3.14.68 \(\int \frac {(1-2 x)^3 (3+5 x)^2}{(2+3 x)^5} \, dx\) [1368]

Optimal. Leaf size=60 \[ -\frac {200 x}{243}-\frac {343}{2916 (2+3 x)^4}+\frac {3724}{2187 (2+3 x)^3}-\frac {11599}{1458 (2+3 x)^2}+\frac {8198}{729 (2+3 x)}+\frac {2180}{729} \log (2+3 x) \]

[Out]

-200/243*x-343/2916/(2+3*x)^4+3724/2187/(2+3*x)^3-11599/1458/(2+3*x)^2+8198/729/(2+3*x)+2180/729*ln(2+3*x)

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Rubi [A]
time = 0.02, antiderivative size = 60, 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 {200 x}{243}+\frac {8198}{729 (3 x+2)}-\frac {11599}{1458 (3 x+2)^2}+\frac {3724}{2187 (3 x+2)^3}-\frac {343}{2916 (3 x+2)^4}+\frac {2180}{729} \log (3 x+2) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-200*x)/243 - 343/(2916*(2 + 3*x)^4) + 3724/(2187*(2 + 3*x)^3) - 11599/(1458*(2 + 3*x)^2) + 8198/(729*(2 + 3*
x)) + (2180*Log[2 + 3*x])/729

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)^3 (3+5 x)^2}{(2+3 x)^5} \, dx &=\int \left (-\frac {200}{243}+\frac {343}{243 (2+3 x)^5}-\frac {3724}{243 (2+3 x)^4}+\frac {11599}{243 (2+3 x)^3}-\frac {8198}{243 (2+3 x)^2}+\frac {2180}{243 (2+3 x)}\right ) \, dx\\ &=-\frac {200 x}{243}-\frac {343}{2916 (2+3 x)^4}+\frac {3724}{2187 (2+3 x)^3}-\frac {11599}{1458 (2+3 x)^2}+\frac {8198}{729 (2+3 x)}+\frac {2180}{729} \log (2+3 x)\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 51, normalized size = 0.85 \begin {gather*} \frac {460595+2175096 x+2957958 x^2+64152 x^3-1944000 x^4-583200 x^5+26160 (2+3 x)^4 \log (20+30 x)}{8748 (2+3 x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(460595 + 2175096*x + 2957958*x^2 + 64152*x^3 - 1944000*x^4 - 583200*x^5 + 26160*(2 + 3*x)^4*Log[20 + 30*x])/(
8748*(2 + 3*x)^4)

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Maple [A]
time = 0.09, size = 49, normalized size = 0.82

method	result	size
risch	\(-\frac {200 x}{243}+\frac {\frac {8198}{27} x^{3}+\frac {86777}{162} x^{2}+\frac {229258}{729} x +\frac {537395}{8748}}{\left (2+3 x \right )^{4}}+\frac {2180 \ln \left (2+3 x \right )}{729}\)	\(37\)
norman	\(\frac {-\frac {455843}{648} x^{3}-\frac {241487}{648} x^{2}-\frac {32693}{486} x -\frac {844595}{1728} x^{4}-\frac {200}{3} x^{5}}{\left (2+3 x \right )^{4}}+\frac {2180 \ln \left (2+3 x \right )}{729}\)	\(42\)
default	\(-\frac {200 x}{243}-\frac {343}{2916 \left (2+3 x \right )^{4}}+\frac {3724}{2187 \left (2+3 x \right )^{3}}-\frac {11599}{1458 \left (2+3 x \right )^{2}}+\frac {8198}{729 \left (2+3 x \right )}+\frac {2180 \ln \left (2+3 x \right )}{729}\)	\(49\)
meijerg	\(\frac {9 x \left (\frac {27}{8} x^{3}+9 x^{2}+9 x +4\right )}{128 \left (1+\frac {3 x}{2}\right )^{4}}-\frac {x^{2} \left (\frac {9}{4} x^{2}+6 x +6\right )}{16 \left (1+\frac {3 x}{2}\right )^{4}}-\frac {47 x^{3} \left (\frac {3 x}{2}+4\right )}{384 \left (1+\frac {3 x}{2}\right )^{4}}+\frac {69 x^{4}}{64 \left (1+\frac {3 x}{2}\right )^{4}}-\frac {x \left (\frac {3375}{8} x^{3}+585 x^{2}+315 x +60\right )}{162 \left (1+\frac {3 x}{2}\right )^{4}}+\frac {2180 \ln \left (1+\frac {3 x}{2}\right )}{729}-\frac {50 x \left (\frac {243}{4} x^{4}+\frac {3375}{8} x^{3}+585 x^{2}+315 x +60\right )}{729 \left (1+\frac {3 x}{2}\right )^{4}}\)	\(141\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-200/243*x-343/2916/(2+3*x)^4+3724/2187/(2+3*x)^3-11599/1458/(2+3*x)^2+8198/729/(2+3*x)+2180/729*ln(2+3*x)

