3.221 \(\int (1+2 x) (x+x^2)^3 (-18+7 (x+x^2)^3)^2 \, dx\)

Optimal. Leaf size=33 \[ \frac{49}{10} x^{10} (x+1)^{10}-36 x^7 (x+1)^7+81 x^4 (x+1)^4 \]

[Out]

81*x^4*(1 + x)^4 - 36*x^7*(1 + x)^7 + (49*x^10*(1 + x)^10)/10

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Rubi [B]  time = 0.197782, antiderivative size = 96, normalized size of antiderivative = 2.91, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {1593, 1612} \[ \frac{49 x^{20}}{10}+49 x^{19}+\frac{441 x^{18}}{2}+588 x^{17}+1029 x^{16}+\frac{6174 x^{15}}{5}+993 x^{14}+336 x^{13}-\frac{1071 x^{12}}{2}-1211 x^{11}-\frac{12551 x^{10}}{10}-756 x^9-171 x^8+288 x^7+486 x^6+324 x^5+81 x^4 \]

Antiderivative was successfully verified.

[In]

Int[(1 + 2*x)*(x + x^2)^3*(-18 + 7*(x + x^2)^3)^2,x]

[Out]

81*x^4 + 324*x^5 + 486*x^6 + 288*x^7 - 171*x^8 - 756*x^9 - (12551*x^10)/10 - 1211*x^11 - (1071*x^12)/2 + 336*x
^13 + 993*x^14 + (6174*x^15)/5 + 1029*x^16 + 588*x^17 + (441*x^18)/2 + 49*x^19 + (49*x^20)/10

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 1612

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[E
xpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Poly
Q[Px, x] && IntegersQ[m, n]

Rubi steps

\begin{align*} \int (1+2 x) \left (x+x^2\right )^3 \left (-18+7 \left (x+x^2\right )^3\right )^2 \, dx &=\int x^3 (1+x)^3 (1+2 x) \left (-18+7 \left (x+x^2\right )^3\right )^2 \, dx\\ &=\int \left (324 x^3+1620 x^4+2916 x^5+2016 x^6-1368 x^7-6804 x^8-12551 x^9-13321 x^{10}-6426 x^{11}+4368 x^{12}+13902 x^{13}+18522 x^{14}+16464 x^{15}+9996 x^{16}+3969 x^{17}+931 x^{18}+98 x^{19}\right ) \, dx\\ &=81 x^4+324 x^5+486 x^6+288 x^7-171 x^8-756 x^9-\frac{12551 x^{10}}{10}-1211 x^{11}-\frac{1071 x^{12}}{2}+336 x^{13}+993 x^{14}+\frac{6174 x^{15}}{5}+1029 x^{16}+588 x^{17}+\frac{441 x^{18}}{2}+49 x^{19}+\frac{49 x^{20}}{10}\\ \end{align*}

Mathematica [B]  time = 0.006101, size = 96, normalized size = 2.91 \[ \frac{49 x^{20}}{10}+49 x^{19}+\frac{441 x^{18}}{2}+588 x^{17}+1029 x^{16}+\frac{6174 x^{15}}{5}+993 x^{14}+336 x^{13}-\frac{1071 x^{12}}{2}-1211 x^{11}-\frac{12551 x^{10}}{10}-756 x^9-171 x^8+288 x^7+486 x^6+324 x^5+81 x^4 \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + 2*x)*(x + x^2)^3*(-18 + 7*(x + x^2)^3)^2,x]

[Out]

81*x^4 + 324*x^5 + 486*x^6 + 288*x^7 - 171*x^8 - 756*x^9 - (12551*x^10)/10 - 1211*x^11 - (1071*x^12)/2 + 336*x
^13 + 993*x^14 + (6174*x^15)/5 + 1029*x^16 + 588*x^17 + (441*x^18)/2 + 49*x^19 + (49*x^20)/10

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Maple [B]  time = 0.002, size = 87, normalized size = 2.6 \begin{align*}{\frac{49\,{x}^{20}}{10}}+49\,{x}^{19}+{\frac{441\,{x}^{18}}{2}}+588\,{x}^{17}+1029\,{x}^{16}+{\frac{6174\,{x}^{15}}{5}}+993\,{x}^{14}+336\,{x}^{13}-{\frac{1071\,{x}^{12}}{2}}-1211\,{x}^{11}-{\frac{12551\,{x}^{10}}{10}}-756\,{x}^{9}-171\,{x}^{8}+288\,{x}^{7}+486\,{x}^{6}+324\,{x}^{5}+81\,{x}^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+2*x)*(x^2+x)^3*(-18+7*(x^2+x)^3)^2,x)

[Out]

