3.16.47 \(\int \frac {64 x^6+480 x^7+1440 x^8+2160 x^9+1620 x^{10}+486 x^{11}+e^{16} (8+24 x)}{32 x^5+240 x^6+720 x^7+1080 x^8+810 x^9+243 x^{10}} \, dx\)

Optimal. Leaf size=20 \[ -1-\frac {e^{16}}{(-2-3 x)^4 x^4}+x^2 \]

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Rubi [B]  time = 0.19, antiderivative size = 100, normalized size of antiderivative = 5.00, number of steps used = 2, number of rules used = 1, integrand size = 74, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {2074} \begin {gather*} -\frac {e^{16}}{16 x^4}+\frac {3 e^{16}}{8 x^3}+x^2-\frac {45 e^{16}}{32 x^2}-\frac {405 e^{16}}{32 (3 x+2)}-\frac {405 e^{16}}{32 (3 x+2)^2}-\frac {81 e^{16}}{8 (3 x+2)^3}-\frac {81 e^{16}}{16 (3 x+2)^4}+\frac {135 e^{16}}{32 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(64*x^6 + 480*x^7 + 1440*x^8 + 2160*x^9 + 1620*x^10 + 486*x^11 + E^16*(8 + 24*x))/(32*x^5 + 240*x^6 + 720*
x^7 + 1080*x^8 + 810*x^9 + 243*x^10),x]

[Out]

-1/16*E^16/x^4 + (3*E^16)/(8*x^3) - (45*E^16)/(32*x^2) + (135*E^16)/(32*x) + x^2 - (81*E^16)/(16*(2 + 3*x)^4)
- (81*E^16)/(8*(2 + 3*x)^3) - (405*E^16)/(32*(2 + 3*x)^2) - (405*E^16)/(32*(2 + 3*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 {e^{16}}{4 x^5}-\frac {9 e^{16}}{8 x^4}+\frac {45 e^{16}}{16 x^3}-\frac {135 e^{16}}{32 x^2}+2 x+\frac {243 e^{16}}{4 (2+3 x)^5}+\frac {729 e^{16}}{8 (2+3 x)^4}+\frac {1215 e^{16}}{16 (2+3 x)^3}+\frac {1215 e^{16}}{32 (2+3 x)^2}\right ) \, dx\\ &=-\frac {e^{16}}{16 x^4}+\frac {3 e^{16}}{8 x^3}-\frac {45 e^{16}}{32 x^2}+\frac {135 e^{16}}{32 x}+x^2-\frac {81 e^{16}}{16 (2+3 x)^4}-\frac {81 e^{16}}{8 (2+3 x)^3}-\frac {405 e^{16}}{32 (2+3 x)^2}-\frac {405 e^{16}}{32 (2+3 x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 28, normalized size = 1.40 \begin {gather*} \frac {-e^{16}+x^6 (2+3 x)^4}{x^4 (2+3 x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(64*x^6 + 480*x^7 + 1440*x^8 + 2160*x^9 + 1620*x^10 + 486*x^11 + E^16*(8 + 24*x))/(32*x^5 + 240*x^6
+ 720*x^7 + 1080*x^8 + 810*x^9 + 243*x^10),x]

[Out]

(-E^16 + x^6*(2 + 3*x)^4)/(x^4*(2 + 3*x)^4)

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fricas [B]  time = 0.64, size = 59, normalized size = 2.95 \begin {gather*} \frac {81 \, x^{10} + 216 \, x^{9} + 216 \, x^{8} + 96 \, x^{7} + 16 \, x^{6} - e^{16}}{81 \, x^{8} + 216 \, x^{7} + 216 \, x^{6} + 96 \, x^{5} + 16 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x+8)*exp(4)^4+486*x^11+1620*x^10+2160*x^9+1440*x^8+480*x^7+64*x^6)/(243*x^10+810*x^9+1080*x^8+7
20*x^7+240*x^6+32*x^5),x, algorithm="fricas")

[Out]

(81*x^10 + 216*x^9 + 216*x^8 + 96*x^7 + 16*x^6 - e^16)/(81*x^8 + 216*x^7 + 216*x^6 + 96*x^5 + 16*x^4)

