3.4.24 \(\int \frac {1}{a x^2+b x^2} \, dx\) [324]

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

[Out]

-1/(a+b)/x

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Rubi [A]
time = 0.00, antiderivative size = 10, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {1}{x (a+b)} \end {gather*}

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*}

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Mathematica [A]
time = 0.00, size = 10, normalized size = 1.00 \begin {gather*} -\frac {1}{(a+b) x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*x^2 + b*x^2)^(-1),x]

[Out]

-(1/((a + b)*x))

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Maple [A]
time = 0.38, size = 11, normalized size = 1.10

method	result	size
gosper	\(-\frac {1}{\left (a +b \right ) x}\)	\(11\)
default	\(-\frac {1}{\left (a +b \right ) x}\)	\(11\)
norman	\(-\frac {1}{\left (a +b \right ) x}\)	\(11\)
risch	\(-\frac {1}{\left (a +b \right ) x}\)	\(11\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(a+b)/x

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Maxima [A]
time = 0.30, size = 10, normalized size = 1.00 \begin {gather*} -\frac {1}{{\left (a + b\right )} x} \end {gather*}

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)

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Fricas [A]
time = 1.55, size = 10, normalized size = 1.00 \begin {gather*} -\frac {1}{{\left (a + b\right )} x} \end {gather*}

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)

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Sympy [A]
time = 0.01, size = 7, normalized size = 0.70 \begin {gather*} - \frac {1}{x \left (a + b\right )} \end {gather*}

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))

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Giac [A]
time = 0.47, size = 10, normalized size = 1.00 \begin {gather*} -\frac {1}{{\left (a + b\right )} x} \end {gather*}

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)

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Mupad [B]
time = 0.03, size = 10, normalized size = 1.00 \begin {gather*} -\frac {1}{x\,\left (a+b\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x^2 + b*x^2),x)

[Out]

-1/(x*(a + b))

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