\(\int \frac {(a+b x)^2}{x^8} \, dx\) [63]

   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 = 11, antiderivative size = 30 \[ \int \frac {(a+b x)^2}{x^8} \, dx=-\frac {a^2}{7 x^7}-\frac {a b}{3 x^6}-\frac {b^2}{5 x^5} \]

[Out]

-1/7*a^2/x^7-1/3*a*b/x^6-1/5*b^2/x^5

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \[ \int \frac {(a+b x)^2}{x^8} \, dx=-\frac {a^2}{7 x^7}-\frac {a b}{3 x^6}-\frac {b^2}{5 x^5} \]

[In]

Int[(a + b*x)^2/x^8,x]

[Out]

-1/7*a^2/x^7 - (a*b)/(3*x^6) - b^2/(5*x^5)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^2}{x^8}+\frac {2 a b}{x^7}+\frac {b^2}{x^6}\right ) \, dx \\ & = -\frac {a^2}{7 x^7}-\frac {a b}{3 x^6}-\frac {b^2}{5 x^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^2}{x^8} \, dx=-\frac {a^2}{7 x^7}-\frac {a b}{3 x^6}-\frac {b^2}{5 x^5} \]

[In]

Integrate[(a + b*x)^2/x^8,x]

[Out]

-1/7*a^2/x^7 - (a*b)/(3*x^6) - b^2/(5*x^5)

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80

method result size
norman \(\frac {-\frac {1}{5} b^{2} x^{2}-\frac {1}{3} a b x -\frac {1}{7} a^{2}}{x^{7}}\) \(24\)
risch \(\frac {-\frac {1}{5} b^{2} x^{2}-\frac {1}{3} a b x -\frac {1}{7} a^{2}}{x^{7}}\) \(24\)
gosper \(-\frac {21 b^{2} x^{2}+35 a b x +15 a^{2}}{105 x^{7}}\) \(25\)
default \(-\frac {a^{2}}{7 x^{7}}-\frac {a b}{3 x^{6}}-\frac {b^{2}}{5 x^{5}}\) \(25\)
parallelrisch \(\frac {-21 b^{2} x^{2}-35 a b x -15 a^{2}}{105 x^{7}}\) \(25\)

[In]

int((b*x+a)^2/x^8,x,method=_RETURNVERBOSE)

[Out]

1/x^7*(-1/5*b^2*x^2-1/3*a*b*x-1/7*a^2)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \frac {(a+b x)^2}{x^8} \, dx=-\frac {21 \, b^{2} x^{2} + 35 \, a b x + 15 \, a^{2}}{105 \, x^{7}} \]

[In]

integrate((b*x+a)^2/x^8,x, algorithm="fricas")

[Out]

-1/105*(21*b^2*x^2 + 35*a*b*x + 15*a^2)/x^7

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {(a+b x)^2}{x^8} \, dx=\frac {- 15 a^{2} - 35 a b x - 21 b^{2} x^{2}}{105 x^{7}} \]

[In]

integrate((b*x+a)**2/x**8,x)

[Out]

(-15*a**2 - 35*a*b*x - 21*b**2*x**2)/(105*x**7)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \frac {(a+b x)^2}{x^8} \, dx=-\frac {21 \, b^{2} x^{2} + 35 \, a b x + 15 \, a^{2}}{105 \, x^{7}} \]

[In]

integrate((b*x+a)^2/x^8,x, algorithm="maxima")

[Out]

-1/105*(21*b^2*x^2 + 35*a*b*x + 15*a^2)/x^7

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \frac {(a+b x)^2}{x^8} \, dx=-\frac {21 \, b^{2} x^{2} + 35 \, a b x + 15 \, a^{2}}{105 \, x^{7}} \]

[In]

integrate((b*x+a)^2/x^8,x, algorithm="giac")

[Out]

-1/105*(21*b^2*x^2 + 35*a*b*x + 15*a^2)/x^7

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \frac {(a+b x)^2}{x^8} \, dx=-\frac {\frac {a^2}{7}+\frac {a\,b\,x}{3}+\frac {b^2\,x^2}{5}}{x^7} \]

[In]

int((a + b*x)^2/x^8,x)

[Out]

-(a^2/7 + (b^2*x^2)/5 + (a*b*x)/3)/x^7