3.87 \(\int \frac {(a+b x)^5}{x^4} \, dx\)

Optimal. Leaf size=60 \[ -\frac {a^5}{3 x^3}-\frac {5 a^4 b}{2 x^2}-\frac {10 a^3 b^2}{x}+10 a^2 b^3 \log (x)+5 a b^4 x+\frac {b^5 x^2}{2} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 60, 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} \[ -\frac {10 a^3 b^2}{x}+10 a^2 b^3 \log (x)-\frac {5 a^4 b}{2 x^2}-\frac {a^5}{3 x^3}+5 a b^4 x+\frac {b^5 x^2}{2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^5/x^4,x]

[Out]

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

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Mathematica [A]  time = 0.00, size = 60, normalized size = 1.00 \[ -\frac {a^5}{3 x^3}-\frac {5 a^4 b}{2 x^2}-\frac {10 a^3 b^2}{x}+10 a^2 b^3 \log (x)+5 a b^4 x+\frac {b^5 x^2}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^5/x^4,x]

[Out]

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

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fricas [A]  time = 0.48, size = 59, normalized size = 0.98 \[ \frac {3 \, b^{5} x^{5} + 30 \, a b^{4} x^{4} + 60 \, a^{2} b^{3} x^{3} \log \relax (x) - 60 \, a^{3} b^{2} x^{2} - 15 \, a^{4} b x - 2 \, a^{5}}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/x^4,x, algorithm="fricas")

[Out]

1/6*(3*b^5*x^5 + 30*a*b^4*x^4 + 60*a^2*b^3*x^3*log(x) - 60*a^3*b^2*x^2 - 15*a^4*b*x - 2*a^5)/x^3

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giac [A]  time = 1.00, size = 56, normalized size = 0.93 \[ \frac {1}{2} \, b^{5} x^{2} + 5 \, a b^{4} x + 10 \, a^{2} b^{3} \log \left ({\left | x \right |}\right ) - \frac {60 \, a^{3} b^{2} x^{2} + 15 \, a^{4} b x + 2 \, a^{5}}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/x^4,x, algorithm="giac")

[Out]

1/2*b^5*x^2 + 5*a*b^4*x + 10*a^2*b^3*log(abs(x)) - 1/6*(60*a^3*b^2*x^2 + 15*a^4*b*x + 2*a^5)/x^3

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maple [A]  time = 0.01, size = 55, normalized size = 0.92 \[ \frac {b^{5} x^{2}}{2}+10 a^{2} b^{3} \ln \relax (x )+5 a \,b^{4} x -\frac {10 a^{3} b^{2}}{x}-\frac {5 a^{4} b}{2 x^{2}}-\frac {a^{5}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^5/x^4,x)

[Out]

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

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maxima [A]  time = 1.41, size = 55, normalized size = 0.92 \[ \frac {1}{2} \, b^{5} x^{2} + 5 \, a b^{4} x + 10 \, a^{2} b^{3} \log \relax (x) - \frac {60 \, a^{3} b^{2} x^{2} + 15 \, a^{4} b x + 2 \, a^{5}}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/x^4,x, algorithm="maxima")

[Out]

1/2*b^5*x^2 + 5*a*b^4*x + 10*a^2*b^3*log(x) - 1/6*(60*a^3*b^2*x^2 + 15*a^4*b*x + 2*a^5)/x^3

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mupad [B]  time = 0.04, size = 55, normalized size = 0.92 \[ \frac {b^5\,x^2}{2}-\frac {\frac {a^5}{3}+\frac {5\,a^4\,b\,x}{2}+10\,a^3\,b^2\,x^2}{x^3}+10\,a^2\,b^3\,\ln \relax (x)+5\,a\,b^4\,x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^5/x^4,x)

[Out]

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

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sympy [A]  time = 0.26, size = 60, normalized size = 1.00 \[ 10 a^{2} b^{3} \log {\relax (x )} + 5 a b^{4} x + \frac {b^{5} x^{2}}{2} + \frac {- 2 a^{5} - 15 a^{4} b x - 60 a^{3} b^{2} x^{2}}{6 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**5/x**4,x)

[Out]

10*a**2*b**3*log(x) + 5*a*b**4*x + b**5*x**2/2 + (-2*a**5 - 15*a**4*b*x - 60*a**3*b**2*x**2)/(6*x**3)

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