3.160 \(\int \frac{-3 x+x^4}{(1+2 x)^5} \, dx\)

Optimal. Leaf size=55 \[ \frac{1}{8 (2 x+1)}-\frac{3}{32 (2 x+1)^2}+\frac{7}{24 (2 x+1)^3}-\frac{25}{128 (2 x+1)^4}+\frac{1}{32} \log (2 x+1) \]

[Out]

-25/(128*(1 + 2*x)^4) + 7/(24*(1 + 2*x)^3) - 3/(32*(1 + 2*x)^2) + 1/(8*(1 + 2*x)) + Log[1 + 2*x]/32

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Rubi [A]  time = 0.0392657, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {1593, 1620} \[ \frac{1}{8 (2 x+1)}-\frac{3}{32 (2 x+1)^2}+\frac{7}{24 (2 x+1)^3}-\frac{25}{128 (2 x+1)^4}+\frac{1}{32} \log (2 x+1) \]

Antiderivative was successfully verified.

[In]

Int[(-3*x + x^4)/(1 + 2*x)^5,x]

[Out]

-25/(128*(1 + 2*x)^4) + 7/(24*(1 + 2*x)^3) - 3/(32*(1 + 2*x)^2) + 1/(8*(1 + 2*x)) + Log[1 + 2*x]/32

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rubi steps

\begin{align*} \int \frac{-3 x+x^4}{(1+2 x)^5} \, dx &=\int \frac{x \left (-3+x^3\right )}{(1+2 x)^5} \, dx\\ &=\int \left (\frac{25}{16 (1+2 x)^5}-\frac{7}{4 (1+2 x)^4}+\frac{3}{8 (1+2 x)^3}-\frac{1}{4 (1+2 x)^2}+\frac{1}{16 (1+2 x)}\right ) \, dx\\ &=-\frac{25}{128 (1+2 x)^4}+\frac{7}{24 (1+2 x)^3}-\frac{3}{32 (1+2 x)^2}+\frac{1}{8 (1+2 x)}+\frac{1}{32} \log (1+2 x)\\ \end{align*}

Mathematica [A]  time = 0.0116806, size = 41, normalized size = 0.75 \[ \frac{384 x^3+432 x^2+368 x+12 (2 x+1)^4 \log (2 x+1)+49}{384 (2 x+1)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(-3*x + x^4)/(1 + 2*x)^5,x]

[Out]

(49 + 368*x + 432*x^2 + 384*x^3 + 12*(1 + 2*x)^4*Log[1 + 2*x])/(384*(1 + 2*x)^4)

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Maple [A]  time = 0.005, size = 46, normalized size = 0.8 \begin{align*} -{\frac{25}{128\, \left ( 1+2\,x \right ) ^{4}}}+{\frac{7}{24\, \left ( 1+2\,x \right ) ^{3}}}-{\frac{3}{32\, \left ( 1+2\,x \right ) ^{2}}}+{\frac{1}{8+16\,x}}+{\frac{\ln \left ( 1+2\,x \right ) }{32}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4-3*x)/(1+2*x)^5,x)

[Out]

-25/128/(1+2*x)^4+7/24/(1+2*x)^3-3/32/(1+2*x)^2+1/8/(1+2*x)+1/32*ln(1+2*x)

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Maxima [A]  time = 0.933696, size = 65, normalized size = 1.18 \begin{align*} \frac{384 \, x^{3} + 432 \, x^{2} + 368 \, x + 49}{384 \,{\left (16 \, x^{4} + 32 \, x^{3} + 24 \, x^{2} + 8 \, x + 1\right )}} + \frac{1}{32} \, \log \left (2 \, x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-3*x)/(1+2*x)^5,x, algorithm="maxima")

[Out]

1/384*(384*x^3 + 432*x^2 + 368*x + 49)/(16*x^4 + 32*x^3 + 24*x^2 + 8*x + 1) + 1/32*log(2*x + 1)

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Fricas [A]  time = 1.9424, size = 178, normalized size = 3.24 \begin{align*} \frac{384 \, x^{3} + 432 \, x^{2} + 12 \,{\left (16 \, x^{4} + 32 \, x^{3} + 24 \, x^{2} + 8 \, x + 1\right )} \log \left (2 \, x + 1\right ) + 368 \, x + 49}{384 \,{\left (16 \, x^{4} + 32 \, x^{3} + 24 \, x^{2} + 8 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-3*x)/(1+2*x)^5,x, algorithm="fricas")

[Out]

1/384*(384*x^3 + 432*x^2 + 12*(16*x^4 + 32*x^3 + 24*x^2 + 8*x + 1)*log(2*x + 1) + 368*x + 49)/(16*x^4 + 32*x^3
+ 24*x^2 + 8*x + 1)

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Sympy [A]  time = 0.11814, size = 42, normalized size = 0.76 \begin{align*} \frac{384 x^{3} + 432 x^{2} + 368 x + 49}{6144 x^{4} + 12288 x^{3} + 9216 x^{2} + 3072 x + 384} + \frac{\log{\left (2 x + 1 \right )}}{32} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4-3*x)/(1+2*x)**5,x)

[Out]

(384*x**3 + 432*x**2 + 368*x + 49)/(6144*x**4 + 12288*x**3 + 9216*x**2 + 3072*x + 384) + log(2*x + 1)/32

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Giac [A]  time = 1.06204, size = 74, normalized size = 1.35 \begin{align*} \frac{1}{8 \,{\left (2 \, x + 1\right )}} - \frac{3}{32 \,{\left (2 \, x + 1\right )}^{2}} + \frac{7}{24 \,{\left (2 \, x + 1\right )}^{3}} - \frac{25}{128 \,{\left (2 \, x + 1\right )}^{4}} - \frac{1}{32} \, \log \left (\frac{{\left | 2 \, x + 1 \right |}}{2 \,{\left (2 \, x + 1\right )}^{2}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-3*x)/(1+2*x)^5,x, algorithm="giac")

[Out]

1/8/(2*x + 1) - 3/32/(2*x + 1)^2 + 7/24/(2*x + 1)^3 - 25/128/(2*x + 1)^4 - 1/32*log(1/2*abs(2*x + 1)/(2*x + 1)
^2)