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

Optimal. Leaf size=27 \[ -\frac {6 x}{25}-\frac {11}{125 (5 x+3)}+\frac {31}{125} \log (5 x+3) \]

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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 = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {77} \begin {gather*} -\frac {6 x}{25}-\frac {11}{125 (5 x+3)}+\frac {31}{125} \log (5 x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*x)/25 - 11/(125*(3 + 5*x)) + (31*Log[3 + 5*x])/125

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin {align*} \int \frac {(1-2 x) (2+3 x)}{(3+5 x)^2} \, dx &=\int \left (-\frac {6}{25}+\frac {11}{25 (3+5 x)^2}+\frac {31}{25 (3+5 x)}\right ) \, dx\\ &=-\frac {6 x}{25}-\frac {11}{125 (3+5 x)}+\frac {31}{125} \log (3+5 x)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 26, normalized size = 0.96 \begin {gather*} \frac {1}{125} \left (-30 x-\frac {11}{5 x+3}+31 \log (5 x+3)-18\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-18 - 30*x - 11/(3 + 5*x) + 31*Log[3 + 5*x])/125

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.02, size = 32, normalized size = 1.19 \begin {gather*} -\frac {150 \, x^{2} - 31 \, {\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) + 90 \, x + 11}{125 \, {\left (5 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/125*(150*x^2 - 31*(5*x + 3)*log(5*x + 3) + 90*x + 11)/(5*x + 3)

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giac [A]  time = 1.15, size = 32, normalized size = 1.19 \begin {gather*} -\frac {6}{25} \, x - \frac {11}{125 \, {\left (5 \, x + 3\right )}} - \frac {31}{125} \, \log \left (\frac {{\left | 5 \, x + 3 \right |}}{5 \, {\left (5 \, x + 3\right )}^{2}}\right ) - \frac {18}{125} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6/25*x - 11/125/(5*x + 3) - 31/125*log(1/5*abs(5*x + 3)/(5*x + 3)^2) - 18/125

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maple [A]  time = 0.01, size = 22, normalized size = 0.81 \begin {gather*} -\frac {6 x}{25}+\frac {31 \ln \left (5 x +3\right )}{125}-\frac {11}{125 \left (5 x +3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6/25*x-11/125/(5*x+3)+31/125*ln(5*x+3)

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maxima [A]  time = 0.53, size = 21, normalized size = 0.78 \begin {gather*} -\frac {6}{25} \, x - \frac {11}{125 \, {\left (5 \, x + 3\right )}} + \frac {31}{125} \, \log \left (5 \, x + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6/25*x - 11/125/(5*x + 3) + 31/125*log(5*x + 3)

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mupad [B]  time = 1.08, size = 19, normalized size = 0.70 \begin {gather*} \frac {31\,\ln \left (x+\frac {3}{5}\right )}{125}-\frac {6\,x}{25}-\frac {11}{625\,\left (x+\frac {3}{5}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(31*log(x + 3/5))/125 - (6*x)/25 - 11/(625*(x + 3/5))

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sympy [A]  time = 0.10, size = 20, normalized size = 0.74 \begin {gather*} - \frac {6 x}{25} + \frac {31 \log {\left (5 x + 3 \right )}}{125} - \frac {11}{625 x + 375} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6*x/25 + 31*log(5*x + 3)/125 - 11/(625*x + 375)

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