3.1228 \(\int \frac {(1-2 x) (2+3 x)}{(3+5 x)^3} \, dx\)

Optimal. Leaf size=33 \[ -\frac {31}{125 (5 x+3)}-\frac {11}{250 (5 x+3)^2}-\frac {6}{125} \log (5 x+3) \]

[Out]

-11/250/(3+5*x)^2-31/125/(3+5*x)-6/125*ln(3+5*x)

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Rubi [A]  time = 0.01, antiderivative size = 33, 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} \[ -\frac {31}{125 (5 x+3)}-\frac {11}{250 (5 x+3)^2}-\frac {6}{125} \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-11/(250*(3 + 5*x)^2) - 31/(125*(3 + 5*x)) - (6*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)^3} \, dx &=\int \left (\frac {11}{25 (3+5 x)^3}+\frac {31}{25 (3+5 x)^2}-\frac {6}{25 (3+5 x)}\right ) \, dx\\ &=-\frac {11}{250 (3+5 x)^2}-\frac {31}{125 (3+5 x)}-\frac {6}{125} \log (3+5 x)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 33, normalized size = 1.00 \[ -\frac {31}{125 (5 x+3)}-\frac {11}{250 (5 x+3)^2}-\frac {6}{125} \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-11/(250*(3 + 5*x)^2) - 31/(125*(3 + 5*x)) - (6*Log[3 + 5*x])/125

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fricas [A]  time = 1.12, size = 37, normalized size = 1.12 \[ -\frac {12 \, {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) + 310 \, x + 197}{250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/250*(12*(25*x^2 + 30*x + 9)*log(5*x + 3) + 310*x + 197)/(25*x^2 + 30*x + 9)

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giac [A]  time = 1.24, size = 24, normalized size = 0.73 \[ -\frac {310 \, x + 197}{250 \, {\left (5 \, x + 3\right )}^{2}} - \frac {6}{125} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/250*(310*x + 197)/(5*x + 3)^2 - 6/125*log(abs(5*x + 3))

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maple [A]  time = 0.01, size = 28, normalized size = 0.85 \[ -\frac {6 \ln \left (5 x +3\right )}{125}-\frac {11}{250 \left (5 x +3\right )^{2}}-\frac {31}{125 \left (5 x +3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-11/250/(5*x+3)^2-31/125/(5*x+3)-6/125*ln(5*x+3)

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maxima [A]  time = 0.66, size = 28, normalized size = 0.85 \[ -\frac {310 \, x + 197}{250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac {6}{125} \, \log \left (5 \, x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/250*(310*x + 197)/(25*x^2 + 30*x + 9) - 6/125*log(5*x + 3)

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mupad [B]  time = 1.11, size = 24, normalized size = 0.73 \[ -\frac {6\,\ln \left (x+\frac {3}{5}\right )}{125}-\frac {\frac {31\,x}{625}+\frac {197}{6250}}{x^2+\frac {6\,x}{5}+\frac {9}{25}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- (6*log(x + 3/5))/125 - ((31*x)/625 + 197/6250)/((6*x)/5 + x^2 + 9/25)

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sympy [A]  time = 0.12, size = 26, normalized size = 0.79 \[ - \frac {310 x + 197}{6250 x^{2} + 7500 x + 2250} - \frac {6 \log {\left (5 x + 3 \right )}}{125} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(310*x + 197)/(6250*x**2 + 7500*x + 2250) - 6*log(5*x + 3)/125

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