3.31.44 \(\int \frac {-10+40 x-x^2+8 x^3+5898240 x^7-11796480 x^8+15040512 x^9-15777792 x^{10}[phone] x^{11}-5999616 x^{12}+1975680 x^{13}-422208 x^{14}+56538 x^{15}-4320 x^{16}+144 x^{17}}{-x^3+4 x^4+589824 x^{10}-1179648 x^{11}+1032192 x^{12}-516096 x^{13}+161280 x^{14}-32256 x^{15}+4032 x^{16}-288 x^{17}+9 x^{18}} \, dx\)

Optimal. Leaf size=25 \[ 4-\frac {5}{x^2}+\log \left (x-4 x^2-9 (-4+x)^8 x^8\right ) \]

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Rubi [B]  time = 1.05, antiderivative size = 59, normalized size of antiderivative = 2.36, number of steps used = 3, number of rules used = 2, integrand size = 129, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.016, Rules used = {2074, 1587} \begin {gather*} -\frac {5}{x^2}+\log \left (-9 x^{15}+288 x^{14}-4032 x^{13}+32256 x^{12}-161280 x^{11}+516096 x^{10}-1032192 x^9+1179648 x^8-589824 x^7-4 x+1\right )+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-10 + 40*x - x^2 + 8*x^3 + 5898240*x^7 - 11796480*x^8 + 15040512*x^9 - 15777792*x^10 + 11934720*x^11 - 59
99616*x^12 + 1975680*x^13 - 422208*x^14 + 56538*x^15 - 4320*x^16 + 144*x^17)/(-x^3 + 4*x^4 + 589824*x^10 - 117
9648*x^11 + 1032192*x^12 - 516096*x^13 + 161280*x^14 - 32256*x^15 + 4032*x^16 - 288*x^17 + 9*x^18),x]

[Out]

-5/x^2 + Log[x] + Log[1 - 4*x - 589824*x^7 + 1179648*x^8 - 1032192*x^9 + 516096*x^10 - 161280*x^11 + 32256*x^1
2 - 4032*x^13 + 288*x^14 - 9*x^15]

Rule 1587

Int[(Pp_)/(Qq_), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*Log[RemoveConte
nt[Qq, x]])/(q*Coeff[Qq, x, q]), x] /; EqQ[p, q - 1] && EqQ[Pp, Simplify[(Coeff[Pp, x, p]*D[Qq, x])/(q*Coeff[Q
q, x, q])]]] /; PolyQ[Pp, x] && PolyQ[Qq, x]

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {10}{x^3}+\frac {1}{x}+\frac {4+4128768 x^6-9437184 x^7+9289728 x^8-5160960 x^9+1774080 x^{10}-387072 x^{11}+52416 x^{12}-4032 x^{13}+135 x^{14}}{-1+4 x+589824 x^7-1179648 x^8+1032192 x^9-516096 x^{10}+161280 x^{11}-32256 x^{12}+4032 x^{13}-288 x^{14}+9 x^{15}}\right ) \, dx\\ &=-\frac {5}{x^2}+\log (x)+\int \frac {4+4128768 x^6-9437184 x^7+9289728 x^8-5160960 x^9+1774080 x^{10}-387072 x^{11}+52416 x^{12}-4032 x^{13}+135 x^{14}}{-1+4 x+589824 x^7-1179648 x^8+1032192 x^9-516096 x^{10}+161280 x^{11}-32256 x^{12}+4032 x^{13}-288 x^{14}+9 x^{15}} \, dx\\ &=-\frac {5}{x^2}+\log (x)+\log \left (1-4 x-589824 x^7+1179648 x^8-1032192 x^9+516096 x^{10}-161280 x^{11}+32256 x^{12}-4032 x^{13}+288 x^{14}-9 x^{15}\right )\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.04, size = 59, normalized size = 2.36 \begin {gather*} -\frac {5}{x^2}+\log (x)+\log \left (1-4 x-589824 x^7+1179648 x^8-1032192 x^9+516096 x^{10}-161280 x^{11}+32256 x^{12}-4032 x^{13}+288 x^{14}-9 x^{15}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-10 + 40*x - x^2 + 8*x^3 + 5898240*x^7 - 11796480*x^8 + 15040512*x^9 - 15777792*x^10 + 11934720*x^1
1 - 5999616*x^12 + 1975680*x^13 - 422208*x^14 + 56538*x^15 - 4320*x^16 + 144*x^17)/(-x^3 + 4*x^4 + 589824*x^10
- 1179648*x^11 + 1032192*x^12 - 516096*x^13 + 161280*x^14 - 32256*x^15 + 4032*x^16 - 288*x^17 + 9*x^18),x]

