3.119 \(\int \frac{7+8 x^3}{(1+x) (1+2 x)^3} \, dx\)

Optimal. Leaf size=23 \[ \frac{3}{2 x+1}-\frac{3}{(2 x+1)^2}+\log (x+1) \]

[Out]

-3/(1 + 2*x)^2 + 3/(1 + 2*x) + Log[1 + x]

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Rubi [A]  time = 0.0263759, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {1620} \[ \frac{3}{2 x+1}-\frac{3}{(2 x+1)^2}+\log (x+1) \]

Antiderivative was successfully verified.

[In]

Int[(7 + 8*x^3)/((1 + x)*(1 + 2*x)^3),x]

[Out]

-3/(1 + 2*x)^2 + 3/(1 + 2*x) + Log[1 + x]

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{7+8 x^3}{(1+x) (1+2 x)^3} \, dx &=\int \left (\frac{1}{1+x}+\frac{12}{(1+2 x)^3}-\frac{6}{(1+2 x)^2}\right ) \, dx\\ &=-\frac{3}{(1+2 x)^2}+\frac{3}{1+2 x}+\log (1+x)\\ \end{align*}

Mathematica [A]  time = 0.0092826, size = 24, normalized size = 1.04 \[ \frac{6 x+(2 x+1)^2 \log (x+1)}{(2 x+1)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(7 + 8*x^3)/((1 + x)*(1 + 2*x)^3),x]

[Out]

(6*x + (1 + 2*x)^2*Log[1 + x])/(1 + 2*x)^2

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Maple [A]  time = 0.007, size = 24, normalized size = 1. \begin{align*} -3\, \left ( 1+2\,x \right ) ^{-2}+3\, \left ( 1+2\,x \right ) ^{-1}+\ln \left ( 1+x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*x^3+7)/(1+x)/(1+2*x)^3,x)

[Out]

-3/(1+2*x)^2+3/(1+2*x)+ln(1+x)

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Maxima [A]  time = 0.938688, size = 27, normalized size = 1.17 \begin{align*} \frac{6 \, x}{4 \, x^{2} + 4 \, x + 1} + \log \left (x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x^3+7)/(1+x)/(1+2*x)^3,x, algorithm="maxima")

[Out]

6*x/(4*x^2 + 4*x + 1) + log(x + 1)

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Fricas [A]  time = 0.74517, size = 76, normalized size = 3.3 \begin{align*} \frac{{\left (4 \, x^{2} + 4 \, x + 1\right )} \log \left (x + 1\right ) + 6 \, x}{4 \, x^{2} + 4 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x^3+7)/(1+x)/(1+2*x)^3,x, algorithm="fricas")

[Out]

((4*x^2 + 4*x + 1)*log(x + 1) + 6*x)/(4*x^2 + 4*x + 1)

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Sympy [A]  time = 0.104271, size = 17, normalized size = 0.74 \begin{align*} \frac{6 x}{4 x^{2} + 4 x + 1} + \log{\left (x + 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x**3+7)/(1+x)/(1+2*x)**3,x)

[Out]

6*x/(4*x**2 + 4*x + 1) + log(x + 1)

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Giac [A]  time = 1.15593, size = 22, normalized size = 0.96 \begin{align*} \frac{6 \, x}{{\left (2 \, x + 1\right )}^{2}} + \log \left ({\left | x + 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x^3+7)/(1+x)/(1+2*x)^3,x, algorithm="giac")

[Out]

6*x/(2*x + 1)^2 + log(abs(x + 1))