3.12.86 \(\int \frac {(1-2 x) (3+5 x)^3}{(2+3 x)^3} \, dx\) [1186]

Optimal. Leaf size=45 \[ \frac {175 x}{27}-\frac {125 x^2}{27}+\frac {7}{486 (2+3 x)^2}-\frac {107}{243 (2+3 x)}-\frac {185}{81} \log (2+3 x) \]

[Out]

175/27*x-125/27*x^2+7/486/(2+3*x)^2-107/243/(2+3*x)-185/81*ln(2+3*x)

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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 {125 x^2}{27}+\frac {175 x}{27}-\frac {107}{243 (3 x+2)}+\frac {7}{486 (3 x+2)^2}-\frac {185}{81} \log (3 x+2) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(175*x)/27 - (125*x^2)/27 + 7/(486*(2 + 3*x)^2) - 107/(243*(2 + 3*x)) - (185*Log[2 + 3*x])/81

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)^3} \, dx &=\int \left (\frac {175}{27}-\frac {250 x}{27}-\frac {7}{81 (2+3 x)^3}+\frac {107}{81 (2+3 x)^2}-\frac {185}{27 (2+3 x)}\right ) \, dx\\ &=\frac {175 x}{27}-\frac {125 x^2}{27}+\frac {7}{486 (2+3 x)^2}-\frac {107}{243 (2+3 x)}-\frac {185}{81} \log (2+3 x)\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 46, normalized size = 1.02 \begin {gather*} \frac {3993+16386 x+18900 x^2+450 x^3-6750 x^4-370 (2+3 x)^2 \log (2+3 x)}{162 (2+3 x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3993 + 16386*x + 18900*x^2 + 450*x^3 - 6750*x^4 - 370*(2 + 3*x)^2*Log[2 + 3*x])/(162*(2 + 3*x)^2)

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Maple [A]
time = 0.08, size = 36, normalized size = 0.80

method	result	size
risch	\(-\frac {125 x^{2}}{27}+\frac {175 x}{27}+\frac {-\frac {107 x}{81}-\frac {421}{486}}{\left (2+3 x \right )^{2}}-\frac {185 \ln \left (2+3 x \right )}{81}\)	\(32\)
default	\(\frac {175 x}{27}-\frac {125 x^{2}}{27}+\frac {7}{486 \left (2+3 x \right )^{2}}-\frac {107}{243 \left (2+3 x \right )}-\frac {185 \ln \left (2+3 x \right )}{81}\)	\(36\)
norman	\(\frac {\frac {1469}{54} x +\frac {1469}{24} x^{2}+\frac {25}{9} x^{3}-\frac {125}{3} x^{4}}{\left (2+3 x \right )^{2}}-\frac {185 \ln \left (2+3 x \right )}{81}\)	\(37\)
meijerg	\(\frac {27 x \left (\frac {3 x}{2}+2\right )}{16 \left (1+\frac {3 x}{2}\right )^{2}}+\frac {81 x^{2}}{16 \left (1+\frac {3 x}{2}\right )^{2}}+\frac {5 x \left (\frac {27 x}{2}+6\right )}{12 \left (1+\frac {3 x}{2}\right )^{2}}-\frac {185 \ln \left (1+\frac {3 x}{2}\right )}{81}-\frac {325 x \left (9 x^{2}+27 x +12\right )}{108 \left (1+\frac {3 x}{2}\right )^{2}}+\frac {50 x \left (-\frac {135}{8} x^{3}+45 x^{2}+135 x +60\right )}{81 \left (1+\frac {3 x}{2}\right )^{2}}\)	\(97\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

175/27*x-125/27*x^2+7/486/(2+3*x)^2-107/243/(2+3*x)-185/81*ln(2+3*x)

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Maxima [A]
time = 0.28, size = 36, normalized size = 0.80 \begin {gather*} -\frac {125}{27} \, x^{2} + \frac {175}{27} \, x - \frac {642 \, x + 421}{486 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {185}{81} \, \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)^3,x, algorithm="maxima")

[Out]

-125/27*x^2 + 175/27*x - 1/486*(642*x + 421)/(9*x^2 + 12*x + 4) - 185/81*log(3*x + 2)

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Fricas [A]
time = 0.69, size = 52, normalized size = 1.16 \begin {gather*} -\frac {20250 \, x^{4} - 1350 \, x^{3} - 28800 \, x^{2} + 1110 \, {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (3 \, x + 2\right ) - 11958 \, x + 421}{486 \, {\left (9 \, x^{2} + 12 \, x + 4\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)^3,x, algorithm="fricas")

[Out]

-1/486*(20250*x^4 - 1350*x^3 - 28800*x^2 + 1110*(9*x^2 + 12*x + 4)*log(3*x + 2) - 11958*x + 421)/(9*x^2 + 12*x
+ 4)

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Sympy [A]
time = 0.04, size = 36, normalized size = 0.80 \begin {gather*} - \frac {125 x^{2}}{27} + \frac {175 x}{27} - \frac {642 x + 421}{4374 x^{2} + 5832 x + 1944} - \frac {185 \log {\left (3 x + 2 \right )}}{81} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-125*x**2/27 + 175*x/27 - (642*x + 421)/(4374*x**2 + 5832*x + 1944) - 185*log(3*x + 2)/81

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Giac [A]
time = 1.27, size = 32, normalized size = 0.71 \begin {gather*} -\frac {125}{27} \, x^{2} + \frac {175}{27} \, x - \frac {642 \, x + 421}{486 \, {\left (3 \, x + 2\right )}^{2}} - \frac {185}{81} \, \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)^3,x, algorithm="giac")

[Out]

-125/27*x^2 + 175/27*x - 1/486*(642*x + 421)/(3*x + 2)^2 - 185/81*log(abs(3*x + 2))

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Mupad [B]
time = 0.03, size = 32, normalized size = 0.71 \begin {gather*} \frac {175\,x}{27}-\frac {185\,\ln \left (x+\frac {2}{3}\right )}{81}-\frac {\frac {107\,x}{729}+\frac {421}{4374}}{x^2+\frac {4\,x}{3}+\frac {4}{9}}-\frac {125\,x^2}{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)^3,x)

[Out]

(175*x)/27 - (185*log(x + 2/3))/81 - ((107*x)/729 + 421/4374)/((4*x)/3 + x^2 + 4/9) - (125*x^2)/27

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