Optimal. Leaf size=58 \[ -\frac {a^5}{x}+10 a^3 b^2 x+5 a^2 b^3 x^2+\frac {5}{3} a b^4 x^3+\frac {b^5 x^4}{4}+5 a^4 b \log (x) \]
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Rubi [A]
time = 0.01, antiderivative size = 58, 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 = {45}
\begin {gather*} -\frac {a^5}{x}+5 a^4 b \log (x)+10 a^3 b^2 x+5 a^2 b^3 x^2+\frac {5}{3} a b^4 x^3+\frac {b^5 x^4}{4} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int \frac {(a+b x)^5}{x^2} \, dx &=\int \left (10 a^3 b^2+\frac {a^5}{x^2}+\frac {5 a^4 b}{x}+10 a^2 b^3 x+5 a b^4 x^2+b^5 x^3\right ) \, dx\\ &=-\frac {a^5}{x}+10 a^3 b^2 x+5 a^2 b^3 x^2+\frac {5}{3} a b^4 x^3+\frac {b^5 x^4}{4}+5 a^4 b \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 58, normalized size = 1.00 \begin {gather*} -\frac {a^5}{x}+10 a^3 b^2 x+5 a^2 b^3 x^2+\frac {5}{3} a b^4 x^3+\frac {b^5 x^4}{4}+5 a^4 b \log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 55, normalized size = 0.95
method | result | size |
default | \(-\frac {a^{5}}{x}+10 a^{3} b^{2} x +5 a^{2} b^{3} x^{2}+\frac {5 a \,b^{4} x^{3}}{3}+\frac {b^{5} x^{4}}{4}+5 a^{4} b \ln \left (x \right )\) | \(55\) |
risch | \(-\frac {a^{5}}{x}+10 a^{3} b^{2} x +5 a^{2} b^{3} x^{2}+\frac {5 a \,b^{4} x^{3}}{3}+\frac {b^{5} x^{4}}{4}+5 a^{4} b \ln \left (x \right )\) | \(55\) |
norman | \(\frac {-a^{5}+\frac {1}{4} b^{5} x^{5}+\frac {5}{3} a \,b^{4} x^{4}+5 a^{2} b^{3} x^{3}+10 a^{3} b^{2} x^{2}}{x}+5 a^{4} b \ln \left (x \right )\) | \(59\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 54, normalized size = 0.93 \begin {gather*} \frac {1}{4} \, b^{5} x^{4} + \frac {5}{3} \, a b^{4} x^{3} + 5 \, a^{2} b^{3} x^{2} + 10 \, a^{3} b^{2} x + 5 \, a^{4} b \log \left (x\right ) - \frac {a^{5}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.77, size = 59, normalized size = 1.02 \begin {gather*} \frac {3 \, b^{5} x^{5} + 20 \, a b^{4} x^{4} + 60 \, a^{2} b^{3} x^{3} + 120 \, a^{3} b^{2} x^{2} + 60 \, a^{4} b x \log \left (x\right ) - 12 \, a^{5}}{12 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 56, normalized size = 0.97 \begin {gather*} - \frac {a^{5}}{x} + 5 a^{4} b \log {\left (x \right )} + 10 a^{3} b^{2} x + 5 a^{2} b^{3} x^{2} + \frac {5 a b^{4} x^{3}}{3} + \frac {b^{5} x^{4}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.21, size = 55, normalized size = 0.95 \begin {gather*} \frac {1}{4} \, b^{5} x^{4} + \frac {5}{3} \, a b^{4} x^{3} + 5 \, a^{2} b^{3} x^{2} + 10 \, a^{3} b^{2} x + 5 \, a^{4} b \log \left ({\left | x \right |}\right ) - \frac {a^{5}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 54, normalized size = 0.93 \begin {gather*} \frac {b^5\,x^4}{4}-\frac {a^5}{x}+10\,a^3\,b^2\,x+\frac {5\,a\,b^4\,x^3}{3}+5\,a^4\,b\,\ln \left (x\right )+5\,a^2\,b^3\,x^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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