### 3.165 $$\int \frac{-x^2+x^3}{(-6+x) (3+5 x)^3} \, dx$$

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

[Out]

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

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Rubi [A]  time = 0.0422541, antiderivative size = 43, normalized size of antiderivative = 1., 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} $\frac{201}{15125 (5 x+3)}-\frac{12}{1375 (5 x+3)^2}+\frac{20 \log (6-x)}{3993}+\frac{1493 \log (5 x+3)}{499125}$

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 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]

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]))

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*}

Mathematica [A]  time = 0.0216479, size = 33, normalized size = 0.77 $\frac{\frac{99 (335 x+157)}{(5 x+3)^2}+2500 \log (x-6)+1493 \log (5 x+3)}{499125}$

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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Maple [A]  time = 0.007, size = 34, normalized size = 0.8 \begin{align*} -{\frac{12}{1375\, \left ( 3+5\,x \right ) ^{2}}}+{\frac{201}{45375+75625\,x}}+{\frac{1493\,\ln \left ( 3+5\,x \right ) }{499125}}+{\frac{20\,\ln \left ( -6+x \right ) }{3993}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.92771, size = 46, normalized size = 1.07 \begin{align*} \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{align*}

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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Fricas [A]  time = 1.92462, size = 170, normalized size = 3.95 \begin{align*} \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{align*}

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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Sympy [A]  time = 0.141443, size = 32, normalized size = 0.74 \begin{align*} \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{align*}

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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Giac [A]  time = 1.07588, size = 42, normalized size = 0.98 \begin{align*} \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{align*}

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))