Integrand size = 22, antiderivative size = 18 \[ \int \frac {-260+2376 (20-9 x)^{22}+117 x}{-20+9 x} \, dx=-2+x+12 \left (x+(5 (4-2 x)+x)^{22}\right ) \]
Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {-260+2376 (20-9 x)^{22}+117 x}{-20+9 x} \, dx=-\frac {260}{9}+12 (20-9 x)^{22}+13 x \]
Leaf count is larger than twice the leaf count of optimal. \(109\) vs. \(2(18)=36\).
Time = 1.15 (sec) , antiderivative size = 109, normalized size of antiderivative = 6.06, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2019, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {2376 (20-9 x)^{22}+117 x-260}{9 x-20} \, dx\) |
| \(\Big \downarrow \) 2019 |
| \(\displaystyle \int \left (259979518176473365480584 x^{21}-12132377514902090389093920 x^{20}+269608389220046453090976000 x^{19}-3794488440874727858317440000 x^{18}+37944884408747278583174400000 x^{17}-286694682199423882628428800000 x^{16}+1698931450070660045205504000000 x^{15}-8090149762241238310502400000000 x^{14}+31461693519827037874176000000000 x^{13}-100988152038950985768960000000000 x^{12}+269301738770535962050560000000000 x^{11}-598448308378968804556800000000000 x^{10}+1108237608109201489920000000000000 x^9-1704980935552617676800000000000000 x^8+2165055156257292288000000000000000 x^7-2245242384266821632000000000000000 x^6+1871035320222351360000000000000000 x^5-1222898902106112000000000000000000 x^4+603900692398080000000000000000000 x^3-211894979788800000000000000000000 x^2+47087773286400000000000000000000 x-4982833151999999999999999999987\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 11817250826203334794572 x^{22}-577732262614385256623520 x^{21}+13480419461002322654548800 x^{20}-199709917940775150437760000 x^{19}+2108049133819293254620800000 x^{18}-16864393070554346036966400000 x^{17}+106183215629416252825344000000 x^{16}-539343317482749220700160000000 x^{15}+2247263822844788419584000000000 x^{14}-7768319387611614289920000000000 x^{13}+22441811564211330170880000000000 x^{12}-54404391670815345868800000000000 x^{11}+110823760810920148992000000000000 x^{10}-189442326172513075200000000000000 x^9+270631894532161536000000000000000 x^8-320748912038117376000000000000000 x^7+311839220037058560000000000000000 x^6-244579780421222400000000000000000 x^5+150975173099520000000000000000000 x^4-70631659929600000000000000000000 x^3+23543886643200000000000000000000 x^2-4982833151999999999999999999987 x\) |
-4982833151999999999999999999987*x + 23543886643200000000000000000000*x^2 - 70631659929600000000000000000000*x^3 + 150975173099520000000000000000000 *x^4 - 244579780421222400000000000000000*x^5 + 311839220037058560000000000 000000*x^6 - 320748912038117376000000000000000*x^7 + 270631894532161536000 000000000000*x^8 - 189442326172513075200000000000000*x^9 + 110823760810920 148992000000000000*x^10 - 54404391670815345868800000000000*x^11 + 22441811 564211330170880000000000*x^12 - 7768319387611614289920000000000*x^13 + 224 7263822844788419584000000000*x^14 - 539343317482749220700160000000*x^15 + 106183215629416252825344000000*x^16 - 16864393070554346036966400000*x^17 + 2108049133819293254620800000*x^18 - 199709917940775150437760000*x^19 + 13 480419461002322654548800*x^20 - 577732262614385256623520*x^21 + 1181725082 6203334794572*x^22
3.17.62.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px , Qx, x]^p*Qx^(p + q), x] /; FreeQ[q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(107\) vs. \(2(14)=28\).
