3.17.62 $\int \frac{-260+2376(20-9x{)}^{22}+117x}{-20+9x}dx$

Optimal. Leaf size=18 $$-2+x+12(x+(5(4-2x)+x{)}^{22})$$

________________________________________________________________________________________

Rubi [B]  time = 0.04, antiderivative size = 109, normalized size of antiderivative = 6.06, 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 = {1586} $$\begin{array}{c}11817250826203334794572x^{22}-577732262614385256623520x^{21}+13480419461002322654548800x^{20}-199709917940775150437760000x^{19}+2108049133819293254620800000x^{18}-16864393070554346036966400000x^{17}+106183215629416252825344000000x^{16}-539343317482749220700160000000x^{15}+2247263822844788419584000000000x^{14}-7768319387611614289920000000000x^{13}+22441811564211330170880000000000x^{12}-54404391670815345868800000000000x^{11}+110823760810920148992000000000000x^{10}-189442326172513075200000000000000x^9+270631894532161536000000000000000x^8-320748912038117376000000000000000x^7+311839220037058560000000000000000x^6-244579780421222400000000000000000x^5+150975173099520000000000000000000x^4-70631659929600000000000000000000x^3+23543886643200000000000000000000x^2-4982833151999999999999999999987x\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(-260 + 2376*(20 - 9*x)^22 + 117*x)/(-20 + 9*x),x]

[Out]

-4982833151999999999999999999987*x + 23543886643200000000000000000000*x^2 - 70631659929600000000000000000000*x
^3 + 150975173099520000000000000000000*x^4 - 244579780421222400000000000000000*x^5 + 3118392200370585600000000
00000000*x^6 - 320748912038117376000000000000000*x^7 + 270631894532161536000000000000000*x^8 - 189442326172513
075200000000000000*x^9 + 110823760810920148992000000000000*x^10 - 54404391670815345868800000000000*x^11 + 2244
1811564211330170880000000000*x^12 - 7768319387611614289920000000000*x^13 + 2247263822844788419584000000000*x^1
4 - 539343317482749220700160000000*x^15 + 106183215629416252825344000000*x^16 - 16864393070554346036966400000*
x^17 + 2108049133819293254620800000*x^18 - 199709917940775150437760000*x^19 + 13480419461002322654548800*x^20
- 577732262614385256623520*x^21 + 11817250826203334794572*x^22

Rule 1586

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]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int (-4982833151999999999999999999987+47087773286400000000000000000000x-211894979788800000000000000000000x^2+603900692398080000000000000000000x^3-1222898902106112000000000000000000x^4+1871035320222351360000000000000000x^5-2245242384266821632000000000000000x^6+2165055156257292288000000000000000x^7-1704980935552617676800000000000000x^8+1108237608109201489920000000000000x^9-598448308378968804556800000000000x^{10}+269301738770535962050560000000000x^{11}-100988152038950985768960000000000x^{12}+31461693519827037874176000000000x^{13}-8090149762241238310502400000000x^{14}+1698931450070660045205504000000x^{15}-286694682199423882628428800000x^{16}+37944884408747278583174400000x^{17}-3794488440874727858317440000x^{18}+269608389220046453090976000x^{19}-12132377514902090389093920x^{20}+259979518176473365480584x^{21})dx\\ & =-4982833151999999999999999999987x+23543886643200000000000000000000x^2-70631659929600000000000000000000x^3+150975173099520000000000000000000x^4-244579780421222400000000000000000x^5+311839220037058560000000000000000x^6-320748912038117376000000000000000x^7+270631894532161536000000000000000x^8-189442326172513075200000000000000x^9+110823760810920148992000000000000x^{10}-54404391670815345868800000000000x^{11}+22441811564211330170880000000000x^{12}-7768319387611614289920000000000x^{13}+2247263822844788419584000000000x^{14}-539343317482749220700160000000x^{15}+106183215629416252825344000000x^{16}-16864393070554346036966400000x^{17}+2108049133819293254620800000x^{18}-199709917940775150437760000x^{19}+13480419461002322654548800x^{20}-577732262614385256623520x^{21}+11817250826203334794572x^{22}\end{array}\end{array}$$

