3.1373 \(\int (1-2 x)^3 (2+3 x)^6 (3+5 x)^3 \, dx\)

Optimal. Leaf size=78 \[ -\frac{1000 (3 x+2)^{13}}{28431}+\frac{925 (3 x+2)^{12}}{2187}-\frac{14390 (3 x+2)^{11}}{8019}+\frac{66193 (3 x+2)^{10}}{21870}-\frac{10073 (3 x+2)^9}{6561}+\frac{1813 (3 x+2)^8}{5832}-\frac{49 (3 x+2)^7}{2187} \]

[Out]

(-49*(2 + 3*x)^7)/2187 + (1813*(2 + 3*x)^8)/5832 - (10073*(2 + 3*x)^9)/6561 + (66193*(2 + 3*x)^10)/21870 - (14
390*(2 + 3*x)^11)/8019 + (925*(2 + 3*x)^12)/2187 - (1000*(2 + 3*x)^13)/28431

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Rubi [A]  time = 0.0345636, antiderivative size = 78, 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} \[ -\frac{1000 (3 x+2)^{13}}{28431}+\frac{925 (3 x+2)^{12}}{2187}-\frac{14390 (3 x+2)^{11}}{8019}+\frac{66193 (3 x+2)^{10}}{21870}-\frac{10073 (3 x+2)^9}{6561}+\frac{1813 (3 x+2)^8}{5832}-\frac{49 (3 x+2)^7}{2187} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-49*(2 + 3*x)^7)/2187 + (1813*(2 + 3*x)^8)/5832 - (10073*(2 + 3*x)^9)/6561 + (66193*(2 + 3*x)^10)/21870 - (14
390*(2 + 3*x)^11)/8019 + (925*(2 + 3*x)^12)/2187 - (1000*(2 + 3*x)^13)/28431

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 (1-2 x)^3 (2+3 x)^6 (3+5 x)^3 \, dx &=\int \left (-\frac{343}{729} (2+3 x)^6+\frac{1813}{243} (2+3 x)^7-\frac{10073}{243} (2+3 x)^8+\frac{66193}{729} (2+3 x)^9-\frac{14390}{243} (2+3 x)^{10}+\frac{3700}{243} (2+3 x)^{11}-\frac{1000}{729} (2+3 x)^{12}\right ) \, dx\\ &=-\frac{49 (2+3 x)^7}{2187}+\frac{1813 (2+3 x)^8}{5832}-\frac{10073 (2+3 x)^9}{6561}+\frac{66193 (2+3 x)^{10}}{21870}-\frac{14390 (2+3 x)^{11}}{8019}+\frac{925 (2+3 x)^{12}}{2187}-\frac{1000 (2+3 x)^{13}}{28431}\\ \end{align*}

Mathematica [A]  time = 0.0026811, size = 74, normalized size = 0.95 \[ -\frac{729000 x^{13}}{13}-261225 x^{12}-\frac{5100570 x^{11}}{11}-\frac{3110589 x^{10}}{10}+122655 x^9+\frac{2623581 x^8}{8}+155453 x^7-51908 x^6-\frac{390396 x^5}{5}-20140 x^4+8688 x^3+6912 x^2+1728 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

1728*x + 6912*x^2 + 8688*x^3 - 20140*x^4 - (390396*x^5)/5 - 51908*x^6 + 155453*x^7 + (2623581*x^8)/8 + 122655*
x^9 - (3110589*x^10)/10 - (5100570*x^11)/11 - 261225*x^12 - (729000*x^13)/13

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Maple [A]  time = 0.002, size = 65, normalized size = 0.8 \begin{align*} -{\frac{729000\,{x}^{13}}{13}}-261225\,{x}^{12}-{\frac{5100570\,{x}^{11}}{11}}-{\frac{3110589\,{x}^{10}}{10}}+122655\,{x}^{9}+{\frac{2623581\,{x}^{8}}{8}}+155453\,{x}^{7}-51908\,{x}^{6}-{\frac{390396\,{x}^{5}}{5}}-20140\,{x}^{4}+8688\,{x}^{3}+6912\,{x}^{2}+1728\,x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^3*(2+3*x)^6*(3+5*x)^3,x)

