3.1208 \(\int \frac{(1-2 x) (2+3 x)^6}{(3+5 x)^2} \, dx\)

Optimal. Leaf size=62 \[ -\frac{243 x^6}{25}-\frac{16767 x^5}{625}-\frac{14094 x^4}{625}+\frac{5553 x^3}{3125}+\frac{40743 x^2}{3125}+\frac{555489 x}{78125}-\frac{11}{390625 (5 x+3)}+\frac{196 \log (5 x+3)}{390625} \]

[Out]

(555489*x)/78125 + (40743*x^2)/3125 + (5553*x^3)/3125 - (14094*x^4)/625 - (16767*x^5)/625 - (243*x^6)/25 - 11/
(390625*(3 + 5*x)) + (196*Log[3 + 5*x])/390625

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Rubi [A]  time = 0.0293556, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {77} \[ -\frac{243 x^6}{25}-\frac{16767 x^5}{625}-\frac{14094 x^4}{625}+\frac{5553 x^3}{3125}+\frac{40743 x^2}{3125}+\frac{555489 x}{78125}-\frac{11}{390625 (5 x+3)}+\frac{196 \log (5 x+3)}{390625} \]

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)*(2 + 3*x)^6)/(3 + 5*x)^2,x]

[Out]

(555489*x)/78125 + (40743*x^2)/3125 + (5553*x^3)/3125 - (14094*x^4)/625 - (16767*x^5)/625 - (243*x^6)/25 - 11/
(390625*(3 + 5*x)) + (196*Log[3 + 5*x])/390625

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{(1-2 x) (2+3 x)^6}{(3+5 x)^2} \, dx &=\int \left (\frac{555489}{78125}+\frac{81486 x}{3125}+\frac{16659 x^2}{3125}-\frac{56376 x^3}{625}-\frac{16767 x^4}{125}-\frac{1458 x^5}{25}+\frac{11}{78125 (3+5 x)^2}+\frac{196}{78125 (3+5 x)}\right ) \, dx\\ &=\frac{555489 x}{78125}+\frac{40743 x^2}{3125}+\frac{5553 x^3}{3125}-\frac{14094 x^4}{625}-\frac{16767 x^5}{625}-\frac{243 x^6}{25}-\frac{11}{390625 (3+5 x)}+\frac{196 \log (3+5 x)}{390625}\\ \end{align*}

Mathematica [A]  time = 0.0176136, size = 59, normalized size = 0.95 \[ \frac{-94921875 x^7-318937500 x^6-377409375 x^5-114778125 x^4+137733750 x^3+145829250 x^2+53832870 x+980 (5 x+3) \log (5 x+3)+7302662}{1953125 (5 x+3)} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)*(2 + 3*x)^6)/(3 + 5*x)^2,x]

[Out]

(7302662 + 53832870*x + 145829250*x^2 + 137733750*x^3 - 114778125*x^4 - 377409375*x^5 - 318937500*x^6 - 949218
75*x^7 + 980*(3 + 5*x)*Log[3 + 5*x])/(1953125*(3 + 5*x))

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Maple [A]  time = 0.006, size = 47, normalized size = 0.8 \begin{align*}{\frac{555489\,x}{78125}}+{\frac{40743\,{x}^{2}}{3125}}+{\frac{5553\,{x}^{3}}{3125}}-{\frac{14094\,{x}^{4}}{625}}-{\frac{16767\,{x}^{5}}{625}}-{\frac{243\,{x}^{6}}{25}}-{\frac{11}{1171875+1953125\,x}}+{\frac{196\,\ln \left ( 3+5\,x \right ) }{390625}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)*(2+3*x)^6/(3+5*x)^2,x)

[Out]

555489/78125*x+40743/3125*x^2+5553/3125*x^3-14094/625*x^4-16767/625*x^5-243/25*x^6-11/390625/(3+5*x)+196/39062
5*ln(3+5*x)

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Maxima [A]  time = 1.12597, size = 62, normalized size = 1. \begin{align*} -\frac{243}{25} \, x^{6} - \frac{16767}{625} \, x^{5} - \frac{14094}{625} \, x^{4} + \frac{5553}{3125} \, x^{3} + \frac{40743}{3125} \, x^{2} + \frac{555489}{78125} \, x - \frac{11}{390625 \,{\left (5 \, x + 3\right )}} + \frac{196}{390625} \, \log \left (5 \, x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(2+3*x)^6/(3+5*x)^2,x, algorithm="maxima")

[Out]

-243/25*x^6 - 16767/625*x^5 - 14094/625*x^4 + 5553/3125*x^3 + 40743/3125*x^2 + 555489/78125*x - 11/390625/(5*x
+ 3) + 196/390625*log(5*x + 3)

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Fricas [A]  time = 1.48571, size = 212, normalized size = 3.42 \begin{align*} -\frac{18984375 \, x^{7} + 63787500 \, x^{6} + 75481875 \, x^{5} + 22955625 \, x^{4} - 27546750 \, x^{3} - 29165850 \, x^{2} - 196 \,{\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) - 8332335 \, x + 11}{390625 \,{\left (5 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(2+3*x)^6/(3+5*x)^2,x, algorithm="fricas")

[Out]

-1/390625*(18984375*x^7 + 63787500*x^6 + 75481875*x^5 + 22955625*x^4 - 27546750*x^3 - 29165850*x^2 - 196*(5*x
+ 3)*log(5*x + 3) - 8332335*x + 11)/(5*x + 3)

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Sympy [A]  time = 0.107375, size = 54, normalized size = 0.87 \begin{align*} - \frac{243 x^{6}}{25} - \frac{16767 x^{5}}{625} - \frac{14094 x^{4}}{625} + \frac{5553 x^{3}}{3125} + \frac{40743 x^{2}}{3125} + \frac{555489 x}{78125} + \frac{196 \log{\left (5 x + 3 \right )}}{390625} - \frac{11}{1953125 x + 1171875} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(2+3*x)**6/(3+5*x)**2,x)

[Out]

-243*x**6/25 - 16767*x**5/625 - 14094*x**4/625 + 5553*x**3/3125 + 40743*x**2/3125 + 555489*x/78125 + 196*log(5
*x + 3)/390625 - 11/(1953125*x + 1171875)

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Giac [A]  time = 2.08506, size = 113, normalized size = 1.82 \begin{align*} \frac{9}{1953125} \,{\left (5 \, x + 3\right )}^{6}{\left (\frac{567}{5 \, x + 3} + \frac{1890}{{\left (5 \, x + 3\right )}^{2}} + \frac{2275}{{\left (5 \, x + 3\right )}^{3}} + \frac{1575}{{\left (5 \, x + 3\right )}^{4}} + \frac{805}{{\left (5 \, x + 3\right )}^{5}} - 135\right )} - \frac{11}{390625 \,{\left (5 \, x + 3\right )}} - \frac{196}{390625} \, \log \left (\frac{{\left | 5 \, x + 3 \right |}}{5 \,{\left (5 \, x + 3\right )}^{2}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(2+3*x)^6/(3+5*x)^2,x, algorithm="giac")

[Out]

9/1953125*(5*x + 3)^6*(567/(5*x + 3) + 1890/(5*x + 3)^2 + 2275/(5*x + 3)^3 + 1575/(5*x + 3)^4 + 805/(5*x + 3)^
5 - 135) - 11/390625/(5*x + 3) - 196/390625*log(1/5*abs(5*x + 3)/(5*x + 3)^2)