3.13.87 \(\int \frac {(1-2 x)^2 (3+5 x)^3}{(2+3 x)^7} \, dx\) [1287]

Optimal. Leaf size=67 \[ \frac {49}{4374 (2+3 x)^6}-\frac {763}{3645 (2+3 x)^5}+\frac {4099}{2916 (2+3 x)^4}-\frac {8285}{2187 (2+3 x)^3}+\frac {1900}{729 (2+3 x)^2}-\frac {500}{729 (2+3 x)} \]

[Out]

49/4374/(2+3*x)^6-763/3645/(2+3*x)^5+4099/2916/(2+3*x)^4-8285/2187/(2+3*x)^3+1900/729/(2+3*x)^2-500/729/(2+3*x
)

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Rubi [A]
time = 0.02, antiderivative size = 67, 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 = {90} \begin {gather*} -\frac {500}{729 (3 x+2)}+\frac {1900}{729 (3 x+2)^2}-\frac {8285}{2187 (3 x+2)^3}+\frac {4099}{2916 (3 x+2)^4}-\frac {763}{3645 (3 x+2)^5}+\frac {49}{4374 (3 x+2)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

49/(4374*(2 + 3*x)^6) - 763/(3645*(2 + 3*x)^5) + 4099/(2916*(2 + 3*x)^4) - 8285/(2187*(2 + 3*x)^3) + 1900/(729
*(2 + 3*x)^2) - 500/(729*(2 + 3*x))

Rule 90

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)^3}{(2+3 x)^7} \, dx &=\int \left (-\frac {49}{243 (2+3 x)^7}+\frac {763}{243 (2+3 x)^6}-\frac {4099}{243 (2+3 x)^5}+\frac {8285}{243 (2+3 x)^4}-\frac {3800}{243 (2+3 x)^3}+\frac {500}{243 (2+3 x)^2}\right ) \, dx\\ &=\frac {49}{4374 (2+3 x)^6}-\frac {763}{3645 (2+3 x)^5}+\frac {4099}{2916 (2+3 x)^4}-\frac {8285}{2187 (2+3 x)^3}+\frac {1900}{729 (2+3 x)^2}-\frac {500}{729 (2+3 x)}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 36, normalized size = 0.54 \begin {gather*} -\frac {233482+1510848 x+5370435 x^2+12249900 x^3+15066000 x^4+7290000 x^5}{43740 (2+3 x)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/43740*(233482 + 1510848*x + 5370435*x^2 + 12249900*x^3 + 15066000*x^4 + 7290000*x^5)/(2 + 3*x)^6

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Maple [A]
time = 0.10, size = 56, normalized size = 0.84

method	result	size
gosper	\(-\frac {7290000 x^{5}[phone] x^{4}[phone] x^{3}+5370435 x^{2}+1510848 x +233482}{43740 \left (2+3 x \right )^{6}}\)	\(35\)
risch	\(\frac {-\frac {500}{3} x^{5}-\frac {3100}{9} x^{4}-\frac {22685}{81} x^{3}-\frac {39781}{324} x^{2}-\frac {41968}{1215} x -\frac {116741}{21870}}{\left (2+3 x \right )^{6}}\)	\(35\)
norman	\(\frac {\frac {1949}{32} x^{4}+\frac {12247}{160} x^{5}+\frac {27}{2} x +\frac {321}{4} x^{3}+\frac {459}{8} x^{2}+\frac {116741}{1920} x^{6}}{\left (2+3 x \right )^{6}}\)	\(38\)
default	\(\frac {49}{4374 \left (2+3 x \right )^{6}}-\frac {763}{3645 \left (2+3 x \right )^{5}}+\frac {4099}{2916 \left (2+3 x \right )^{4}}-\frac {8285}{2187 \left (2+3 x \right )^{3}}+\frac {1900}{729 \left (2+3 x \right )^{2}}-\frac {500}{729 \left (2+3 x \right )}\)	\(56\)
meijerg	\(\frac {9 x \left (\frac {243}{32} x^{5}+\frac {243}{8} x^{4}+\frac {405}{8} x^{3}+45 x^{2}+\frac {45}{2} x +6\right )}{256 \left (1+\frac {3 x}{2}\right )^{6}}+\frac {9 x^{2} \left (\frac {81}{16} x^{4}+\frac {81}{4} x^{3}+\frac {135}{4} x^{2}+30 x +15\right )}{1280 \left (1+\frac {3 x}{2}\right )^{6}}-\frac {69 x^{3} \left (\frac {27}{8} x^{3}+\frac {27}{2} x^{2}+\frac {45}{2} x +20\right )}{2560 \left (1+\frac {3 x}{2}\right )^{6}}-\frac {47 x^{4} \left (\frac {9}{4} x^{2}+9 x +15\right )}{1536 \left (1+\frac {3 x}{2}\right )^{6}}+\frac {5 x^{5} \left (\frac {3 x}{2}+6\right )}{48 \left (1+\frac {3 x}{2}\right )^{6}}+\frac {125 x^{6}}{192 \left (1+\frac {3 x}{2}\right )^{6}}\)	\(147\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