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 16, normalized size = 0.89 $$\begin{array}{c}-\frac{260}{9}+12(20-9x{)}^{22}+13x\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(-260 + 2376*(20 - 9*x)^22 + 117*x)/(-20 + 9*x),x]

[Out]

-260/9 + 12*(20 - 9*x)^22 + 13*x

________________________________________________________________________________________

fricas [B]  time = 0.64, size = 109, normalized size = 6.06 $$\begin{array}{c}11817250826203334794572x^{22}-577732262614385256623520x^{21}+13480419461002322654548800x^{20}-199709917940775150437760000x^{19}+2108049133819293254620800000x^{18}-16864393070554346036966400000x^{17}+106183215629416252825344000000x^{16}-539343317482749220700160000000x^{15}+2247263822844788419584000000000x^{14}-7768319387611614289920000000000x^{13}+22441811564211330170880000000000x^{12}-54404391670815345868800000000000x^{11}+110823760810920148992000000000000x^{10}-189442326172513075200000000000000x^9+270631894532161536000000000000000x^8-320748912038117376000000000000000x^7+311839220037058560000000000000000x^6-244579780421222400000000000000000x^5+150975173099520000000000000000000x^4-70631659929600000000000000000000x^3+23543886643200000000000000000000x^2-4982833151999999999999999999987x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2376*(-9*x+20)^22+117*x-260)/(9*x-20),x, algorithm="fricas")

[Out]

11817250826203334794572*x^22 - 577732262614385256623520*x^21 + 13480419461002322654548800*x^20 - 1997099179407
75150437760000*x^19 + 2108049133819293254620800000*x^18 - 16864393070554346036966400000*x^17 + 106183215629416
252825344000000*x^16 - 539343317482749220700160000000*x^15 + 2247263822844788419584000000000*x^14 - 7768319387
611614289920000000000*x^13 + 22441811564211330170880000000000*x^12 - 54404391670815345868800000000000*x^11 + 1
10823760810920148992000000000000*x^10 - 189442326172513075200000000000000*x^9 + 270631894532161536000000000000
000*x^8 - 320748912038117376000000000000000*x^7 + 311839220037058560000000000000000*x^6 - 24457978042122240000
0000000000000*x^5 + 150975173099520000000000000000000*x^4 - 70631659929600000000000000000000*x^3 + 23543886643
200000000000000000000*x^2 - 4982833151999999999999999999987*x

________________________________________________________________________________________

giac [B]  time = 0.17, size = 109, normalized size = 6.06 $$\begin{array}{c}11817250826203334794572x^{22}-577732262614385256623520x^{21}+13480419461002322654548800x^{20}-199709917940775150437760000x^{19}+2108049133819293254620800000x^{18}-16864393070554346036966400000x^{17}+106183215629416252825344000000x^{16}-539343317482749220700160000000x^{15}+2247263822844788419584000000000x^{14}-7768319387611614289920000000000x^{13}+22441811564211330170880000000000x^{12}-54404391670815345868800000000000x^{11}+110823760810920148992000000000000x^{10}-189442326172513075200000000000000x^9+270631894532161536000000000000000x^8-320748912038117376000000000000000x^7+311839220037058560000000000000000x^6-244579780421222400000000000000000x^5+150975173099520000000000000000000x^4-70631659929600000000000000000000x^3+23543886643200000000000000000000x^2-4982833151999999999999999999987x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2376*(-9*x+20)^22+117*x-260)/(9*x-20),x, algorithm="giac")

[Out]

