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

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

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Rubi [A]  time = 0.04, antiderivative size = 73, 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} \begin {gather*} -\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} \end {gather*}

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*}

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Mathematica [A]  time = 0.03, size = 64, normalized size = 0.88 \begin {gather*} \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} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(1-2 x) (2+3 x)^7}{(3+5 x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.08, size = 72, normalized size = 0.99 \begin {gather*} -\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 {gather*}

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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giac [A]  time = 1.19, size = 52, normalized size = 0.71 \begin {gather*} -\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 {gather*}

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))

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maple [A]  time = 0.01, size = 56, normalized size = 0.77 \begin {gather*} -\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 {2037 \ln \left (5 x +3\right )}{1953125}-\frac {11}{3906250 \left (5 x +3\right )^{2}}-\frac {229}{1953125 \left (5 x +3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)*(3*x+2)^7/(5*x+3)^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/(5*x+3)
^2-229/1953125/(5*x+3)+2037/1953125*ln(5*x+3)

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maxima [A]  time = 0.48, size = 56, normalized size = 0.77 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 0.04, size = 52, normalized size = 0.71 \begin {gather*} \frac {1851147\,x}{390625}+\frac {2037\,\ln \left (x+\frac {3}{5}\right )}{1953125}-\frac {\frac {229\,x}{9765625}+\frac {277}{19531250}}{x^2+\frac {6\,x}{5}+\frac {9}{25}}+\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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1851147*x)/390625 + (2037*log(x + 3/5))/1953125 - ((229*x)/9765625 + 277/19531250)/((6*x)/5 + x^2 + 9/25) + (
129654*x^2)/15625 + (2052*x^3)/3125 - (181521*x^4)/12500 - (51759*x^5)/3125 - (729*x^6)/125

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sympy [A]  time = 0.14, size = 63, normalized size = 0.86 \begin {gather*} - \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 {gather*}

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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