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giac [A]  time = 0.16, size = 19, normalized size = 0.95 \begin {gather*} x^{2} - \frac {e^{16}}{{\left (3 \, x^{2} + 2 \, x\right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x+8)*exp(4)^4+486*x^11+1620*x^10+2160*x^9+1440*x^8+480*x^7+64*x^6)/(243*x^10+810*x^9+1080*x^8+7
20*x^7+240*x^6+32*x^5),x, algorithm="giac")

[Out]

x^2 - e^16/(3*x^2 + 2*x)^4

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maple [A]  time = 0.09, size = 34, normalized size = 1.70

method	result	size
risch	\(x^{2}-\frac {{\mathrm e}^{16}}{x^{4} \left (81 x^{4}+216 x^{3}+216 x^{2}+96 x +16\right )}\)	\(34\)
norman	\(\frac {16 x^{6}+96 x^{7}+216 x^{8}+216 x^{9}+81 x^{10}-{\mathrm e}^{16}}{x^{4} \left (3 x +2\right )^{4}}\)	\(44\)
gosper	\(-\frac {-243 x^{10}-648 x^{9}+1440 x^{7}+1680 x^{6}+768 x^{5}+3 \,{\mathrm e}^{16}+128 x^{4}}{3 x^{4} \left (81 x^{4}+216 x^{3}+216 x^{2}+96 x +16\right )}\)	\(65\)
default	\(x^{2}-\frac {81 \,{\mathrm e}^{16}}{16 \left (3 x +2\right )^{4}}-\frac {81 \,{\mathrm e}^{16}}{8 \left (3 x +2\right )^{3}}-\frac {405 \,{\mathrm e}^{16}}{32 \left (3 x +2\right )^{2}}-\frac {405 \,{\mathrm e}^{16}}{32 \left (3 x +2\right )}-\frac {{\mathrm e}^{16}}{16 x^{4}}+\frac {3 \,{\mathrm e}^{16}}{8 x^{3}}-\frac {45 \,{\mathrm e}^{16}}{32 x^{2}}+\frac {135 \,{\mathrm e}^{16}}{32 x}\)	\(77\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((24*x+8)*exp(4)^4+486*x^11+1620*x^10+2160*x^9+1440*x^8+480*x^7+64*x^6)/(243*x^10+810*x^9+1080*x^8+720*x^7
+240*x^6+32*x^5),x,method=_RETURNVERBOSE)

[Out]

x^2-exp(16)/x^4/(81*x^4+216*x^3+216*x^2+96*x+16)

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maxima [A]  time = 0.39, size = 36, normalized size = 1.80 \begin {gather*} x^{2} - \frac {e^{16}}{81 \, x^{8} + 216 \, x^{7} + 216 \, x^{6} + 96 \, x^{5} + 16 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x+8)*exp(4)^4+486*x^11+1620*x^10+2160*x^9+1440*x^8+480*x^7+64*x^6)/(243*x^10+810*x^9+1080*x^8+7
20*x^7+240*x^6+32*x^5),x, algorithm="maxima")

[Out]

x^2 - e^16/(81*x^8 + 216*x^7 + 216*x^6 + 96*x^5 + 16*x^4)

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mupad [B]  time = 1.04, size = 18, normalized size = 0.90 \begin {gather*} x^2-\frac {{\mathrm {e}}^{16}}{x^4\,{\left (3\,x+2\right )}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((64*x^6 + 480*x^7 + 1440*x^8 + 2160*x^9 + 1620*x^10 + 486*x^11 + exp(16)*(24*x + 8))/(32*x^5 + 240*x^6 + 7
20*x^7 + 1080*x^8 + 810*x^9 + 243*x^10),x)

[Out]

x^2 - exp(16)/(x^4*(3*x + 2)^4)

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sympy [A]  time = 0.28, size = 31, normalized size = 1.55 \begin {gather*} x^{2} - \frac {e^{16}}{81 x^{8} + 216 x^{7} + 216 x^{6} + 96 x^{5} + 16 x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x+8)*exp(4)**4+486*x**11+1620*x**10+2160*x**9+1440*x**8+480*x**7+64*x**6)/(243*x**10+810*x**9+1
080*x**8+720*x**7+240*x**6+32*x**5),x)

[Out]

x**2 - exp(16)/(81*x**8 + 216*x**7 + 216*x**6 + 96*x**5 + 16*x**4)

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