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Maxima [A]
time = 0.41, size = 51, normalized size = 0.85 \begin {gather*} -\frac {200}{243} \, x + \frac {2656152 \, x^{3} + 4685958 \, x^{2} + 2751096 \, x + 537395}{8748 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {2180}{729} \, \log \left (3 \, x + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-200/243*x + 1/8748*(2656152*x^3 + 4685958*x^2 + 2751096*x + 537395)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)
+ 2180/729*log(3*x + 2)

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Fricas [A]
time = 0.46, size = 77, normalized size = 1.28 \begin {gather*} -\frac {583200 \, x^{5} + 1555200 \, x^{4} - 1100952 \, x^{3} - 3994758 \, x^{2} - 26160 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (3 \, x + 2\right ) - 2635896 \, x - 537395}{8748 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8748*(583200*x^5 + 1555200*x^4 - 1100952*x^3 - 3994758*x^2 - 26160*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)
*log(3*x + 2) - 2635896*x - 537395)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)

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Sympy [A]
time = 0.07, size = 51, normalized size = 0.85 \begin {gather*} - \frac {200 x}{243} - \frac {- 2656152 x^{3} - 4685958 x^{2} - 2751096 x - 537395}{708588 x^{4} + 1889568 x^{3} + 1889568 x^{2} + 839808 x + 139968} + \frac {2180 \log {\left (3 x + 2 \right )}}{729} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-200*x/243 - (-2656152*x**3 - 4685958*x**2 - 2751096*x - 537395)/(708588*x**4 + 1889568*x**3 + 1889568*x**2 +
839808*x + 139968) + 2180*log(3*x + 2)/729

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Giac [A]
time = 1.69, size = 59, normalized size = 0.98 \begin {gather*} -\frac {200}{243} \, x + \frac {8198}{729 \, {\left (3 \, x + 2\right )}} - \frac {11599}{1458 \, {\left (3 \, x + 2\right )}^{2}} + \frac {3724}{2187 \, {\left (3 \, x + 2\right )}^{3}} - \frac {343}{2916 \, {\left (3 \, x + 2\right )}^{4}} - \frac {2180}{729} \, \log \left (\frac {{\left | 3 \, x + 2 \right |}}{3 \, {\left (3 \, x + 2\right )}^{2}}\right ) - \frac {400}{729} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-200/243*x + 8198/729/(3*x + 2) - 11599/1458/(3*x + 2)^2 + 3724/2187/(3*x + 2)^3 - 343/2916/(3*x + 2)^4 - 2180
/729*log(1/3*abs(3*x + 2)/(3*x + 2)^2) - 400/729

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Mupad [B]
time = 1.11, size = 46, normalized size = 0.77 \begin {gather*} \frac {2180\,\ln \left (x+\frac {2}{3}\right )}{729}-\frac {200\,x}{243}+\frac {\frac {8198\,x^3}{2187}+\frac {86777\,x^2}{13122}+\frac {229258\,x}{59049}+\frac {537395}{708588}}{x^4+\frac {8\,x^3}{3}+\frac {8\,x^2}{3}+\frac {32\,x}{27}+\frac {16}{81}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2180*log(x + 2/3))/729 - (200*x)/243 + ((229258*x)/59049 + (86777*x^2)/13122 + (8198*x^3)/2187 + 537395/70858
8)/((32*x)/27 + (8*x^2)/3 + (8*x^3)/3 + x^4 + 16/81)

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