3.1418 \(\int \frac{(1-2 x)^3 (2+3 x)^7}{(3+5 x)^3} \, dx\)

Optimal. Leaf size=87 \[ -\frac{2187 x^8}{125}-\frac{119556 x^7}{4375}+\frac{33291 x^6}{3125}+\frac{491913 x^5}{15625}+\frac{6507 x^4}{62500}-\frac{5918904 x^3}{390625}-\frac{2300646 x^2}{1953125}+\frac{46214407 x}{9765625}-\frac{1089}{1953125 (5 x+3)}-\frac{1331}{97656250 (5 x+3)^2}+\frac{47289 \log (5 x+3)}{9765625} \]

[Out]

(46214407*x)/9765625 - (2300646*x^2)/1953125 - (5918904*x^3)/390625 + (6507*x^4)/62500 + (491913*x^5)/15625 +
(33291*x^6)/3125 - (119556*x^7)/4375 - (2187*x^8)/125 - 1331/(97656250*(3 + 5*x)^2) - 1089/(1953125*(3 + 5*x))
+ (47289*Log[3 + 5*x])/9765625

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Rubi [A]  time = 0.0478469, antiderivative size = 87, normalized size of antiderivative = 1., 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 = {88} \[ -\frac{2187 x^8}{125}-\frac{119556 x^7}{4375}+\frac{33291 x^6}{3125}+\frac{491913 x^5}{15625}+\frac{6507 x^4}{62500}-\frac{5918904 x^3}{390625}-\frac{2300646 x^2}{1953125}+\frac{46214407 x}{9765625}-\frac{1089}{1953125 (5 x+3)}-\frac{1331}{97656250 (5 x+3)^2}+\frac{47289 \log (5 x+3)}{9765625} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(46214407*x)/9765625 - (2300646*x^2)/1953125 - (5918904*x^3)/390625 + (6507*x^4)/62500 + (491913*x^5)/15625 +
(33291*x^6)/3125 - (119556*x^7)/4375 - (2187*x^8)/125 - 1331/(97656250*(3 + 5*x)^2) - 1089/(1953125*(3 + 5*x))
+ (47289*Log[3 + 5*x])/9765625

Rule 88

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)^7}{(3+5 x)^3} \, dx &=\int \left (\frac{46214407}{9765625}-\frac{4601292 x}{1953125}-\frac{17756712 x^2}{390625}+\frac{6507 x^3}{15625}+\frac{491913 x^4}{3125}+\frac{199746 x^5}{3125}-\frac{119556 x^6}{625}-\frac{17496 x^7}{125}+\frac{1331}{9765625 (3+5 x)^3}+\frac{1089}{390625 (3+5 x)^2}+\frac{47289}{1953125 (3+5 x)}\right ) \, dx\\ &=\frac{46214407 x}{9765625}-\frac{2300646 x^2}{1953125}-\frac{5918904 x^3}{390625}+\frac{6507 x^4}{62500}+\frac{491913 x^5}{15625}+\frac{33291 x^6}{3125}-\frac{119556 x^7}{4375}-\frac{2187 x^8}{125}-\frac{1331}{97656250 (3+5 x)^2}-\frac{1089}{1953125 (3+5 x)}+\frac{47289 \log (3+5 x)}{9765625}\\ \end{align*}

Mathematica [A]  time = 0.0306184, size = 76, normalized size = 0.87 \[ \frac{-598007812500 x^{10}-1651640625000 x^9-972000000000 x^8+1176752812500 x^7+1425913453125 x^6-126252393750 x^5-660465159375 x^4-73008617500 x^3+229405636575 x^2+117985377690 x+6620460 (5 x+3)^2 \log (5 x+3)[phone]}{1367187500 (5 x+3)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(17925405377 + 117985377690*x + 229405636575*x^2 - 73008617500*x^3 - 660465159375*x^4 - 126252393750*x^5 + 142
5913453125*x^6 + 1176752812500*x^7 - 972000000000*x^8 - 1651640625000*x^9 - 598007812500*x^10 + 6620460*(3 + 5
*x)^2*Log[3 + 5*x])/(1367187500*(3 + 5*x)^2)

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Maple [A]  time = 0.006, size = 66, normalized size = 0.8 \begin{align*}{\frac{46214407\,x}{9765625}}-{\frac{2300646\,{x}^{2}}{1953125}}-{\frac{5918904\,{x}^{3}}{390625}}+{\frac{6507\,{x}^{4}}{62500}}+{\frac{491913\,{x}^{5}}{15625}}+{\frac{33291\,{x}^{6}}{3125}}-{\frac{119556\,{x}^{7}}{4375}}-{\frac{2187\,{x}^{8}}{125}}-{\frac{1331}{97656250\, \left ( 3+5\,x \right ) ^{2}}}-{\frac{1089}{5859375+9765625\,x}}+{\frac{47289\,\ln \left ( 3+5\,x \right ) }{9765625}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

