3.90 \(\int \frac{(a+b x)^5}{x^7} \, dx\)

Optimal. Leaf size=17 \[ -\frac{(a+b x)^6}{6 a x^6} \]

[Out]

-(a + b*x)^6/(6*a*x^6)

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Rubi [A]  time = 0.0016275, 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} \[ -\frac{(a+b x)^6}{6 a x^6} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^5/x^7,x]

[Out]

-(a + b*x)^6/(6*a*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]

Rubi steps

\begin{align*} \int \frac{(a+b x)^5}{x^7} \, dx &=-\frac{(a+b x)^6}{6 a x^6}\\ \end{align*}

Mathematica [B]  time = 0.0087569, size = 65, normalized size = 3.82 \[ -\frac{5 a^3 b^2}{2 x^4}-\frac{10 a^2 b^3}{3 x^3}-\frac{a^4 b}{x^5}-\frac{a^5}{6 x^6}-\frac{5 a b^4}{2 x^2}-\frac{b^5}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^5/x^7,x]

[Out]

-a^5/(6*x^6) - (a^4*b)/x^5 - (5*a^3*b^2)/(2*x^4) - (10*a^2*b^3)/(3*x^3) - (5*a*b^4)/(2*x^2) - b^5/x

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Maple [B]  time = 0.006, size = 58, normalized size = 3.4 \begin{align*} -{\frac{10\,{a}^{2}{b}^{3}}{3\,{x}^{3}}}-{\frac{{a}^{4}b}{{x}^{5}}}-{\frac{5\,{a}^{3}{b}^{2}}{2\,{x}^{4}}}-{\frac{{a}^{5}}{6\,{x}^{6}}}-{\frac{5\,a{b}^{4}}{2\,{x}^{2}}}-{\frac{{b}^{5}}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^5/x^7,x)

[Out]

-10/3*a^2*b^3/x^3-a^4*b/x^5-5/2*a^3*b^2/x^4-1/6*a^5/x^6-5/2*a*b^4/x^2-b^5/x

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Maxima [B]  time = 1.05717, size = 74, normalized size = 4.35 \begin{align*} -\frac{6 \, b^{5} x^{5} + 15 \, a b^{4} x^{4} + 20 \, a^{2} b^{3} x^{3} + 15 \, a^{3} b^{2} x^{2} + 6 \, a^{4} b x + a^{5}}{6 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/x^7,x, algorithm="maxima")

[Out]

-1/6*(6*b^5*x^5 + 15*a*b^4*x^4 + 20*a^2*b^3*x^3 + 15*a^3*b^2*x^2 + 6*a^4*b*x + a^5)/x^6

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Fricas [B]  time = 1.51836, size = 120, normalized size = 7.06 \begin{align*} -\frac{6 \, b^{5} x^{5} + 15 \, a b^{4} x^{4} + 20 \, a^{2} b^{3} x^{3} + 15 \, a^{3} b^{2} x^{2} + 6 \, a^{4} b x + a^{5}}{6 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/x^7,x, algorithm="fricas")

[Out]

-1/6*(6*b^5*x^5 + 15*a*b^4*x^4 + 20*a^2*b^3*x^3 + 15*a^3*b^2*x^2 + 6*a^4*b*x + a^5)/x^6

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Sympy [B]  time = 0.613611, size = 60, normalized size = 3.53 \begin{align*} - \frac{a^{5} + 6 a^{4} b x + 15 a^{3} b^{2} x^{2} + 20 a^{2} b^{3} x^{3} + 15 a b^{4} x^{4} + 6 b^{5} x^{5}}{6 x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**5/x**7,x)

[Out]

-(a**5 + 6*a**4*b*x + 15*a**3*b**2*x**2 + 20*a**2*b**3*x**3 + 15*a*b**4*x**4 + 6*b**5*x**5)/(6*x**6)

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Giac [B]  time = 1.20617, size = 74, normalized size = 4.35 \begin{align*} -\frac{6 \, b^{5} x^{5} + 15 \, a b^{4} x^{4} + 20 \, a^{2} b^{3} x^{3} + 15 \, a^{3} b^{2} x^{2} + 6 \, a^{4} b x + a^{5}}{6 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/x^7,x, algorithm="giac")

[Out]

-1/6*(6*b^5*x^5 + 15*a*b^4*x^4 + 20*a^2*b^3*x^3 + 15*a^3*b^2*x^2 + 6*a^4*b*x + a^5)/x^6