\(\int \frac {7442-3904 x+3278 x^3-4590 x^4+2496 x^5-1408 x^6+320 x^7-96 x^8}{x^3} \, dx\) [5038]

Optimal result

Integrand size = 39, antiderivative size = 33 \[ \int \frac {7442-3904 x+3278 x^3-4590 x^4+2496 x^5-1408 x^6+320 x^7-96 x^8}{x^3} \, dx=5 e^3-2 x+x^2-\frac {\left (3-4 \left (4-x+x^2\right )^2\right )^2}{x^2} \]

[Out]

5*exp(3)-2*x+x^2-(3-4*(x^2-x+4)^2)^2/x^2

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.18, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {14} \[ \int \frac {7442-3904 x+3278 x^3-4590 x^4+2496 x^5-1408 x^6+320 x^7-96 x^8}{x^3} \, dx=-16 x^6+64 x^5-352 x^4+832 x^3-2295 x^2-\frac {3721}{x^2}+3278 x+\frac {3904}{x} \]

[In]

Int[(7442 - 3904*x + 3278*x^3 - 4590*x^4 + 2496*x^5 - 1408*x^6 + 320*x^7 - 96*x^8)/x^3,x]

[Out]

-3721/x^2 + 3904/x + 3278*x - 2295*x^2 + 832*x^3 - 352*x^4 + 64*x^5 - 16*x^6

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps \begin{align*} \text {integral}& = \int \left (3278+\frac {7442}{x^3}-\frac {3904}{x^2}-4590 x+2496 x^2-1408 x^3+320 x^4-96 x^5\right ) \, dx \\ & = -\frac {3721}{x^2}+\frac {3904}{x}+3278 x-2295 x^2+832 x^3-352 x^4+64 x^5-16 x^6 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.18 \[ \int \frac {7442-3904 x+3278 x^3-4590 x^4+2496 x^5-1408 x^6+320 x^7-96 x^8}{x^3} \, dx=-\frac {3721}{x^2}+\frac {3904}{x}+3278 x-2295 x^2+832 x^3-352 x^4+64 x^5-16 x^6 \]

[In]

Integrate[(7442 - 3904*x + 3278*x^3 - 4590*x^4 + 2496*x^5 - 1408*x^6 + 320*x^7 - 96*x^8)/x^3,x]

[Out]

-3721/x^2 + 3904/x + 3278*x - 2295*x^2 + 832*x^3 - 352*x^4 + 64*x^5 - 16*x^6

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.18

method	result	size
risch	\(-16 x^{6}+64 x^{5}-352 x^{4}+832 x^{3}-2295 x^{2}+3278 x +\frac {3904 x -3721}{x^{2}}\)	\(39\)
default	\(-16 x^{6}+64 x^{5}-352 x^{4}+832 x^{3}-2295 x^{2}+3278 x +\frac {3904}{x}-\frac {3721}{x^{2}}\)	\(40\)
norman	\(\frac {-16 x^{8}+64 x^{7}-352 x^{6}+832 x^{5}-2295 x^{4}+3278 x^{3}+3904 x -3721}{x^{2}}\)	\(40\)
gosper	\(-\frac {16 x^{8}-64 x^{7}+352 x^{6}-832 x^{5}+2295 x^{4}-3278 x^{3}-3904 x +3721}{x^{2}}\)	\(41\)
parallelrisch	\(-\frac {16 x^{8}-64 x^{7}+352 x^{6}-832 x^{5}+2295 x^{4}-3278 x^{3}-3904 x +3721}{x^{2}}\)	\(41\)

[In]

int((-96*x^8+320*x^7-1408*x^6+2496*x^5-4590*x^4+3278*x^3-3904*x+7442)/x^3,x,method=_RETURNVERBOSE)

[Out]

-16*x^6+64*x^5-352*x^4+832*x^3-2295*x^2+3278*x+(3904*x-3721)/x^2

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.21 \[ \int \frac {7442-3904 x+3278 x^3-4590 x^4+2496 x^5-1408 x^6+320 x^7-96 x^8}{x^3} \, dx=-\frac {16 \, x^{8} - 64 \, x^{7} + 352 \, x^{6} - 832 \, x^{5} + 2295 \, x^{4} - 3278 \, x^{3} - 3904 \, x + 3721}{x^{2}} \]

[In]

integrate((-96*x^8+320*x^7-1408*x^6+2496*x^5-4590*x^4+3278*x^3-3904*x+7442)/x^3,x, algorithm="fricas")

[Out]

-(16*x^8 - 64*x^7 + 352*x^6 - 832*x^5 + 2295*x^4 - 3278*x^3 - 3904*x + 3721)/x^2

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09 \[ \int \frac {7442-3904 x+3278 x^3-4590 x^4+2496 x^5-1408 x^6+320 x^7-96 x^8}{x^3} \, dx=- 16 x^{6} + 64 x^{5} - 352 x^{4} + 832 x^{3} - 2295 x^{2} + 3278 x - \frac {3721 - 3904 x}{x^{2}} \]

[In]

integrate((-96*x**8+320*x**7-1408*x**6+2496*x**5-4590*x**4+3278*x**3-3904*x+7442)/x**3,x)

[Out]

-16*x**6 + 64*x**5 - 352*x**4 + 832*x**3 - 2295*x**2 + 3278*x - (3721 - 3904*x)/x**2

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.18 \[ \int \frac {7442-3904 x+3278 x^3-4590 x^4+2496 x^5-1408 x^6+320 x^7-96 x^8}{x^3} \, dx=-16 \, x^{6} + 64 \, x^{5} - 352 \, x^{4} + 832 \, x^{3} - 2295 \, x^{2} + 3278 \, x + \frac {61 \, {\left (64 \, x - 61\right )}}{x^{2}} \]

[In]

integrate((-96*x^8+320*x^7-1408*x^6+2496*x^5-4590*x^4+3278*x^3-3904*x+7442)/x^3,x, algorithm="maxima")

[Out]

-16*x^6 + 64*x^5 - 352*x^4 + 832*x^3 - 2295*x^2 + 3278*x + 61*(64*x - 61)/x^2

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.18 \[ \int \frac {7442-3904 x+3278 x^3-4590 x^4+2496 x^5-1408 x^6+320 x^7-96 x^8}{x^3} \, dx=-16 \, x^{6} + 64 \, x^{5} - 352 \, x^{4} + 832 \, x^{3} - 2295 \, x^{2} + 3278 \, x + \frac {61 \, {\left (64 \, x - 61\right )}}{x^{2}} \]

[In]

integrate((-96*x^8+320*x^7-1408*x^6+2496*x^5-4590*x^4+3278*x^3-3904*x+7442)/x^3,x, algorithm="giac")

[Out]

-16*x^6 + 64*x^5 - 352*x^4 + 832*x^3 - 2295*x^2 + 3278*x + 61*(64*x - 61)/x^2

Mupad [B] (verification not implemented)

Time = 11.37 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.15 \[ \int \frac {7442-3904 x+3278 x^3-4590 x^4+2496 x^5-1408 x^6+320 x^7-96 x^8}{x^3} \, dx=3278\,x+\frac {3904\,x-3721}{x^2}-2295\,x^2+832\,x^3-352\,x^4+64\,x^5-16\,x^6 \]

[In]

int(-(3904*x - 3278*x^3 + 4590*x^4 - 2496*x^5 + 1408*x^6 - 320*x^7 + 96*x^8 - 7442)/x^3,x)

[Out]

3278*x + (3904*x - 3721)/x^2 - 2295*x^2 + 832*x^3 - 352*x^4 + 64*x^5 - 16*x^6