(sqrt(a)*x**(-n/2)/sqrt(b))/(sqrt(a)*c*n)

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