3.12.87 \(\int \frac {(1-2 x) (3+5 x)^3}{(2+3 x)^4} \, dx\) [1187]

Optimal. Leaf size=49 \[ -\frac {250 x}{81}+\frac {7}{729 (2+3 x)^3}-\frac {107}{486 (2+3 x)^2}+\frac {185}{81 (2+3 x)}+\frac {1025}{243} \log (2+3 x) \]

[Out]

-250/81*x+7/729/(2+3*x)^3-107/486/(2+3*x)^2+185/81/(2+3*x)+1025/243*ln(2+3*x)

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Rubi [A]
time = 0.01, antiderivative size = 49, 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 {250 x}{81}+\frac {185}{81 (3 x+2)}-\frac {107}{486 (3 x+2)^2}+\frac {7}{729 (3 x+2)^3}+\frac {1025}{243} \log (3 x+2) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)*(3 + 5*x)^3)/(2 + 3*x)^4,x]

[Out]

(-250*x)/81 + 7/(729*(2 + 3*x)^3) - 107/(486*(2 + 3*x)^2) + 185/(81*(2 + 3*x)) + (1025*Log[2 + 3*x])/243

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)^3}{(2+3 x)^4} \, dx &=\int \left (-\frac {250}{81}-\frac {7}{81 (2+3 x)^4}+\frac {107}{81 (2+3 x)^3}-\frac {185}{27 (2+3 x)^2}+\frac {1025}{81 (2+3 x)}\right ) \, dx\\ &=-\frac {250 x}{81}+\frac {7}{729 (2+3 x)^3}-\frac {107}{486 (2+3 x)^2}+\frac {185}{81 (2+3 x)}+\frac {1025}{243} \log (2+3 x)\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 47, normalized size = 0.96 \begin {gather*} \frac {\frac {14}{(2+3 x)^3}-\frac {321}{(2+3 x)^2}+\frac {3330}{2+3 x}-1500 (2+3 x)+6150 \log (2+3 x)}{1458} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)*(3 + 5*x)^3)/(2 + 3*x)^4,x]

[Out]

(14/(2 + 3*x)^3 - 321/(2 + 3*x)^2 + 3330/(2 + 3*x) - 1500*(2 + 3*x) + 6150*Log[2 + 3*x])/1458

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Maple [A]
time = 0.09, size = 40, normalized size = 0.82

method	result	size
risch	\(-\frac {250 x}{81}+\frac {\frac {185}{9} x^{2}+\frac {4333}{162} x +\frac {6346}{729}}{\left (2+3 x \right )^{3}}+\frac {1025 \ln \left (2+3 x \right )}{243}\)	\(32\)
norman	\(\frac {-\frac {8063}{54} x^{2}-\frac {6013}{162} x -\frac {21173}{108} x^{3}-\frac {250}{3} x^{4}}{\left (2+3 x \right )^{3}}+\frac {1025 \ln \left (2+3 x \right )}{243}\)	\(37\)
default	\(-\frac {250 x}{81}+\frac {7}{729 \left (2+3 x \right )^{3}}-\frac {107}{486 \left (2+3 x \right )^{2}}+\frac {185}{81 \left (2+3 x \right )}+\frac {1025 \ln \left (2+3 x \right )}{243}\)	\(40\)
meijerg	\(\frac {9 x \left (\frac {9}{4} x^{2}+\frac {9}{2} x +3\right )}{16 \left (1+\frac {3 x}{2}\right )^{3}}+\frac {27 x^{2} \left (3+\frac {3 x}{2}\right )}{32 \left (1+\frac {3 x}{2}\right )^{3}}-\frac {15 x^{3}}{16 \left (1+\frac {3 x}{2}\right )^{3}}+\frac {325 x \left (\frac {99}{2} x^{2}+45 x +12\right )}{648 \left (1+\frac {3 x}{2}\right )^{3}}+\frac {1025 \ln \left (1+\frac {3 x}{2}\right )}{243}-\frac {50 x \left (\frac {405}{8} x^{3}+\frac {495}{2} x^{2}+225 x +60\right )}{243 \left (1+\frac {3 x}{2}\right )^{3}}\)	\(104\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)*(3+5*x)^3/(2+3*x)^4,x,method=_RETURNVERBOSE)

[Out]

-250/81*x+7/729/(2+3*x)^3-107/486/(2+3*x)^2+185/81/(2+3*x)+1025/243*ln(2+3*x)

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Maxima [A]
time = 0.28, size = 41, normalized size = 0.84 \begin {gather*} -\frac {250}{81} \, x + \frac {29970 \, x^{2} + 38997 \, x + 12692}{1458 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {1025}{243} \, \log \left (3 \, x + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(3+5*x)^3/(2+3*x)^4,x, algorithm="maxima")

[Out]

-250/81*x + 1/1458*(29970*x^2 + 38997*x + 12692)/(27*x^3 + 54*x^2 + 36*x + 8) + 1025/243*log(3*x + 2)

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Fricas [A]
time = 0.78, size = 62, normalized size = 1.27 \begin {gather*} -\frac {121500 \, x^{4} + 243000 \, x^{3} + 132030 \, x^{2} - 6150 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (3 \, x + 2\right ) - 2997 \, x - 12692}{1458 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(3+5*x)^3/(2+3*x)^4,x, algorithm="fricas")

[Out]

-1/1458*(121500*x^4 + 243000*x^3 + 132030*x^2 - 6150*(27*x^3 + 54*x^2 + 36*x + 8)*log(3*x + 2) - 2997*x - 1269
2)/(27*x^3 + 54*x^2 + 36*x + 8)

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Sympy [A]
time = 0.05, size = 41, normalized size = 0.84 \begin {gather*} - \frac {250 x}{81} - \frac {- 29970 x^{2} - 38997 x - 12692}{39366 x^{3} + 78732 x^{2} + 52488 x + 11664} + \frac {1025 \log {\left (3 x + 2 \right )}}{243} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(3+5*x)**3/(2+3*x)**4,x)

[Out]

-250*x/81 - (-29970*x**2 - 38997*x - 12692)/(39366*x**3 + 78732*x**2 + 52488*x + 11664) + 1025*log(3*x + 2)/24
3

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Giac [A]
time = 1.42, size = 32, normalized size = 0.65 \begin {gather*} -\frac {250}{81} \, x + \frac {29970 \, x^{2} + 38997 \, x + 12692}{1458 \, {\left (3 \, x + 2\right )}^{3}} + \frac {1025}{243} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(3+5*x)^3/(2+3*x)^4,x, algorithm="giac")

[Out]

-250/81*x + 1/1458*(29970*x^2 + 38997*x + 12692)/(3*x + 2)^3 + 1025/243*log(abs(3*x + 2))

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Mupad [B]
time = 0.04, size = 36, normalized size = 0.73 \begin {gather*} \frac {1025\,\ln \left (x+\frac {2}{3}\right )}{243}-\frac {250\,x}{81}+\frac {\frac {185\,x^2}{243}+\frac {4333\,x}{4374}+\frac {6346}{19683}}{x^3+2\,x^2+\frac {4\,x}{3}+\frac {8}{27}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((2*x - 1)*(5*x + 3)^3)/(3*x + 2)^4,x)

[Out]

(1025*log(x + 2/3))/243 - (250*x)/81 + ((4333*x)/4374 + (185*x^2)/243 + 6346/19683)/((4*x)/3 + 2*x^2 + x^3 + 8
/27)

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