\(\int \frac {(a+b x)^5}{x^9} \, dx\) [92]

Optimal result

Integrand size = 11, antiderivative size = 56 \[ \int \frac {(a+b x)^5}{x^9} \, dx=-\frac {(a+b x)^6}{8 a x^8}+\frac {b (a+b x)^6}{28 a^2 x^7}-\frac {b^2 (a+b x)^6}{168 a^3 x^6} \]

[Out]

-1/8*(b*x+a)^6/a/x^8+1/28*b*(b*x+a)^6/a^2/x^7-1/168*b^2*(b*x+a)^6/a^3/x^6

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {47, 37} \[ \int \frac {(a+b x)^5}{x^9} \, dx=-\frac {b^2 (a+b x)^6}{168 a^3 x^6}+\frac {b (a+b x)^6}{28 a^2 x^7}-\frac {(a+b x)^6}{8 a x^8} \]

[In]

Int[(a + b*x)^5/x^9,x]

[Out]

-1/8*(a + b*x)^6/(a*x^8) + (b*(a + b*x)^6)/(28*a^2*x^7) - (b^2*(a + b*x)^6)/(168*a^3*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}{8 a x^8}-\frac {b \int \frac {(a+b x)^5}{x^8} \, dx}{4 a} \\ & = -\frac {(a+b x)^6}{8 a x^8}+\frac {b (a+b x)^6}{28 a^2 x^7}+\frac {b^2 \int \frac {(a+b x)^5}{x^7} \, dx}{28 a^2} \\ & = -\frac {(a+b x)^6}{8 a x^8}+\frac {b (a+b x)^6}{28 a^2 x^7}-\frac {b^2 (a+b x)^6}{168 a^3 x^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.20 \[ \int \frac {(a+b x)^5}{x^9} \, dx=-\frac {a^5}{8 x^8}-\frac {5 a^4 b}{7 x^7}-\frac {5 a^3 b^2}{3 x^6}-\frac {2 a^2 b^3}{x^5}-\frac {5 a b^4}{4 x^4}-\frac {b^5}{3 x^3} \]

[In]

Integrate[(a + b*x)^5/x^9,x]

[Out]

-1/8*a^5/x^8 - (5*a^4*b)/(7*x^7) - (5*a^3*b^2)/(3*x^6) - (2*a^2*b^3)/x^5 - (5*a*b^4)/(4*x^4) - b^5/(3*x^3)

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.02

method	result	size
norman	\(\frac {-\frac {1}{3} b^{5} x^{5}-\frac {5}{4} a \,b^{4} x^{4}-2 a^{2} b^{3} x^{3}-\frac {5}{3} a^{3} b^{2} x^{2}-\frac {5}{7} a^{4} b x -\frac {1}{8} a^{5}}{x^{8}}\)	\(57\)
risch	\(\frac {-\frac {1}{3} b^{5} x^{5}-\frac {5}{4} a \,b^{4} x^{4}-2 a^{2} b^{3} x^{3}-\frac {5}{3} a^{3} b^{2} x^{2}-\frac {5}{7} a^{4} b x -\frac {1}{8} a^{5}}{x^{8}}\)	\(57\)
gosper	\(-\frac {56 b^{5} x^{5}+210 a \,b^{4} x^{4}+336 a^{2} b^{3} x^{3}+280 a^{3} b^{2} x^{2}+120 a^{4} b x +21 a^{5}}{168 x^{8}}\)	\(58\)
default	\(-\frac {5 a^{3} b^{2}}{3 x^{6}}-\frac {5 a^{4} b}{7 x^{7}}-\frac {b^{5}}{3 x^{3}}-\frac {5 a \,b^{4}}{4 x^{4}}-\frac {2 a^{2} b^{3}}{x^{5}}-\frac {a^{5}}{8 x^{8}}\)	\(58\)
parallelrisch	\(\frac {-56 b^{5} x^{5}-210 a \,b^{4} x^{4}-336 a^{2} b^{3} x^{3}-280 a^{3} b^{2} x^{2}-120 a^{4} b x -21 a^{5}}{168 x^{8}}\)	\(58\)