49/4374/(2+3*x)^6-763/3645/(2+3*x)^5+4099/2916/(2+3*x)^4-8285/2187/(2+3*x)^3+1900/729/(2+3*x)^2-500/729/(2+3*x
)

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Maxima [A]
time = 0.34, size = 59, normalized size = 0.88 \begin {gather*} -\frac {7290000 \, x^{5} + 15066000 \, x^{4} + 12249900 \, x^{3} + 5370435 \, x^{2} + 1510848 \, x + 233482}{43740 \, {\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)^3/(2+3*x)^7,x, algorithm="maxima")

[Out]

-1/43740*(7290000*x^5 + 15066000*x^4 + 12249900*x^3 + 5370435*x^2 + 1510848*x + 233482)/(729*x^6 + 2916*x^5 +
4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)

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Fricas [A]
time = 0.41, size = 59, normalized size = 0.88 \begin {gather*} -\frac {7290000 \, x^{5} + 15066000 \, x^{4} + 12249900 \, x^{3} + 5370435 \, x^{2} + 1510848 \, x + 233482}{43740 \, {\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)^3/(2+3*x)^7,x, algorithm="fricas")

[Out]

-1/43740*(7290000*x^5 + 15066000*x^4 + 12249900*x^3 + 5370435*x^2 + 1510848*x + 233482)/(729*x^6 + 2916*x^5 +
4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)

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Sympy [A]
time = 0.07, size = 56, normalized size = 0.84 \begin {gather*} \frac {- 7290000 x^{5} - 15066000 x^{4} - 12249900 x^{3} - 5370435 x^{2} - 1510848 x - 233482}{31886460 x^{6} + 127545840 x^{5} + 212576400 x^{4} + 188956800 x^{3} + 94478400 x^{2} + 25194240 x + 2799360} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-7290000*x**5 - 15066000*x**4 - 12249900*x**3 - 5370435*x**2 - 1510848*x - 233482)/(31886460*x**6 + 127545840
*x**5 + 212576400*x**4 + 188956800*x**3 + 94478400*x**2 + 25194240*x + 2799360)

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Giac [A]
time = 1.10, size = 34, normalized size = 0.51 \begin {gather*} -\frac {7290000 \, x^{5} + 15066000 \, x^{4} + 12249900 \, x^{3} + 5370435 \, x^{2} + 1510848 \, x + 233482}{43740 \, {\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)^3/(2+3*x)^7,x, algorithm="giac")

[Out]

-1/43740*(7290000*x^5 + 15066000*x^4 + 12249900*x^3 + 5370435*x^2 + 1510848*x + 233482)/(3*x + 2)^6

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Mupad [B]
time = 1.10, size = 55, normalized size = 0.82 \begin {gather*} \frac {1900}{729\,{\left (3\,x+2\right )}^2}-\frac {500}{729\,\left (3\,x+2\right )}-\frac {8285}{2187\,{\left (3\,x+2\right )}^3}+\frac {4099}{2916\,{\left (3\,x+2\right )}^4}-\frac {763}{3645\,{\left (3\,x+2\right )}^5}+\frac {49}{4374\,{\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)^3)/(3*x + 2)^7,x)

[Out]

1900/(729*(3*x + 2)^2) - 500/(729*(3*x + 2)) - 8285/(2187*(3*x + 2)^3) + 4099/(2916*(3*x + 2)^4) - 763/(3645*(
3*x + 2)^5) + 49/(4374*(3*x + 2)^6)

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