3.1223 \(\int \frac{(1-2 x) (2+3 x)^6}{(3+5 x)^3} \, dx\)

Optimal. Leaf size=66 \[ -\frac{1458 x^5}{625}-\frac{12393 x^4}{2500}-\frac{6399 x^3}{3125}+\frac{297 x^2}{125}+\frac{36936 x}{15625}-\frac{196}{390625 (5 x+3)}-\frac{11}{781250 (5 x+3)^2}+\frac{1449 \log (5 x+3)}{390625} \]

[Out]

(36936*x)/15625 + (297*x^2)/125 - (6399*x^3)/3125 - (12393*x^4)/2500 - (1458*x^5)/625 - 11/(781250*(3 + 5*x)^2
) - 196/(390625*(3 + 5*x)) + (1449*Log[3 + 5*x])/390625

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Rubi [A]  time = 0.0331897, antiderivative size = 66, 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} \[ -\frac{1458 x^5}{625}-\frac{12393 x^4}{2500}-\frac{6399 x^3}{3125}+\frac{297 x^2}{125}+\frac{36936 x}{15625}-\frac{196}{390625 (5 x+3)}-\frac{11}{781250 (5 x+3)^2}+\frac{1449 \log (5 x+3)}{390625} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(36936*x)/15625 + (297*x^2)/125 - (6399*x^3)/3125 - (12393*x^4)/2500 - (1458*x^5)/625 - 11/(781250*(3 + 5*x)^2
) - 196/(390625*(3 + 5*x)) + (1449*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)^3} \, dx &=\int \left (\frac{36936}{15625}+\frac{594 x}{125}-\frac{19197 x^2}{3125}-\frac{12393 x^3}{625}-\frac{1458 x^4}{125}+\frac{11}{78125 (3+5 x)^3}+\frac{196}{78125 (3+5 x)^2}+\frac{1449}{78125 (3+5 x)}\right ) \, dx\\ &=\frac{36936 x}{15625}+\frac{297 x^2}{125}-\frac{6399 x^3}{3125}-\frac{12393 x^4}{2500}-\frac{1458 x^5}{625}-\frac{11}{781250 (3+5 x)^2}-\frac{196}{390625 (3+5 x)}+\frac{1449 \log (3+5 x)}{390625}\\ \end{align*}

Mathematica [A]  time = 0.0188367, size = 61, normalized size = 0.92 \[ \frac{-455625000 x^7-1514953125 x^6-1725806250 x^5-364415625 x^4+874597500 x^3+834723225 x^2+302537270 x+28980 (5 x+3)^2 \log (5 x+3)[phone]}{7812500 (5 x+3)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(40891591 + 302537270*x + 834723225*x^2 + 874597500*x^3 - 364415625*x^4 - 1725806250*x^5 - 1514953125*x^6 - 45
5625000*x^7 + 28980*(3 + 5*x)^2*Log[3 + 5*x])/(7812500*(3 + 5*x)^2)

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Maple [A]  time = 0.006, size = 51, normalized size = 0.8 \begin{align*}{\frac{36936\,x}{15625}}+{\frac{297\,{x}^{2}}{125}}-{\frac{6399\,{x}^{3}}{3125}}-{\frac{12393\,{x}^{4}}{2500}}-{\frac{1458\,{x}^{5}}{625}}-{\frac{11}{781250\, \left ( 3+5\,x \right ) ^{2}}}-{\frac{196}{1171875+1953125\,x}}+{\frac{1449\,\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)^3,x)

[Out]

36936/15625*x+297/125*x^2-6399/3125*x^3-12393/2500*x^4-1458/625*x^5-11/781250/(3+5*x)^2-196/390625/(3+5*x)+144
9/390625*ln(3+5*x)

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Maxima [A]  time = 1.06637, size = 69, normalized size = 1.05 \begin{align*} -\frac{1458}{625} \, x^{5} - \frac{12393}{2500} \, x^{4} - \frac{6399}{3125} \, x^{3} + \frac{297}{125} \, x^{2} + \frac{36936}{15625} \, x - \frac{1960 \, x + 1187}{781250 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{1449}{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)^3,x, algorithm="maxima")

[Out]

-1458/625*x^5 - 12393/2500*x^4 - 6399/3125*x^3 + 297/125*x^2 + 36936/15625*x - 1/781250*(1960*x + 1187)/(25*x^
2 + 30*x + 9) + 1449/390625*log(5*x + 3)

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Fricas [A]  time = 1.49571, size = 251, normalized size = 3.8 \begin{align*} -\frac{91125000 \, x^{7} + 302990625 \, x^{6} + 345161250 \, x^{5} + 72883125 \, x^{4} - 174919500 \, x^{3} - 144220500 \, x^{2} - 5796 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) - 33238480 \, x + 2374}{1562500 \,{\left (25 \, x^{2} + 30 \, x + 9\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)^3,x, algorithm="fricas")

[Out]

-1/1562500*(91125000*x^7 + 302990625*x^6 + 345161250*x^5 + 72883125*x^4 - 174919500*x^3 - 144220500*x^2 - 5796
*(25*x^2 + 30*x + 9)*log(5*x + 3) - 33238480*x + 2374)/(25*x^2 + 30*x + 9)

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Sympy [A]  time = 0.123635, size = 56, normalized size = 0.85 \begin{align*} - \frac{1458 x^{5}}{625} - \frac{12393 x^{4}}{2500} - \frac{6399 x^{3}}{3125} + \frac{297 x^{2}}{125} + \frac{36936 x}{15625} - \frac{1960 x + 1187}{19531250 x^{2} + 23437500 x + 7031250} + \frac{1449 \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)**3,x)

[Out]

-1458*x**5/625 - 12393*x**4/2500 - 6399*x**3/3125 + 297*x**2/125 + 36936*x/15625 - (1960*x + 1187)/(19531250*x
**2 + 23437500*x + 7031250) + 1449*log(5*x + 3)/390625

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Giac [A]  time = 2.97934, size = 63, normalized size = 0.95 \begin{align*} -\frac{1458}{625} \, x^{5} - \frac{12393}{2500} \, x^{4} - \frac{6399}{3125} \, x^{3} + \frac{297}{125} \, x^{2} + \frac{36936}{15625} \, x - \frac{1960 \, x + 1187}{781250 \,{\left (5 \, x + 3\right )}^{2}} + \frac{1449}{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)^3,x, algorithm="giac")

[Out]

-1458/625*x^5 - 12393/2500*x^4 - 6399/3125*x^3 + 297/125*x^2 + 36936/15625*x - 1/781250*(1960*x + 1187)/(5*x +
3)^2 + 1449/390625*log(abs(5*x + 3))