3.1222 \(\int \frac{(1-2 x) (2+3 x)^7}{(3+5 x)^3} \, dx\)

Optimal. Leaf size=73 \[ -\frac{729 x^6}{125}-\frac{51759 x^5}{3125}-\frac{181521 x^4}{12500}+\frac{2052 x^3}{3125}+\frac{129654 x^2}{15625}+\frac{1851147 x}{390625}-\frac{229}{1953125 (5 x+3)}-\frac{11}{3906250 (5 x+3)^2}+\frac{2037 \log (5 x+3)}{1953125} \]

[Out]

(1851147*x)/390625 + (129654*x^2)/15625 + (2052*x^3)/3125 - (181521*x^4)/12500 - (51759*x^5)/3125 - (729*x^6)/
125 - 11/(3906250*(3 + 5*x)^2) - 229/(1953125*(3 + 5*x)) + (2037*Log[3 + 5*x])/1953125

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Rubi [A]  time = 0.040904, antiderivative size = 73, 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} \[ -\frac{729 x^6}{125}-\frac{51759 x^5}{3125}-\frac{181521 x^4}{12500}+\frac{2052 x^3}{3125}+\frac{129654 x^2}{15625}+\frac{1851147 x}{390625}-\frac{229}{1953125 (5 x+3)}-\frac{11}{3906250 (5 x+3)^2}+\frac{2037 \log (5 x+3)}{1953125} \]

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)*(2 + 3*x)^7)/(3 + 5*x)^3,x]

[Out]

(1851147*x)/390625 + (129654*x^2)/15625 + (2052*x^3)/3125 - (181521*x^4)/12500 - (51759*x^5)/3125 - (729*x^6)/
125 - 11/(3906250*(3 + 5*x)^2) - 229/(1953125*(3 + 5*x)) + (2037*Log[3 + 5*x])/1953125

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)^7}{(3+5 x)^3} \, dx &=\int \left (\frac{1851147}{390625}+\frac{259308 x}{15625}+\frac{6156 x^2}{3125}-\frac{181521 x^3}{3125}-\frac{51759 x^4}{625}-\frac{4374 x^5}{125}+\frac{11}{390625 (3+5 x)^3}+\frac{229}{390625 (3+5 x)^2}+\frac{2037}{390625 (3+5 x)}\right ) \, dx\\ &=\frac{1851147 x}{390625}+\frac{129654 x^2}{15625}+\frac{2052 x^3}{3125}-\frac{181521 x^4}{12500}-\frac{51759 x^5}{3125}-\frac{729 x^6}{125}-\frac{11}{3906250 (3+5 x)^2}-\frac{229}{1953125 (3+5 x)}+\frac{2037 \log (3+5 x)}{1953125}\\ \end{align*}

Mathematica [A]  time = 0.0237194, size = 64, normalized size = 0.88 \[ \frac{8148 \log (-3 (5 x+3))-\frac{5 \left (227812500 x^8+920362500 x^7+1425650625 x^6+887969250 x^5-150703875 x^4-583310700 x^3-372626040 x^2-107200136 x-12167374\right )}{(5 x+3)^2}}{7812500} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)*(2 + 3*x)^7)/(3 + 5*x)^3,x]

[Out]

((-5*(-12167374 - 107200136*x - 372626040*x^2 - 583310700*x^3 - 150703875*x^4 + 887969250*x^5 + 1425650625*x^6
+ 920362500*x^7 + 227812500*x^8))/(3 + 5*x)^2 + 8148*Log[-3*(3 + 5*x)])/7812500

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Maple [A]  time = 0.007, size = 56, normalized size = 0.8 \begin{align*}{\frac{1851147\,x}{390625}}+{\frac{129654\,{x}^{2}}{15625}}+{\frac{2052\,{x}^{3}}{3125}}-{\frac{181521\,{x}^{4}}{12500}}-{\frac{51759\,{x}^{5}}{3125}}-{\frac{729\,{x}^{6}}{125}}-{\frac{11}{3906250\, \left ( 3+5\,x \right ) ^{2}}}-{\frac{229}{5859375+9765625\,x}}+{\frac{2037\,\ln \left ( 3+5\,x \right ) }{1953125}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)*(2+3*x)^7/(3+5*x)^3,x)

