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

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

[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])

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Rubi [A]  time = 0.0339088, antiderivative size = 80, normalized size of antiderivative = 1., 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} \[ \frac{\log (x) \left (a+b x^3\right )}{a \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (a+b x^3\right ) \log \left (a+b x^3\right )}{3 a \sqrt{a^2+2 a b x^3+b^2 x^6}} \]

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 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]

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 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 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]

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*}

Mathematica [A]  time = 0.011428, size = 42, normalized size = 0.52 \[ \frac{\left (a+b x^3\right ) \left (3 \log (x)-\log \left (a+b x^3\right )\right )}{3 a \sqrt{\left (a+b x^3\right )^2}} \]

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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Maple [A]  time = 0.007, size = 39, normalized size = 0.5 \begin{align*}{\frac{ \left ( b{x}^{3}+a \right ) \left ( 3\,\ln \left ( x \right ) -\ln \left ( b{x}^{3}+a \right ) \right ) }{3\,a}{\frac{1}{\sqrt{ \left ( b{x}^{3}+a \right ) ^{2}}}}} \end{align*}

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 [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [A]  time = 1.88243, size = 49, normalized size = 0.61 \begin{align*} -\frac{\log \left (b x^{3} + a\right ) - 3 \, \log \left (x\right )}{3 \, a} \end{align*}

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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Sympy [A]  time = 0.262047, size = 15, normalized size = 0.19 \begin{align*} \frac{\log{\left (x \right )}}{a} - \frac{\log{\left (\frac{a}{b} + x^{3} \right )}}{3 a} \end{align*}

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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Giac [A]  time = 1.1237, size = 43, normalized size = 0.54 \begin{align*} -\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{align*}

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)