3.15.23 \(\int \frac {(3+5 x)^2}{(1-2 x)^2} \, dx\)

Optimal. Leaf size=27 \[ \frac {25 x}{4}+\frac {121}{8 (1-2 x)}+\frac {55}{4} \log (1-2 x) \]

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Rubi [A]  time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {43} \begin {gather*} \frac {25 x}{4}+\frac {121}{8 (1-2 x)}+\frac {55}{4} \log (1-2 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)^2/(1 - 2*x)^2,x]

[Out]

121/(8*(1 - 2*x)) + (25*x)/4 + (55*Log[1 - 2*x])/4

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])

Rubi steps

\begin {align*} \int \frac {(3+5 x)^2}{(1-2 x)^2} \, dx &=\int \left (\frac {25}{4}+\frac {121}{4 (-1+2 x)^2}+\frac {55}{2 (-1+2 x)}\right ) \, dx\\ &=\frac {121}{8 (1-2 x)}+\frac {25 x}{4}+\frac {55}{4} \log (1-2 x)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 26, normalized size = 0.96 \begin {gather*} \frac {1}{8} \left (50 x+\frac {121}{1-2 x}+110 \log (1-2 x)-25\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + 5*x)^2/(1 - 2*x)^2,x]

[Out]

(-25 + 121/(1 - 2*x) + 50*x + 110*Log[1 - 2*x])/8

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(3+5 x)^2}{(1-2 x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(3 + 5*x)^2/(1 - 2*x)^2,x]

[Out]

IntegrateAlgebraic[(3 + 5*x)^2/(1 - 2*x)^2, x]

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fricas [A]  time = 1.29, size = 32, normalized size = 1.19 \begin {gather*} \frac {100 \, x^{2} + 110 \, {\left (2 \, x - 1\right )} \log \left (2 \, x - 1\right ) - 50 \, x - 121}{8 \, {\left (2 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^2/(1-2*x)^2,x, algorithm="fricas")

[Out]

1/8*(100*x^2 + 110*(2*x - 1)*log(2*x - 1) - 50*x - 121)/(2*x - 1)

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giac [A]  time = 0.99, size = 32, normalized size = 1.19 \begin {gather*} \frac {25}{4} \, x - \frac {121}{8 \, {\left (2 \, x - 1\right )}} - \frac {55}{4} \, \log \left (\frac {{\left | 2 \, x - 1 \right |}}{2 \, {\left (2 \, x - 1\right )}^{2}}\right ) - \frac {25}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^2/(1-2*x)^2,x, algorithm="giac")

[Out]

25/4*x - 121/8/(2*x - 1) - 55/4*log(1/2*abs(2*x - 1)/(2*x - 1)^2) - 25/8

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maple [A]  time = 0.01, size = 22, normalized size = 0.81 \begin {gather*} \frac {25 x}{4}+\frac {55 \ln \left (2 x -1\right )}{4}-\frac {121}{8 \left (2 x -1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x+3)^2/(1-2*x)^2,x)

[Out]

25/4*x-121/8/(2*x-1)+55/4*ln(2*x-1)

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maxima [A]  time = 0.60, size = 21, normalized size = 0.78 \begin {gather*} \frac {25}{4} \, x - \frac {121}{8 \, {\left (2 \, x - 1\right )}} + \frac {55}{4} \, \log \left (2 \, x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^2/(1-2*x)^2,x, algorithm="maxima")

[Out]

25/4*x - 121/8/(2*x - 1) + 55/4*log(2*x - 1)

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mupad [B]  time = 1.13, size = 19, normalized size = 0.70 \begin {gather*} \frac {25\,x}{4}+\frac {55\,\ln \left (x-\frac {1}{2}\right )}{4}-\frac {121}{16\,\left (x-\frac {1}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x + 3)^2/(2*x - 1)^2,x)

[Out]

(25*x)/4 + (55*log(x - 1/2))/4 - 121/(16*(x - 1/2))

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sympy [A]  time = 0.10, size = 20, normalized size = 0.74 \begin {gather*} \frac {25 x}{4} + \frac {55 \log {\left (2 x - 1 \right )}}{4} - \frac {121}{16 x - 8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)**2/(1-2*x)**2,x)

[Out]

25*x/4 + 55*log(2*x - 1)/4 - 121/(16*x - 8)

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