Optimal. Leaf size=57 \[ -\frac {a^3 x}{b^4}+\frac {a^2 x^2}{2 b^3}-\frac {a x^3}{3 b^2}+\frac {x^4}{4 b}+\frac {a^4 \log (a+b x)}{b^5} \]
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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 = {45}
\begin {gather*} \frac {a^4 \log (a+b x)}{b^5}-\frac {a^3 x}{b^4}+\frac {a^2 x^2}{2 b^3}-\frac {a x^3}{3 b^2}+\frac {x^4}{4 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int \frac {x^4}{a+b x} \, dx &=\int \left (-\frac {a^3}{b^4}+\frac {a^2 x}{b^3}-\frac {a x^2}{b^2}+\frac {x^3}{b}+\frac {a^4}{b^4 (a+b x)}\right ) \, dx\\ &=-\frac {a^3 x}{b^4}+\frac {a^2 x^2}{2 b^3}-\frac {a x^3}{3 b^2}+\frac {x^4}{4 b}+\frac {a^4 \log (a+b x)}{b^5}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 57, normalized size = 1.00 \begin {gather*} -\frac {a^3 x}{b^4}+\frac {a^2 x^2}{2 b^3}-\frac {a x^3}{3 b^2}+\frac {x^4}{4 b}+\frac {a^4 \log (a+b x)}{b^5} \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 1.92, size = 50, normalized size = 0.88 \begin {gather*} \frac {a^4 \text {Log}\left [a+b x\right ]-a^3 b x+\frac {a^2 b^2 x^2}{2}-\frac {a b^3 x^3}{3}+\frac {b^4 x^4}{4}}{b^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 52, normalized size = 0.91
method | result | size |
default | \(-\frac {-\frac {1}{4} b^{3} x^{4}+\frac {1}{3} a \,b^{2} x^{3}-\frac {1}{2} a^{2} b \,x^{2}+a^{3} x}{b^{4}}+\frac {a^{4} \ln \left (b x +a \right )}{b^{5}}\) | \(52\) |
norman | \(-\frac {a^{3} x}{b^{4}}+\frac {a^{2} x^{2}}{2 b^{3}}-\frac {a \,x^{3}}{3 b^{2}}+\frac {x^{4}}{4 b}+\frac {a^{4} \ln \left (b x +a \right )}{b^{5}}\) | \(52\) |
risch | \(-\frac {a^{3} x}{b^{4}}+\frac {a^{2} x^{2}}{2 b^{3}}-\frac {a \,x^{3}}{3 b^{2}}+\frac {x^{4}}{4 b}+\frac {a^{4} \ln \left (b x +a \right )}{b^{5}}\) | \(52\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.24, size = 52, normalized size = 0.91 \begin {gather*} \frac {a^{4} \log \left (b x + a\right )}{b^{5}} + \frac {3 \, b^{3} x^{4} - 4 \, a b^{2} x^{3} + 6 \, a^{2} b x^{2} - 12 \, a^{3} x}{12 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.30, size = 52, normalized size = 0.91 \begin {gather*} \frac {3 \, b^{4} x^{4} - 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} - 12 \, a^{3} b x + 12 \, a^{4} \log \left (b x + a\right )}{12 \, b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 49, normalized size = 0.86 \begin {gather*} \frac {a^{4} \log {\left (a + b x \right )}}{b^{5}} - \frac {a^{3} x}{b^{4}} + \frac {a^{2} x^{2}}{2 b^{3}} - \frac {a x^{3}}{3 b^{2}} + \frac {x^{4}}{4 b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 61, normalized size = 1.07 \begin {gather*} \frac {\frac {1}{4} x^{4} b^{3}-\frac {1}{3} x^{3} b^{2} a+\frac {1}{2} x^{2} b a^{2}-x a^{3}}{b^{4}}+\frac {a^{4} \ln \left |x b+a\right |}{b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.10, size = 51, normalized size = 0.89 \begin {gather*} \frac {x^4}{4\,b}+\frac {a^4\,\ln \left (a+b\,x\right )}{b^5}-\frac {a\,x^3}{3\,b^2}-\frac {a^3\,x}{b^4}+\frac {a^2\,x^2}{2\,b^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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