3.1.94 \(\int \frac {1}{x \sqrt {a^2+2 a b x^3+b^2 x^6}} \, dx\)

Optimal. Leaf size=80 \[ \frac {\log (x) \left (a+b x^3\right )}{a \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (a+b x^3\right ) \log \left (a+b x^3\right )}{3 a \sqrt {a^2+2 a b x^3+b^2 x^6}} \]

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Rubi [A]  time = 0.03, antiderivative size = 80, 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 = {1355, 266, 36, 29, 31} \begin {gather*} \frac {\log (x) \left (a+b x^3\right )}{a \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (a+b x^3\right ) \log \left (a+b x^3\right )}{3 a \sqrt {a^2+2 a b x^3+b^2 x^6}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]),x]

[Out]

((a + b*x^3)*Log[x])/(a*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) - ((a + b*x^3)*Log[a + b*x^3])/(3*a*Sqrt[a^2 + 2*a*b*
x^3 + b^2*x^6])

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

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

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]]

Rule 1355

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_), x_Symbol] :> Dist[(a + b*x^n + c*x^
(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*FracPart[p])), Int[(d*x)^m*(b/2 + c*x^n)^(2*p), x], x] /; Fr
eeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]

Rubi steps

\begin {align*} \int \frac {1}{x \sqrt {a^2+2 a b x^3+b^2 x^6}} \, dx &=\frac {\left (a b+b^2 x^3\right ) \int \frac {1}{x \left (a b+b^2 x^3\right )} \, dx}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\\ &=\frac {\left (a b+b^2 x^3\right ) \operatorname {Subst}\left (\int \frac {1}{x \left (a b+b^2 x\right )} \, dx,x,x^3\right )}{3 \sqrt {a^2+2 a b x^3+b^2 x^6}}\\ &=\frac {\left (a b+b^2 x^3\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^3\right )}{3 a b \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (b \left (a b+b^2 x^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a b+b^2 x} \, dx,x,x^3\right )}{3 a \sqrt {a^2+2 a b x^3+b^2 x^6}}\\ &=\frac {\left (a+b x^3\right ) \log (x)}{a \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (a+b x^3\right ) \log \left (a+b x^3\right )}{3 a \sqrt {a^2+2 a b x^3+b^2 x^6}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 42, normalized size = 0.52 \begin {gather*} \frac {\left (a+b x^3\right ) \left (3 \log (x)-\log \left (a+b x^3\right )\right )}{3 a \sqrt {\left (a+b x^3\right )^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.20, size = 94, normalized size = 1.18 \begin {gather*} \frac {\log \left (-a \sqrt {a^2+2 a b x^3+b^2 x^6}+a^2+a \sqrt {b^2} x^3\right )}{3 a}-\frac {\log \left (\sqrt {a^2+2 a b x^3+b^2 x^6}+a-\sqrt {b^2} x^3\right )}{3 a} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]),x]

[Out]

-1/3*Log[a - Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]]/a + Log[a^2 + a*Sqrt[b^2]*x^3 - a*Sqrt[a^2 + 2*a
*b*x^3 + b^2*x^6]]/(3*a)

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fricas [A]  time = 1.12, size = 18, normalized size = 0.22 \begin {gather*} -\frac {\log \left (b x^{3} + a\right ) - 3 \, \log \relax (x)}{3 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/((b*x^3+a)^2)^(1/2),x, algorithm="fricas")

[Out]

-1/3*(log(b*x^3 + a) - 3*log(x))/a

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giac [A]  time = 0.34, size = 32, normalized size = 0.40 \begin {gather*} -\frac {1}{3} \, {\left (\frac {\log \left ({\left | b x^{3} + a \right |}\right )}{a} - \frac {3 \, \log \left ({\left | x \right |}\right )}{a}\right )} \mathrm {sgn}\left (b x^{3} + a\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/((b*x^3+a)^2)^(1/2),x, algorithm="giac")

[Out]

-1/3*(log(abs(b*x^3 + a))/a - 3*log(abs(x))/a)*sgn(b*x^3 + a)

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maple [A]  time = 0.01, size = 39, normalized size = 0.49 \begin {gather*} \frac {\left (b \,x^{3}+a \right ) \left (3 \ln \relax (x )-\ln \left (b \,x^{3}+a \right )\right )}{3 \sqrt {\left (b \,x^{3}+a \right )^{2}}\, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/((b*x^3+a)^2)^(1/2),x)

[Out]

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

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maxima [A]  time = 1.15, size = 43, normalized size = 0.54 \begin {gather*} -\frac {\left (-1\right )^{2 \, a b x^{3} + 2 \, a^{2}} \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{x^{2} {\left | x \right |}}\right )}{3 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/((b*x^3+a)^2)^(1/2),x, algorithm="maxima")

[Out]

-1/3*(-1)^(2*a*b*x^3 + 2*a^2)*log(2*a*b*x/abs(x) + 2*a^2/(x^2*abs(x)))/a

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mupad [B]  time = 1.39, size = 48, normalized size = 0.60 \begin {gather*} -\frac {\ln \left (a\,b+\frac {a^2}{x^3}+\frac {\sqrt {a^2}\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{x^3}\right )}{3\,\sqrt {a^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*((a + b*x^3)^2)^(1/2)),x)

[Out]

-log(a*b + a^2/x^3 + ((a^2)^(1/2)*(a^2 + b^2*x^6 + 2*a*b*x^3)^(1/2))/x^3)/(3*(a^2)^(1/2))

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sympy [A]  time = 0.28, size = 15, normalized size = 0.19 \begin {gather*} \frac {\log {\relax (x )}}{a} - \frac {\log {\left (\frac {a}{b} + x^{3} \right )}}{3 a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/((b*x**3+a)**2)**(1/2),x)

[Out]

log(x)/a - log(a/b + x**3)/(3*a)

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