Optimal. Leaf size=72 \[ -\frac {a^5 c^4}{4 x^4}+\frac {a^4 b c^4}{x^3}-\frac {a^3 b^2 c^4}{x^2}-\frac {2 a^2 b^3 c^4}{x}+b^5 c^4 x-3 a b^4 c^4 \log (x) \]
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Rubi [A]
time = 0.02, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {76}
\begin {gather*} -\frac {a^5 c^4}{4 x^4}+\frac {a^4 b c^4}{x^3}-\frac {a^3 b^2 c^4}{x^2}-\frac {2 a^2 b^3 c^4}{x}-3 a b^4 c^4 \log (x)+b^5 c^4 x \end {gather*}
Antiderivative was successfully verified.
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Rule 76
Rubi steps
\begin {align*} \int \frac {(a+b x) (a c-b c x)^4}{x^5} \, dx &=\int \left (b^5 c^4+\frac {a^5 c^4}{x^5}-\frac {3 a^4 b c^4}{x^4}+\frac {2 a^3 b^2 c^4}{x^3}+\frac {2 a^2 b^3 c^4}{x^2}-\frac {3 a b^4 c^4}{x}\right ) \, dx\\ &=-\frac {a^5 c^4}{4 x^4}+\frac {a^4 b c^4}{x^3}-\frac {a^3 b^2 c^4}{x^2}-\frac {2 a^2 b^3 c^4}{x}+b^5 c^4 x-3 a b^4 c^4 \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 72, normalized size = 1.00 \begin {gather*} -\frac {a^5 c^4}{4 x^4}+\frac {a^4 b c^4}{x^3}-\frac {a^3 b^2 c^4}{x^2}-\frac {2 a^2 b^3 c^4}{x}+b^5 c^4 x-3 a b^4 c^4 \log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 57, normalized size = 0.79
method | result | size |
default | \(c^{4} \left (b^{5} x -\frac {2 a^{2} b^{3}}{x}-\frac {a^{5}}{4 x^{4}}-\frac {a^{3} b^{2}}{x^{2}}-3 a \,b^{4} \ln \left (x \right )+\frac {a^{4} b}{x^{3}}\right )\) | \(57\) |
risch | \(b^{5} c^{4} x +\frac {-2 a^{2} b^{3} c^{4} x^{3}-a^{3} b^{2} c^{4} x^{2}+a^{4} b \,c^{4} x -\frac {1}{4} a^{5} c^{4}}{x^{4}}-3 a \,b^{4} c^{4} \ln \left (x \right )\) | \(71\) |
norman | \(\frac {b^{5} c^{4} x^{5}+a^{4} b \,c^{4} x -\frac {1}{4} a^{5} c^{4}-2 a^{2} b^{3} c^{4} x^{3}-a^{3} b^{2} c^{4} x^{2}}{x^{4}}-3 a \,b^{4} c^{4} \ln \left (x \right )\) | \(73\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 71, normalized size = 0.99 \begin {gather*} b^{5} c^{4} x - 3 \, a b^{4} c^{4} \log \left (x\right ) - \frac {8 \, a^{2} b^{3} c^{4} x^{3} + 4 \, a^{3} b^{2} c^{4} x^{2} - 4 \, a^{4} b c^{4} x + a^{5} c^{4}}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.96, size = 77, normalized size = 1.07 \begin {gather*} \frac {4 \, b^{5} c^{4} x^{5} - 12 \, a b^{4} c^{4} x^{4} \log \left (x\right ) - 8 \, a^{2} b^{3} c^{4} x^{3} - 4 \, a^{3} b^{2} c^{4} x^{2} + 4 \, a^{4} b c^{4} x - a^{5} c^{4}}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.14, size = 75, normalized size = 1.04 \begin {gather*} - 3 a b^{4} c^{4} \log {\left (x \right )} + b^{5} c^{4} x + \frac {- a^{5} c^{4} + 4 a^{4} b c^{4} x - 4 a^{3} b^{2} c^{4} x^{2} - 8 a^{2} b^{3} c^{4} x^{3}}{4 x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.73, size = 72, normalized size = 1.00 \begin {gather*} b^{5} c^{4} x - 3 \, a b^{4} c^{4} \log \left ({\left | x \right |}\right ) - \frac {8 \, a^{2} b^{3} c^{4} x^{3} + 4 \, a^{3} b^{2} c^{4} x^{2} - 4 \, a^{4} b c^{4} x + a^{5} c^{4}}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.29, size = 60, normalized size = 0.83 \begin {gather*} -\frac {c^4\,\left (a^5-4\,b^5\,x^5+4\,a^3\,b^2\,x^2+8\,a^2\,b^3\,x^3-4\,a^4\,b\,x+12\,a\,b^4\,x^4\,\ln \left (x\right )\right )}{4\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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