11817250826203334794572*x^22 - 577732262614385256623520*x^21 + 13480419461002322654548800*x^20 - 1997099179407
75150437760000*x^19 + 2108049133819293254620800000*x^18 - 16864393070554346036966400000*x^17 + 106183215629416
252825344000000*x^16 - 539343317482749220700160000000*x^15 + 2247263822844788419584000000000*x^14 - 7768319387
611614289920000000000*x^13 + 22441811564211330170880000000000*x^12 - 54404391670815345868800000000000*x^11 + 1
10823760810920148992000000000000*x^10 - 189442326172513075200000000000000*x^9 + 270631894532161536000000000000
000*x^8 - 320748912038117376000000000000000*x^7 + 311839220037058560000000000000000*x^6 - 24457978042122240000
0000000000000*x^5 + 150975173099520000000000000000000*x^4 - 70631659929600000000000000000000*x^3 + 23543886643
200000000000000000000*x^2 - 4982833151999999999999999999987*x

________________________________________________________________________________________

maple [B]  time = 0.47, size = 108, normalized size = 6.00

method	result	size
gosper	$x(11817250826203334794572x^{21}-577732262614385256623520x^{20}+13480419461002322654548800x^{19}-199709917940775150437760000x^{18}+2108049133819293254620800000x^{17}-16864393070554346036966400000x^{16}+106183215629416252825344000000x^{15}-539343317482749220700160000000x^{14}+2247263822844788419584000000000x^{13}-7768319387611614289920000000000x^{12}+22441811564211330170880000000000x^{11}-54404391670815345868800000000000x^{10}+110823760810920148992000000000000x^9-189442326172513075200000000000000x^8+270631894532161536000000000000000x^7-320748912038117376000000000000000x^6+311839220037058560000000000000000x^5-244579780421222400000000000000000x^4+150975173099520000000000000000000x^3-70631659929600000000000000000000x^2+23543886643200000000000000000000x-4982833151999999999999999999987)$	$108$
default	$11817250826203334794572x^{22}-577732262614385256623520x^{21}+13480419461002322654548800x^{20}-199709917940775150437760000x^{19}+2108049133819293254620800000x^{18}-16864393070554346036966400000x^{17}+106183215629416252825344000000x^{16}-539343317482749220700160000000x^{15}+2247263822844788419584000000000x^{14}-7768319387611614289920000000000x^{13}+22441811564211330170880000000000x^{12}-54404391670815345868800000000000x^{11}+110823760810920148992000000000000x^{10}-189442326172513075200000000000000x^9+270631894532161536000000000000000x^8-320748912038117376000000000000000x^7+311839220037058560000000000000000x^6-244579780421222400000000000000000x^5+150975173099520000000000000000000x^4-70631659929600000000000000000000x^3+23543886643200000000000000000000x^2-4982833151999999999999999999987x$	$110$
norman	$11817250826203334794572x^{22}-577732262614385256623520x^{21}+13480419461002322654548800x^{20}-199709917940775150437760000x^{19}+2108049133819293254620800000x^{18}-16864393070554346036966400000x^{17}+106183215629416252825344000000x^{16}-539343317482749220700160000000x^{15}+2247263822844788419584000000000x^{14}-7768319387611614289920000000000x^{13}+22441811564211330170880000000000x^{12}-54404391670815345868800000000000x^{11}+110823760810920148992000000000000x^{10}-189442326172513075200000000000000x^9+270631894532161536000000000000000x^8-320748912038117376000000000000000x^7+311839220037058560000000000000000x^6-244579780421222400000000000000000x^5+150975173099520000000000000000000x^4-70631659929600000000000000000000x^3+23543886643200000000000000000000x^2-4982833151999999999999999999987x$	$110$
risch	$11817250826203334794572x^{22}-577732262614385256623520x^{21}+13480419461002322654548800x^{20}-199709917940775150437760000x^{19}+2108049133819293254620800000x^{18}-16864393070554346036966400000x^{17}+106183215629416252825344000000x^{16}-539343317482749220700160000000x^{15}+2247263822844788419584000000000x^{14}-7768319387611614289920000000000x^{13}+22441811564211330170880000000000x^{12}-54404391670815345868800000000000x^{11}+110823760810920148992000000000000x^{10}-189442326172513075200000000000000x^9+270631894532161536000000000000000x^8-320748912038117376000000000000000x^7+311839220037058560000000000000000x^6-244579780421222400000000000000000x^5+150975173099520000000000000000000x^4-70631659929600000000000000000000x^3+23543886643200000000000000000000x^2-4982833151999999999999999999987x$	$110$
meijerg	$\text{Expression too large to display}$	$1223$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2376*(-9*x+20)^22+117*x-260)/(9*x-20),x,method=_RETURNVERBOSE)