[Out]

-729000/13*x^13-261225*x^12-5100570/11*x^11-3110589/10*x^10+122655*x^9+2623581/8*x^8+155453*x^7-51908*x^6-3903
96/5*x^5-20140*x^4+8688*x^3+6912*x^2+1728*x

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Maxima [A]  time = 1.90446, size = 86, normalized size = 1.1 \begin{align*} -\frac{729000}{13} \, x^{13} - 261225 \, x^{12} - \frac{5100570}{11} \, x^{11} - \frac{3110589}{10} \, x^{10} + 122655 \, x^{9} + \frac{2623581}{8} \, x^{8} + 155453 \, x^{7} - 51908 \, x^{6} - \frac{390396}{5} \, x^{5} - 20140 \, x^{4} + 8688 \, x^{3} + 6912 \, x^{2} + 1728 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^3*(2+3*x)^6*(3+5*x)^3,x, algorithm="maxima")

[Out]

-729000/13*x^13 - 261225*x^12 - 5100570/11*x^11 - 3110589/10*x^10 + 122655*x^9 + 2623581/8*x^8 + 155453*x^7 -
51908*x^6 - 390396/5*x^5 - 20140*x^4 + 8688*x^3 + 6912*x^2 + 1728*x

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Fricas [A]  time = 1.16562, size = 242, normalized size = 3.1 \begin{align*} -\frac{729000}{13} x^{13} - 261225 x^{12} - \frac{5100570}{11} x^{11} - \frac{3110589}{10} x^{10} + 122655 x^{9} + \frac{2623581}{8} x^{8} + 155453 x^{7} - 51908 x^{6} - \frac{390396}{5} x^{5} - 20140 x^{4} + 8688 x^{3} + 6912 x^{2} + 1728 x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^3*(2+3*x)^6*(3+5*x)^3,x, algorithm="fricas")

[Out]

-729000/13*x^13 - 261225*x^12 - 5100570/11*x^11 - 3110589/10*x^10 + 122655*x^9 + 2623581/8*x^8 + 155453*x^7 -
51908*x^6 - 390396/5*x^5 - 20140*x^4 + 8688*x^3 + 6912*x^2 + 1728*x

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Sympy [A]  time = 0.074929, size = 71, normalized size = 0.91 \begin{align*} - \frac{729000 x^{13}}{13} - 261225 x^{12} - \frac{5100570 x^{11}}{11} - \frac{3110589 x^{10}}{10} + 122655 x^{9} + \frac{2623581 x^{8}}{8} + 155453 x^{7} - 51908 x^{6} - \frac{390396 x^{5}}{5} - 20140 x^{4} + 8688 x^{3} + 6912 x^{2} + 1728 x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**3*(2+3*x)**6*(3+5*x)**3,x)

[Out]

-729000*x**13/13 - 261225*x**12 - 5100570*x**11/11 - 3110589*x**10/10 + 122655*x**9 + 2623581*x**8/8 + 155453*
x**7 - 51908*x**6 - 390396*x**5/5 - 20140*x**4 + 8688*x**3 + 6912*x**2 + 1728*x

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Giac [A]  time = 2.74099, size = 86, normalized size = 1.1 \begin{align*} -\frac{729000}{13} \, x^{13} - 261225 \, x^{12} - \frac{5100570}{11} \, x^{11} - \frac{3110589}{10} \, x^{10} + 122655 \, x^{9} + \frac{2623581}{8} \, x^{8} + 155453 \, x^{7} - 51908 \, x^{6} - \frac{390396}{5} \, x^{5} - 20140 \, x^{4} + 8688 \, x^{3} + 6912 \, x^{2} + 1728 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^3*(2+3*x)^6*(3+5*x)^3,x, algorithm="giac")

[Out]

-729000/13*x^13 - 261225*x^12 - 5100570/11*x^11 - 3110589/10*x^10 + 122655*x^9 + 2623581/8*x^8 + 155453*x^7 -
51908*x^6 - 390396/5*x^5 - 20140*x^4 + 8688*x^3 + 6912*x^2 + 1728*x