3.17.62 \(\int \frac {-260+2376 (20-9 x)^{22}+117 x}{-20+9 x} \, dx\) [1662]

Optimal. Leaf size=18 \[ -2+x+12 \left (x+(5 (4-2 x)+x)^{22}\right ) \]

[Out]

13*x-2+12*(-9*x+20)^22

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(109\) vs. \(2(18)=36\).
time = 0.03, 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 = {1600} \begin {gather*} 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 \end {gather*}

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 1600

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 {gather*} \begin {aligned} \text {integral} &=\int \left (-4982833151999999999999999999987+47087773286400000000000000000000 x-211894979788800000000000000000000 x^2+603900692398080000000000000000000 x^3-1222898902106112000000000000000000 x^4+1871035320222351360000000000000000 x^5-2245242384266821632000000000000000 x^6+2165055156257292288000000000000000 x^7-1704980935552617676800000000000000 x^8+1108237608109201489920000000000000 x^9-598448308378968804556800000000000 x^{10}+269301738770535962050560000000000 x^{11}-100988152038950985768960000000000 x^{12}+31461693519827037874176000000000 x^{13}-8090149762241238310502400000000 x^{14}+1698931450070660045205504000000 x^{15}-286694682199423882628428800000 x^{16}+37944884408747278583174400000 x^{17}-3794488440874727858317440000 x^{18}+269608389220046453090976000 x^{19}-12132377514902090389093920 x^{20}+259979518176473365480584 x^{21}\right ) \, dx\\ &=-4982833151999999999999999999987 x+23543886643200000000000000000000 x^2-70631659929600000000000000000000 x^3+150975173099520000000000000000000 x^4-244579780421222400000000000000000 x^5+311839220037058560000000000000000 x^6-320748912038117376000000000000000 x^7+270631894532161536000000000000000 x^8-189442326172513075200000000000000 x^9+110823760810920148992000000000000 x^{10}-54404391670815345868800000000000 x^{11}+22441811564211330170880000000000 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}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

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

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

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(109\) vs. \(2(14)=28\).
time = 0.35, size = 110, normalized size = 6.11

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	\(11817250826203334794572 x^{22}-577732262614385256623520 x^{21}+13480419461002322654548800 x^{20}-199709917940775150437760000 x^{19}+2108049133819293254620800000 x^{18}-16864393070554346036966400000 x^{17}+106183215629416252825344000000 x^{16}-539343317482749220700160000000 x^{15}+2247263822844788419584000000000 x^{14}-7768319387611614289920000000000 x^{13}+22441811564211330170880000000000 x^{12}-54404391670815345868800000000000 x^{11}+110823760810920148992000000000000 x^{10}-189442326172513075200000000000000 x^{9}+270631894532161536000000000000000 x^{8}-320748912038117376000000000000000 x^{7}+311839220037058560000000000000000 x^{6}-244579780421222400000000000000000 x^{5}+150975173099520000000000000000000 x^{4}-70631659929600000000000000000000 x^{3}+23543886643200000000000000000000 x^{2}-4982833151999999999999999999987 x\)	\(110\)
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]

11817250826203334794572*x^22-577732262614385256623520*x^21+13480419461002322654548800*x^20-1997099179407751504
37760000*x^19+2108049133819293254620800000*x^18-16864393070554346036966400000*x^17+106183215629416252825344000
000*x^16-539343317482749220700160000000*x^15+2247263822844788419584000000000*x^14-7768319387611614289920000000
000*x^13+22441811564211330170880000000000*x^12-54404391670815345868800000000000*x^11+1108237608109201489920000
00000000*x^10-189442326172513075200000000000000*x^9+270631894532161536000000000000000*x^8-32074891203811737600
0000000000000*x^7+311839220037058560000000000000000*x^6-244579780421222400000000000000000*x^5+1509751730995200
00000000000000000*x^4-70631659929600000000000000000000*x^3+23543886643200000000000000000000*x^2-49828331519999
99999999999999987*x

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (14) = 28\).
time = 0.26, size = 109, normalized size = 6.06 \begin {gather*} 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 \end {gather*}

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (14) = 28\).
time = 0.35, size = 109, normalized size = 6.06 \begin {gather*} 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 \end {gather*}

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

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (12) = 24\).
time = 0.05, size = 109, normalized size = 6.06 \begin {gather*} 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 \end {gather*}

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

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (14) = 28\).
time = 0.40, size = 109, normalized size = 6.06 \begin {gather*} 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 \end {gather*}

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

________________________________________________________________________________________

Mupad [B]
time = 1.46, size = 109, normalized size = 6.06 \begin {gather*} 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 \end {gather*}

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

________________________________________________________________________________________