Time = 0.33 (sec) , antiderivative size = 108, normalized size of antiderivative = 6.00
| method | result | size |
| gosper | \(x \left (11817250826203334794572 x^{21}-577732262614385256623520 x^{20}+13480419461002322654548800 x^{19}-199709917940775150437760000 x^{18}+2108049133819293254620800000 x^{17}-16864393070554346036966400000 x^{16}+106183215629416252825344000000 x^{15}-539343317482749220700160000000 x^{14}+2247263822844788419584000000000 x^{13}-7768319387611614289920000000000 x^{12}+22441811564211330170880000000000 x^{11}-54404391670815345868800000000000 x^{10}+110823760810920148992000000000000 x^{9}-189442326172513075200000000000000 x^{8}+270631894532161536000000000000000 x^{7}-320748912038117376000000000000000 x^{6}+311839220037058560000000000000000 x^{5}-244579780421222400000000000000000 x^{4}+150975173099520000000000000000000 x^{3}-70631659929600000000000000000000 x^{2}+23543886643200000000000000000000 x -4982833151999999999999999999987\right )\) | \(108\) |
| default | \(11817250826203334794572 x^{22}-577732262614385256623520 x^{21}+13480419461002322654548800 x^{20}-199709917940775150437760000 x^{19}+2108049133819293254620800000 x^{18}-16864393070554346036966400000 x^{17}+106183215629416252825344000000 x^{16}-539343317482749220700160000000 x^{15}+2247263822844788419584000000000 x^{14}-7768319387611614289920000000000 x^{13}+22441811564211330170880000000000 x^{12}-54404391670815345868800000000000 x^{11}+110823760810920148992000000000000 x^{10}-189442326172513075200000000000000 x^{9}+270631894532161536000000000000000 x^{8}-320748912038117376000000000000000 x^{7}+311839220037058560000000000000000 x^{6}-244579780421222400000000000000000 x^{5}+150975173099520000000000000000000 x^{4}-70631659929600000000000000000000 x^{3}+23543886643200000000000000000000 x^{2}-4982833151999999999999999999987 x\) | \(110\) |
| norman | \(11817250826203334794572 x^{22}-577732262614385256623520 x^{21}+13480419461002322654548800 x^{20}-199709917940775150437760000 x^{19}+2108049133819293254620800000 x^{18}-16864393070554346036966400000 x^{17}+106183215629416252825344000000 x^{16}-539343317482749220700160000000 x^{15}+2247263822844788419584000000000 x^{14}-7768319387611614289920000000000 x^{13}+22441811564211330170880000000000 x^{12}-54404391670815345868800000000000 x^{11}+110823760810920148992000000000000 x^{10}-189442326172513075200000000000000 x^{9}+270631894532161536000000000000000 x^{8}-320748912038117376000000000000000 x^{7}+311839220037058560000000000000000 x^{6}-244579780421222400000000000000000 x^{5}+150975173099520000000000000000000 x^{4}-70631659929600000000000000000000 x^{3}+23543886643200000000000000000000 x^{2}-4982833151999999999999999999987 x\) | \(110\) |
| risch | \(11817250826203334794572 x^{22}-577732262614385256623520 x^{21}+13480419461002322654548800 x^{20}-199709917940775150437760000 x^{19}+2108049133819293254620800000 x^{18}-16864393070554346036966400000 x^{17}+106183215629416252825344000000 x^{16}-539343317482749220700160000000 x^{15}+2247263822844788419584000000000 x^{14}-7768319387611614289920000000000 x^{13}+22441811564211330170880000000000 x^{12}-54404391670815345868800000000000 x^{11}+110823760810920148992000000000000 x^{10}-189442326172513075200000000000000 x^{9}+270631894532161536000000000000000 x^{8}-320748912038117376000000000000000 x^{7}+311839220037058560000000000000000 x^{6}-244579780421222400000000000000000 x^{5}+150975173099520000000000000000000 x^{4}-70631659929600000000000000000000 x^{3}+23543886643200000000000000000000 x^{2}-4982833151999999999999999999987 x\) | \(110\) |
| parallelrisch | \(11817250826203334794572 x^{22}-577732262614385256623520 x^{21}+13480419461002322654548800 x^{20}-199709917940775150437760000 x^{19}+2108049133819293254620800000 x^{18}-16864393070554346036966400000 x^{17}+106183215629416252825344000000 x^{16}-539343317482749220700160000000 x^{15}+2247263822844788419584000000000 x^{14}-7768319387611614289920000000000 x^{13}+22441811564211330170880000000000 x^{12}-54404391670815345868800000000000 x^{11}+110823760810920148992000000000000 x^{10}-189442326172513075200000000000000 x^{9}+270631894532161536000000000000000 x^{8}-320748912038117376000000000000000 x^{7}+311839220037058560000000000000000 x^{6}-244579780421222400000000000000000 x^{5}+150975173099520000000000000000000 x^{4}-70631659929600000000000000000000 x^{3}+23543886643200000000000000000000 x^{2}-4982833151999999999999999999987 x\) | \(110\) |