[Out]

1851147/390625*x+129654/15625*x^2+2052/3125*x^3-181521/12500*x^4-51759/3125*x^5-729/125*x^6-11/3906250/(3+5*x)
^2-229/1953125/(3+5*x)+2037/1953125*ln(3+5*x)

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Maxima [A]  time = 1.05701, size = 76, normalized size = 1.04 \begin{align*} -\frac{729}{125} \, x^{6} - \frac{51759}{3125} \, x^{5} - \frac{181521}{12500} \, x^{4} + \frac{2052}{3125} \, x^{3} + \frac{129654}{15625} \, x^{2} + \frac{1851147}{390625} \, x - \frac{458 \, x + 277}{781250 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{2037}{1953125} \, \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)^7/(3+5*x)^3,x, algorithm="maxima")

[Out]

-729/125*x^6 - 51759/3125*x^5 - 181521/12500*x^4 + 2052/3125*x^3 + 129654/15625*x^2 + 1851147/390625*x - 1/781
250*(458*x + 277)/(25*x^2 + 30*x + 9) + 2037/1953125*log(5*x + 3)

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Fricas [A]  time = 1.41086, size = 285, normalized size = 3.9 \begin{align*} -\frac{1139062500 \, x^{8} + 4601812500 \, x^{7} + 7128253125 \, x^{6} + 4439846250 \, x^{5} - 753519375 \, x^{4} - 2916553500 \, x^{3} - 1694131200 \, x^{2} - 8148 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) - 333201880 \, x + 2770}{7812500 \,{\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)^7/(3+5*x)^3,x, algorithm="fricas")

[Out]

-1/7812500*(1139062500*x^8 + 4601812500*x^7 + 7128253125*x^6 + 4439846250*x^5 - 753519375*x^4 - 2916553500*x^3
- 1694131200*x^2 - 8148*(25*x^2 + 30*x + 9)*log(5*x + 3) - 333201880*x + 2770)/(25*x^2 + 30*x + 9)

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Sympy [A]  time = 0.128747, size = 63, normalized size = 0.86 \begin{align*} - \frac{729 x^{6}}{125} - \frac{51759 x^{5}}{3125} - \frac{181521 x^{4}}{12500} + \frac{2052 x^{3}}{3125} + \frac{129654 x^{2}}{15625} + \frac{1851147 x}{390625} - \frac{458 x + 277}{19531250 x^{2} + 23437500 x + 7031250} + \frac{2037 \log{\left (5 x + 3 \right )}}{1953125} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(2+3*x)**7/(3+5*x)**3,x)

[Out]

-729*x**6/125 - 51759*x**5/3125 - 181521*x**4/12500 + 2052*x**3/3125 + 129654*x**2/15625 + 1851147*x/390625 -
(458*x + 277)/(19531250*x**2 + 23437500*x + 7031250) + 2037*log(5*x + 3)/1953125

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Giac [A]  time = 1.67741, size = 70, normalized size = 0.96 \begin{align*} -\frac{729}{125} \, x^{6} - \frac{51759}{3125} \, x^{5} - \frac{181521}{12500} \, x^{4} + \frac{2052}{3125} \, x^{3} + \frac{129654}{15625} \, x^{2} + \frac{1851147}{390625} \, x - \frac{458 \, x + 277}{781250 \,{\left (5 \, x + 3\right )}^{2}} + \frac{2037}{1953125} \, \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)^7/(3+5*x)^3,x, algorithm="giac")

[Out]

-729/125*x^6 - 51759/3125*x^5 - 181521/12500*x^4 + 2052/3125*x^3 + 129654/15625*x^2 + 1851147/390625*x - 1/781
250*(458*x + 277)/(5*x + 3)^2 + 2037/1953125*log(abs(5*x + 3))