3.15.21 \(\int \frac {(1-2 x)^3 (2+3 x)^4}{(3+5 x)^3} \, dx\) [1421]

Optimal. Leaf size=66 \[ \frac {4691 x}{15625}-\frac {7617 x^2}{6250}+\frac {2826 x^3}{3125}+\frac {513 x^4}{625}-\frac {648 x^5}{625}-\frac {1331}{781250 (3+5 x)^2}-\frac {15246}{390625 (3+5 x)}+\frac {63294 \log (3+5 x)}{390625} \]

[Out]

4691/15625*x-7617/6250*x^2+2826/3125*x^3+513/625*x^4-648/625*x^5-1331/781250/(3+5*x)^2-15246/390625/(3+5*x)+63
294/390625*ln(3+5*x)

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Rubi [A]
time = 0.02, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {90} \begin {gather*} -\frac {648 x^5}{625}+\frac {513 x^4}{625}+\frac {2826 x^3}{3125}-\frac {7617 x^2}{6250}+\frac {4691 x}{15625}-\frac {15246}{390625 (5 x+3)}-\frac {1331}{781250 (5 x+3)^2}+\frac {63294 \log (5 x+3)}{390625} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^3*(2 + 3*x)^4)/(3 + 5*x)^3,x]

[Out]

(4691*x)/15625 - (7617*x^2)/6250 + (2826*x^3)/3125 + (513*x^4)/625 - (648*x^5)/625 - 1331/(781250*(3 + 5*x)^2)
- 15246/(390625*(3 + 5*x)) + (63294*Log[3 + 5*x])/390625

Rule 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin {align*} \int \frac {(1-2 x)^3 (2+3 x)^4}{(3+5 x)^3} \, dx &=\int \left (\frac {4691}{15625}-\frac {7617 x}{3125}+\frac {8478 x^2}{3125}+\frac {2052 x^3}{625}-\frac {648 x^4}{125}+\frac {1331}{78125 (3+5 x)^3}+\frac {15246}{78125 (3+5 x)^2}+\frac {63294}{78125 (3+5 x)}\right ) \, dx\\ &=\frac {4691 x}{15625}-\frac {7617 x^2}{6250}+\frac {2826 x^3}{3125}+\frac {513 x^4}{625}-\frac {648 x^5}{625}-\frac {1331}{781250 (3+5 x)^2}-\frac {15246}{390625 (3+5 x)}+\frac {63294 \log (3+5 x)}{390625}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 63, normalized size = 0.95 \begin {gather*} \frac {21586298+83293560 x+53587800 x^2-81707500 x^3+15815625 x^4+148050000 x^5-41343750 x^6-101250000 x^7+632940 (3+5 x)^2 \log (6 (3+5 x))}{3906250 (3+5 x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^3*(2 + 3*x)^4)/(3 + 5*x)^3,x]

[Out]

(21586298 + 83293560*x + 53587800*x^2 - 81707500*x^3 + 15815625*x^4 + 148050000*x^5 - 41343750*x^6 - 101250000
*x^7 + 632940*(3 + 5*x)^2*Log[6*(3 + 5*x)])/(3906250*(3 + 5*x)^2)

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Maple [A]
time = 0.11, size = 51, normalized size = 0.77

method	result	size
risch	\(-\frac {648 x^{5}}{625}+\frac {513 x^{4}}{625}+\frac {2826 x^{3}}{3125}-\frac {7617 x^{2}}{6250}+\frac {4691 x}{15625}+\frac {-\frac {15246 x}{78125}-\frac {92807}{781250}}{\left (3+5 x \right )^{2}}+\frac {63294 \ln \left (3+5 x \right )}{390625}\)	\(47\)
default	\(\frac {4691 x}{15625}-\frac {7617 x^{2}}{6250}+\frac {2826 x^{3}}{3125}+\frac {513 x^{4}}{625}-\frac {648 x^{5}}{625}-\frac {1331}{781250 \left (3+5 x \right )^{2}}-\frac {15246}{390625 \left (3+5 x \right )}+\frac {63294 \ln \left (3+5 x \right )}{390625}\)	\(51\)
norman	\(\frac {\frac {680354}{234375} x -\frac {229469}{140625} x^{2}-\frac {65366}{3125} x^{3}+\frac {5061}{1250} x^{4}+\frac {23688}{625} x^{5}-\frac {1323}{125} x^{6}-\frac {648}{25} x^{7}}{\left (3+5 x \right )^{2}}+\frac {63294 \ln \left (3+5 x \right )}{390625}\)	\(52\)
meijerg	\(\frac {8 x \left (\frac {5 x}{3}+2\right )}{27 \left (1+\frac {5 x}{3}\right )^{2}}+\frac {28 x \left (15 x +6\right )}{75 \left (1+\frac {5 x}{3}\right )^{2}}+\frac {63294 \ln \left (1+\frac {5 x}{3}\right )}{390625}-\frac {14 x \left (\frac {100}{9} x^{2}+30 x +12\right )}{125 \left (1+\frac {5 x}{3}\right )^{2}}-\frac {1827 x \left (-\frac {625}{27} x^{3}+\frac {500}{9} x^{2}+150 x +60\right )}{6250 \left (1+\frac {5 x}{3}\right )^{2}}+\frac {567 x \left (\frac {1250}{81} x^{4}-\frac {625}{27} x^{3}+\frac {500}{9} x^{2}+150 x +60\right )}{3125 \left (1+\frac {5 x}{3}\right )^{2}}+\frac {729 x \left (-\frac {21875}{243} x^{5}+\frac {8750}{81} x^{4}-\frac {4375}{27} x^{3}+\frac {3500}{9} x^{2}+1050 x +420\right )}{15625 \left (1+\frac {5 x}{3}\right )^{2}}-\frac {6561 x \left (\frac {125000}{729} x^{6}-\frac {43750}{243} x^{5}+\frac {17500}{81} x^{4}-\frac {8750}{27} x^{3}+\frac {7000}{9} x^{2}+2100 x +840\right )}{390625 \left (1+\frac {5 x}{3}\right )^{2}}\)	\(190\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^3*(2+3*x)^4/(3+5*x)^3,x,method=_RETURNVERBOSE)

