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]

555489/78125*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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Rubi [A]  time = 0.03, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, 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*}

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Mathematica [A]  time = 0.02, 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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fricas [A]  time = 0.65, size = 57, normalized size = 0.92 \[ -\frac {18984375 \, x^{7} + 63787500 \, x^{6} + 75481875 \, x^{5} + 22955625 \, x^{4} - 27546750 \, x^{3} - 29165850 \, x^{2} - 196 \, {\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) - 8332335 \, x + 11}{390625 \, {\left (5 \, x + 3\right )}} \]

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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giac [A]  time = 1.20, size = 84, normalized size = 1.35 \[ \frac {9}{1953125} \, {\left (5 \, x + 3\right )}^{6} {\left (\frac {567}{5 \, x + 3} + \frac {1890}{{\left (5 \, x + 3\right )}^{2}} + \frac {2275}{{\left (5 \, x + 3\right )}^{3}} + \frac {1575}{{\left (5 \, x + 3\right )}^{4}} + \frac {805}{{\left (5 \, x + 3\right )}^{5}} - 135\right )} - \frac {11}{390625 \, {\left (5 \, x + 3\right )}} - \frac {196}{390625} \, \log \left (\frac {{\left | 5 \, x + 3 \right |}}{5 \, {\left (5 \, x + 3\right )}^{2}}\right ) \]

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)

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maple [A]  time = 0.01, size = 47, normalized size = 0.76 \[ -\frac {243 x^{6}}{25}-\frac {16767 x^{5}}{625}-\frac {14094 x^{4}}{625}+\frac {5553 x^{3}}{3125}+\frac {40743 x^{2}}{3125}+\frac {555489 x}{78125}+\frac {196 \ln \left (5 x +3\right )}{390625}-\frac {11}{390625 \left (5 x +3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)*(3*x+2)^6/(5*x+3)^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/(5*x+3)+196/39062
5*ln(5*x+3)

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maxima [A]  time = 0.57, size = 46, normalized size = 0.74 \[ -\frac {243}{25} \, x^{6} - \frac {16767}{625} \, x^{5} - \frac {14094}{625} \, x^{4} + \frac {5553}{3125} \, x^{3} + \frac {40743}{3125} \, x^{2} + \frac {555489}{78125} \, x - \frac {11}{390625 \, {\left (5 \, x + 3\right )}} + \frac {196}{390625} \, \log \left (5 \, x + 3\right ) \]

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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mupad [B]  time = 0.03, size = 44, normalized size = 0.71 \[ \frac {555489\,x}{78125}+\frac {196\,\ln \left (x+\frac {3}{5}\right )}{390625}-\frac {11}{1953125\,\left (x+\frac {3}{5}\right )}+\frac {40743\,x^2}{3125}+\frac {5553\,x^3}{3125}-\frac {14094\,x^4}{625}-\frac {16767\,x^5}{625}-\frac {243\,x^6}{25} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.12, size = 54, normalized size = 0.87 \[ - \frac {243 x^{6}}{25} - \frac {16767 x^{5}}{625} - \frac {14094 x^{4}}{625} + \frac {5553 x^{3}}{3125} + \frac {40743 x^{2}}{3125} + \frac {555489 x}{78125} + \frac {196 \log {\left (5 x + 3 \right )}}{390625} - \frac {11}{1953125 x + 1171875} \]

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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