\(\int (1+2 x) (x+x^2)^3 (-18+7 (x+x^2)^3)^2 \, dx\) [221]

Optimal result

Integrand size = 26, antiderivative size = 33 \[ \int (1+2 x) \left (x+x^2\right )^3 \left (-18+7 \left (x+x^2\right )^3\right )^2 \, dx=81 x^4 (1+x)^4-36 x^7 (1+x)^7+\frac {49}{10} x^{10} (1+x)^{10} \]

[Out]

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

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(96\) vs. \(2(33)=66\).

Time = 0.13 (sec) , antiderivative size = 96, normalized size of antiderivative = 2.91, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1607, 1626} \[ \int (1+2 x) \left (x+x^2\right )^3 \left (-18+7 \left (x+x^2\right )^3\right )^2 \, dx=\frac {49 x^{20}}{10}+49 x^{19}+\frac {441 x^{18}}{2}+588 x^{17}+1029 x^{16}+\frac {6174 x^{15}}{5}+993 x^{14}+336 x^{13}-\frac {1071 x^{12}}{2}-1211 x^{11}-\frac {12551 x^{10}}{10}-756 x^9-171 x^8+288 x^7+486 x^6+324 x^5+81 x^4 \]

[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 1607

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 1626

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*} \text {integral}& = \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] (verified)

Leaf count is larger than twice the leaf count of optimal. \(96\) vs. \(2(33)=66\).

Time = 0.01 (sec) , antiderivative size = 96, normalized size of antiderivative = 2.91 \[ \int (1+2 x) \left (x+x^2\right )^3 \left (-18+7 \left (x+x^2\right )^3\right )^2 \, dx=81 x^4+324 x^5+486 x^6+288 x^7-171 x^8-756 x^9-\frac {12551 x^{10}}{10}-1211 x^{11}-\frac {1071 x^{12}}{2}+336 x^{13}+993 x^{14}+\frac {6174 x^{15}}{5}+1029 x^{16}+588 x^{17}+\frac {441 x^{18}}{2}+49 x^{19}+\frac {49 x^{20}}{10} \]

[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

Maple [A] (verified)

Time = 0.78 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.88

method	result	size
default	\(\frac {49 \left (x^{2}+x \right )^{10}}{10}-36 \left (x^{2}+x \right )^{7}+81 \left (x^{2}+x \right )^{4}\)	\(29\)
gosper	\(\frac {\left (x +1\right )^{3} \left (49 x^{13}+343 x^{12}+1029 x^{11}+1715 x^{10}+1715 x^{9}+1029 x^{8}-17 x^{7}-1391 x^{6}-2160 x^{5}-1440 x^{4}-360 x^{3}+810 x +810\right ) x^{4}}{10}\)	\(71\)
norman	\(81 x^{4}+324 x^{5}+486 x^{6}+288 x^{7}-171 x^{8}-756 x^{9}-\frac {12551}{10} x^{10}-1211 x^{11}-\frac {1071}{2} x^{12}+336 x^{13}+993 x^{14}+\frac {6174}{5} x^{15}+1029 x^{16}+588 x^{17}+\frac {441}{2} x^{18}+49 x^{19}+\frac {49}{10} x^{20}\)	\(87\)
risch	\(81 x^{4}+324 x^{5}+486 x^{6}+288 x^{7}-171 x^{8}-756 x^{9}-\frac {12551}{10} x^{10}-1211 x^{11}-\frac {1071}{2} x^{12}+336 x^{13}+993 x^{14}+\frac {6174}{5} x^{15}+1029 x^{16}+588 x^{17}+\frac {441}{2} x^{18}+49 x^{19}+\frac {49}{10} x^{20}\)	\(87\)
parallelrisch	\(81 x^{4}+324 x^{5}+486 x^{6}+288 x^{7}-171 x^{8}-756 x^{9}-\frac {12551}{10} x^{10}-1211 x^{11}-\frac {1071}{2} x^{12}+336 x^{13}+993 x^{14}+\frac {6174}{5} x^{15}+1029 x^{16}+588 x^{17}+\frac {441}{2} x^{18}+49 x^{19}+\frac {49}{10} x^{20}\)	\(87\)

[In]

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

[Out]

49/10*(x^2+x)^10-36*(x^2+x)^7+81*(x^2+x)^4

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (31) = 62\).

