3.324 \(\int \frac{1}{a x^2+b x^2} \, dx\)

Optimal. Leaf size=10 \[ -\frac{1}{x (a+b)} \]

[Out]

-(1/((a + b)*x))

________________________________________________________________________________________

Rubi [A]  time = 0.0016056, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {6, 12, 30} \[ -\frac{1}{x (a+b)} \]

Antiderivative was successfully verified.

[In]

Int[(a*x^2 + b*x^2)^(-1),x]

[Out]

-(1/((a + b)*x))

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{1}{a x^2+b x^2} \, dx &=\int \frac{1}{(a+b) x^2} \, dx\\ &=\frac{\int \frac{1}{x^2} \, dx}{a+b}\\ &=-\frac{1}{(a+b) x}\\ \end{align*}

Mathematica [A]  time = 0.0006774, size = 10, normalized size = 1. \[ -\frac{1}{x (a+b)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*x^2 + b*x^2)^(-1),x]

[Out]

-(1/((a + b)*x))

________________________________________________________________________________________

Maple [A]  time = 0., size = 11, normalized size = 1.1 \begin{align*} -{\frac{1}{ \left ( a+b \right ) x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(a+b)/x

________________________________________________________________________________________

Maxima [A]  time = 1.08684, size = 14, normalized size = 1.4 \begin{align*} -\frac{1}{{\left (a + b\right )} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/((a + b)*x)

________________________________________________________________________________________

Fricas [A]  time = 0.639769, size = 22, normalized size = 2.2 \begin{align*} -\frac{1}{{\left (a + b\right )} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/((a + b)*x)

________________________________________________________________________________________

Sympy [A]  time = 0.081292, size = 7, normalized size = 0.7 \begin{align*} - \frac{1}{x \left (a + b\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(x*(a + b))

________________________________________________________________________________________

Giac [A]  time = 1.13651, size = 14, normalized size = 1.4 \begin{align*} -\frac{1}{{\left (a + b\right )} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/((a + b)*x)