| parts | \(11817250826203334794572 x^{22}-577732262614385256623520 x^{21}+13480419461002322654548800 x^{20}-199709917940775150437760000 x^{19}+2108049133819293254620800000 x^{18}-16864393070554346036966400000 x^{17}+106183215629416252825344000000 x^{16}-539343317482749220700160000000 x^{15}+2247263822844788419584000000000 x^{14}-7768319387611614289920000000000 x^{13}+22441811564211330170880000000000 x^{12}-54404391670815345868800000000000 x^{11}+110823760810920148992000000000000 x^{10}-189442326172513075200000000000000 x^{9}+270631894532161536000000000000000 x^{8}-320748912038117376000000000000000 x^{7}+311839220037058560000000000000000 x^{6}-244579780421222400000000000000000 x^{5}+150975173099520000000000000000000 x^{4}-70631659929600000000000000000000 x^{3}+23543886643200000000000000000000 x^{2}-4982833151999999999999999999987 x\) | \(110\) |
| meijerg | \(\text {Expression too large to display}\) | \(1223\) |
x*(11817250826203334794572*x^21-577732262614385256623520*x^20+134804194610 02322654548800*x^19-199709917940775150437760000*x^18+210804913381929325462 0800000*x^17-16864393070554346036966400000*x^16+10618321562941625282534400 0000*x^15-539343317482749220700160000000*x^14+2247263822844788419584000000 000*x^13-7768319387611614289920000000000*x^12+2244181156421133017088000000 0000*x^11-54404391670815345868800000000000*x^10+11082376081092014899200000 0000000*x^9-189442326172513075200000000000000*x^8+270631894532161536000000 000000000*x^7-320748912038117376000000000000000*x^6+3118392200370585600000 00000000000*x^5-244579780421222400000000000000000*x^4+15097517309952000000 0000000000000*x^3-70631659929600000000000000000000*x^2+2354388664320000000 0000000000000*x-4982833151999999999999999999987)
Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (14) = 28\).
Time = 0.24 (sec) , antiderivative size = 109, normalized size of antiderivative = 6.06 \[ \int \frac {-260+2376 (20-9 x)^{22}+117 x}{-20+9 x} \, dx=11817250826203334794572 \, x^{22} - 577732262614385256623520 \, x^{21} + 13480419461002322654548800 \, x^{20} - 199709917940775150437760000 \, x^{19} + 2108049133819293254620800000 \, x^{18} - 16864393070554346036966400000 \, x^{17} + 106183215629416252825344000000 \, x^{16} - 539343317482749220700160000000 \, x^{15} + 2247263822844788419584000000000 \, x^{14} - 7768319387611614289920000000000 \, x^{13} + 22441811564211330170880000000000 \, x^{12} - 54404391670815345868800000000000 \, x^{11} + 110823760810920148992000000000000 \, x^{10} - 189442326172513075200000000000000 \, x^{9} + 270631894532161536000000000000000 \, x^{8} - 320748912038117376000000000000000 \, x^{7} + 311839220037058560000000000000000 \, x^{6} - 244579780421222400000000000000000 \, x^{5} + 150975173099520000000000000000000 \, x^{4} - 70631659929600000000000000000000 \, x^{3} + 23543886643200000000000000000000 \, x^{2} - 4982833151999999999999999999987 \, x \]
11817250826203334794572*x^22 - 577732262614385256623520*x^21 + 13480419461 002322654548800*x^20 - 199709917940775150437760000*x^19 + 2108049133819293 254620800000*x^18 - 16864393070554346036966400000*x^17 + 10618321562941625 2825344000000*x^16 - 539343317482749220700160000000*x^15 + 224726382284478 8419584000000000*x^14 - 7768319387611614289920000000000*x^13 + 22441811564 211330170880000000000*x^12 - 54404391670815345868800000000000*x^11 + 11082 3760810920148992000000000000*x^10 - 189442326172513075200000000000000*x^9 + 270631894532161536000000000000000*x^8 - 32074891203811737600000000000000 0*x^7 + 311839220037058560000000000000000*x^6 - 24457978042122240000000000 0000000*x^5 + 150975173099520000000000000000000*x^4 - 70631659929600000000 000000000000*x^3 + 23543886643200000000000000000000*x^2 - 4982833151999999 999999999999987*x
Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (12) = 24\).
