3.1268 \(\int \frac{(1-2 x)^2 (3+5 x)^2}{(2+3 x)^7} \, dx\)

Optimal. Leaf size=56 \[ -\frac{50}{243 (3 x+2)^2}+\frac{740}{729 (3 x+2)^3}-\frac{503}{324 (3 x+2)^4}+\frac{518}{1215 (3 x+2)^5}-\frac{49}{1458 (3 x+2)^6} \]

[Out]

-49/(1458*(2 + 3*x)^6) + 518/(1215*(2 + 3*x)^5) - 503/(324*(2 + 3*x)^4) + 740/(729*(2 + 3*x)^3) - 50/(243*(2 +
3*x)^2)

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Rubi [A]  time = 0.0190698, antiderivative size = 56, 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} \[ -\frac{50}{243 (3 x+2)^2}+\frac{740}{729 (3 x+2)^3}-\frac{503}{324 (3 x+2)^4}+\frac{518}{1215 (3 x+2)^5}-\frac{49}{1458 (3 x+2)^6} \]

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^2*(3 + 5*x)^2)/(2 + 3*x)^7,x]

[Out]

-49/(1458*(2 + 3*x)^6) + 518/(1215*(2 + 3*x)^5) - 503/(324*(2 + 3*x)^4) + 740/(729*(2 + 3*x)^3) - 50/(243*(2 +
3*x)^2)

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)^7} \, dx &=\int \left (\frac{49}{81 (2+3 x)^7}-\frac{518}{81 (2+3 x)^6}+\frac{503}{27 (2+3 x)^5}-\frac{740}{81 (2+3 x)^4}+\frac{100}{81 (2+3 x)^3}\right ) \, dx\\ &=-\frac{49}{1458 (2+3 x)^6}+\frac{518}{1215 (2+3 x)^5}-\frac{503}{324 (2+3 x)^4}+\frac{740}{729 (2+3 x)^3}-\frac{50}{243 (2+3 x)^2}\\ \end{align*}

Mathematica [A]  time = 0.0104087, size = 31, normalized size = 0.55 \[ -\frac{243000 x^4+248400 x^3+52515 x^2+8172 x+8198}{14580 (3 x+2)^6} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^2*(3 + 5*x)^2)/(2 + 3*x)^7,x]

[Out]

-(8198 + 8172*x + 52515*x^2 + 248400*x^3 + 243000*x^4)/(14580*(2 + 3*x)^6)

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Maple [A]  time = 0.005, size = 47, normalized size = 0.8 \begin{align*} -{\frac{49}{1458\, \left ( 2+3\,x \right ) ^{6}}}+{\frac{518}{1215\, \left ( 2+3\,x \right ) ^{5}}}-{\frac{503}{324\, \left ( 2+3\,x \right ) ^{4}}}+{\frac{740}{729\, \left ( 2+3\,x \right ) ^{3}}}-{\frac{50}{243\, \left ( 2+3\,x \right ) ^{2}}} \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)^7,x)

[Out]

-49/1458/(2+3*x)^6+518/1215/(2+3*x)^5-503/324/(2+3*x)^4+740/729/(2+3*x)^3-50/243/(2+3*x)^2

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Maxima [A]  time = 2.26662, size = 73, normalized size = 1.3 \begin{align*} -\frac{243000 \, x^{4} + 248400 \, x^{3} + 52515 \, x^{2} + 8172 \, x + 8198}{14580 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\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)^7,x, algorithm="maxima")

[Out]

-1/14580*(243000*x^4 + 248400*x^3 + 52515*x^2 + 8172*x + 8198)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 216
0*x^2 + 576*x + 64)

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Fricas [A]  time = 1.4553, size = 177, normalized size = 3.16 \begin{align*} -\frac{243000 \, x^{4} + 248400 \, x^{3} + 52515 \, x^{2} + 8172 \, x + 8198}{14580 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\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)^7,x, algorithm="fricas")

[Out]

-1/14580*(243000*x^4 + 248400*x^3 + 52515*x^2 + 8172*x + 8198)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 216
0*x^2 + 576*x + 64)

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Sympy [A]  time = 0.158872, size = 51, normalized size = 0.91 \begin{align*} - \frac{243000 x^{4} + 248400 x^{3} + 52515 x^{2} + 8172 x + 8198}{10628820 x^{6} + 42515280 x^{5} + 70858800 x^{4} + 62985600 x^{3} + 31492800 x^{2} + 8398080 x + 933120} \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)**7,x)

[Out]

-(243000*x**4 + 248400*x**3 + 52515*x**2 + 8172*x + 8198)/(10628820*x**6 + 42515280*x**5 + 70858800*x**4 + 629
85600*x**3 + 31492800*x**2 + 8398080*x + 933120)

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Giac [A]  time = 4.53511, size = 39, normalized size = 0.7 \begin{align*} -\frac{243000 \, x^{4} + 248400 \, x^{3} + 52515 \, x^{2} + 8172 \, x + 8198}{14580 \,{\left (3 \, x + 2\right )}^{6}} \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)^7,x, algorithm="giac")

[Out]

-1/14580*(243000*x^4 + 248400*x^3 + 52515*x^2 + 8172*x + 8198)/(3*x + 2)^6