\(\int \frac {1}{x^2 (a x+b x^3)} \, dx\) [18]

Optimal result

Integrand size = 15, antiderivative size = 35 \[ \int \frac {1}{x^2 \left (a x+b x^3\right )} \, dx=-\frac {1}{2 a x^2}-\frac {b \log (x)}{a^2}+\frac {b \log \left (a+b x^2\right )}{2 a^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1598, 272, 46} \[ \int \frac {1}{x^2 \left (a x+b x^3\right )} \, dx=\frac {b \log \left (a+b x^2\right )}{2 a^2}-\frac {b \log (x)}{a^2}-\frac {1}{2 a x^2} \]

[In]

Int[1/(x^2*(a*x + b*x^3)),x]

[Out]

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

Rule 46

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x
)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

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]]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \left (a x+b x^3\right )} \, dx=-\frac {1}{2 a x^2}-\frac {b \log (x)}{a^2}+\frac {b \log \left (a+b x^2\right )}{2 a^2} \]

[In]

Integrate[1/(x^2*(a*x + b*x^3)),x]

[Out]

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

Maple [A] (verified)

Time = 2.06 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.91

method	result	size
default	\(-\frac {1}{2 a \,x^{2}}-\frac {b \ln \left (x \right )}{a^{2}}+\frac {b \ln \left (b \,x^{2}+a \right )}{2 a^{2}}\)	\(32\)
norman	\(-\frac {1}{2 a \,x^{2}}-\frac {b \ln \left (x \right )}{a^{2}}+\frac {b \ln \left (b \,x^{2}+a \right )}{2 a^{2}}\)	\(32\)
parallelrisch	\(-\frac {2 b \ln \left (x \right ) x^{2}-b \ln \left (b \,x^{2}+a \right ) x^{2}+a}{2 x^{2} a^{2}}\)	\(33\)
risch	\(-\frac {1}{2 a \,x^{2}}-\frac {b \ln \left (x \right )}{a^{2}}+\frac {b \ln \left (-b \,x^{2}-a \right )}{2 a^{2}}\)	\(35\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.94 \[ \int \frac {1}{x^2 \left (a x+b x^3\right )} \, dx=\frac {b x^{2} \log \left (b x^{2} + a\right ) - 2 \, b x^{2} \log \left (x\right ) - a}{2 \, a^{2} x^{2}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x^2 \left (a x+b x^3\right )} \, dx=- \frac {1}{2 a x^{2}} - \frac {b \log {\left (x \right )}}{a^{2}} + \frac {b \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{2}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x^2 \left (a x+b x^3\right )} \, dx=\frac {b \log \left (b x^{2} + a\right )}{2 \, a^{2}} - \frac {b \log \left (x\right )}{a^{2}} - \frac {1}{2 \, a x^{2}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.23 \[ \int \frac {1}{x^2 \left (a x+b x^3\right )} \, dx=-\frac {b \log \left (x^{2}\right )}{2 \, a^{2}} + \frac {b \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{2}} + \frac {b x^{2} - a}{2 \, a^{2} x^{2}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 10.66 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x^2 \left (a x+b x^3\right )} \, dx=\frac {b\,\ln \left (b\,x^2+a\right )}{2\,a^2}-\frac {1}{2\,a\,x^2}-\frac {b\,\ln \left (x\right )}{a^2} \]

[In]

int(1/(x^2*(a*x + b*x^3)),x)

[Out]

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