3.89 \(\int \frac{(a+b x)^5}{x^6} \, dx\)

Optimal. Leaf size=61 \[ -\frac{10 a^3 b^2}{3 x^3}-\frac{5 a^2 b^3}{x^2}-\frac{5 a^4 b}{4 x^4}-\frac{a^5}{5 x^5}-\frac{5 a b^4}{x}+b^5 \log (x) \]

[Out]

-a^5/(5*x^5) - (5*a^4*b)/(4*x^4) - (10*a^3*b^2)/(3*x^3) - (5*a^2*b^3)/x^2 - (5*a*b^4)/x + b^5*Log[x]

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Rubi [A]  time = 0.0209423, antiderivative size = 61, 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} \[ -\frac{10 a^3 b^2}{3 x^3}-\frac{5 a^2 b^3}{x^2}-\frac{5 a^4 b}{4 x^4}-\frac{a^5}{5 x^5}-\frac{5 a b^4}{x}+b^5 \log (x) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^5/x^6,x]

[Out]

-a^5/(5*x^5) - (5*a^4*b)/(4*x^4) - (10*a^3*b^2)/(3*x^3) - (5*a^2*b^3)/x^2 - (5*a*b^4)/x + b^5*Log[x]

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)^5}{x^6} \, dx &=\int \left (\frac{a^5}{x^6}+\frac{5 a^4 b}{x^5}+\frac{10 a^3 b^2}{x^4}+\frac{10 a^2 b^3}{x^3}+\frac{5 a b^4}{x^2}+\frac{b^5}{x}\right ) \, dx\\ &=-\frac{a^5}{5 x^5}-\frac{5 a^4 b}{4 x^4}-\frac{10 a^3 b^2}{3 x^3}-\frac{5 a^2 b^3}{x^2}-\frac{5 a b^4}{x}+b^5 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0053105, size = 61, normalized size = 1. \[ -\frac{10 a^3 b^2}{3 x^3}-\frac{5 a^2 b^3}{x^2}-\frac{5 a^4 b}{4 x^4}-\frac{a^5}{5 x^5}-\frac{5 a b^4}{x}+b^5 \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^5/x^6,x]

[Out]

-a^5/(5*x^5) - (5*a^4*b)/(4*x^4) - (10*a^3*b^2)/(3*x^3) - (5*a^2*b^3)/x^2 - (5*a*b^4)/x + b^5*Log[x]

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Maple [A]  time = 0.006, size = 56, normalized size = 0.9 \begin{align*} -{\frac{{a}^{5}}{5\,{x}^{5}}}-{\frac{5\,{a}^{4}b}{4\,{x}^{4}}}-{\frac{10\,{a}^{3}{b}^{2}}{3\,{x}^{3}}}-5\,{\frac{{a}^{2}{b}^{3}}{{x}^{2}}}-5\,{\frac{a{b}^{4}}{x}}+{b}^{5}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^5/x^6,x)

[Out]

-1/5*a^5/x^5-5/4*a^4*b/x^4-10/3*a^3*b^2/x^3-5*a^2*b^3/x^2-5*a*b^4/x+b^5*ln(x)

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Maxima [A]  time = 1.02984, size = 76, normalized size = 1.25 \begin{align*} b^{5} \log \left (x\right ) - \frac{300 \, a b^{4} x^{4} + 300 \, a^{2} b^{3} x^{3} + 200 \, a^{3} b^{2} x^{2} + 75 \, a^{4} b x + 12 \, a^{5}}{60 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/x^6,x, algorithm="maxima")

[Out]

b^5*log(x) - 1/60*(300*a*b^4*x^4 + 300*a^2*b^3*x^3 + 200*a^3*b^2*x^2 + 75*a^4*b*x + 12*a^5)/x^5

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Fricas [A]  time = 1.62891, size = 140, normalized size = 2.3 \begin{align*} \frac{60 \, b^{5} x^{5} \log \left (x\right ) - 300 \, a b^{4} x^{4} - 300 \, a^{2} b^{3} x^{3} - 200 \, a^{3} b^{2} x^{2} - 75 \, a^{4} b x - 12 \, a^{5}}{60 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/x^6,x, algorithm="fricas")

[Out]

1/60*(60*b^5*x^5*log(x) - 300*a*b^4*x^4 - 300*a^2*b^3*x^3 - 200*a^3*b^2*x^2 - 75*a^4*b*x - 12*a^5)/x^5

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Sympy [A]  time = 0.620298, size = 58, normalized size = 0.95 \begin{align*} b^{5} \log{\left (x \right )} - \frac{12 a^{5} + 75 a^{4} b x + 200 a^{3} b^{2} x^{2} + 300 a^{2} b^{3} x^{3} + 300 a b^{4} x^{4}}{60 x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**5/x**6,x)

[Out]

b**5*log(x) - (12*a**5 + 75*a**4*b*x + 200*a**3*b**2*x**2 + 300*a**2*b**3*x**3 + 300*a*b**4*x**4)/(60*x**5)

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Giac [A]  time = 1.20627, size = 77, normalized size = 1.26 \begin{align*} b^{5} \log \left ({\left | x \right |}\right ) - \frac{300 \, a b^{4} x^{4} + 300 \, a^{2} b^{3} x^{3} + 200 \, a^{3} b^{2} x^{2} + 75 \, a^{4} b x + 12 \, a^{5}}{60 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/x^6,x, algorithm="giac")

[Out]

b^5*log(abs(x)) - 1/60*(300*a*b^4*x^4 + 300*a^2*b^3*x^3 + 200*a^3*b^2*x^2 + 75*a^4*b*x + 12*a^5)/x^5