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

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

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Rubi [A]  time = 0.02, antiderivative size = 56, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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} \end {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 31, normalized size = 0.55 \begin {gather*} -\frac {243000 x^4+248400 x^3+52515 x^2+8172 x+8198}{14580 (3 x+2)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(1-2 x)^2 (3+5 x)^2}{(2+3 x)^7} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.64, size = 54, normalized size = 0.96 \begin {gather*} -\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 {gather*}

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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giac [A]  time = 0.99, size = 29, normalized size = 0.52 \begin {gather*} -\frac {243000 \, x^{4} + 248400 \, x^{3} + 52515 \, x^{2} + 8172 \, x + 8198}{14580 \, {\left (3 \, x + 2\right )}^{6}} \end {gather*}

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

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maple [A]  time = 0.01, size = 47, normalized size = 0.84 \begin {gather*} -\frac {49}{1458 \left (3 x +2\right )^{6}}+\frac {518}{1215 \left (3 x +2\right )^{5}}-\frac {503}{324 \left (3 x +2\right )^{4}}+\frac {740}{729 \left (3 x +2\right )^{3}}-\frac {50}{243 \left (3 x +2\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^2*(5*x+3)^2/(3*x+2)^7,x)

[Out]

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

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maxima [A]  time = 0.52, size = 54, normalized size = 0.96 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 0.03, size = 46, normalized size = 0.82 \begin {gather*} \frac {740}{729\,{\left (3\,x+2\right )}^3}-\frac {50}{243\,{\left (3\,x+2\right )}^2}-\frac {503}{324\,{\left (3\,x+2\right )}^4}+\frac {518}{1215\,{\left (3\,x+2\right )}^5}-\frac {49}{1458\,{\left (3\,x+2\right )}^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x - 1)^2*(5*x + 3)^2)/(3*x + 2)^7,x)

[Out]

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

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sympy [A]  time = 0.17, size = 51, normalized size = 0.91 \begin {gather*} \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 {gather*}

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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