3.7.3 \(\int \frac {(a+b x)^2}{(a^2-b^2 x^2)^2} \, dx\)

Optimal. Leaf size=12 \[ \frac {1}{b (a-b x)} \]

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Rubi [A]  time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {627, 32} \begin {gather*} \frac {1}{b (a-b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2/(a^2 - b^2*x^2)^2,x]

[Out]

1/(b*(a - b*x))

Rule 32

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

Rule 627

Int[((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a/d + (c*x)/e)^
p, x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && I
ntegerQ[m + p]))

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 12, normalized size = 1.00 \begin {gather*} \frac {1}{b (a-b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^2/(a^2 - b^2*x^2)^2,x]

[Out]

1/(b*(a - b*x))

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^2}{\left (a^2-b^2 x^2\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)^2/(a^2 - b^2*x^2)^2,x]

[Out]

IntegrateAlgebraic[(a + b*x)^2/(a^2 - b^2*x^2)^2, x]

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fricas [A]  time = 0.39, size = 14, normalized size = 1.17 \begin {gather*} -\frac {1}{b^{2} x - a b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(b^2*x - a*b)

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giac [A]  time = 0.15, size = 14, normalized size = 1.17 \begin {gather*} -\frac {1}{{\left (b x - a\right )} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/((b*x - a)*b)

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maple [A]  time = 0.03, size = 15, normalized size = 1.25 \begin {gather*} -\frac {1}{\left (b x -a \right ) b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.33, size = 14, normalized size = 1.17 \begin {gather*} -\frac {1}{b^{2} x - a b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(b^2*x - a*b)

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mupad [B]  time = 0.41, size = 12, normalized size = 1.00 \begin {gather*} \frac {1}{b\,\left (a-b\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^2/(a^2 - b^2*x^2)^2,x)

[Out]

1/(b*(a - b*x))

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sympy [A]  time = 0.19, size = 10, normalized size = 0.83 \begin {gather*} - \frac {1}{- a b + b^{2} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(-a*b + b**2*x)

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