Optimal. Leaf size=76 \[ -3 a^4 b c^4 x+a^3 b^2 c^4 x^2+\frac {2}{3} a^2 b^3 c^4 x^3-\frac {3}{4} a b^4 c^4 x^4+\frac {1}{5} b^5 c^4 x^5+a^5 c^4 \log (x) \]
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Rubi [A]
time = 0.02, antiderivative size = 76, 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*} a^5 c^4 \log (x)-3 a^4 b c^4 x+a^3 b^2 c^4 x^2+\frac {2}{3} a^2 b^3 c^4 x^3-\frac {3}{4} a b^4 c^4 x^4+\frac {1}{5} b^5 c^4 x^5 \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} \, dx &=\int \left (-3 a^4 b c^4+\frac {a^5 c^4}{x}+2 a^3 b^2 c^4 x+2 a^2 b^3 c^4 x^2-3 a b^4 c^4 x^3+b^5 c^4 x^4\right ) \, dx\\ &=-3 a^4 b c^4 x+a^3 b^2 c^4 x^2+\frac {2}{3} a^2 b^3 c^4 x^3-\frac {3}{4} a b^4 c^4 x^4+\frac {1}{5} b^5 c^4 x^5+a^5 c^4 \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 70, normalized size = 0.92 \begin {gather*} \frac {1}{60} c^4 \left (113 a^5-180 a^4 b x+60 a^3 b^2 x^2+40 a^2 b^3 x^3-45 a b^4 x^4+12 b^5 x^5+60 a^5 \log (-b c x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 57, normalized size = 0.75
method | result | size |
default | \(c^{4} \left (\frac {b^{5} x^{5}}{5}-\frac {3 a \,b^{4} x^{4}}{4}+\frac {2 a^{2} b^{3} x^{3}}{3}+a^{3} b^{2} x^{2}-3 a^{4} b x +a^{5} \ln \left (x \right )\right )\) | \(57\) |
norman | \(-3 a^{4} b \,c^{4} x +a^{3} b^{2} c^{4} x^{2}+\frac {2 a^{2} b^{3} c^{4} x^{3}}{3}-\frac {3 a \,b^{4} c^{4} x^{4}}{4}+\frac {b^{5} c^{4} x^{5}}{5}+a^{5} c^{4} \ln \left (x \right )\) | \(71\) |
risch | \(-3 a^{4} b \,c^{4} x +a^{3} b^{2} c^{4} x^{2}+\frac {2 a^{2} b^{3} c^{4} x^{3}}{3}-\frac {3 a \,b^{4} c^{4} x^{4}}{4}+\frac {b^{5} c^{4} x^{5}}{5}+a^{5} c^{4} \ln \left (x \right )\) | \(71\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 70, normalized size = 0.92 \begin {gather*} \frac {1}{5} \, b^{5} c^{4} x^{5} - \frac {3}{4} \, a b^{4} c^{4} x^{4} + \frac {2}{3} \, a^{2} b^{3} c^{4} x^{3} + a^{3} b^{2} c^{4} x^{2} - 3 \, a^{4} b c^{4} x + a^{5} c^{4} \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.12, size = 70, normalized size = 0.92 \begin {gather*} \frac {1}{5} \, b^{5} c^{4} x^{5} - \frac {3}{4} \, a b^{4} c^{4} x^{4} + \frac {2}{3} \, a^{2} b^{3} c^{4} x^{3} + a^{3} b^{2} c^{4} x^{2} - 3 \, a^{4} b c^{4} x + a^{5} c^{4} \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 78, normalized size = 1.03 \begin {gather*} a^{5} c^{4} \log {\left (x \right )} - 3 a^{4} b c^{4} x + a^{3} b^{2} c^{4} x^{2} + \frac {2 a^{2} b^{3} c^{4} x^{3}}{3} - \frac {3 a b^{4} c^{4} x^{4}}{4} + \frac {b^{5} c^{4} x^{5}}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.95, size = 71, normalized size = 0.93 \begin {gather*} \frac {1}{5} \, b^{5} c^{4} x^{5} - \frac {3}{4} \, a b^{4} c^{4} x^{4} + \frac {2}{3} \, a^{2} b^{3} c^{4} x^{3} + a^{3} b^{2} c^{4} x^{2} - 3 \, a^{4} b c^{4} x + a^{5} c^{4} \log \left ({\left | x \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 70, normalized size = 0.92 \begin {gather*} \frac {b^5\,c^4\,x^5}{5}+a^5\,c^4\,\ln \left (x\right )-\frac {3\,a\,b^4\,c^4\,x^4}{4}+a^3\,b^2\,c^4\,x^2+\frac {2\,a^2\,b^3\,c^4\,x^3}{3}-3\,a^4\,b\,c^4\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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