3.1401 \(\int \frac{(1-2 x)^3}{(2+3 x)^7 (3+5 x)} \, dx\)

Optimal. Leaf size=81 \[ \frac{33275}{3 x+2}+\frac{6655}{2 (3 x+2)^2}+\frac{1331}{3 (3 x+2)^3}+\frac{7189}{108 (3 x+2)^4}+\frac{1421}{135 (3 x+2)^5}+\frac{343}{162 (3 x+2)^6}-166375 \log (3 x+2)+166375 \log (5 x+3) \]

[Out]

343/(162*(2 + 3*x)^6) + 1421/(135*(2 + 3*x)^5) + 7189/(108*(2 + 3*x)^4) + 1331/(3*(2 + 3*x)^3) + 6655/(2*(2 +
3*x)^2) + 33275/(2 + 3*x) - 166375*Log[2 + 3*x] + 166375*Log[3 + 5*x]

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Rubi [A]  time = 0.0336496, antiderivative size = 81, 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} \[ \frac{33275}{3 x+2}+\frac{6655}{2 (3 x+2)^2}+\frac{1331}{3 (3 x+2)^3}+\frac{7189}{108 (3 x+2)^4}+\frac{1421}{135 (3 x+2)^5}+\frac{343}{162 (3 x+2)^6}-166375 \log (3 x+2)+166375 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

343/(162*(2 + 3*x)^6) + 1421/(135*(2 + 3*x)^5) + 7189/(108*(2 + 3*x)^4) + 1331/(3*(2 + 3*x)^3) + 6655/(2*(2 +
3*x)^2) + 33275/(2 + 3*x) - 166375*Log[2 + 3*x] + 166375*Log[3 + 5*x]

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}{(2+3 x)^7 (3+5 x)} \, dx &=\int \left (-\frac{343}{9 (2+3 x)^7}-\frac{1421}{9 (2+3 x)^6}-\frac{7189}{9 (2+3 x)^5}-\frac{3993}{(2+3 x)^4}-\frac{19965}{(2+3 x)^3}-\frac{99825}{(2+3 x)^2}-\frac{499125}{2+3 x}+\frac{831875}{3+5 x}\right ) \, dx\\ &=\frac{343}{162 (2+3 x)^6}+\frac{1421}{135 (2+3 x)^5}+\frac{7189}{108 (2+3 x)^4}+\frac{1331}{3 (2+3 x)^3}+\frac{6655}{2 (2+3 x)^2}+\frac{33275}{2+3 x}-166375 \log (2+3 x)+166375 \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.055875, size = 75, normalized size = 0.93 \[ \frac{53905500 (3 x+2)^5+5390550 (3 x+2)^4+718740 (3 x+2)^3+107835 (3 x+2)^2+17052 (3 x+2)+3430}{1620 (3 x+2)^6}-166375 \log (5 (3 x+2))+166375 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3430 + 17052*(2 + 3*x) + 107835*(2 + 3*x)^2 + 718740*(2 + 3*x)^3 + 5390550*(2 + 3*x)^4 + 53905500*(2 + 3*x)^5
)/(1620*(2 + 3*x)^6) - 166375*Log[5*(2 + 3*x)] + 166375*Log[3 + 5*x]

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Maple [A]  time = 0.009, size = 72, normalized size = 0.9 \begin{align*}{\frac{343}{162\, \left ( 2+3\,x \right ) ^{6}}}+{\frac{1421}{135\, \left ( 2+3\,x \right ) ^{5}}}+{\frac{7189}{108\, \left ( 2+3\,x \right ) ^{4}}}+{\frac{1331}{3\, \left ( 2+3\,x \right ) ^{3}}}+{\frac{6655}{2\, \left ( 2+3\,x \right ) ^{2}}}+33275\, \left ( 2+3\,x \right ) ^{-1}-166375\,\ln \left ( 2+3\,x \right ) +166375\,\ln \left ( 3+5\,x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

343/162/(2+3*x)^6+1421/135/(2+3*x)^5+7189/108/(2+3*x)^4+1331/3/(2+3*x)^3+6655/2/(2+3*x)^2+33275/(2+3*x)-166375
*ln(2+3*x)+166375*ln(3+5*x)

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Maxima [A]  time = 1.03953, size = 103, normalized size = 1.27 \begin{align*} \frac{13099036500 \, x^{5} + 44100089550 \, x^{4} + 59401704780 \, x^{3} + 40016101275 \, x^{2} + 13482032616 \, x + 1817443594}{1620 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + 166375 \, \log \left (5 \, x + 3\right ) - 166375 \, \log \left (3 \, x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1620*(13099036500*x^5 + 44100089550*x^4 + 59401704780*x^3 + 40016101275*x^2 + 13482032616*x + 1817443594)/(7
29*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64) + 166375*log(5*x + 3) - 166375*log(3*x + 2)

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Fricas [A]  time = 1.21099, size = 486, normalized size = 6. \begin{align*} \frac{13099036500 \, x^{5} + 44100089550 \, x^{4} + 59401704780 \, x^{3} + 40016101275 \, x^{2} + 269527500 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (5 \, x + 3\right ) - 269527500 \,{\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 ) + 13482032616 \, x + 1817443594}{1620 \,{\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/(2+3*x)^7/(3+5*x),x, algorithm="fricas")

[Out]

1/1620*(13099036500*x^5 + 44100089550*x^4 + 59401704780*x^3 + 40016101275*x^2 + 269527500*(729*x^6 + 2916*x^5
+ 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*log(5*x + 3) - 269527500*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*
x^3 + 2160*x^2 + 576*x + 64)*log(3*x + 2) + 13482032616*x + 1817443594)/(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 = 0.194197, size = 71, normalized size = 0.88 \begin{align*} \frac{13099036500 x^{5} + 44100089550 x^{4} + 59401704780 x^{3} + 40016101275 x^{2} + 13482032616 x + 1817443594}{1180980 x^{6} + 4723920 x^{5} + 7873200 x^{4} + 6998400 x^{3} + 3499200 x^{2} + 933120 x + 103680} + 166375 \log{\left (x + \frac{3}{5} \right )} - 166375 \log{\left (x + \frac{2}{3} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(13099036500*x**5 + 44100089550*x**4 + 59401704780*x**3 + 40016101275*x**2 + 13482032616*x + 1817443594)/(1180
980*x**6 + 4723920*x**5 + 7873200*x**4 + 6998400*x**3 + 3499200*x**2 + 933120*x + 103680) + 166375*log(x + 3/5
) - 166375*log(x + 2/3)

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Giac [A]  time = 2.38606, size = 72, normalized size = 0.89 \begin{align*} \frac{13099036500 \, x^{5} + 44100089550 \, x^{4} + 59401704780 \, x^{3} + 40016101275 \, x^{2} + 13482032616 \, x + 1817443594}{1620 \,{\left (3 \, x + 2\right )}^{6}} + 166375 \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - 166375 \, \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/(2+3*x)^7/(3+5*x),x, algorithm="giac")

[Out]

1/1620*(13099036500*x^5 + 44100089550*x^4 + 59401704780*x^3 + 40016101275*x^2 + 13482032616*x + 1817443594)/(3
*x + 2)^6 + 166375*log(abs(5*x + 3)) - 166375*log(abs(3*x + 2))