49/10*x^20+49*x^19+441/2*x^18+588*x^17+1029*x^16+6174/5*x^15+993*x^14+336*x^13-1071/2*x^12-1211*x^11-12551/10*
x^10-756*x^9-171*x^8+288*x^7+486*x^6+324*x^5+81*x^4

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Maxima [B]  time = 0.992962, size = 116, normalized size = 3.52 \begin{align*} \frac{49}{10} \, x^{20} + 49 \, x^{19} + \frac{441}{2} \, x^{18} + 588 \, x^{17} + 1029 \, x^{16} + \frac{6174}{5} \, x^{15} + 993 \, x^{14} + 336 \, x^{13} - \frac{1071}{2} \, x^{12} - 1211 \, x^{11} - \frac{12551}{10} \, x^{10} - 756 \, x^{9} - 171 \, x^{8} + 288 \, x^{7} + 486 \, x^{6} + 324 \, x^{5} + 81 \, x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)*(x^2+x)^3*(-18+7*(x^2+x)^3)^2,x, algorithm="maxima")

[Out]

49/10*x^20 + 49*x^19 + 441/2*x^18 + 588*x^17 + 1029*x^16 + 6174/5*x^15 + 993*x^14 + 336*x^13 - 1071/2*x^12 - 1
211*x^11 - 12551/10*x^10 - 756*x^9 - 171*x^8 + 288*x^7 + 486*x^6 + 324*x^5 + 81*x^4

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Fricas [B]  time = 1.1462, size = 263, normalized size = 7.97 \begin{align*} \frac{49}{10} x^{20} + 49 x^{19} + \frac{441}{2} x^{18} + 588 x^{17} + 1029 x^{16} + \frac{6174}{5} x^{15} + 993 x^{14} + 336 x^{13} - \frac{1071}{2} x^{12} - 1211 x^{11} - \frac{12551}{10} x^{10} - 756 x^{9} - 171 x^{8} + 288 x^{7} + 486 x^{6} + 324 x^{5} + 81 x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)*(x^2+x)^3*(-18+7*(x^2+x)^3)^2,x, algorithm="fricas")

[Out]

49/10*x^20 + 49*x^19 + 441/2*x^18 + 588*x^17 + 1029*x^16 + 6174/5*x^15 + 993*x^14 + 336*x^13 - 1071/2*x^12 - 1
211*x^11 - 12551/10*x^10 - 756*x^9 - 171*x^8 + 288*x^7 + 486*x^6 + 324*x^5 + 81*x^4

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Sympy [B]  time = 0.07515, size = 94, normalized size = 2.85 \begin{align*} \frac{49 x^{20}}{10} + 49 x^{19} + \frac{441 x^{18}}{2} + 588 x^{17} + 1029 x^{16} + \frac{6174 x^{15}}{5} + 993 x^{14} + 336 x^{13} - \frac{1071 x^{12}}{2} - 1211 x^{11} - \frac{12551 x^{10}}{10} - 756 x^{9} - 171 x^{8} + 288 x^{7} + 486 x^{6} + 324 x^{5} + 81 x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)*(x**2+x)**3*(-18+7*(x**2+x)**3)**2,x)

[Out]

49*x**20/10 + 49*x**19 + 441*x**18/2 + 588*x**17 + 1029*x**16 + 6174*x**15/5 + 993*x**14 + 336*x**13 - 1071*x*
*12/2 - 1211*x**11 - 12551*x**10/10 - 756*x**9 - 171*x**8 + 288*x**7 + 486*x**6 + 324*x**5 + 81*x**4

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Giac [B]  time = 1.22729, size = 116, normalized size = 3.52 \begin{align*} \frac{49}{10} \, x^{20} + 49 \, x^{19} + \frac{441}{2} \, x^{18} + 588 \, x^{17} + 1029 \, x^{16} + \frac{6174}{5} \, x^{15} + 993 \, x^{14} + 336 \, x^{13} - \frac{1071}{2} \, x^{12} - 1211 \, x^{11} - \frac{12551}{10} \, x^{10} - 756 \, x^{9} - 171 \, x^{8} + 288 \, x^{7} + 486 \, x^{6} + 324 \, x^{5} + 81 \, x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)*(x^2+x)^3*(-18+7*(x^2+x)^3)^2,x, algorithm="giac")

[Out]

49/10*x^20 + 49*x^19 + 441/2*x^18 + 588*x^17 + 1029*x^16 + 6174/5*x^15 + 993*x^14 + 336*x^13 - 1071/2*x^12 - 1
211*x^11 - 12551/10*x^10 - 756*x^9 - 171*x^8 + 288*x^7 + 486*x^6 + 324*x^5 + 81*x^4