3.1193 \(\int \frac{(1-2 x) (2+3 x)^6}{3+5 x} \, dx\)

Optimal. Leaf size=58 \[ -\frac{1458 x^7}{35}-\frac{7047 x^6}{50}-\frac{106677 x^5}{625}-\frac{152469 x^4}{2500}+\frac{152469 x^3}{3125}+\frac{1777779 x^2}{31250}+\frac{1666663 x}{78125}+\frac{11 \log (5 x+3)}{390625} \]

[Out]

(1666663*x)/78125 + (1777779*x^2)/31250 + (152469*x^3)/3125 - (152469*x^4)/2500 - (106677*x^5)/625 - (7047*x^6
)/50 - (1458*x^7)/35 + (11*Log[3 + 5*x])/390625

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Rubi [A]  time = 0.0212677, antiderivative size = 58, 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} \[ -\frac{1458 x^7}{35}-\frac{7047 x^6}{50}-\frac{106677 x^5}{625}-\frac{152469 x^4}{2500}+\frac{152469 x^3}{3125}+\frac{1777779 x^2}{31250}+\frac{1666663 x}{78125}+\frac{11 \log (5 x+3)}{390625} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1666663*x)/78125 + (1777779*x^2)/31250 + (152469*x^3)/3125 - (152469*x^4)/2500 - (106677*x^5)/625 - (7047*x^6
)/50 - (1458*x^7)/35 + (11*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} \, dx &=\int \left (\frac{1666663}{78125}+\frac{1777779 x}{15625}+\frac{457407 x^2}{3125}-\frac{152469 x^3}{625}-\frac{106677 x^4}{125}-\frac{21141 x^5}{25}-\frac{1458 x^6}{5}+\frac{11}{78125 (3+5 x)}\right ) \, dx\\ &=\frac{1666663 x}{78125}+\frac{1777779 x^2}{31250}+\frac{152469 x^3}{3125}-\frac{152469 x^4}{2500}-\frac{106677 x^5}{625}-\frac{7047 x^6}{50}-\frac{1458 x^7}{35}+\frac{11 \log (3+5 x)}{390625}\\ \end{align*}

Mathematica [A]  time = 0.0151661, size = 47, normalized size = 0.81 \[ \frac{-2278125000 x^7-7707656250 x^6-9334237500 x^5-3335259375 x^4+2668207500 x^3+3111113250 x^2+1166664100 x+1540 \log (5 x+3)[phone]}{54687500} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(158585307 + 1166664100*x + 3111113250*x^2 + 2668207500*x^3 - 3335259375*x^4 - 9334237500*x^5 - 7707656250*x^6
- 2278125000*x^7 + 1540*Log[3 + 5*x])/54687500

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Maple [A]  time = 0.003, size = 43, normalized size = 0.7 \begin{align*}{\frac{1666663\,x}{78125}}+{\frac{1777779\,{x}^{2}}{31250}}+{\frac{152469\,{x}^{3}}{3125}}-{\frac{152469\,{x}^{4}}{2500}}-{\frac{106677\,{x}^{5}}{625}}-{\frac{7047\,{x}^{6}}{50}}-{\frac{1458\,{x}^{7}}{35}}+{\frac{11\,\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),x)

[Out]

1666663/78125*x+1777779/31250*x^2+152469/3125*x^3-152469/2500*x^4-106677/625*x^5-7047/50*x^6-1458/35*x^7+11/39
0625*ln(3+5*x)

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Maxima [A]  time = 1.51867, size = 57, normalized size = 0.98 \begin{align*} -\frac{1458}{35} \, x^{7} - \frac{7047}{50} \, x^{6} - \frac{106677}{625} \, x^{5} - \frac{152469}{2500} \, x^{4} + \frac{152469}{3125} \, x^{3} + \frac{1777779}{31250} \, x^{2} + \frac{1666663}{78125} \, x + \frac{11}{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),x, algorithm="maxima")

[Out]

-1458/35*x^7 - 7047/50*x^6 - 106677/625*x^5 - 152469/2500*x^4 + 152469/3125*x^3 + 1777779/31250*x^2 + 1666663/
78125*x + 11/390625*log(5*x + 3)

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Fricas [A]  time = 1.51813, size = 194, normalized size = 3.34 \begin{align*} -\frac{1458}{35} \, x^{7} - \frac{7047}{50} \, x^{6} - \frac{106677}{625} \, x^{5} - \frac{152469}{2500} \, x^{4} + \frac{152469}{3125} \, x^{3} + \frac{1777779}{31250} \, x^{2} + \frac{1666663}{78125} \, x + \frac{11}{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),x, algorithm="fricas")

[Out]

-1458/35*x^7 - 7047/50*x^6 - 106677/625*x^5 - 152469/2500*x^4 + 152469/3125*x^3 + 1777779/31250*x^2 + 1666663/
78125*x + 11/390625*log(5*x + 3)

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Sympy [A]  time = 0.094373, size = 54, normalized size = 0.93 \begin{align*} - \frac{1458 x^{7}}{35} - \frac{7047 x^{6}}{50} - \frac{106677 x^{5}}{625} - \frac{152469 x^{4}}{2500} + \frac{152469 x^{3}}{3125} + \frac{1777779 x^{2}}{31250} + \frac{1666663 x}{78125} + \frac{11 \log{\left (5 x + 3 \right )}}{390625} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1458*x**7/35 - 7047*x**6/50 - 106677*x**5/625 - 152469*x**4/2500 + 152469*x**3/3125 + 1777779*x**2/31250 + 16
66663*x/78125 + 11*log(5*x + 3)/390625

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Giac [A]  time = 2.69911, size = 58, normalized size = 1. \begin{align*} -\frac{1458}{35} \, x^{7} - \frac{7047}{50} \, x^{6} - \frac{106677}{625} \, x^{5} - \frac{152469}{2500} \, x^{4} + \frac{152469}{3125} \, x^{3} + \frac{1777779}{31250} \, x^{2} + \frac{1666663}{78125} \, x + \frac{11}{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)*(2+3*x)^6/(3+5*x),x, algorithm="giac")

[Out]

-1458/35*x^7 - 7047/50*x^6 - 106677/625*x^5 - 152469/2500*x^4 + 152469/3125*x^3 + 1777779/31250*x^2 + 1666663/
78125*x + 11/390625*log(abs(5*x + 3))