Time = 0.05 (sec) , antiderivative size = 109, normalized size of antiderivative = 6.06 \[ \int \frac {-260+2376 (20-9 x)^{22}+117 x}{-20+9 x} \, dx=11817250826203334794572 x^{22} - 577732262614385256623520 x^{21} + 13480419461002322654548800 x^{20} - 199709917940775150437760000 x^{19} + 2108049133819293254620800000 x^{18} - 16864393070554346036966400000 x^{17} + 106183215629416252825344000000 x^{16} - 539343317482749220700160000000 x^{15} + 2247263822844788419584000000000 x^{14} - 7768319387611614289920000000000 x^{13} + 22441811564211330170880000000000 x^{12} - 54404391670815345868800000000000 x^{11} + 110823760810920148992000000000000 x^{10} - 189442326172513075200000000000000 x^{9} + 270631894532161536000000000000000 x^{8} - 320748912038117376000000000000000 x^{7} + 311839220037058560000000000000000 x^{6} - 244579780421222400000000000000000 x^{5} + 150975173099520000000000000000000 x^{4} - 70631659929600000000000000000000 x^{3} + 23543886643200000000000000000000 x^{2} - 4982833151999999999999999999987 x \]
11817250826203334794572*x**22 - 577732262614385256623520*x**21 + 134804194 61002322654548800*x**20 - 199709917940775150437760000*x**19 + 210804913381 9293254620800000*x**18 - 16864393070554346036966400000*x**17 + 10618321562 9416252825344000000*x**16 - 539343317482749220700160000000*x**15 + 2247263 822844788419584000000000*x**14 - 7768319387611614289920000000000*x**13 + 2 2441811564211330170880000000000*x**12 - 54404391670815345868800000000000*x **11 + 110823760810920148992000000000000*x**10 - 1894423261725130752000000 00000000*x**9 + 270631894532161536000000000000000*x**8 - 32074891203811737 6000000000000000*x**7 + 311839220037058560000000000000000*x**6 - 244579780 421222400000000000000000*x**5 + 150975173099520000000000000000000*x**4 - 7 0631659929600000000000000000000*x**3 + 23543886643200000000000000000000*x* *2 - 4982833151999999999999999999987*x
Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (14) = 28\).
Time = 0.18 (sec) , antiderivative size = 109, normalized size of antiderivative = 6.06 \[ \int \frac {-260+2376 (20-9 x)^{22}+117 x}{-20+9 x} \, dx=11817250826203334794572 \, x^{22} - 577732262614385256623520 \, x^{21} + 13480419461002322654548800 \, x^{20} - 199709917940775150437760000 \, x^{19} + 2108049133819293254620800000 \, x^{18} - 16864393070554346036966400000 \, x^{17} + 106183215629416252825344000000 \, x^{16} - 539343317482749220700160000000 \, x^{15} + 2247263822844788419584000000000 \, x^{14} - 7768319387611614289920000000000 \, x^{13} + 22441811564211330170880000000000 \, x^{12} - 54404391670815345868800000000000 \, x^{11} + 110823760810920148992000000000000 \, x^{10} - 189442326172513075200000000000000 \, x^{9} + 270631894532161536000000000000000 \, x^{8} - 320748912038117376000000000000000 \, x^{7} + 311839220037058560000000000000000 \, x^{6} - 244579780421222400000000000000000 \, x^{5} + 150975173099520000000000000000000 \, x^{4} - 70631659929600000000000000000000 \, x^{3} + 23543886643200000000000000000000 \, x^{2} - 4982833151999999999999999999987 \, x \]
11817250826203334794572*x^22 - 577732262614385256623520*x^21 + 13480419461 002322654548800*x^20 - 199709917940775150437760000*x^19 + 2108049133819293 254620800000*x^18 - 16864393070554346036966400000*x^17 + 10618321562941625 2825344000000*x^16 - 539343317482749220700160000000*x^15 + 224726382284478 8419584000000000*x^14 - 7768319387611614289920000000000*x^13 + 22441811564 211330170880000000000*x^12 - 54404391670815345868800000000000*x^11 + 11082 3760810920148992000000000000*x^10 - 189442326172513075200000000000000*x^9 + 270631894532161536000000000000000*x^8 - 32074891203811737600000000000000 0*x^7 + 311839220037058560000000000000000*x^6 - 24457978042122240000000000 0000000*x^5 + 150975173099520000000000000000000*x^4 - 70631659929600000000 000000000000*x^3 + 23543886643200000000000000000000*x^2 - 4982833151999999 999999999999987*x
Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (14) = 28\).
