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

Optimal. Leaf size=57 \[ -\frac {a^5}{4 x^4}-\frac {5 a^4 b}{3 x^3}-\frac {5 a^3 b^2}{x^2}-\frac {10 a^2 b^3}{x}+5 a b^4 \log (x)+b^5 x \]

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Rubi [A]  time = 0.02, antiderivative size = 57, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {5 a^3 b^2}{x^2}-\frac {10 a^2 b^3}{x}-\frac {5 a^4 b}{3 x^3}-\frac {a^5}{4 x^4}+5 a b^4 \log (x)+b^5 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^5/x^5,x]

[Out]

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

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Mathematica [A]  time = 0.01, size = 57, normalized size = 1.00 \begin {gather*} -\frac {a^5}{4 x^4}-\frac {5 a^4 b}{3 x^3}-\frac {5 a^3 b^2}{x^2}-\frac {10 a^2 b^3}{x}+5 a b^4 \log (x)+b^5 x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^5/x^5,x]

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^5}{x^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)^5/x^5,x]

[Out]

IntegrateAlgebraic[(a + b*x)^5/x^5, x]

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fricas [A]  time = 0.81, size = 59, normalized size = 1.04 \begin {gather*} \frac {12 \, b^{5} x^{5} + 60 \, a b^{4} x^{4} \log \relax (x) - 120 \, a^{2} b^{3} x^{3} - 60 \, a^{3} b^{2} x^{2} - 20 \, a^{4} b x - 3 \, a^{5}}{12 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(12*b^5*x^5 + 60*a*b^4*x^4*log(x) - 120*a^2*b^3*x^3 - 60*a^3*b^2*x^2 - 20*a^4*b*x - 3*a^5)/x^4

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giac [A]  time = 1.38, size = 55, normalized size = 0.96 \begin {gather*} b^{5} x + 5 \, a b^{4} \log \left ({\left | x \right |}\right ) - \frac {120 \, a^{2} b^{3} x^{3} + 60 \, a^{3} b^{2} x^{2} + 20 \, a^{4} b x + 3 \, a^{5}}{12 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^5*x + 5*a*b^4*log(abs(x)) - 1/12*(120*a^2*b^3*x^3 + 60*a^3*b^2*x^2 + 20*a^4*b*x + 3*a^5)/x^4

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maple [A]  time = 0.01, size = 54, normalized size = 0.95 \begin {gather*} 5 a \,b^{4} \ln \relax (x )+b^{5} x -\frac {10 a^{2} b^{3}}{x}-\frac {5 a^{3} b^{2}}{x^{2}}-\frac {5 a^{4} b}{3 x^{3}}-\frac {a^{5}}{4 x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.37, size = 54, normalized size = 0.95 \begin {gather*} b^{5} x + 5 \, a b^{4} \log \relax (x) - \frac {120 \, a^{2} b^{3} x^{3} + 60 \, a^{3} b^{2} x^{2} + 20 \, a^{4} b x + 3 \, a^{5}}{12 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^5*x + 5*a*b^4*log(x) - 1/12*(120*a^2*b^3*x^3 + 60*a^3*b^2*x^2 + 20*a^4*b*x + 3*a^5)/x^4

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mupad [B]  time = 0.08, size = 54, normalized size = 0.95 \begin {gather*} b^5\,x-\frac {\frac {a^5}{4}+\frac {5\,a^4\,b\,x}{3}+5\,a^3\,b^2\,x^2+10\,a^2\,b^3\,x^3}{x^4}+5\,a\,b^4\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^5/x^5,x)

[Out]

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

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sympy [A]  time = 0.29, size = 58, normalized size = 1.02 \begin {gather*} 5 a b^{4} \log {\relax (x )} + b^{5} x + \frac {- 3 a^{5} - 20 a^{4} b x - 60 a^{3} b^{2} x^{2} - 120 a^{2} b^{3} x^{3}}{12 x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5*a*b**4*log(x) + b**5*x + (-3*a**5 - 20*a**4*b*x - 60*a**3*b**2*x**2 - 120*a**2*b**3*x**3)/(12*x**4)

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