[Out]

4691/15625*x-7617/6250*x^2+2826/3125*x^3+513/625*x^4-648/625*x^5-1331/781250/(3+5*x)^2-15246/390625/(3+5*x)+63
294/390625*ln(3+5*x)

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Maxima [A]
time = 0.28, size = 51, normalized size = 0.77 \begin {gather*} -\frac {648}{625} \, x^{5} + \frac {513}{625} \, x^{4} + \frac {2826}{3125} \, x^{3} - \frac {7617}{6250} \, x^{2} + \frac {4691}{15625} \, x - \frac {121 \, {\left (1260 \, x + 767\right )}}{781250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {63294}{390625} \, \log \left (5 \, x + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^3*(2+3*x)^4/(3+5*x)^3,x, algorithm="maxima")

[Out]

-648/625*x^5 + 513/625*x^4 + 2826/3125*x^3 - 7617/6250*x^2 + 4691/15625*x - 121/781250*(1260*x + 767)/(25*x^2
+ 30*x + 9) + 63294/390625*log(5*x + 3)

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Fricas [A]
time = 0.38, size = 67, normalized size = 1.02 \begin {gather*} -\frac {20250000 \, x^{7} + 8268750 \, x^{6} - 29610000 \, x^{5} - 3163125 \, x^{4} + 16341500 \, x^{3} + 1532625 \, x^{2} - 126588 \, {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) - 1958490 \, x + 92807}{781250 \, {\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)^3*(2+3*x)^4/(3+5*x)^3,x, algorithm="fricas")

[Out]

-1/781250*(20250000*x^7 + 8268750*x^6 - 29610000*x^5 - 3163125*x^4 + 16341500*x^3 + 1532625*x^2 - 126588*(25*x
^2 + 30*x + 9)*log(5*x + 3) - 1958490*x + 92807)/(25*x^2 + 30*x + 9)

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Sympy [A]
time = 0.05, size = 56, normalized size = 0.85 \begin {gather*} - \frac {648 x^{5}}{625} + \frac {513 x^{4}}{625} + \frac {2826 x^{3}}{3125} - \frac {7617 x^{2}}{6250} + \frac {4691 x}{15625} - \frac {152460 x + 92807}{19531250 x^{2} + 23437500 x + 7031250} + \frac {63294 \log {\left (5 x + 3 \right )}}{390625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**3*(2+3*x)**4/(3+5*x)**3,x)

[Out]

-648*x**5/625 + 513*x**4/625 + 2826*x**3/3125 - 7617*x**2/6250 + 4691*x/15625 - (152460*x + 92807)/(19531250*x
**2 + 23437500*x + 7031250) + 63294*log(5*x + 3)/390625

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Giac [A]
time = 0.72, size = 47, normalized size = 0.71 \begin {gather*} -\frac {648}{625} \, x^{5} + \frac {513}{625} \, x^{4} + \frac {2826}{3125} \, x^{3} - \frac {7617}{6250} \, x^{2} + \frac {4691}{15625} \, x - \frac {121 \, {\left (1260 \, x + 767\right )}}{781250 \, {\left (5 \, x + 3\right )}^{2}} + \frac {63294}{390625} \, \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)^3*(2+3*x)^4/(3+5*x)^3,x, algorithm="giac")

[Out]

-648/625*x^5 + 513/625*x^4 + 2826/3125*x^3 - 7617/6250*x^2 + 4691/15625*x - 121/781250*(1260*x + 767)/(5*x + 3
)^2 + 63294/390625*log(abs(5*x + 3))

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Mupad [B]
time = 0.03, size = 47, normalized size = 0.71 \begin {gather*} \frac {4691\,x}{15625}+\frac {63294\,\ln \left (x+\frac {3}{5}\right )}{390625}-\frac {\frac {15246\,x}{1953125}+\frac {92807}{19531250}}{x^2+\frac {6\,x}{5}+\frac {9}{25}}-\frac {7617\,x^2}{6250}+\frac {2826\,x^3}{3125}+\frac {513\,x^4}{625}-\frac {648\,x^5}{625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((2*x - 1)^3*(3*x + 2)^4)/(5*x + 3)^3,x)

[Out]

(4691*x)/15625 + (63294*log(x + 3/5))/390625 - ((15246*x)/1953125 + 92807/19531250)/((6*x)/5 + x^2 + 9/25) - (
7617*x^2)/6250 + (2826*x^3)/3125 + (513*x^4)/625 - (648*x^5)/625

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