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

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

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Rubi [A]  time = 0.03, antiderivative size = 58, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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} \end {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 47, normalized size = 0.81 \begin {gather*} \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} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(1-2 x)^3 (2+3 x)^4}{3+5 x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.83, size = 42, normalized size = 0.72 \begin {gather*} -\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 {gather*}

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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giac [A]  time = 0.83, size = 43, normalized size = 0.74 \begin {gather*} -\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 {gather*}

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))

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maple [A]  time = 0.00, size = 43, normalized size = 0.74 \begin {gather*} -\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 \ln \left (5 x +3\right )}{390625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maxima [A]  time = 0.47, size = 42, normalized size = 0.72 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 0.04, size = 40, normalized size = 0.69 \begin {gather*} \frac {416223\,x}{78125}+\frac {1331\,\ln \left (x+\frac {3}{5}\right )}{390625}-\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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(416223*x)/78125 + (1331*log(x + 3/5))/390625 - (138741*x^2)/31250 - (128753*x^3)/9375 + (31251*x^4)/2500 + (1
4958*x^5)/625 - (306*x^6)/25 - (648*x^7)/35

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sympy [A]  time = 0.11, size = 54, normalized size = 0.93 \begin {gather*} - \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 {gather*}

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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