Optimal. Leaf size=65 \[ \frac {x}{b^4}-\frac {a^4}{3 b^5 (a+b x)^3}+\frac {2 a^3}{b^5 (a+b x)^2}-\frac {6 a^2}{b^5 (a+b x)}-\frac {4 a \log (a+b x)}{b^5} \]
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Rubi [A]
time = 0.03, antiderivative size = 65, 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}{3 b^5 (a+b x)^3}+\frac {2 a^3}{b^5 (a+b x)^2}-\frac {6 a^2}{b^5 (a+b x)}-\frac {4 a \log (a+b x)}{b^5}+\frac {x}{b^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int \frac {x^4}{(a+b x)^4} \, dx &=\int \left (\frac {1}{b^4}+\frac {a^4}{b^4 (a+b x)^4}-\frac {4 a^3}{b^4 (a+b x)^3}+\frac {6 a^2}{b^4 (a+b x)^2}-\frac {4 a}{b^4 (a+b x)}\right ) \, dx\\ &=\frac {x}{b^4}-\frac {a^4}{3 b^5 (a+b x)^3}+\frac {2 a^3}{b^5 (a+b x)^2}-\frac {6 a^2}{b^5 (a+b x)}-\frac {4 a \log (a+b x)}{b^5}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 51, normalized size = 0.78 \begin {gather*} -\frac {-3 b x+\frac {a^2 \left (13 a^2+30 a b x+18 b^2 x^2\right )}{(a+b x)^3}+12 a \log (a+b x)}{3 b^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 64, normalized size = 0.98
method | result | size |
risch | \(\frac {x}{b^{4}}+\frac {-6 a^{2} b \,x^{2}-10 a^{3} x -\frac {13 a^{4}}{3 b}}{b^{4} \left (b x +a \right )^{3}}-\frac {4 a \ln \left (b x +a \right )}{b^{5}}\) | \(54\) |
norman | \(\frac {\frac {x^{4}}{b}-\frac {22 a^{4}}{3 b^{5}}-\frac {12 a^{2} x^{2}}{b^{3}}-\frac {18 a^{3} x}{b^{4}}}{\left (b x +a \right )^{3}}-\frac {4 a \ln \left (b x +a \right )}{b^{5}}\) | \(58\) |
default | \(\frac {x}{b^{4}}-\frac {a^{4}}{3 b^{5} \left (b x +a \right )^{3}}+\frac {2 a^{3}}{b^{5} \left (b x +a \right )^{2}}-\frac {6 a^{2}}{b^{5} \left (b x +a \right )}-\frac {4 a \ln \left (b x +a \right )}{b^{5}}\) | \(64\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 79, normalized size = 1.22 \begin {gather*} -\frac {18 \, a^{2} b^{2} x^{2} + 30 \, a^{3} b x + 13 \, a^{4}}{3 \, {\left (b^{8} x^{3} + 3 \, a b^{7} x^{2} + 3 \, a^{2} b^{6} x + a^{3} b^{5}\right )}} + \frac {x}{b^{4}} - \frac {4 \, a \log \left (b x + a\right )}{b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.71, size = 116, normalized size = 1.78 \begin {gather*} \frac {3 \, b^{4} x^{4} + 9 \, a b^{3} x^{3} - 9 \, a^{2} b^{2} x^{2} - 27 \, a^{3} b x - 13 \, a^{4} - 12 \, {\left (a b^{3} x^{3} + 3 \, a^{2} b^{2} x^{2} + 3 \, a^{3} b x + a^{4}\right )} \log \left (b x + a\right )}{3 \, {\left (b^{8} x^{3} + 3 \, a b^{7} x^{2} + 3 \, a^{2} b^{6} x + a^{3} b^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.16, size = 82, normalized size = 1.26 \begin {gather*} - \frac {4 a \log {\left (a + b x \right )}}{b^{5}} + \frac {- 13 a^{4} - 30 a^{3} b x - 18 a^{2} b^{2} x^{2}}{3 a^{3} b^{5} + 9 a^{2} b^{6} x + 9 a b^{7} x^{2} + 3 b^{8} x^{3}} + \frac {x}{b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.96, size = 55, normalized size = 0.85 \begin {gather*} \frac {x}{b^{4}} - \frac {4 \, a \log \left ({\left | b x + a \right |}\right )}{b^{5}} - \frac {18 \, a^{2} b^{2} x^{2} + 30 \, a^{3} b x + 13 \, a^{4}}{3 \, {\left (b x + a\right )}^{3} b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.17, size = 55, normalized size = 0.85 \begin {gather*} -\frac {4\,a\,\ln \left (a+b\,x\right )-b\,x+\frac {6\,a^2}{a+b\,x}-\frac {2\,a^3}{{\left (a+b\,x\right )}^2}+\frac {a^4}{3\,{\left (a+b\,x\right )}^3}}{b^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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