3.1390 \(\int \frac{(1-2 x)^3 (2+3 x)^4}{3+5 x} \, dx\)

Optimal. Leaf size=58 \[ -\frac{648 x^7}{35}-\frac{306 x^6}{25}+\frac{14958 x^5}{625}+\frac{31251 x^4}{2500}-\frac{128753 x^3}{9375}-\frac{138741 x^2}{31250}+\frac{416223 x}{78125}+\frac{1331 \log (5 x+3)}{390625} \]

[Out]

(416223*x)/78125 - (138741*x^2)/31250 - (128753*x^3)/9375 + (31251*x^4)/2500 + (14958*x^5)/625 - (306*x^6)/25
- (648*x^7)/35 + (1331*Log[3 + 5*x])/390625

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Rubi [A]  time = 0.0252993, antiderivative size = 58, 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} \[ -\frac{648 x^7}{35}-\frac{306 x^6}{25}+\frac{14958 x^5}{625}+\frac{31251 x^4}{2500}-\frac{128753 x^3}{9375}-\frac{138741 x^2}{31250}+\frac{416223 x}{78125}+\frac{1331 \log (5 x+3)}{390625} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(416223*x)/78125 - (138741*x^2)/31250 - (128753*x^3)/9375 + (31251*x^4)/2500 + (14958*x^5)/625 - (306*x^6)/25
- (648*x^7)/35 + (1331*Log[3 + 5*x])/390625

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)^4}{3+5 x} \, dx &=\int \left (\frac{416223}{78125}-\frac{138741 x}{15625}-\frac{128753 x^2}{3125}+\frac{31251 x^3}{625}+\frac{14958 x^4}{125}-\frac{1836 x^5}{25}-\frac{648 x^6}{5}+\frac{1331}{78125 (3+5 x)}\right ) \, dx\\ &=\frac{416223 x}{78125}-\frac{138741 x^2}{31250}-\frac{128753 x^3}{9375}+\frac{31251 x^4}{2500}+\frac{14958 x^5}{625}-\frac{306 x^6}{25}-\frac{648 x^7}{35}+\frac{1331 \log (3+5 x)}{390625}\\ \end{align*}

Mathematica [A]  time = 0.0122657, size = 47, normalized size = 0.81 \[ \frac{-3037500000 x^7-2008125000 x^6+3926475000 x^5+2050846875 x^4-2253177500 x^3-728390250 x^2+874068300 x+559020 \log (5 x+3)[phone]}{164062500} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(348168591 + 874068300*x - 728390250*x^2 - 2253177500*x^3 + 2050846875*x^4 + 3926475000*x^5 - 2008125000*x^6 -
3037500000*x^7 + 559020*Log[3 + 5*x])/164062500

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Maple [A]  time = 0.003, size = 43, normalized size = 0.7 \begin{align*}{\frac{416223\,x}{78125}}-{\frac{138741\,{x}^{2}}{31250}}-{\frac{128753\,{x}^{3}}{9375}}+{\frac{31251\,{x}^{4}}{2500}}+{\frac{14958\,{x}^{5}}{625}}-{\frac{306\,{x}^{6}}{25}}-{\frac{648\,{x}^{7}}{35}}+{\frac{1331\,\ln \left ( 3+5\,x \right ) }{390625}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

416223/78125*x-138741/31250*x^2-128753/9375*x^3+31251/2500*x^4+14958/625*x^5-306/25*x^6-648/35*x^7+1331/390625
*ln(3+5*x)

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Maxima [A]  time = 1.07682, size = 57, normalized size = 0.98 \begin{align*} -\frac{648}{35} \, x^{7} - \frac{306}{25} \, x^{6} + \frac{14958}{625} \, x^{5} + \frac{31251}{2500} \, x^{4} - \frac{128753}{9375} \, x^{3} - \frac{138741}{31250} \, x^{2} + \frac{416223}{78125} \, x + \frac{1331}{390625} \, \log \left (5 \, x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-648/35*x^7 - 306/25*x^6 + 14958/625*x^5 + 31251/2500*x^4 - 128753/9375*x^3 - 138741/31250*x^2 + 416223/78125*
x + 1331/390625*log(5*x + 3)

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Fricas [A]  time = 1.48992, size = 189, normalized size = 3.26 \begin{align*} -\frac{648}{35} \, x^{7} - \frac{306}{25} \, x^{6} + \frac{14958}{625} \, x^{5} + \frac{31251}{2500} \, x^{4} - \frac{128753}{9375} \, x^{3} - \frac{138741}{31250} \, x^{2} + \frac{416223}{78125} \, x + \frac{1331}{390625} \, \log \left (5 \, x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-648/35*x^7 - 306/25*x^6 + 14958/625*x^5 + 31251/2500*x^4 - 128753/9375*x^3 - 138741/31250*x^2 + 416223/78125*
x + 1331/390625*log(5*x + 3)

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Sympy [A]  time = 0.094408, size = 54, normalized size = 0.93 \begin{align*} - \frac{648 x^{7}}{35} - \frac{306 x^{6}}{25} + \frac{14958 x^{5}}{625} + \frac{31251 x^{4}}{2500} - \frac{128753 x^{3}}{9375} - \frac{138741 x^{2}}{31250} + \frac{416223 x}{78125} + \frac{1331 \log{\left (5 x + 3 \right )}}{390625} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-648*x**7/35 - 306*x**6/25 + 14958*x**5/625 + 31251*x**4/2500 - 128753*x**3/9375 - 138741*x**2/31250 + 416223*
x/78125 + 1331*log(5*x + 3)/390625

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Giac [A]  time = 2.16958, size = 58, normalized size = 1. \begin{align*} -\frac{648}{35} \, x^{7} - \frac{306}{25} \, x^{6} + \frac{14958}{625} \, x^{5} + \frac{31251}{2500} \, x^{4} - \frac{128753}{9375} \, x^{3} - \frac{138741}{31250} \, x^{2} + \frac{416223}{78125} \, x + \frac{1331}{390625} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-648/35*x^7 - 306/25*x^6 + 14958/625*x^5 + 31251/2500*x^4 - 128753/9375*x^3 - 138741/31250*x^2 + 416223/78125*
x + 1331/390625*log(abs(5*x + 3))