3.14 \(\int \frac {x^2}{a x+b x^3} \, dx\)

Optimal. Leaf size=15 \[ \frac {\log \left (a+b x^2\right )}{2 b} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1584, 260} \[ \frac {\log \left (a+b x^2\right )}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a*x + b*x^3),x]

[Out]

Log[a + b*x^2]/(2*b)

Rule 260

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

Rule 1584

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

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 15, normalized size = 1.00 \[ \frac {\log \left (a+b x^2\right )}{2 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a*x + b*x^3),x]

[Out]

Log[a + b*x^2]/(2*b)

________________________________________________________________________________________

fricas [A]  time = 0.88, size = 13, normalized size = 0.87 \[ \frac {\log \left (b x^{2} + a\right )}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(b*x^2 + a)/b

________________________________________________________________________________________

giac [A]  time = 0.19, size = 14, normalized size = 0.93 \[ \frac {\log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(abs(b*x^2 + a))/b

________________________________________________________________________________________

maple [A]  time = 0.04, size = 14, normalized size = 0.93 \[ \frac {\ln \left (b \,x^{2}+a \right )}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.42, size = 13, normalized size = 0.87 \[ \frac {\log \left (b x^{2} + a\right )}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(b*x^2 + a)/b

________________________________________________________________________________________

mupad [B]  time = 4.92, size = 13, normalized size = 0.87 \[ \frac {\ln \left (b\,x^2+a\right )}{2\,b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a*x + b*x^3),x)

[Out]

log(a + b*x^2)/(2*b)

________________________________________________________________________________________

sympy [A]  time = 0.13, size = 10, normalized size = 0.67 \[ \frac {\log {\left (a + b x^{2} \right )}}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(a + b*x**2)/(2*b)

________________________________________________________________________________________