3.1266 \(\int \frac{(1-2 x)^2 (3+5 x)^2}{(2+3 x)^5} \, dx\)

Optimal. Leaf size=55 \[ \frac{740}{243 (3 x+2)}-\frac{503}{162 (3 x+2)^2}+\frac{518}{729 (3 x+2)^3}-\frac{49}{972 (3 x+2)^4}+\frac{100}{243} \log (3 x+2) \]

[Out]

-49/(972*(2 + 3*x)^4) + 518/(729*(2 + 3*x)^3) - 503/(162*(2 + 3*x)^2) + 740/(243*(2 + 3*x)) + (100*Log[2 + 3*x
])/243

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Rubi [A]  time = 0.0204819, antiderivative size = 55, 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} \[ \frac{740}{243 (3 x+2)}-\frac{503}{162 (3 x+2)^2}+\frac{518}{729 (3 x+2)^3}-\frac{49}{972 (3 x+2)^4}+\frac{100}{243} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-49/(972*(2 + 3*x)^4) + 518/(729*(2 + 3*x)^3) - 503/(162*(2 + 3*x)^2) + 740/(243*(2 + 3*x)) + (100*Log[2 + 3*x
])/243

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{(1-2 x)^2 (3+5 x)^2}{(2+3 x)^5} \, dx &=\int \left (\frac{49}{81 (2+3 x)^5}-\frac{518}{81 (2+3 x)^4}+\frac{503}{27 (2+3 x)^3}-\frac{740}{81 (2+3 x)^2}+\frac{100}{81 (2+3 x)}\right ) \, dx\\ &=-\frac{49}{972 (2+3 x)^4}+\frac{518}{729 (2+3 x)^3}-\frac{503}{162 (2+3 x)^2}+\frac{740}{243 (2+3 x)}+\frac{100}{243} \log (2+3 x)\\ \end{align*}

Mathematica [A]  time = 0.0126728, size = 41, normalized size = 0.75 \[ \frac{239760 x^3+398034 x^2+217248 x+1200 (3 x+2)^4 \log (3 x+2)+38821}{2916 (3 x+2)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(38821 + 217248*x + 398034*x^2 + 239760*x^3 + 1200*(2 + 3*x)^4*Log[2 + 3*x])/(2916*(2 + 3*x)^4)

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Maple [A]  time = 0.006, size = 46, normalized size = 0.8 \begin{align*} -{\frac{49}{972\, \left ( 2+3\,x \right ) ^{4}}}+{\frac{518}{729\, \left ( 2+3\,x \right ) ^{3}}}-{\frac{503}{162\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{740}{486+729\,x}}+{\frac{100\,\ln \left ( 2+3\,x \right ) }{243}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-49/972/(2+3*x)^4+518/729/(2+3*x)^3-503/162/(2+3*x)^2+740/243/(2+3*x)+100/243*ln(2+3*x)

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Maxima [A]  time = 1.128, size = 65, normalized size = 1.18 \begin{align*} \frac{239760 \, x^{3} + 398034 \, x^{2} + 217248 \, x + 38821}{2916 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{100}{243} \, \log \left (3 \, x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2916*(239760*x^3 + 398034*x^2 + 217248*x + 38821)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 100/243*log(3*x
+ 2)

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Fricas [A]  time = 1.53777, size = 209, normalized size = 3.8 \begin{align*} \frac{239760 \, x^{3} + 398034 \, x^{2} + 1200 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (3 \, x + 2\right ) + 217248 \, x + 38821}{2916 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2916*(239760*x^3 + 398034*x^2 + 1200*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log(3*x + 2) + 217248*x + 3882
1)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)

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Sympy [A]  time = 0.14222, size = 44, normalized size = 0.8 \begin{align*} \frac{239760 x^{3} + 398034 x^{2} + 217248 x + 38821}{236196 x^{4} + 629856 x^{3} + 629856 x^{2} + 279936 x + 46656} + \frac{100 \log{\left (3 x + 2 \right )}}{243} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(239760*x**3 + 398034*x**2 + 217248*x + 38821)/(236196*x**4 + 629856*x**3 + 629856*x**2 + 279936*x + 46656) +
100*log(3*x + 2)/243

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Giac [A]  time = 1.752, size = 74, normalized size = 1.35 \begin{align*} \frac{740}{243 \,{\left (3 \, x + 2\right )}} - \frac{503}{162 \,{\left (3 \, x + 2\right )}^{2}} + \frac{518}{729 \,{\left (3 \, x + 2\right )}^{3}} - \frac{49}{972 \,{\left (3 \, x + 2\right )}^{4}} - \frac{100}{243} \, \log \left (\frac{{\left | 3 \, x + 2 \right |}}{3 \,{\left (3 \, x + 2\right )}^{2}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

740/243/(3*x + 2) - 503/162/(3*x + 2)^2 + 518/729/(3*x + 2)^3 - 49/972/(3*x + 2)^4 - 100/243*log(1/3*abs(3*x +
2)/(3*x + 2)^2)