[Out]

x*(11817250826203334794572*x^21-577732262614385256623520*x^20+13480419461002322654548800*x^19-1997099179407751
50437760000*x^18+2108049133819293254620800000*x^17-16864393070554346036966400000*x^16+106183215629416252825344
000000*x^15-539343317482749220700160000000*x^14+2247263822844788419584000000000*x^13-7768319387611614289920000
000000*x^12+22441811564211330170880000000000*x^11-54404391670815345868800000000000*x^10+1108237608109201489920
00000000000*x^9-189442326172513075200000000000000*x^8+270631894532161536000000000000000*x^7-320748912038117376
000000000000000*x^6+311839220037058560000000000000000*x^5-244579780421222400000000000000000*x^4+15097517309952
0000000000000000000*x^3-70631659929600000000000000000000*x^2+23543886643200000000000000000000*x-49828331519999
99999999999999987)

________________________________________________________________________________________

maxima [B]  time = 0.42, size = 109, normalized size = 6.06 $$\begin{array}{c}11817250826203334794572x^{22}-577732262614385256623520x^{21}+13480419461002322654548800x^{20}-199709917940775150437760000x^{19}+2108049133819293254620800000x^{18}-16864393070554346036966400000x^{17}+106183215629416252825344000000x^{16}-539343317482749220700160000000x^{15}+2247263822844788419584000000000x^{14}-7768319387611614289920000000000x^{13}+22441811564211330170880000000000x^{12}-54404391670815345868800000000000x^{11}+110823760810920148992000000000000x^{10}-189442326172513075200000000000000x^9+270631894532161536000000000000000x^8-320748912038117376000000000000000x^7+311839220037058560000000000000000x^6-244579780421222400000000000000000x^5+150975173099520000000000000000000x^4-70631659929600000000000000000000x^3+23543886643200000000000000000000x^2-4982833151999999999999999999987x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2376*(-9*x+20)^22+117*x-260)/(9*x-20),x, algorithm="maxima")

[Out]

11817250826203334794572*x^22 - 577732262614385256623520*x^21 + 13480419461002322654548800*x^20 - 1997099179407
75150437760000*x^19 + 2108049133819293254620800000*x^18 - 16864393070554346036966400000*x^17 + 106183215629416
252825344000000*x^16 - 539343317482749220700160000000*x^15 + 2247263822844788419584000000000*x^14 - 7768319387
611614289920000000000*x^13 + 22441811564211330170880000000000*x^12 - 54404391670815345868800000000000*x^11 + 1
10823760810920148992000000000000*x^10 - 189442326172513075200000000000000*x^9 + 270631894532161536000000000000
000*x^8 - 320748912038117376000000000000000*x^7 + 311839220037058560000000000000000*x^6 - 24457978042122240000
0000000000000*x^5 + 150975173099520000000000000000000*x^4 - 70631659929600000000000000000000*x^3 + 23543886643
200000000000000000000*x^2 - 4982833151999999999999999999987*x