Time = 0.25 (sec) , antiderivative size = 109, normalized size of antiderivative = 6.06 \[ \int \frac {-260+2376 (20-9 x)^{22}+117 x}{-20+9 x} \, dx=11817250826203334794572 \, x^{22} - 577732262614385256623520 \, x^{21} + 13480419461002322654548800 \, x^{20} - 199709917940775150437760000 \, x^{19} + 2108049133819293254620800000 \, x^{18} - 16864393070554346036966400000 \, x^{17} + 106183215629416252825344000000 \, x^{16} - 539343317482749220700160000000 \, x^{15} + 2247263822844788419584000000000 \, x^{14} - 7768319387611614289920000000000 \, x^{13} + 22441811564211330170880000000000 \, x^{12} - 54404391670815345868800000000000 \, x^{11} + 110823760810920148992000000000000 \, x^{10} - 189442326172513075200000000000000 \, x^{9} + 270631894532161536000000000000000 \, x^{8} - 320748912038117376000000000000000 \, x^{7} + 311839220037058560000000000000000 \, x^{6} - 244579780421222400000000000000000 \, x^{5} + 150975173099520000000000000000000 \, x^{4} - 70631659929600000000000000000000 \, x^{3} + 23543886643200000000000000000000 \, x^{2} - 4982833151999999999999999999987 \, x \]
11817250826203334794572*x^22 - 577732262614385256623520*x^21 + 13480419461 002322654548800*x^20 - 199709917940775150437760000*x^19 + 2108049133819293 254620800000*x^18 - 16864393070554346036966400000*x^17 + 10618321562941625 2825344000000*x^16 - 539343317482749220700160000000*x^15 + 224726382284478 8419584000000000*x^14 - 7768319387611614289920000000000*x^13 + 22441811564 211330170880000000000*x^12 - 54404391670815345868800000000000*x^11 + 11082 3760810920148992000000000000*x^10 - 189442326172513075200000000000000*x^9 + 270631894532161536000000000000000*x^8 - 32074891203811737600000000000000 0*x^7 + 311839220037058560000000000000000*x^6 - 24457978042122240000000000 0000000*x^5 + 150975173099520000000000000000000*x^4 - 70631659929600000000 000000000000*x^3 + 23543886643200000000000000000000*x^2 - 4982833151999999 999999999999987*x
Time = 8.69 (sec) , antiderivative size = 109, normalized size of antiderivative = 6.06 \[ \int \frac {-260+2376 (20-9 x)^{22}+117 x}{-20+9 x} \, dx=11817250826203334794572\,x^{22}-577732262614385256623520\,x^{21}+13480419461002322654548800\,x^{20}-199709917940775150437760000\,x^{19}+2108049133819293254620800000\,x^{18}-16864393070554346036966400000\,x^{17}+106183215629416252825344000000\,x^{16}-539343317482749220700160000000\,x^{15}+2247263822844788419584000000000\,x^{14}-7768319387611614289920000000000\,x^{13}+22441811564211330170880000000000\,x^{12}-54404391670815345868800000000000\,x^{11}+110823760810920148992000000000000\,x^{10}-189442326172513075200000000000000\,x^9+270631894532161536000000000000000\,x^8-320748912038117376000000000000000\,x^7+311839220037058560000000000000000\,x^6-244579780421222400000000000000000\,x^5+150975173099520000000000000000000\,x^4-70631659929600000000000000000000\,x^3+23543886643200000000000000000000\,x^2-4982833151999999999999999999987\,x \]
23543886643200000000000000000000*x^2 - 4982833151999999999999999999987*x - 70631659929600000000000000000000*x^3 + 150975173099520000000000000000000* x^4 - 244579780421222400000000000000000*x^5 + 3118392200370585600000000000 00000*x^6 - 320748912038117376000000000000000*x^7 + 2706318945321615360000 00000000000*x^8 - 189442326172513075200000000000000*x^9 + 1108237608109201 48992000000000000*x^10 - 54404391670815345868800000000000*x^11 + 224418115 64211330170880000000000*x^12 - 7768319387611614289920000000000*x^13 + 2247 263822844788419584000000000*x^14 - 539343317482749220700160000000*x^15 + 1 06183215629416252825344000000*x^16 - 16864393070554346036966400000*x^17 + 2108049133819293254620800000*x^18 - 199709917940775150437760000*x^19 + 134 80419461002322654548800*x^20 - 577732262614385256623520*x^21 + 11817250826 203334794572*x^22