3.76 \(\int \frac{(a+b x)^3}{x^8} \, dx\)

Optimal. Leaf size=43 \[ -\frac{a^2 b}{2 x^6}-\frac{a^3}{7 x^7}-\frac{3 a b^2}{5 x^5}-\frac{b^3}{4 x^4} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0129022, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {43} \[ -\frac{a^2 b}{2 x^6}-\frac{a^3}{7 x^7}-\frac{3 a b^2}{5 x^5}-\frac{b^3}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^3/x^8,x]

[Out]

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

Rule 43

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

Mathematica [A]  time = 0.0037817, size = 43, normalized size = 1. \[ -\frac{a^2 b}{2 x^6}-\frac{a^3}{7 x^7}-\frac{3 a b^2}{5 x^5}-\frac{b^3}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^3/x^8,x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 36, normalized size = 0.8 \begin{align*} -{\frac{{a}^{3}}{7\,{x}^{7}}}-{\frac{{a}^{2}b}{2\,{x}^{6}}}-{\frac{3\,{b}^{2}a}{5\,{x}^{5}}}-{\frac{{b}^{3}}{4\,{x}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3/x^8,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.11632, size = 47, normalized size = 1.09 \begin{align*} -\frac{35 \, b^{3} x^{3} + 84 \, a b^{2} x^{2} + 70 \, a^{2} b x + 20 \, a^{3}}{140 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/x^8,x, algorithm="maxima")

[Out]

-1/140*(35*b^3*x^3 + 84*a*b^2*x^2 + 70*a^2*b*x + 20*a^3)/x^7

________________________________________________________________________________________

Fricas [A]  time = 1.62208, size = 84, normalized size = 1.95 \begin{align*} -\frac{35 \, b^{3} x^{3} + 84 \, a b^{2} x^{2} + 70 \, a^{2} b x + 20 \, a^{3}}{140 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/x^8,x, algorithm="fricas")

[Out]

-1/140*(35*b^3*x^3 + 84*a*b^2*x^2 + 70*a^2*b*x + 20*a^3)/x^7

________________________________________________________________________________________

Sympy [A]  time = 0.559754, size = 37, normalized size = 0.86 \begin{align*} - \frac{20 a^{3} + 70 a^{2} b x + 84 a b^{2} x^{2} + 35 b^{3} x^{3}}{140 x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3/x**8,x)

[Out]

-(20*a**3 + 70*a**2*b*x + 84*a*b**2*x**2 + 35*b**3*x**3)/(140*x**7)

________________________________________________________________________________________

Giac [A]  time = 1.16648, size = 47, normalized size = 1.09 \begin{align*} -\frac{35 \, b^{3} x^{3} + 84 \, a b^{2} x^{2} + 70 \, a^{2} b x + 20 \, a^{3}}{140 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/x^8,x, algorithm="giac")

[Out]

-1/140*(35*b^3*x^3 + 84*a*b^2*x^2 + 70*a^2*b*x + 20*a^3)/x^7