3.1689 \(\int \frac{(2+3 x)^3}{(1-2 x)^3 (3+5 x)^2} \, dx\)

Optimal. Leaf size=54 \[ -\frac{1421}{5324 (1-2 x)}-\frac{1}{6655 (5 x+3)}+\frac{343}{968 (1-2 x)^2}-\frac{21 \log (1-2 x)}{14641}+\frac{21 \log (5 x+3)}{14641} \]

[Out]

343/(968*(1 - 2*x)^2) - 1421/(5324*(1 - 2*x)) - 1/(6655*(3 + 5*x)) - (21*Log[1 - 2*x])/14641 + (21*Log[3 + 5*x
])/14641

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Rubi [A]  time = 0.0228212, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {88} \[ -\frac{1421}{5324 (1-2 x)}-\frac{1}{6655 (5 x+3)}+\frac{343}{968 (1-2 x)^2}-\frac{21 \log (1-2 x)}{14641}+\frac{21 \log (5 x+3)}{14641} \]

Antiderivative was successfully verified.

[In]

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

[Out]

343/(968*(1 - 2*x)^2) - 1421/(5324*(1 - 2*x)) - 1/(6655*(3 + 5*x)) - (21*Log[1 - 2*x])/14641 + (21*Log[3 + 5*x
])/14641

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int \frac{(2+3 x)^3}{(1-2 x)^3 (3+5 x)^2} \, dx &=\int \left (-\frac{343}{242 (-1+2 x)^3}-\frac{1421}{2662 (-1+2 x)^2}-\frac{42}{14641 (-1+2 x)}+\frac{1}{1331 (3+5 x)^2}+\frac{105}{14641 (3+5 x)}\right ) \, dx\\ &=\frac{343}{968 (1-2 x)^2}-\frac{1421}{5324 (1-2 x)}-\frac{1}{6655 (3+5 x)}-\frac{21 \log (1-2 x)}{14641}+\frac{21 \log (3+5 x)}{14641}\\ \end{align*}

Mathematica [A]  time = 0.032758, size = 47, normalized size = 0.87 \[ \frac{\frac{11 \left (142068 x^2+108567 x+13957\right )}{(1-2 x)^2 (5 x+3)}-840 \log (1-2 x)+840 \log (10 x+6)}{585640} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((11*(13957 + 108567*x + 142068*x^2))/((1 - 2*x)^2*(3 + 5*x)) - 840*Log[1 - 2*x] + 840*Log[6 + 10*x])/585640

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Maple [A]  time = 0.01, size = 45, normalized size = 0.8 \begin{align*}{\frac{343}{968\, \left ( 2\,x-1 \right ) ^{2}}}+{\frac{1421}{10648\,x-5324}}-{\frac{21\,\ln \left ( 2\,x-1 \right ) }{14641}}-{\frac{1}{19965+33275\,x}}+{\frac{21\,\ln \left ( 3+5\,x \right ) }{14641}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

343/968/(2*x-1)^2+1421/5324/(2*x-1)-21/14641*ln(2*x-1)-1/6655/(3+5*x)+21/14641*ln(3+5*x)

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Maxima [A]  time = 2.35598, size = 62, normalized size = 1.15 \begin{align*} \frac{142068 \, x^{2} + 108567 \, x + 13957}{53240 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} + \frac{21}{14641} \, \log \left (5 \, x + 3\right ) - \frac{21}{14641} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/53240*(142068*x^2 + 108567*x + 13957)/(20*x^3 - 8*x^2 - 7*x + 3) + 21/14641*log(5*x + 3) - 21/14641*log(2*x
- 1)

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Fricas [A]  time = 1.5288, size = 221, normalized size = 4.09 \begin{align*} \frac{1562748 \, x^{2} + 840 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (5 \, x + 3\right ) - 840 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (2 \, x - 1\right ) + 1194237 \, x + 153527}{585640 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/585640*(1562748*x^2 + 840*(20*x^3 - 8*x^2 - 7*x + 3)*log(5*x + 3) - 840*(20*x^3 - 8*x^2 - 7*x + 3)*log(2*x -
1) + 1194237*x + 153527)/(20*x^3 - 8*x^2 - 7*x + 3)

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Sympy [A]  time = 0.16474, size = 44, normalized size = 0.81 \begin{align*} \frac{142068 x^{2} + 108567 x + 13957}{1064800 x^{3} - 425920 x^{2} - 372680 x + 159720} - \frac{21 \log{\left (x - \frac{1}{2} \right )}}{14641} + \frac{21 \log{\left (x + \frac{3}{5} \right )}}{14641} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(142068*x**2 + 108567*x + 13957)/(1064800*x**3 - 425920*x**2 - 372680*x + 159720) - 21*log(x - 1/2)/14641 + 21
*log(x + 3/5)/14641

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Giac [A]  time = 4.34742, size = 69, normalized size = 1.28 \begin{align*} -\frac{1}{6655 \,{\left (5 \, x + 3\right )}} + \frac{245 \,{\left (\frac{66}{5 \, x + 3} + 23\right )}}{29282 \,{\left (\frac{11}{5 \, x + 3} - 2\right )}^{2}} - \frac{21}{14641} \, \log \left ({\left | -\frac{11}{5 \, x + 3} + 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6655/(5*x + 3) + 245/29282*(66/(5*x + 3) + 23)/(11/(5*x + 3) - 2)^2 - 21/14641*log(abs(-11/(5*x + 3) + 2))