[Out]

-5/x^2 + Log[x] + Log[1 - 4*x - 589824*x^7 + 1179648*x^8 - 1032192*x^9 + 516096*x^10 - 161280*x^11 + 32256*x^1
2 - 4032*x^13 + 288*x^14 - 9*x^15]

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fricas [B]  time = 0.69, size = 65, normalized size = 2.60 \begin {gather*} \frac {x^{2} \log \left (9 \, x^{16} - 288 \, x^{15} + 4032 \, x^{14} - 32256 \, x^{13} + 161280 \, x^{12} - 516096 \, x^{11} + 1032192 \, x^{10} - 1179648 \, x^{9} + 589824 \, x^{8} + 4 \, x^{2} - x\right ) - 5}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((144*x^17-4320*x^16+56538*x^15-422208*x^14+1975680*x^13-5999616*x^12+11934720*x^11-15777792*x^10+150
40512*x^9-11796480*x^8+5898240*x^7+8*x^3-x^2+40*x-10)/(9*x^18-288*x^17+4032*x^16-32256*x^15+161280*x^14-516096
*x^13+1032192*x^12-1179648*x^11+589824*x^10+4*x^4-x^3),x, algorithm="fricas")

[Out]

(x^2*log(9*x^16 - 288*x^15 + 4032*x^14 - 32256*x^13 + 161280*x^12 - 516096*x^11 + 1032192*x^10 - 1179648*x^9 +
589824*x^8 + 4*x^2 - x) - 5)/x^2

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giac [B]  time = 0.28, size = 61, normalized size = 2.44 \begin {gather*} -\frac {5}{x^{2}} + \log \left ({\left | 9 \, x^{15} - 288 \, x^{14} + 4032 \, x^{13} - 32256 \, x^{12} + 161280 \, x^{11} - 516096 \, x^{10} + 1032192 \, x^{9} - 1179648 \, x^{8} + 589824 \, x^{7} + 4 \, x - 1 \right |}\right ) + \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((144*x^17-4320*x^16+56538*x^15-422208*x^14+1975680*x^13-5999616*x^12+11934720*x^11-15777792*x^10+150
40512*x^9-11796480*x^8+5898240*x^7+8*x^3-x^2+40*x-10)/(9*x^18-288*x^17+4032*x^16-32256*x^15+161280*x^14-516096
*x^13+1032192*x^12-1179648*x^11+589824*x^10+4*x^4-x^3),x, algorithm="giac")

[Out]

-5/x^2 + log(abs(9*x^15 - 288*x^14 + 4032*x^13 - 32256*x^12 + 161280*x^11 - 516096*x^10 + 1032192*x^9 - 117964
8*x^8 + 589824*x^7 + 4*x - 1)) + log(abs(x))

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maple [B]  time = 0.07, size = 60, normalized size = 2.40

method	result	size
default	\(\ln \relax (x )-\frac {5}{x^{2}}+\ln \left (9 x^{15}-288 x^{14}+4032 x^{13}-32256 x^{12}+161280 x^{11}-516096 x^{10}+1032192 x^{9}-1179648 x^{8}+589824 x^{7}+4 x -1\right )\)	\(60\)
norman	\(\ln \relax (x )-\frac {5}{x^{2}}+\ln \left (9 x^{15}-288 x^{14}+4032 x^{13}-32256 x^{12}+161280 x^{11}-516096 x^{10}+1032192 x^{9}-1179648 x^{8}+589824 x^{7}+4 x -1\right )\)	\(60\)
risch	\(-\frac {5}{x^{2}}+\ln \left (9 x^{16}-288 x^{15}+4032 x^{14}-32256 x^{13}+161280 x^{12}-516096 x^{11}+1032192 x^{10}-1179648 x^{9}+589824 x^{8}+4 x^{2}-x \right )\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((144*x^17-4320*x^16+56538*x^15-422208*x^14+1975680*x^13-5999616*x^12+11934720*x^11-15777792*x^10+15040512*
x^9-11796480*x^8+5898240*x^7+8*x^3-x^2+40*x-10)/(9*x^18-288*x^17+4032*x^16-32256*x^15+161280*x^14-516096*x^13+
1032192*x^12-1179648*x^11+589824*x^10+4*x^4-x^3),x,method=_RETURNVERBOSE)

