3.1.17 \(\int \frac {1}{x (a x+b x^3)} \, dx\) [17]

Optimal. Leaf size=34 \[ -\frac {1}{a x}-\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1598, 331, 211} \begin {gather*} -\frac {\sqrt {b} \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2}}-\frac {1}{a x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*(a*x + b*x^3)),x]

[Out]

-(1/(a*x)) - (Sqrt[b]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(3/2)

Rule 211

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

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 1598

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.01, size = 34, normalized size = 1.00 \begin {gather*} -\frac {1}{a x}-\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(a*x + b*x^3)),x]

[Out]

-(1/(a*x)) - (Sqrt[b]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(3/2)

________________________________________________________________________________________

Maple [A]
time = 0.35, size = 30, normalized size = 0.88

method	result	size
default	\(-\frac {b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{a \sqrt {a b}}-\frac {1}{a x}\)	\(30\)
risch	\(-\frac {1}{a x}+\frac {\sqrt {-a b}\, \ln \left (-b x +\sqrt {-a b}\right )}{2 a^{2}}-\frac {\sqrt {-a b}\, \ln \left (-b x -\sqrt {-a b}\right )}{2 a^{2}}\)	\(58\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(b*x^3+a*x),x,method=_RETURNVERBOSE)

[Out]

-b/a/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))-1/a/x

________________________________________________________________________________________

Maxima [A]
time = 0.51, size = 29, normalized size = 0.85 \begin {gather*} -\frac {b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a} - \frac {1}{a x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a) - 1/(a*x)

________________________________________________________________________________________

Fricas [A]
time = 1.34, size = 82, normalized size = 2.41 \begin {gather*} \left [\frac {x \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) - 2}{2 \, a x}, -\frac {x \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + 1}{a x}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(x*sqrt(-b/a)*log((b*x^2 - 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) - 2)/(a*x), -(x*sqrt(b/a)*arctan(x*sqrt(b/a
)) + 1)/(a*x)]

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (29) = 58\).
time = 0.07, size = 65, normalized size = 1.91 \begin {gather*} \frac {\sqrt {- \frac {b}{a^{3}}} \log {\left (- \frac {a^{2} \sqrt {- \frac {b}{a^{3}}}}{b} + x \right )}}{2} - \frac {\sqrt {- \frac {b}{a^{3}}} \log {\left (\frac {a^{2} \sqrt {- \frac {b}{a^{3}}}}{b} + x \right )}}{2} - \frac {1}{a x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 2.57, size = 29, normalized size = 0.85 \begin {gather*} -\frac {b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a} - \frac {1}{a x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a) - 1/(a*x)

________________________________________________________________________________________

Mupad [B]
time = 4.96, size = 26, normalized size = 0.76 \begin {gather*} -\frac {1}{a\,x}-\frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{a^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(a*x + b*x^3)),x)

[Out]

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

________________________________________________________________________________________