3.12.28 \(\int \frac {(1-2 x)^2 (3+5 x)^2}{(2+3 x)^6} \, dx\)

Optimal. Leaf size=56 \[ -\frac {100}{243 (3 x+2)}+\frac {370}{243 (3 x+2)^2}-\frac {503}{243 (3 x+2)^3}+\frac {259}{486 (3 x+2)^4}-\frac {49}{1215 (3 x+2)^5} \]

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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 {100}{243 (3 x+2)}+\frac {370}{243 (3 x+2)^2}-\frac {503}{243 (3 x+2)^3}+\frac {259}{486 (3 x+2)^4}-\frac {49}{1215 (3 x+2)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^2*(3 + 5*x)^2)/(2 + 3*x)^6,x]

[Out]

-49/(1215*(2 + 3*x)^5) + 259/(486*(2 + 3*x)^4) - 503/(243*(2 + 3*x)^3) + 370/(243*(2 + 3*x)^2) - 100/(243*(2 +
3*x))

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)^6} \, dx &=\int \left (\frac {49}{81 (2+3 x)^6}-\frac {518}{81 (2+3 x)^5}+\frac {503}{27 (2+3 x)^4}-\frac {740}{81 (2+3 x)^3}+\frac {100}{81 (2+3 x)^2}\right ) \, dx\\ &=-\frac {49}{1215 (2+3 x)^5}+\frac {259}{486 (2+3 x)^4}-\frac {503}{243 (2+3 x)^3}+\frac {370}{243 (2+3 x)^2}-\frac {100}{243 (2+3 x)}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 31, normalized size = 0.55 \begin {gather*} -\frac {81000 x^4+116100 x^3+61470 x^2+19275 x+4028}{2430 (3 x+2)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^2*(3 + 5*x)^2)/(2 + 3*x)^6,x]

[Out]

-1/2430*(4028 + 19275*x + 61470*x^2 + 116100*x^3 + 81000*x^4)/(2 + 3*x)^5

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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)^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((1 - 2*x)^2*(3 + 5*x)^2)/(2 + 3*x)^6,x]

[Out]

IntegrateAlgebraic[((1 - 2*x)^2*(3 + 5*x)^2)/(2 + 3*x)^6, x]

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fricas [A]  time = 1.05, size = 49, normalized size = 0.88 \begin {gather*} -\frac {81000 \, x^{4} + 116100 \, x^{3} + 61470 \, x^{2} + 19275 \, x + 4028}{2430 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\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)^6,x, algorithm="fricas")

[Out]

-1/2430*(81000*x^4 + 116100*x^3 + 61470*x^2 + 19275*x + 4028)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x
+ 32)

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giac [A]  time = 0.92, size = 29, normalized size = 0.52 \begin {gather*} -\frac {81000 \, x^{4} + 116100 \, x^{3} + 61470 \, x^{2} + 19275 \, x + 4028}{2430 \, {\left (3 \, x + 2\right )}^{5}} \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)^6,x, algorithm="giac")

[Out]

-1/2430*(81000*x^4 + 116100*x^3 + 61470*x^2 + 19275*x + 4028)/(3*x + 2)^5

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maple [A]  time = 0.01, size = 47, normalized size = 0.84 \begin {gather*} -\frac {49}{1215 \left (3 x +2\right )^{5}}+\frac {259}{486 \left (3 x +2\right )^{4}}-\frac {503}{243 \left (3 x +2\right )^{3}}+\frac {370}{243 \left (3 x +2\right )^{2}}-\frac {100}{243 \left (3 x +2\right )} \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)^6,x)

[Out]

-49/1215/(3*x+2)^5+259/486/(3*x+2)^4-503/243/(3*x+2)^3+370/243/(3*x+2)^2-100/243/(3*x+2)

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maxima [A]  time = 0.65, size = 49, normalized size = 0.88 \begin {gather*} -\frac {81000 \, x^{4} + 116100 \, x^{3} + 61470 \, x^{2} + 19275 \, x + 4028}{2430 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\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)^6,x, algorithm="maxima")

[Out]

-1/2430*(81000*x^4 + 116100*x^3 + 61470*x^2 + 19275*x + 4028)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x
+ 32)

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mupad [B]  time = 1.12, size = 46, normalized size = 0.82 \begin {gather*} \frac {370}{243\,{\left (3\,x+2\right )}^2}-\frac {100}{243\,\left (3\,x+2\right )}-\frac {503}{243\,{\left (3\,x+2\right )}^3}+\frac {259}{486\,{\left (3\,x+2\right )}^4}-\frac {49}{1215\,{\left (3\,x+2\right )}^5} \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)^6,x)

[Out]

370/(243*(3*x + 2)^2) - 100/(243*(3*x + 2)) - 503/(243*(3*x + 2)^3) + 259/(486*(3*x + 2)^4) - 49/(1215*(3*x +
2)^5)

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sympy [A]  time = 0.16, size = 46, normalized size = 0.82 \begin {gather*} \frac {- 81000 x^{4} - 116100 x^{3} - 61470 x^{2} - 19275 x - 4028}{590490 x^{5} + 1968300 x^{4} + 2624400 x^{3} + 1749600 x^{2} + 583200 x + 77760} \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)**6,x)

[Out]

(-81000*x**4 - 116100*x**3 - 61470*x**2 - 19275*x - 4028)/(590490*x**5 + 1968300*x**4 + 2624400*x**3 + 1749600
*x**2 + 583200*x + 77760)

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