3.75 \(\int \frac{(a+b x)^3}{x^7} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0126824, 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} \[ -\frac{3 a^2 b}{5 x^5}-\frac{a^3}{6 x^6}-\frac{3 a b^2}{4 x^4}-\frac{b^3}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^3/x^7,x]

[Out]

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

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

Mathematica [A]  time = 0.003698, size = 43, normalized size = 1. \[ -\frac{3 a^2 b}{5 x^5}-\frac{a^3}{6 x^6}-\frac{3 a b^2}{4 x^4}-\frac{b^3}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^3/x^7,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3/x^7,x)

[Out]

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

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Maxima [A]  time = 1.04416, size = 47, normalized size = 1.09 \begin{align*} -\frac{20 \, b^{3} x^{3} + 45 \, a b^{2} x^{2} + 36 \, a^{2} b x + 10 \, a^{3}}{60 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/x^7,x, algorithm="maxima")

[Out]

-1/60*(20*b^3*x^3 + 45*a*b^2*x^2 + 36*a^2*b*x + 10*a^3)/x^6

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Fricas [A]  time = 1.51823, size = 82, normalized size = 1.91 \begin{align*} -\frac{20 \, b^{3} x^{3} + 45 \, a b^{2} x^{2} + 36 \, a^{2} b x + 10 \, a^{3}}{60 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/x^7,x, algorithm="fricas")

[Out]

-1/60*(20*b^3*x^3 + 45*a*b^2*x^2 + 36*a^2*b*x + 10*a^3)/x^6

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Sympy [A]  time = 0.552533, size = 37, normalized size = 0.86 \begin{align*} - \frac{10 a^{3} + 36 a^{2} b x + 45 a b^{2} x^{2} + 20 b^{3} x^{3}}{60 x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3/x**7,x)

[Out]

-(10*a**3 + 36*a**2*b*x + 45*a*b**2*x**2 + 20*b**3*x**3)/(60*x**6)

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Giac [A]  time = 1.10676, size = 47, normalized size = 1.09 \begin{align*} -\frac{20 \, b^{3} x^{3} + 45 \, a b^{2} x^{2} + 36 \, a^{2} b x + 10 \, a^{3}}{60 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/x^7,x, algorithm="giac")

[Out]

-1/60*(20*b^3*x^3 + 45*a*b^2*x^2 + 36*a^2*b*x + 10*a^3)/x^6