3.1188 \(\int \frac{(1-2 x) (3+5 x)^3}{(2+3 x)^5} \, dx\)

Optimal. Leaf size=55 \[ -\frac{1025}{243 (3 x+2)}+\frac{185}{162 (3 x+2)^2}-\frac{107}{729 (3 x+2)^3}+\frac{7}{972 (3 x+2)^4}-\frac{250}{243} \log (3 x+2) \]

[Out]

7/(972*(2 + 3*x)^4) - 107/(729*(2 + 3*x)^3) + 185/(162*(2 + 3*x)^2) - 1025/(243*(2 + 3*x)) - (250*Log[2 + 3*x]
)/243

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Rubi [A]  time = 0.0199059, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {77} \[ -\frac{1025}{243 (3 x+2)}+\frac{185}{162 (3 x+2)^2}-\frac{107}{729 (3 x+2)^3}+\frac{7}{972 (3 x+2)^4}-\frac{250}{243} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

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

[Out]

7/(972*(2 + 3*x)^4) - 107/(729*(2 + 3*x)^3) + 185/(162*(2 + 3*x)^2) - 1025/(243*(2 + 3*x)) - (250*Log[2 + 3*x]
)/243

Rule 77

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)^5} \, dx &=\int \left (-\frac{7}{81 (2+3 x)^5}+\frac{107}{81 (2+3 x)^4}-\frac{185}{27 (2+3 x)^3}+\frac{1025}{81 (2+3 x)^2}-\frac{250}{81 (2+3 x)}\right ) \, dx\\ &=\frac{7}{972 (2+3 x)^4}-\frac{107}{729 (2+3 x)^3}+\frac{185}{162 (2+3 x)^2}-\frac{1025}{243 (2+3 x)}-\frac{250}{243} \log (2+3 x)\\ \end{align*}

Mathematica [A]  time = 0.0133431, size = 41, normalized size = 0.75 \[ -\frac{332100 x^3+634230 x^2+404124 x+3000 (3 x+2)^4 \log (3 x+2)+85915}{2916 (3 x+2)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(85915 + 404124*x + 634230*x^2 + 332100*x^3 + 3000*(2 + 3*x)^4*Log[2 + 3*x])/(2916*(2 + 3*x)^4)

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Maple [A]  time = 0.006, size = 46, normalized size = 0.8 \begin{align*}{\frac{7}{972\, \left ( 2+3\,x \right ) ^{4}}}-{\frac{107}{729\, \left ( 2+3\,x \right ) ^{3}}}+{\frac{185}{162\, \left ( 2+3\,x \right ) ^{2}}}-{\frac{1025}{486+729\,x}}-{\frac{250\,\ln \left ( 2+3\,x \right ) }{243}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

7/972/(2+3*x)^4-107/729/(2+3*x)^3+185/162/(2+3*x)^2-1025/243/(2+3*x)-250/243*ln(2+3*x)

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Maxima [A]  time = 1.10932, size = 65, normalized size = 1.18 \begin{align*} -\frac{332100 \, x^{3} + 634230 \, x^{2} + 404124 \, x + 85915}{2916 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} - \frac{250}{243} \, \log \left (3 \, x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2916*(332100*x^3 + 634230*x^2 + 404124*x + 85915)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) - 250/243*log(3*
x + 2)

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Fricas [A]  time = 1.64937, size = 211, normalized size = 3.84 \begin{align*} -\frac{332100 \, x^{3} + 634230 \, x^{2} + 3000 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (3 \, x + 2\right ) + 404124 \, x + 85915}{2916 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2916*(332100*x^3 + 634230*x^2 + 3000*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log(3*x + 2) + 404124*x + 859
15)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)

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Sympy [A]  time = 0.142802, size = 46, normalized size = 0.84 \begin{align*} - \frac{332100 x^{3} + 634230 x^{2} + 404124 x + 85915}{236196 x^{4} + 629856 x^{3} + 629856 x^{2} + 279936 x + 46656} - \frac{250 \log{\left (3 x + 2 \right )}}{243} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(332100*x**3 + 634230*x**2 + 404124*x + 85915)/(236196*x**4 + 629856*x**3 + 629856*x**2 + 279936*x + 46656) -
250*log(3*x + 2)/243

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Giac [A]  time = 2.28923, size = 74, normalized size = 1.35 \begin{align*} -\frac{1025}{243 \,{\left (3 \, x + 2\right )}} + \frac{185}{162 \,{\left (3 \, x + 2\right )}^{2}} - \frac{107}{729 \,{\left (3 \, x + 2\right )}^{3}} + \frac{7}{972 \,{\left (3 \, x + 2\right )}^{4}} + \frac{250}{243} \, \log \left (\frac{{\left | 3 \, x + 2 \right |}}{3 \,{\left (3 \, x + 2\right )}^{2}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1025/243/(3*x + 2) + 185/162/(3*x + 2)^2 - 107/729/(3*x + 2)^3 + 7/972/(3*x + 2)^4 + 250/243*log(1/3*abs(3*x
+ 2)/(3*x + 2)^2)