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

Optimal. Leaf size=56 \[ -\frac{b^2 (a+b x)^6}{168 a^3 x^6}+\frac{b (a+b x)^6}{28 a^2 x^7}-\frac{(a+b x)^6}{8 a x^8} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0096565, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {45, 37} \[ -\frac{b^2 (a+b x)^6}{168 a^3 x^6}+\frac{b (a+b x)^6}{28 a^2 x^7}-\frac{(a+b x)^6}{8 a x^8} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^5/x^9,x]

[Out]

-(a + b*x)^6/(8*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 45

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])

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]

Rubi steps

\begin{align*} \int \frac{(a+b x)^5}{x^9} \, dx &=-\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]  time = 0.006202, size = 67, normalized size = 1.2 \[ -\frac{5 a^3 b^2}{3 x^6}-\frac{2 a^2 b^3}{x^5}-\frac{5 a^4 b}{7 x^7}-\frac{a^5}{8 x^8}-\frac{5 a b^4}{4 x^4}-\frac{b^5}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^5/x^9,x]

[Out]

-a^5/(8*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]  time = 0.007, size = 58, normalized size = 1. \begin{align*} -{\frac{{b}^{5}}{3\,{x}^{3}}}-2\,{\frac{{a}^{2}{b}^{3}}{{x}^{5}}}-{\frac{5\,a{b}^{4}}{4\,{x}^{4}}}-{\frac{5\,{a}^{3}{b}^{2}}{3\,{x}^{6}}}-{\frac{{a}^{5}}{8\,{x}^{8}}}-{\frac{5\,{a}^{4}b}{7\,{x}^{7}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.09121, size = 77, normalized size = 1.38 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

Fricas [A]  time = 1.46598, size = 135, normalized size = 2.41 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]  time = 0.727871, size = 61, normalized size = 1.09 \begin{align*} - \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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
)

________________________________________________________________________________________

Giac [A]  time = 1.19054, size = 77, normalized size = 1.38 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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