[In]

int((b*x+a)^5/x^9,x,method=_RETURNVERBOSE)

[Out]

1/x^8*(-1/3*b^5*x^5-5/4*a*b^4*x^4-2*a^2*b^3*x^3-5/3*a^3*b^2*x^2-5/7*a^4*b*x-1/8*a^5)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.02 \[ \int \frac {(a+b x)^5}{x^9} \, dx=-\frac {56 \, b^{5} x^{5} + 210 \, a b^{4} x^{4} + 336 \, a^{2} b^{3} x^{3} + 280 \, a^{3} b^{2} x^{2} + 120 \, a^{4} b x + 21 \, a^{5}}{168 \, x^{8}} \]

[In]

integrate((b*x+a)^5/x^9,x, algorithm="fricas")

[Out]

-1/168*(56*b^5*x^5 + 210*a*b^4*x^4 + 336*a^2*b^3*x^3 + 280*a^3*b^2*x^2 + 120*a^4*b*x + 21*a^5)/x^8

Sympy [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.09 \[ \int \frac {(a+b x)^5}{x^9} \, dx=\frac {- 21 a^{5} - 120 a^{4} b x - 280 a^{3} b^{2} x^{2} - 336 a^{2} b^{3} x^{3} - 210 a b^{4} x^{4} - 56 b^{5} x^{5}}{168 x^{8}} \]

[In]

integrate((b*x+a)**5/x**9,x)

[Out]

(-21*a**5 - 120*a**4*b*x - 280*a**3*b**2*x**2 - 336*a**2*b**3*x**3 - 210*a*b**4*x**4 - 56*b**5*x**5)/(168*x**8
)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.02 \[ \int \frac {(a+b x)^5}{x^9} \, dx=-\frac {56 \, b^{5} x^{5} + 210 \, a b^{4} x^{4} + 336 \, a^{2} b^{3} x^{3} + 280 \, a^{3} b^{2} x^{2} + 120 \, a^{4} b x + 21 \, a^{5}}{168 \, x^{8}} \]

[In]

integrate((b*x+a)^5/x^9,x, algorithm="maxima")

[Out]

-1/168*(56*b^5*x^5 + 210*a*b^4*x^4 + 336*a^2*b^3*x^3 + 280*a^3*b^2*x^2 + 120*a^4*b*x + 21*a^5)/x^8

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.02 \[ \int \frac {(a+b x)^5}{x^9} \, dx=-\frac {56 \, b^{5} x^{5} + 210 \, a b^{4} x^{4} + 336 \, a^{2} b^{3} x^{3} + 280 \, a^{3} b^{2} x^{2} + 120 \, a^{4} b x + 21 \, a^{5}}{168 \, x^{8}} \]

[In]

integrate((b*x+a)^5/x^9,x, algorithm="giac")

[Out]

-1/168*(56*b^5*x^5 + 210*a*b^4*x^4 + 336*a^2*b^3*x^3 + 280*a^3*b^2*x^2 + 120*a^4*b*x + 21*a^5)/x^8

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.02 \[ \int \frac {(a+b x)^5}{x^9} \, dx=-\frac {\frac {a^5}{8}+\frac {5\,a^4\,b\,x}{7}+\frac {5\,a^3\,b^2\,x^2}{3}+2\,a^2\,b^3\,x^3+\frac {5\,a\,b^4\,x^4}{4}+\frac {b^5\,x^5}{3}}{x^8} \]

[In]

int((a + b*x)^5/x^9,x)

[Out]

-(a^5/8 + (b^5*x^5)/3 + (5*a*b^4*x^4)/4 + (5*a^3*b^2*x^2)/3 + 2*a^2*b^3*x^3 + (5*a^4*b*x)/7)/x^8