3.1.7 \(\int \frac {-x^2+x^3}{(-6+x) (3+5 x)^3} \, dx\)

Optimal. Leaf size=43 \[ \frac {201}{15125 (5 x+3)}-\frac {12}{1375 (5 x+3)^2}+\frac {20 \log (6-x)}{3993}+\frac {1493 \log (5 x+3)}{499125} \]

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Rubi [A]  time = 0.04, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1593, 148} \begin {gather*} \frac {201}{15125 (5 x+3)}-\frac {12}{1375 (5 x+3)^2}+\frac {20 \log (6-x)}{3993}+\frac {1493 \log (5 x+3)}{499125} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-x^2 + x^3)/((-6 + x)*(3 + 5*x)^3),x]

[Out]

-12/(1375*(3 + 5*x)^2) + 201/(15125*(3 + 5*x)) + (20*Log[6 - x])/3993 + (1493*Log[3 + 5*x])/499125

Rule 148

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x), x], x] /; FreeQ[{a, b, c, d, e, f, g
, h, m}, x] && (IntegersQ[m, n, p] || (IGtQ[n, 0] && IGtQ[p, 0]))

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rubi steps

\begin {align*} \int \frac {-x^2+x^3}{(-6+x) (3+5 x)^3} \, dx &=\int \frac {(-1+x) x^2}{(-6+x) (3+5 x)^3} \, dx\\ &=\int \left (\frac {20}{3993 (-6+x)}+\frac {24}{275 (3+5 x)^3}-\frac {201}{3025 (3+5 x)^2}+\frac {1493}{99825 (3+5 x)}\right ) \, dx\\ &=-\frac {12}{1375 (3+5 x)^2}+\frac {201}{15125 (3+5 x)}+\frac {20 \log (6-x)}{3993}+\frac {1493 \log (3+5 x)}{499125}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 33, normalized size = 0.77 \begin {gather*} \frac {\frac {99 (335 x+157)}{(5 x+3)^2}+2500 \log (x-6)+1493 \log (5 x+3)}{499125} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-x^2 + x^3)/((-6 + x)*(3 + 5*x)^3),x]

[Out]

((99*(157 + 335*x))/(3 + 5*x)^2 + 2500*Log[-6 + x] + 1493*Log[3 + 5*x])/499125

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-x^2+x^3}{(-6+x) (3+5 x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(-x^2 + x^3)/((-6 + x)*(3 + 5*x)^3),x]

[Out]

IntegrateAlgebraic[(-x^2 + x^3)/((-6 + x)*(3 + 5*x)^3), x]

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fricas [A]  time = 1.44, size = 53, normalized size = 1.23 \begin {gather*} \frac {1493 \, {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) + 2500 \, {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (x - 6\right ) + 33165 \, x + 15543}{499125 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-x^2)/(-6+x)/(3+5*x)^3,x, algorithm="fricas")

[Out]

1/499125*(1493*(25*x^2 + 30*x + 9)*log(5*x + 3) + 2500*(25*x^2 + 30*x + 9)*log(x - 6) + 33165*x + 15543)/(25*x
^2 + 30*x + 9)

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giac [A]  time = 1.18, size = 31, normalized size = 0.72 \begin {gather*} \frac {3 \, {\left (335 \, x + 157\right )}}{15125 \, {\left (5 \, x + 3\right )}^{2}} + \frac {1493}{499125} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) + \frac {20}{3993} \, \log \left ({\left | x - 6 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-x^2)/(-6+x)/(3+5*x)^3,x, algorithm="giac")

[Out]

3/15125*(335*x + 157)/(5*x + 3)^2 + 1493/499125*log(abs(5*x + 3)) + 20/3993*log(abs(x - 6))

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maple [A]  time = 0.01, size = 34, normalized size = 0.79 \begin {gather*} \frac {1493 \ln \left (5 x +3\right )}{499125}+\frac {20 \ln \left (x -6\right )}{3993}-\frac {12}{1375 \left (5 x +3\right )^{2}}+\frac {201}{15125 \left (5 x +3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3-x^2)/(x-6)/(5*x+3)^3,x)

[Out]

-12/1375/(5*x+3)^2+201/15125/(5*x+3)+1493/499125*ln(5*x+3)+20/3993*ln(x-6)

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maxima [A]  time = 0.43, size = 34, normalized size = 0.79 \begin {gather*} \frac {3 \, {\left (335 \, x + 157\right )}}{15125 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {1493}{499125} \, \log \left (5 \, x + 3\right ) + \frac {20}{3993} \, \log \left (x - 6\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-x^2)/(-6+x)/(3+5*x)^3,x, algorithm="maxima")

[Out]

3/15125*(335*x + 157)/(25*x^2 + 30*x + 9) + 1493/499125*log(5*x + 3) + 20/3993*log(x - 6)

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mupad [B]  time = 0.12, size = 29, normalized size = 0.67 \begin {gather*} \frac {20\,\ln \left (x-6\right )}{3993}+\frac {1493\,\ln \left (x+\frac {3}{5}\right )}{499125}+\frac {\frac {201\,x}{75625}+\frac {471}{378125}}{x^2+\frac {6\,x}{5}+\frac {9}{25}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^2 - x^3)/((5*x + 3)^3*(x - 6)),x)

[Out]

(20*log(x - 6))/3993 + (1493*log(x + 3/5))/499125 + ((201*x)/75625 + 471/378125)/((6*x)/5 + x^2 + 9/25)

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sympy [A]  time = 0.16, size = 32, normalized size = 0.74 \begin {gather*} \frac {1005 x + 471}{378125 x^{2} + 453750 x + 136125} + \frac {20 \log {\left (x - 6 \right )}}{3993} + \frac {1493 \log {\left (x + \frac {3}{5} \right )}}{499125} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3-x**2)/(-6+x)/(3+5*x)**3,x)

[Out]

(1005*x + 471)/(378125*x**2 + 453750*x + 136125) + 20*log(x - 6)/3993 + 1493*log(x + 3/5)/499125

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