[Out]

ln(x)-5/x^2+ln(9*x^15-288*x^14+4032*x^13-32256*x^12+161280*x^11-516096*x^10+1032192*x^9-1179648*x^8+589824*x^7
+4*x-1)

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maxima [B]  time = 0.38, size = 59, normalized size = 2.36 \begin {gather*} -\frac {5}{x^{2}} + \log \left (9 \, x^{15} - 288 \, x^{14} + 4032 \, x^{13} - 32256 \, x^{12} + 161280 \, x^{11} - 516096 \, x^{10} + 1032192 \, x^{9} - 1179648 \, x^{8} + 589824 \, x^{7} + 4 \, x - 1\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((144*x^17-4320*x^16+56538*x^15-422208*x^14+1975680*x^13-5999616*x^12+11934720*x^11-15777792*x^10+150
40512*x^9-11796480*x^8+5898240*x^7+8*x^3-x^2+40*x-10)/(9*x^18-288*x^17+4032*x^16-32256*x^15+161280*x^14-516096
*x^13+1032192*x^12-1179648*x^11+589824*x^10+4*x^4-x^3),x, algorithm="maxima")

[Out]

-5/x^2 + log(9*x^15 - 288*x^14 + 4032*x^13 - 32256*x^12 + 161280*x^11 - 516096*x^10 + 1032192*x^9 - 1179648*x^
8 + 589824*x^7 + 4*x - 1) + log(x)

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mupad [B]  time = 0.24, size = 59, normalized size = 2.36 \begin {gather*} \ln \left (x^{16}-32\,x^{15}+448\,x^{14}-3584\,x^{13}+17920\,x^{12}-57344\,x^{11}+114688\,x^{10}-131072\,x^9+65536\,x^8+\frac {4\,x^2}{9}-\frac {x}{9}\right )-\frac {5}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((40*x - x^2 + 8*x^3 + 5898240*x^7 - 11796480*x^8 + 15040512*x^9 - 15777792*x^10 + 11934720*x^11 - 5999616*
x^12 + 1975680*x^13 - 422208*x^14 + 56538*x^15 - 4320*x^16 + 144*x^17 - 10)/(4*x^4 - x^3 + 589824*x^10 - 11796
48*x^11 + 1032192*x^12 - 516096*x^13 + 161280*x^14 - 32256*x^15 + 4032*x^16 - 288*x^17 + 9*x^18),x)

[Out]

log((4*x^2)/9 - x/9 + 65536*x^8 - 131072*x^9 + 114688*x^10 - 57344*x^11 + 17920*x^12 - 3584*x^13 + 448*x^14 -
32*x^15 + x^16) - 5/x^2

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sympy [B]  time = 0.24, size = 58, normalized size = 2.32 \begin {gather*} \log {\left (9 x^{16} - 288 x^{15} + 4032 x^{14} - 32256 x^{13} + 161280 x^{12} - 516096 x^{11} + 1032192 x^{10} - 1179648 x^{9} + 589824 x^{8} + 4 x^{2} - x \right )} - \frac {5}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((144*x**17-4320*x**16+56538*x**15-422208*x**14+1975680*x**13-5999616*x**12+11934720*x**11-15777792*x
**10+15040512*x**9-11796480*x**8+5898240*x**7+8*x**3-x**2+40*x-10)/(9*x**18-288*x**17+4032*x**16-32256*x**15+1
61280*x**14-516096*x**13+1032192*x**12-1179648*x**11+589824*x**10+4*x**4-x**3),x)

[Out]

log(9*x**16 - 288*x**15 + 4032*x**14 - 32256*x**13 + 161280*x**12 - 516096*x**11 + 1032192*x**10 - 1179648*x**
9 + 589824*x**8 + 4*x**2 - x) - 5/x**2

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