3.73 \(\int \frac{(a+b x)^3}{x^5} \, dx\)

Optimal. Leaf size=17 \[ -\frac{(a+b x)^4}{4 a x^4} \]

[Out]

-(a + b*x)^4/(4*a*x^4)

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Rubi [A]  time = 0.0016722, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {37} \[ -\frac{(a+b x)^4}{4 a x^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^3/x^5,x]

[Out]

-(a + b*x)^4/(4*a*x^4)

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)^3}{x^5} \, dx &=-\frac{(a+b x)^4}{4 a x^4}\\ \end{align*}

Mathematica [B]  time = 0.0046872, size = 39, normalized size = 2.29 \[ -\frac{a^2 b}{x^3}-\frac{a^3}{4 x^4}-\frac{3 a b^2}{2 x^2}-\frac{b^3}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^3/x^5,x]

[Out]

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

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Maple [B]  time = 0.005, size = 36, normalized size = 2.1 \begin{align*} -{\frac{{a}^{2}b}{{x}^{3}}}-{\frac{{a}^{3}}{4\,{x}^{4}}}-{\frac{3\,{b}^{2}a}{2\,{x}^{2}}}-{\frac{{b}^{3}}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3/x^5,x)

[Out]

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

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Maxima [B]  time = 1.01065, size = 45, normalized size = 2.65 \begin{align*} -\frac{4 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 4 \, a^{2} b x + a^{3}}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/x^5,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.61322, size = 73, normalized size = 4.29 \begin{align*} -\frac{4 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 4 \, a^{2} b x + a^{3}}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/x^5,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.448039, size = 36, normalized size = 2.12 \begin{align*} - \frac{a^{3} + 4 a^{2} b x + 6 a b^{2} x^{2} + 4 b^{3} x^{3}}{4 x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3/x**5,x)

[Out]

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

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Giac [B]  time = 1.1902, size = 45, normalized size = 2.65 \begin{align*} -\frac{4 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 4 \, a^{2} b x + a^{3}}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/x^5,x, algorithm="giac")

[Out]

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