\(\int \frac {(a+b x)^5}{x^8} \, dx\) [91]

Optimal result

Integrand size = 11, antiderivative size = 36 \[ \int \frac {(a+b x)^5}{x^8} \, dx=-\frac {(a+b x)^6}{7 a x^7}+\frac {b (a+b x)^6}{42 a^2 x^6} \]

[Out]

-1/7*(b*x+a)^6/a/x^7+1/42*b*(b*x+a)^6/a^2/x^6

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {47, 37} \[ \int \frac {(a+b x)^5}{x^8} \, dx=\frac {b (a+b x)^6}{42 a^2 x^6}-\frac {(a+b x)^6}{7 a x^7} \]

[In]

Int[(a + b*x)^5/x^8,x]

[Out]

-1/7*(a + b*x)^6/(a*x^7) + (b*(a + b*x)^6)/(42*a^2*x^6)

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*(Simplify[m + n + 2]/((b*c - a*d)*(m + 1))), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b x)^6}{7 a x^7}-\frac {b \int \frac {(a+b x)^5}{x^7} \, dx}{7 a} \\ & = -\frac {(a+b x)^6}{7 a x^7}+\frac {b (a+b x)^6}{42 a^2 x^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.86 \[ \int \frac {(a+b x)^5}{x^8} \, dx=-\frac {a^5}{7 x^7}-\frac {5 a^4 b}{6 x^6}-\frac {2 a^3 b^2}{x^5}-\frac {5 a^2 b^3}{2 x^4}-\frac {5 a b^4}{3 x^3}-\frac {b^5}{2 x^2} \]

[In]

Integrate[(a + b*x)^5/x^8,x]

[Out]

-1/7*a^5/x^7 - (5*a^4*b)/(6*x^6) - (2*a^3*b^2)/x^5 - (5*a^2*b^3)/(2*x^4) - (5*a*b^4)/(3*x^3) - b^5/(2*x^2)

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.58

method	result	size
norman	\(\frac {-\frac {1}{2} b^{5} x^{5}-\frac {5}{3} a \,b^{4} x^{4}-\frac {5}{2} a^{2} b^{3} x^{3}-2 a^{3} b^{2} x^{2}-\frac {5}{6} a^{4} b x -\frac {1}{7} a^{5}}{x^{7}}\)	\(57\)
risch	\(\frac {-\frac {1}{2} b^{5} x^{5}-\frac {5}{3} a \,b^{4} x^{4}-\frac {5}{2} a^{2} b^{3} x^{3}-2 a^{3} b^{2} x^{2}-\frac {5}{6} a^{4} b x -\frac {1}{7} a^{5}}{x^{7}}\)	\(57\)
gosper	\(-\frac {21 b^{5} x^{5}+70 a \,b^{4} x^{4}+105 a^{2} b^{3} x^{3}+84 a^{3} b^{2} x^{2}+35 a^{4} b x +6 a^{5}}{42 x^{7}}\)	\(58\)
default	\(-\frac {5 a^{4} b}{6 x^{6}}-\frac {a^{5}}{7 x^{7}}-\frac {5 a \,b^{4}}{3 x^{3}}-\frac {b^{5}}{2 x^{2}}-\frac {5 a^{2} b^{3}}{2 x^{4}}-\frac {2 a^{3} b^{2}}{x^{5}}\)	\(58\)
parallelrisch	\(\frac {-21 b^{5} x^{5}-70 a \,b^{4} x^{4}-105 a^{2} b^{3} x^{3}-84 a^{3} b^{2} x^{2}-35 a^{4} b x -6 a^{5}}{42 x^{7}}\)	\(58\)

[In]

int((b*x+a)^5/x^8,x,method=_RETURNVERBOSE)

[Out]

1/x^7*(-1/2*b^5*x^5-5/3*a*b^4*x^4-5/2*a^2*b^3*x^3-2*a^3*b^2*x^2-5/6*a^4*b*x-1/7*a^5)

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.58 \[ \int \frac {(a+b x)^5}{x^8} \, dx=-\frac {21 \, b^{5} x^{5} + 70 \, a b^{4} x^{4} + 105 \, a^{2} b^{3} x^{3} + 84 \, a^{3} b^{2} x^{2} + 35 \, a^{4} b x + 6 \, a^{5}}{42 \, x^{7}} \]

[In]

integrate((b*x+a)^5/x^8,x, algorithm="fricas")

[Out]

-1/42*(21*b^5*x^5 + 70*a*b^4*x^4 + 105*a^2*b^3*x^3 + 84*a^3*b^2*x^2 + 35*a^4*b*x + 6*a^5)/x^7

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (29) = 58\).

Time = 0.22 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.69 \[ \int \frac {(a+b x)^5}{x^8} \, dx=\frac {- 6 a^{5} - 35 a^{4} b x - 84 a^{3} b^{2} x^{2} - 105 a^{2} b^{3} x^{3} - 70 a b^{4} x^{4} - 21 b^{5} x^{5}}{42 x^{7}} \]

[In]

integrate((b*x+a)**5/x**8,x)

[Out]

(-6*a**5 - 35*a**4*b*x - 84*a**3*b**2*x**2 - 105*a**2*b**3*x**3 - 70*a*b**4*x**4 - 21*b**5*x**5)/(42*x**7)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.58 \[ \int \frac {(a+b x)^5}{x^8} \, dx=-\frac {21 \, b^{5} x^{5} + 70 \, a b^{4} x^{4} + 105 \, a^{2} b^{3} x^{3} + 84 \, a^{3} b^{2} x^{2} + 35 \, a^{4} b x + 6 \, a^{5}}{42 \, x^{7}} \]

[In]

integrate((b*x+a)^5/x^8,x, algorithm="maxima")

[Out]

-1/42*(21*b^5*x^5 + 70*a*b^4*x^4 + 105*a^2*b^3*x^3 + 84*a^3*b^2*x^2 + 35*a^4*b*x + 6*a^5)/x^7

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.58 \[ \int \frac {(a+b x)^5}{x^8} \, dx=-\frac {21 \, b^{5} x^{5} + 70 \, a b^{4} x^{4} + 105 \, a^{2} b^{3} x^{3} + 84 \, a^{3} b^{2} x^{2} + 35 \, a^{4} b x + 6 \, a^{5}}{42 \, x^{7}} \]

[In]

integrate((b*x+a)^5/x^8,x, algorithm="giac")

[Out]

-1/42*(21*b^5*x^5 + 70*a*b^4*x^4 + 105*a^2*b^3*x^3 + 84*a^3*b^2*x^2 + 35*a^4*b*x + 6*a^5)/x^7

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.58 \[ \int \frac {(a+b x)^5}{x^8} \, dx=-\frac {\frac {a^5}{7}+\frac {5\,a^4\,b\,x}{6}+2\,a^3\,b^2\,x^2+\frac {5\,a^2\,b^3\,x^3}{2}+\frac {5\,a\,b^4\,x^4}{3}+\frac {b^5\,x^5}{2}}{x^7} \]

[In]

int((a + b*x)^5/x^8,x)

[Out]

-(a^5/7 + (b^5*x^5)/2 + (5*a*b^4*x^4)/3 + 2*a^3*b^2*x^2 + (5*a^2*b^3*x^3)/2 + (5*a^4*b*x)/6)/x^7