________________________________________________________________________________________

mupad [B]  time = 1.46, size = 109, normalized size = 6.06 $$\begin{array}{c}11817250826203334794572x^{22}-577732262614385256623520x^{21}+13480419461002322654548800x^{20}-199709917940775150437760000x^{19}+2108049133819293254620800000x^{18}-16864393070554346036966400000x^{17}+106183215629416252825344000000x^{16}-539343317482749220700160000000x^{15}+2247263822844788419584000000000x^{14}-7768319387611614289920000000000x^{13}+22441811564211330170880000000000x^{12}-54404391670815345868800000000000x^{11}+110823760810920148992000000000000x^{10}-189442326172513075200000000000000x^9+270631894532161536000000000000000x^8-320748912038117376000000000000000x^7+311839220037058560000000000000000x^6-244579780421222400000000000000000x^5+150975173099520000000000000000000x^4-70631659929600000000000000000000x^3+23543886643200000000000000000000x^2-4982833151999999999999999999987x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((117*x + 2376*(9*x - 20)^22 - 260)/(9*x - 20),x)

[Out]

23543886643200000000000000000000*x^2 - 4982833151999999999999999999987*x - 70631659929600000000000000000000*x^
3 + 150975173099520000000000000000000*x^4 - 244579780421222400000000000000000*x^5 + 31183922003705856000000000
0000000*x^6 - 320748912038117376000000000000000*x^7 + 270631894532161536000000000000000*x^8 - 1894423261725130
75200000000000000*x^9 + 110823760810920148992000000000000*x^10 - 54404391670815345868800000000000*x^11 + 22441
811564211330170880000000000*x^12 - 7768319387611614289920000000000*x^13 + 2247263822844788419584000000000*x^14
- 539343317482749220700160000000*x^15 + 106183215629416252825344000000*x^16 - 16864393070554346036966400000*x
^17 + 2108049133819293254620800000*x^18 - 199709917940775150437760000*x^19 + 13480419461002322654548800*x^20 -
577732262614385256623520*x^21 + 11817250826203334794572*x^22

________________________________________________________________________________________

sympy [B]  time = 0.11, size = 109, normalized size = 6.06 $$\begin{array}{c}11817250826203334794572x^{22}-577732262614385256623520x^{21}+13480419461002322654548800x^{20}-199709917940775150437760000x^{19}+2108049133819293254620800000x^{18}-16864393070554346036966400000x^{17}+106183215629416252825344000000x^{16}-539343317482749220700160000000x^{15}+2247263822844788419584000000000x^{14}-7768319387611614289920000000000x^{13}+22441811564211330170880000000000x^{12}-54404391670815345868800000000000x^{11}+110823760810920148992000000000000x^{10}-189442326172513075200000000000000x^9+270631894532161536000000000000000x^8-320748912038117376000000000000000x^7+311839220037058560000000000000000x^6-244579780421222400000000000000000x^5+150975173099520000000000000000000x^4-70631659929600000000000000000000x^3+23543886643200000000000000000000x^2-4982833151999999999999999999987x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2376*(-9*x+20)**22+117*x-260)/(9*x-20),x)

[Out]

11817250826203334794572*x**22 - 577732262614385256623520*x**21 + 13480419461002322654548800*x**20 - 1997099179
40775150437760000*x**19 + 2108049133819293254620800000*x**18 - 16864393070554346036966400000*x**17 + 106183215
629416252825344000000*x**16 - 539343317482749220700160000000*x**15 + 2247263822844788419584000000000*x**14 - 7
768319387611614289920000000000*x**13 + 22441811564211330170880000000000*x**12 - 544043916708153458688000000000
00*x**11 + 110823760810920148992000000000000*x**10 - 189442326172513075200000000000000*x**9 + 2706318945321615
36000000000000000*x**8 - 320748912038117376000000000000000*x**7 + 311839220037058560000000000000000*x**6 - 244
579780421222400000000000000000*x**5 + 150975173099520000000000000000000*x**4 - 7063165992960000000000000000000
0*x**3 + 23543886643200000000000000000000*x**2 - 4982833151999999999999999999987*x

________________________________________________________________________________________