3.7.1 \(\int \frac {(a+b x)^4}{(a^2-b^2 x^2)^2} \, dx\)

Optimal. Leaf size=31 \[ \frac {4 a^2}{b (a-b x)}+\frac {4 a \log (a-b x)}{b}+x \]

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Rubi [A]  time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {627, 43} \begin {gather*} \frac {4 a^2}{b (a-b x)}+\frac {4 a \log (a-b x)}{b}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^4/(a^2 - b^2*x^2)^2,x]

[Out]

x + (4*a^2)/(b*(a - b*x)) + (4*a*Log[a - b*x])/b

Rule 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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)^4}{\left (a^2-b^2 x^2\right )^2} \, dx &=\int \frac {(a+b x)^2}{(a-b x)^2} \, dx\\ &=\int \left (1+\frac {4 a^2}{(a-b x)^2}-\frac {4 a}{a-b x}\right ) \, dx\\ &=x+\frac {4 a^2}{b (a-b x)}+\frac {4 a \log (a-b x)}{b}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 32, normalized size = 1.03 \begin {gather*} -\frac {4 a^2}{b (b x-a)}+\frac {4 a \log (a-b x)}{b}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^4/(a^2 - b^2*x^2)^2,x]

[Out]

x - (4*a^2)/(b*(-a + b*x)) + (4*a*Log[a - b*x])/b

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^4}{\left (a^2-b^2 x^2\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)^4/(a^2 - b^2*x^2)^2,x]

[Out]

IntegrateAlgebraic[(a + b*x)^4/(a^2 - b^2*x^2)^2, x]

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fricas [A]  time = 0.40, size = 51, normalized size = 1.65 \begin {gather*} \frac {b^{2} x^{2} - a b x - 4 \, a^{2} + 4 \, {\left (a b x - a^{2}\right )} \log \left (b x - a\right )}{b^{2} x - a b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4/(-b^2*x^2+a^2)^2,x, algorithm="fricas")

[Out]

(b^2*x^2 - a*b*x - 4*a^2 + 4*(a*b*x - a^2)*log(b*x - a))/(b^2*x - a*b)

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giac [A]  time = 0.15, size = 34, normalized size = 1.10 \begin {gather*} x + \frac {4 \, a \log \left ({\left | b x - a \right |}\right )}{b} - \frac {4 \, a^{2}}{{\left (b x - a\right )} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4/(-b^2*x^2+a^2)^2,x, algorithm="giac")

[Out]

x + 4*a*log(abs(b*x - a))/b - 4*a^2/((b*x - a)*b)

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maple [A]  time = 0.05, size = 34, normalized size = 1.10 \begin {gather*} -\frac {4 a^{2}}{\left (b x -a \right ) b}+\frac {4 a \ln \left (b x -a \right )}{b}+x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^4/(-b^2*x^2+a^2)^2,x)

[Out]

x+4*a/b*ln(b*x-a)-4*a^2/b/(b*x-a)

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maxima [A]  time = 1.34, size = 33, normalized size = 1.06 \begin {gather*} -\frac {4 \, a^{2}}{b^{2} x - a b} + x + \frac {4 \, a \log \left (b x - a\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4/(-b^2*x^2+a^2)^2,x, algorithm="maxima")

[Out]

-4*a^2/(b^2*x - a*b) + x + 4*a*log(b*x - a)/b

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mupad [B]  time = 0.04, size = 31, normalized size = 1.00 \begin {gather*} x+\frac {4\,a^2}{b\,\left (a-b\,x\right )}+\frac {4\,a\,\ln \left (a-b\,x\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^4/(a^2 - b^2*x^2)^2,x)

[Out]

x + (4*a^2)/(b*(a - b*x)) + (4*a*log(a - b*x))/b

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sympy [A]  time = 0.20, size = 26, normalized size = 0.84 \begin {gather*} - \frac {4 a^{2}}{- a b + b^{2} x} + \frac {4 a \log {\left (- a + b x \right )}}{b} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**4/(-b**2*x**2+a**2)**2,x)

[Out]

-4*a**2/(-a*b + b**2*x) + 4*a*log(-a + b*x)/b + x

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