\(\int x^3 (1+x)^3 (1+2 x) (-18+7 x^3 (1+x)^3)^2 \, dx\) [222]

Optimal result

Integrand size = 28, antiderivative size = 33 \[ \int x^3 (1+x)^3 (1+2 x) \left (-18+7 x^3 (1+x)^3\right )^2 \, dx=81 x^4 (1+x)^4-36 x^7 (1+x)^7+\frac {49}{10} x^{10} (1+x)^{10} \]

[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 = 0.10 (sec) , antiderivative size = 96, normalized size of antiderivative = 2.91, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {1626} \[ \int x^3 (1+x)^3 (1+2 x) \left (-18+7 x^3 (1+x)^3\right )^2 \, dx=\frac {49 x^{20}}{10}+49 x^{19}+\frac {441 x^{18}}{2}+588 x^{17}+1029 x^{16}+\frac {6174 x^{15}}{5}+993 x^{14}+336 x^{13}-\frac {1071 x^{12}}{2}-1211 x^{11}-\frac {12551 x^{10}}{10}-756 x^9-171 x^8+288 x^7+486 x^6+324 x^5+81 x^4 \]

[In]

Int[x^3*(1 + x)^3*(1 + 2*x)*(-18 + 7*x^3*(1 + x)^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 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 \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 = 0.00 (sec) , antiderivative size = 96, normalized size of antiderivative = 2.91 \[ \int x^3 (1+x)^3 (1+2 x) \left (-18+7 x^3 (1+x)^3\right )^2 \, dx=81 x^4+324 x^5+486 x^6+288 x^7-171 x^8-756 x^9-\frac {12551 x^{10}}{10}-1211 x^{11}-\frac {1071 x^{12}}{2}+336 x^{13}+993 x^{14}+\frac {6174 x^{15}}{5}+1029 x^{16}+588 x^{17}+\frac {441 x^{18}}{2}+49 x^{19}+\frac {49 x^{20}}{10} \]

[In]

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

Leaf count of result is larger than twice the leaf count of optimal. \(85\) vs. \(2(31)=62\).

Time = 0.79 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.61

method	result	size
gosper	\(\frac {x^{4} \left (49 x^{16}+490 x^{15}+2205 x^{14}+5880 x^{13}+10290 x^{12}+12348 x^{11}+9930 x^{10}+3360 x^{9}-5355 x^{8}-12110 x^{7}-12551 x^{6}-7560 x^{5}-1710 x^{4}+2880 x^{3}+4860 x^{2}+3240 x +810\right )}{10}\)	\(86\)
default	\(81 x^{4}+324 x^{5}+486 x^{6}+288 x^{7}-171 x^{8}-756 x^{9}-\frac {12551}{10} x^{10}-1211 x^{11}-\frac {1071}{2} x^{12}+336 x^{13}+993 x^{14}+\frac {6174}{5} x^{15}+1029 x^{16}+588 x^{17}+\frac {441}{2} x^{18}+49 x^{19}+\frac {49}{10} x^{20}\)	\(87\)
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(x^3*(x+1)^3*(1+2*x)*(-18+7*x^3*(x+1)^3)^2,x,method=_RETURNVERBOSE)

[Out]

1/10*x^4*(49*x^16+490*x^15+2205*x^14+5880*x^13+10290*x^12+12348*x^11+9930*x^10+3360*x^9-5355*x^8-12110*x^7-125
51*x^6-7560*x^5-1710*x^4+2880*x^3+4860*x^2+3240*x+810)

Fricas [B] (verification not implemented)

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

Time = 0.26 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.61 \[ \int x^3 (1+x)^3 (1+2 x) \left (-18+7 x^3 (1+x)^3\right )^2 \, dx=\frac {49}{10} \, x^{20} + 49 \, x^{19} + \frac {441}{2} \, x^{18} + 588 \, x^{17} + 1029 \, x^{16} + \frac {6174}{5} \, x^{15} + 993 \, x^{14} + 336 \, x^{13} - \frac {1071}{2} \, x^{12} - 1211 \, x^{11} - \frac {12551}{10} \, x^{10} - 756 \, x^{9} - 171 \, x^{8} + 288 \, x^{7} + 486 \, x^{6} + 324 \, x^{5} + 81 \, x^{4} \]

[In]

integrate(x^3*(1+x)^3*(1+2*x)*(-18+7*x^3*(1+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 = 0.04 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.85 \[ \int x^3 (1+x)^3 (1+2 x) \left (-18+7 x^3 (1+x)^3\right )^2 \, dx=\frac {49 x^{20}}{10} + 49 x^{19} + \frac {441 x^{18}}{2} + 588 x^{17} + 1029 x^{16} + \frac {6174 x^{15}}{5} + 993 x^{14} + 336 x^{13} - \frac {1071 x^{12}}{2} - 1211 x^{11} - \frac {12551 x^{10}}{10} - 756 x^{9} - 171 x^{8} + 288 x^{7} + 486 x^{6} + 324 x^{5} + 81 x^{4} \]

[In]

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

[In]

integrate(x^3*(1+x)^3*(1+2*x)*(-18+7*x^3*(1+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 = 0.31 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85 \[ \int x^3 (1+x)^3 (1+2 x) \left (-18+7 x^3 (1+x)^3\right )^2 \, dx=\frac {49}{10} \, {\left (x^{2} + x\right )}^{10} - 36 \, {\left (x^{2} + x\right )}^{7} + 81 \, {\left (x^{2} + x\right )}^{4} \]

[In]

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

[In]

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