3.12.72 \(\int \frac {(1-2 x) (3+5 x)^2}{(2+3 x)^6} \, dx\) [1172]

Optimal. Leaf size=45 \[ -\frac {7}{405 (2+3 x)^5}+\frac {2}{9 (2+3 x)^4}-\frac {65}{81 (2+3 x)^3}+\frac {25}{81 (2+3 x)^2} \]

[Out]

-7/405/(2+3*x)^5+2/9/(2+3*x)^4-65/81/(2+3*x)^3+25/81/(2+3*x)^2

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Rubi [A]
time = 0.01, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {78} \begin {gather*} \frac {25}{81 (3 x+2)^2}-\frac {65}{81 (3 x+2)^3}+\frac {2}{9 (3 x+2)^4}-\frac {7}{405 (3 x+2)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-7/(405*(2 + 3*x)^5) + 2/(9*(2 + 3*x)^4) - 65/(81*(2 + 3*x)^3) + 25/(81*(2 + 3*x)^2)

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin {align*} \int \frac {(1-2 x) (3+5 x)^2}{(2+3 x)^6} \, dx &=\int \left (\frac {7}{27 (2+3 x)^6}-\frac {8}{3 (2+3 x)^5}+\frac {65}{9 (2+3 x)^4}-\frac {50}{27 (2+3 x)^3}\right ) \, dx\\ &=-\frac {7}{405 (2+3 x)^5}+\frac {2}{9 (2+3 x)^4}-\frac {65}{81 (2+3 x)^3}+\frac {25}{81 (2+3 x)^2}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 26, normalized size = 0.58 \begin {gather*} \frac {-127+870 x+3825 x^2+3375 x^3}{405 (2+3 x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-127 + 870*x + 3825*x^2 + 3375*x^3)/(405*(2 + 3*x)^5)

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Maple [A]
time = 0.10, size = 38, normalized size = 0.84

method	result	size
gosper	\(\frac {3375 x^{3}+3825 x^{2}+870 x -127}{405 \left (2+3 x \right )^{5}}\)	\(25\)
risch	\(\frac {\frac {25}{3} x^{3}+\frac {85}{9} x^{2}+\frac {58}{27} x -\frac {127}{405}}{\left (2+3 x \right )^{5}}\)	\(25\)
norman	\(\frac {\frac {227}{12} x^{3}+\frac {9}{2} x +\frac {33}{2} x^{2}+\frac {381}{160} x^{5}+\frac {127}{16} x^{4}}{\left (2+3 x \right )^{5}}\)	\(33\)
default	\(-\frac {7}{405 \left (2+3 x \right )^{5}}+\frac {2}{9 \left (2+3 x \right )^{4}}-\frac {65}{81 \left (2+3 x \right )^{3}}+\frac {25}{81 \left (2+3 x \right )^{2}}\)	\(38\)
meijerg	\(\frac {9 x \left (\frac {81}{16} x^{4}+\frac {135}{8} x^{3}+\frac {45}{2} x^{2}+15 x +5\right )}{320 \left (1+\frac {3 x}{2}\right )^{5}}+\frac {3 x^{2} \left (\frac {27}{8} x^{3}+\frac {45}{4} x^{2}+15 x +10\right )}{320 \left (1+\frac {3 x}{2}\right )^{5}}-\frac {7 x^{3} \left (\frac {9}{4} x^{2}+\frac {15}{2} x +10\right )}{384 \left (1+\frac {3 x}{2}\right )^{5}}-\frac {5 x^{4} \left (\frac {3 x}{2}+5\right )}{128 \left (1+\frac {3 x}{2}\right )^{5}}\)	\(98\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)*(3+5*x)^2/(2+3*x)^6,x,method=_RETURNVERBOSE)

[Out]

-7/405/(2+3*x)^5+2/9/(2+3*x)^4-65/81/(2+3*x)^3+25/81/(2+3*x)^2

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Maxima [A]
time = 0.30, size = 44, normalized size = 0.98 \begin {gather*} \frac {3375 \, x^{3} + 3825 \, x^{2} + 870 \, x - 127}{405 \, {\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)*(3+5*x)^2/(2+3*x)^6,x, algorithm="maxima")

[Out]

1/405*(3375*x^3 + 3825*x^2 + 870*x - 127)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)

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Fricas [A]
time = 0.59, size = 44, normalized size = 0.98 \begin {gather*} \frac {3375 \, x^{3} + 3825 \, x^{2} + 870 \, x - 127}{405 \, {\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)*(3+5*x)^2/(2+3*x)^6,x, algorithm="fricas")

[Out]

1/405*(3375*x^3 + 3825*x^2 + 870*x - 127)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)

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Sympy [A]
time = 0.07, size = 41, normalized size = 0.91 \begin {gather*} - \frac {- 3375 x^{3} - 3825 x^{2} - 870 x + 127}{98415 x^{5} + 328050 x^{4} + 437400 x^{3} + 291600 x^{2} + 97200 x + 12960} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(3+5*x)**2/(2+3*x)**6,x)

[Out]

-(-3375*x**3 - 3825*x**2 - 870*x + 127)/(98415*x**5 + 328050*x**4 + 437400*x**3 + 291600*x**2 + 97200*x + 1296
0)

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Giac [A]
time = 1.24, size = 24, normalized size = 0.53 \begin {gather*} \frac {3375 \, x^{3} + 3825 \, x^{2} + 870 \, x - 127}{405 \, {\left (3 \, x + 2\right )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(3+5*x)^2/(2+3*x)^6,x, algorithm="giac")

[Out]

1/405*(3375*x^3 + 3825*x^2 + 870*x - 127)/(3*x + 2)^5

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Mupad [B]
time = 1.10, size = 37, normalized size = 0.82 \begin {gather*} \frac {25}{81\,{\left (3\,x+2\right )}^2}-\frac {65}{81\,{\left (3\,x+2\right )}^3}+\frac {2}{9\,{\left (3\,x+2\right )}^4}-\frac {7}{405\,{\left (3\,x+2\right )}^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((2*x - 1)*(5*x + 3)^2)/(3*x + 2)^6,x)

[Out]

25/(81*(3*x + 2)^2) - 65/(81*(3*x + 2)^3) + 2/(9*(3*x + 2)^4) - 7/(405*(3*x + 2)^5)

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