\(\int \frac {-15-24 x^6}{32 x^6} \, dx\) [6413]

   Optimal result
   Rubi [A] (verified)
   Mathematica [A] (verified)
   Maple [A] (verified)
   Fricas [A] (verification not implemented)
   Sympy [A] (verification not implemented)
   Maxima [A] (verification not implemented)
   Giac [A] (verification not implemented)
   Mupad [B] (verification not implemented)

Optimal result

Integrand size = 14, antiderivative size = 19 \[ \int \frac {-15-24 x^6}{32 x^6} \, dx=-2+5 e^4+\frac {3}{32 x^5}-\frac {3 x}{4} \]

[Out]

5*exp(4)+3/32/x^5-3/4*x-2

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {12, 14} \[ \int \frac {-15-24 x^6}{32 x^6} \, dx=\frac {3}{32 x^5}-\frac {3 x}{4} \]

[In]

Int[(-15 - 24*x^6)/(32*x^6),x]

[Out]

3/(32*x^5) - (3*x)/4

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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}& = \frac {1}{32} \int \frac {-15-24 x^6}{x^6} \, dx \\ & = \frac {1}{32} \int \left (-24-\frac {15}{x^6}\right ) \, dx \\ & = \frac {3}{32 x^5}-\frac {3 x}{4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.58 \[ \int \frac {-15-24 x^6}{32 x^6} \, dx=\frac {3}{32} \left (\frac {1}{x^5}-8 x\right ) \]

[In]

Integrate[(-15 - 24*x^6)/(32*x^6),x]

[Out]

(3*(x^(-5) - 8*x))/32

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.53

method result size
default \(-\frac {3 x}{4}+\frac {3}{32 x^{5}}\) \(10\)
risch \(-\frac {3 x}{4}+\frac {3}{32 x^{5}}\) \(10\)
norman \(\frac {\frac {3}{32}-\frac {3 x^{6}}{4}}{x^{5}}\) \(12\)
gosper \(-\frac {3 \left (8 x^{6}-1\right )}{32 x^{5}}\) \(13\)
parallelrisch \(-\frac {24 x^{6}-3}{32 x^{5}}\) \(13\)

[In]

int(1/32*(-24*x^6-15)/x^6,x,method=_RETURNVERBOSE)

[Out]

-3/4*x+3/32/x^5

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.63 \[ \int \frac {-15-24 x^6}{32 x^6} \, dx=-\frac {3 \, {\left (8 \, x^{6} - 1\right )}}{32 \, x^{5}} \]

[In]

integrate(1/32*(-24*x^6-15)/x^6,x, algorithm="fricas")

[Out]

-3/32*(8*x^6 - 1)/x^5

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.53 \[ \int \frac {-15-24 x^6}{32 x^6} \, dx=- \frac {3 x}{4} + \frac {3}{32 x^{5}} \]

[In]

integrate(1/32*(-24*x**6-15)/x**6,x)

[Out]

-3*x/4 + 3/(32*x**5)

Maxima [A] (verification not implemented)

none

Time = 0.17 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.47 \[ \int \frac {-15-24 x^6}{32 x^6} \, dx=-\frac {3}{4} \, x + \frac {3}{32 \, x^{5}} \]

[In]

integrate(1/32*(-24*x^6-15)/x^6,x, algorithm="maxima")

[Out]

-3/4*x + 3/32/x^5

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.47 \[ \int \frac {-15-24 x^6}{32 x^6} \, dx=-\frac {3}{4} \, x + \frac {3}{32 \, x^{5}} \]

[In]

integrate(1/32*(-24*x^6-15)/x^6,x, algorithm="giac")

[Out]

-3/4*x + 3/32/x^5

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.47 \[ \int \frac {-15-24 x^6}{32 x^6} \, dx=\frac {3}{32\,x^5}-\frac {3\,x}{4} \]

[In]

int(-((3*x^6)/4 + 15/32)/x^6,x)

[Out]

3/(32*x^5) - (3*x)/4