3.1386 \(\int \frac{(1-2 x)^3 (3+5 x)^3}{(2+3 x)^7} \, dx\)

Optimal. Leaf size=77 \[ -\frac{3700}{729 (3 x+2)}+\frac{7195}{729 (3 x+2)^2}-\frac{66193}{6561 (3 x+2)^3}+\frac{10073}{2916 (3 x+2)^4}-\frac{1813}{3645 (3 x+2)^5}+\frac{343}{13122 (3 x+2)^6}-\frac{1000 \log (3 x+2)}{2187} \]

[Out]

343/(13122*(2 + 3*x)^6) - 1813/(3645*(2 + 3*x)^5) + 10073/(2916*(2 + 3*x)^4) - 66193/(6561*(2 + 3*x)^3) + 7195
/(729*(2 + 3*x)^2) - 3700/(729*(2 + 3*x)) - (1000*Log[2 + 3*x])/2187

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Rubi [A]  time = 0.0291057, antiderivative size = 77, normalized size of antiderivative = 1., 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 = {88} \[ -\frac{3700}{729 (3 x+2)}+\frac{7195}{729 (3 x+2)^2}-\frac{66193}{6561 (3 x+2)^3}+\frac{10073}{2916 (3 x+2)^4}-\frac{1813}{3645 (3 x+2)^5}+\frac{343}{13122 (3 x+2)^6}-\frac{1000 \log (3 x+2)}{2187} \]

Antiderivative was successfully verified.

[In]

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

[Out]

343/(13122*(2 + 3*x)^6) - 1813/(3645*(2 + 3*x)^5) + 10073/(2916*(2 + 3*x)^4) - 66193/(6561*(2 + 3*x)^3) + 7195
/(729*(2 + 3*x)^2) - 3700/(729*(2 + 3*x)) - (1000*Log[2 + 3*x])/2187

Rule 88

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)^3}{(2+3 x)^7} \, dx &=\int \left (-\frac{343}{729 (2+3 x)^7}+\frac{1813}{243 (2+3 x)^6}-\frac{10073}{243 (2+3 x)^5}+\frac{66193}{729 (2+3 x)^4}-\frac{14390}{243 (2+3 x)^3}+\frac{3700}{243 (2+3 x)^2}-\frac{1000}{729 (2+3 x)}\right ) \, dx\\ &=\frac{343}{13122 (2+3 x)^6}-\frac{1813}{3645 (2+3 x)^5}+\frac{10073}{2916 (2+3 x)^4}-\frac{66193}{6561 (2+3 x)^3}+\frac{7195}{729 (2+3 x)^2}-\frac{3700}{729 (2+3 x)}-\frac{1000 \log (2+3 x)}{2187}\\ \end{align*}

Mathematica [A]  time = 0.0149417, size = 51, normalized size = 0.66 \[ -\frac{53946000 x^5+144852300 x^4+158427540 x^3+89062425 x^2+25975248 x+20000 (3 x+2)^6 \log (3 x+2)+3165082}{43740 (3 x+2)^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(3165082 + 25975248*x + 89062425*x^2 + 158427540*x^3 + 144852300*x^4 + 53946000*x^5 + 20000*(2 + 3*x)^6*Log[2
+ 3*x])/(43740*(2 + 3*x)^6)

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Maple [A]  time = 0.006, size = 64, normalized size = 0.8 \begin{align*}{\frac{343}{13122\, \left ( 2+3\,x \right ) ^{6}}}-{\frac{1813}{3645\, \left ( 2+3\,x \right ) ^{5}}}+{\frac{10073}{2916\, \left ( 2+3\,x \right ) ^{4}}}-{\frac{66193}{6561\, \left ( 2+3\,x \right ) ^{3}}}+{\frac{7195}{729\, \left ( 2+3\,x \right ) ^{2}}}-{\frac{3700}{1458+2187\,x}}-{\frac{1000\,\ln \left ( 2+3\,x \right ) }{2187}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

343/13122/(2+3*x)^6-1813/3645/(2+3*x)^5+10073/2916/(2+3*x)^4-66193/6561/(2+3*x)^3+7195/729/(2+3*x)^2-3700/729/
(2+3*x)-1000/2187*ln(2+3*x)

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Maxima [A]  time = 1.09247, size = 92, normalized size = 1.19 \begin{align*} -\frac{53946000 \, x^{5} + 144852300 \, x^{4} + 158427540 \, x^{3} + 89062425 \, x^{2} + 25975248 \, x + 3165082}{43740 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} - \frac{1000}{2187} \, \log \left (3 \, x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^3*(3+5*x)^3/(2+3*x)^7,x, algorithm="maxima")

[Out]

-1/43740*(53946000*x^5 + 144852300*x^4 + 158427540*x^3 + 89062425*x^2 + 25975248*x + 3165082)/(729*x^6 + 2916*
x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64) - 1000/2187*log(3*x + 2)

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Fricas [A]  time = 1.26656, size = 338, normalized size = 4.39 \begin{align*} -\frac{53946000 \, x^{5} + 144852300 \, x^{4} + 158427540 \, x^{3} + 89062425 \, x^{2} + 20000 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (3 \, x + 2\right ) + 25975248 \, x + 3165082}{43740 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^3*(3+5*x)^3/(2+3*x)^7,x, algorithm="fricas")

[Out]

-1/43740*(53946000*x^5 + 144852300*x^4 + 158427540*x^3 + 89062425*x^2 + 20000*(729*x^6 + 2916*x^5 + 4860*x^4 +
4320*x^3 + 2160*x^2 + 576*x + 64)*log(3*x + 2) + 25975248*x + 3165082)/(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 = 0.173971, size = 66, normalized size = 0.86 \begin{align*} - \frac{53946000 x^{5} + 144852300 x^{4} + 158427540 x^{3} + 89062425 x^{2} + 25975248 x + 3165082}{31886460 x^{6} + 127545840 x^{5} + 212576400 x^{4} + 188956800 x^{3} + 94478400 x^{2} + 25194240 x + 2799360} - \frac{1000 \log{\left (3 x + 2 \right )}}{2187} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(53946000*x**5 + 144852300*x**4 + 158427540*x**3 + 89062425*x**2 + 25975248*x + 3165082)/(31886460*x**6 + 127
545840*x**5 + 212576400*x**4 + 188956800*x**3 + 94478400*x**2 + 25194240*x + 2799360) - 1000*log(3*x + 2)/2187

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Giac [A]  time = 2.78377, size = 59, normalized size = 0.77 \begin{align*} -\frac{53946000 \, x^{5} + 144852300 \, x^{4} + 158427540 \, x^{3} + 89062425 \, x^{2} + 25975248 \, x + 3165082}{43740 \,{\left (3 \, x + 2\right )}^{6}} - \frac{1000}{2187} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^3*(3+5*x)^3/(2+3*x)^7,x, algorithm="giac")

[Out]

-1/43740*(53946000*x^5 + 144852300*x^4 + 158427540*x^3 + 89062425*x^2 + 25975248*x + 3165082)/(3*x + 2)^6 - 10
00/2187*log(abs(3*x + 2))