46214407/9765625*x-2300646/1953125*x^2-5918904/390625*x^3+6507/62500*x^4+491913/15625*x^5+33291/3125*x^6-11955
6/4375*x^7-2187/125*x^8-1331/97656250/(3+5*x)^2-1089/1953125/(3+5*x)+47289/9765625*ln(3+5*x)

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Maxima [A]  time = 1.01859, size = 89, normalized size = 1.02 \begin{align*} -\frac{2187}{125} \, x^{8} - \frac{119556}{4375} \, x^{7} + \frac{33291}{3125} \, x^{6} + \frac{491913}{15625} \, x^{5} + \frac{6507}{62500} \, x^{4} - \frac{5918904}{390625} \, x^{3} - \frac{2300646}{1953125} \, x^{2} + \frac{46214407}{9765625} \, x - \frac{121 \,{\left (2250 \, x + 1361\right )}}{97656250 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{47289}{9765625} \, \log \left (5 \, x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^3*(2+3*x)^7/(3+5*x)^3,x, algorithm="maxima")

[Out]

-2187/125*x^8 - 119556/4375*x^7 + 33291/3125*x^6 + 491913/15625*x^5 + 6507/62500*x^4 - 5918904/390625*x^3 - 23
00646/1953125*x^2 + 46214407/9765625*x - 121/97656250*(2250*x + 1361)/(25*x^2 + 30*x + 9) + 47289/9765625*log(
5*x + 3)

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Fricas [A]  time = 1.3692, size = 375, normalized size = 4.31 \begin{align*} -\frac{598007812500 \, x^{10} + 1651640625000 \, x^{9} + 972000000000 \, x^{8} - 1176752812500 \, x^{7} - 1425913453125 \, x^{6} + 126252393750 \, x^{5} + 660465159375 \, x^{4} + 73008617500 \, x^{3} - 179606439600 \, x^{2} - 6620460 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) - 58226341320 \, x + 2305534}{1367187500 \,{\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)^3*(2+3*x)^7/(3+5*x)^3,x, algorithm="fricas")

[Out]

-1/1367187500*(598007812500*x^10 + 1651640625000*x^9 + 972000000000*x^8 - 1176752812500*x^7 - 1425913453125*x^
6 + 126252393750*x^5 + 660465159375*x^4 + 73008617500*x^3 - 179606439600*x^2 - 6620460*(25*x^2 + 30*x + 9)*log
(5*x + 3) - 58226341320*x + 2305534)/(25*x^2 + 30*x + 9)

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Sympy [A]  time = 0.132983, size = 76, normalized size = 0.87 \begin{align*} - \frac{2187 x^{8}}{125} - \frac{119556 x^{7}}{4375} + \frac{33291 x^{6}}{3125} + \frac{491913 x^{5}}{15625} + \frac{6507 x^{4}}{62500} - \frac{5918904 x^{3}}{390625} - \frac{2300646 x^{2}}{1953125} + \frac{46214407 x}{9765625} - \frac{272250 x + 164681}{2441406250 x^{2} + 2929687500 x + 878906250} + \frac{47289 \log{\left (5 x + 3 \right )}}{9765625} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2187*x**8/125 - 119556*x**7/4375 + 33291*x**6/3125 + 491913*x**5/15625 + 6507*x**4/62500 - 5918904*x**3/39062
5 - 2300646*x**2/1953125 + 46214407*x/9765625 - (272250*x + 164681)/(2441406250*x**2 + 2929687500*x + 87890625
0) + 47289*log(5*x + 3)/9765625

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Giac [A]  time = 1.6004, size = 84, normalized size = 0.97 \begin{align*} -\frac{2187}{125} \, x^{8} - \frac{119556}{4375} \, x^{7} + \frac{33291}{3125} \, x^{6} + \frac{491913}{15625} \, x^{5} + \frac{6507}{62500} \, x^{4} - \frac{5918904}{390625} \, x^{3} - \frac{2300646}{1953125} \, x^{2} + \frac{46214407}{9765625} \, x - \frac{121 \,{\left (2250 \, x + 1361\right )}}{97656250 \,{\left (5 \, x + 3\right )}^{2}} + \frac{47289}{9765625} \, \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)^3*(2+3*x)^7/(3+5*x)^3,x, algorithm="giac")

[Out]

-2187/125*x^8 - 119556/4375*x^7 + 33291/3125*x^6 + 491913/15625*x^5 + 6507/62500*x^4 - 5918904/390625*x^3 - 23
00646/1953125*x^2 + 46214407/9765625*x - 121/97656250*(2250*x + 1361)/(5*x + 3)^2 + 47289/9765625*log(abs(5*x
+ 3))