Time = 0.25 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.61 \[ \int (1+2 x) \left (x+x^2\right )^3 \left (-18+7 \left (x+x^2\right )^3\right )^2 \, dx=\frac {49}{10} \, x^{20} + 49 \, x^{19} + \frac {441}{2} \, x^{18} + 588 \, x^{17} + 1029 \, x^{16} + \frac {6174}{5} \, x^{15} + 993 \, x^{14} + 336 \, x^{13} - \frac {1071}{2} \, x^{12} - 1211 \, x^{11} - \frac {12551}{10} \, x^{10} - 756 \, x^{9} - 171 \, x^{8} + 288 \, x^{7} + 486 \, x^{6} + 324 \, x^{5} + 81 \, x^{4} \]

[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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (31) = 62\).

Time = 0.03 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.85 \[ \int (1+2 x) \left (x+x^2\right )^3 \left (-18+7 \left (x+x^2\right )^3\right )^2 \, dx=\frac {49 x^{20}}{10} + 49 x^{19} + \frac {441 x^{18}}{2} + 588 x^{17} + 1029 x^{16} + \frac {6174 x^{15}}{5} + 993 x^{14} + 336 x^{13} - \frac {1071 x^{12}}{2} - 1211 x^{11} - \frac {12551 x^{10}}{10} - 756 x^{9} - 171 x^{8} + 288 x^{7} + 486 x^{6} + 324 x^{5} + 81 x^{4} \]

[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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (31) = 62\).

Time = 0.19 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.61 \[ \int (1+2 x) \left (x+x^2\right )^3 \left (-18+7 \left (x+x^2\right )^3\right )^2 \, dx=\frac {49}{10} \, x^{20} + 49 \, x^{19} + \frac {441}{2} \, x^{18} + 588 \, x^{17} + 1029 \, x^{16} + \frac {6174}{5} \, x^{15} + 993 \, x^{14} + 336 \, x^{13} - \frac {1071}{2} \, x^{12} - 1211 \, x^{11} - \frac {12551}{10} \, x^{10} - 756 \, x^{9} - 171 \, x^{8} + 288 \, x^{7} + 486 \, x^{6} + 324 \, x^{5} + 81 \, x^{4} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85 \[ \int (1+2 x) \left (x+x^2\right )^3 \left (-18+7 \left (x+x^2\right )^3\right )^2 \, dx=\frac {49}{10} \, {\left (x^{2} + x\right )}^{10} - 36 \, {\left (x^{2} + x\right )}^{7} + 81 \, {\left (x^{2} + x\right )}^{4} \]

[In]

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

[Out]

49/10*(x^2 + x)^10 - 36*(x^2 + x)^7 + 81*(x^2 + x)^4

Mupad [B] (verification not implemented)

Time = 9.26 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.61 \[ \int (1+2 x) \left (x+x^2\right )^3 \left (-18+7 \left (x+x^2\right )^3\right )^2 \, dx=\frac {49\,x^{20}}{10}+49\,x^{19}+\frac {441\,x^{18}}{2}+588\,x^{17}+1029\,x^{16}+\frac {6174\,x^{15}}{5}+993\,x^{14}+336\,x^{13}-\frac {1071\,x^{12}}{2}-1211\,x^{11}-\frac {12551\,x^{10}}{10}-756\,x^9-171\,x^8+288\,x^7+486\,x^6+324\,x^5+81\,x^4 \]

[In]

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