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

Optimal. Leaf size=33 \[ \frac {44}{125 (5 x+3)}-\frac {121}{250 (5 x+3)^2}+\frac {4}{125} \log (5 x+3) \]

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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 = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {43} \begin {gather*} \frac {44}{125 (5 x+3)}-\frac {121}{250 (5 x+3)^2}+\frac {4}{125} \log (5 x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^2/(3 + 5*x)^3,x]

[Out]

-121/(250*(3 + 5*x)^2) + 44/(125*(3 + 5*x)) + (4*Log[3 + 5*x])/125

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 {(1-2 x)^2}{(3+5 x)^3} \, dx &=\int \left (\frac {121}{25 (3+5 x)^3}-\frac {44}{25 (3+5 x)^2}+\frac {4}{25 (3+5 x)}\right ) \, dx\\ &=-\frac {121}{250 (3+5 x)^2}+\frac {44}{125 (3+5 x)}+\frac {4}{125} \log (3+5 x)\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 31, normalized size = 0.94 \begin {gather*} \frac {440 x+8 (5 x+3)^2 \log (10 x+6)+143}{250 (5 x+3)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)^2/(3 + 5*x)^3,x]

[Out]

(143 + 440*x + 8*(3 + 5*x)^2*Log[6 + 10*x])/(250*(3 + 5*x)^2)

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.20, size = 37, normalized size = 1.12 \begin {gather*} \frac {8 \, {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) + 440 \, x + 143}{250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/250*(8*(25*x^2 + 30*x + 9)*log(5*x + 3) + 440*x + 143)/(25*x^2 + 30*x + 9)

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giac [A]  time = 0.87, size = 24, normalized size = 0.73 \begin {gather*} \frac {11 \, {\left (40 \, x + 13\right )}}{250 \, {\left (5 \, x + 3\right )}^{2}} + \frac {4}{125} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

11/250*(40*x + 13)/(5*x + 3)^2 + 4/125*log(abs(5*x + 3))

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maple [A]  time = 0.00, size = 28, normalized size = 0.85 \begin {gather*} \frac {4 \ln \left (5 x +3\right )}{125}-\frac {121}{250 \left (5 x +3\right )^{2}}+\frac {44}{125 \left (5 x +3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-121/250/(5*x+3)^2+44/125/(5*x+3)+4/125*ln(5*x+3)

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maxima [A]  time = 0.46, size = 28, normalized size = 0.85 \begin {gather*} \frac {11 \, {\left (40 \, x + 13\right )}}{250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {4}{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+5*x)^3,x, algorithm="maxima")

[Out]

11/250*(40*x + 13)/(25*x^2 + 30*x + 9) + 4/125*log(5*x + 3)

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mupad [B]  time = 0.03, size = 23, normalized size = 0.70 \begin {gather*} \frac {4\,\ln \left (x+\frac {3}{5}\right )}{125}+\frac {\frac {44\,x}{625}+\frac {143}{6250}}{x^2+\frac {6\,x}{5}+\frac {9}{25}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x - 1)^2/(5*x + 3)^3,x)

[Out]

(4*log(x + 3/5))/125 + ((44*x)/625 + 143/6250)/((6*x)/5 + x^2 + 9/25)

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sympy [A]  time = 0.12, size = 24, normalized size = 0.73 \begin {gather*} \frac {440 x + 143}{6250 x^{2} + 7500 x + 2250} + \frac {4 \log {\left (5 x + 3 \right )}}{125} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(440*x + 143)/(6250*x**2 + 7500*x + 2250) + 4*log(5*x + 3)/125

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