Optimal. Leaf size=58 \[ \frac {3 a^2 x}{b^4}-\frac {a x^2}{b^3}+\frac {x^3}{3 b^2}-\frac {a^4}{b^5 (a+b x)}-\frac {4 a^3 \log (a+b x)}{b^5} \]
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Rubi [A]
time = 0.03, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {1598, 45}
\begin {gather*} -\frac {a^4}{b^5 (a+b x)}-\frac {4 a^3 \log (a+b x)}{b^5}+\frac {3 a^2 x}{b^4}-\frac {a x^2}{b^3}+\frac {x^3}{3 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 1598
Rubi steps
\begin {align*} \int \frac {x^8}{\left (a x^2+b x^3\right )^2} \, dx &=\int \frac {x^4}{(a+b x)^2} \, dx\\ &=\int \left (\frac {3 a^2}{b^4}-\frac {2 a x}{b^3}+\frac {x^2}{b^2}+\frac {a^4}{b^4 (a+b x)^2}-\frac {4 a^3}{b^4 (a+b x)}\right ) \, dx\\ &=\frac {3 a^2 x}{b^4}-\frac {a x^2}{b^3}+\frac {x^3}{3 b^2}-\frac {a^4}{b^5 (a+b x)}-\frac {4 a^3 \log (a+b x)}{b^5}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 54, normalized size = 0.93 \begin {gather*} \frac {9 a^2 b x-3 a b^2 x^2+b^3 x^3-\frac {3 a^4}{a+b x}-12 a^3 \log (a+b x)}{3 b^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.37, size = 57, normalized size = 0.98
method | result | size |
default | \(\frac {\frac {1}{3} b^{2} x^{3}-a b \,x^{2}+3 a^{2} x}{b^{4}}-\frac {a^{4}}{b^{5} \left (b x +a \right )}-\frac {4 a^{3} \ln \left (b x +a \right )}{b^{5}}\) | \(57\) |
risch | \(\frac {3 a^{2} x}{b^{4}}-\frac {a \,x^{2}}{b^{3}}+\frac {x^{3}}{3 b^{2}}-\frac {a^{4}}{b^{5} \left (b x +a \right )}-\frac {4 a^{3} \ln \left (b x +a \right )}{b^{5}}\) | \(57\) |
norman | \(\frac {\frac {x^{7}}{3 b}-\frac {2 a \,x^{6}}{3 b^{2}}+\frac {2 a^{2} x^{5}}{b^{3}}-\frac {4 a^{4} x^{3}}{b^{5}}}{x^{3} \left (b x +a \right )}-\frac {4 a^{3} \ln \left (b x +a \right )}{b^{5}}\) | \(67\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 59, normalized size = 1.02 \begin {gather*} -\frac {a^{4}}{b^{6} x + a b^{5}} - \frac {4 \, a^{3} \log \left (b x + a\right )}{b^{5}} + \frac {b^{2} x^{3} - 3 \, a b x^{2} + 9 \, a^{2} x}{3 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.39, size = 73, normalized size = 1.26 \begin {gather*} \frac {b^{4} x^{4} - 2 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 9 \, a^{3} b x - 3 \, a^{4} - 12 \, {\left (a^{3} b x + a^{4}\right )} \log \left (b x + a\right )}{3 \, {\left (b^{6} x + a b^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 54, normalized size = 0.93 \begin {gather*} - \frac {a^{4}}{a b^{5} + b^{6} x} - \frac {4 a^{3} \log {\left (a + b x \right )}}{b^{5}} + \frac {3 a^{2} x}{b^{4}} - \frac {a x^{2}}{b^{3}} + \frac {x^{3}}{3 b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.12, size = 62, normalized size = 1.07 \begin {gather*} -\frac {4 \, a^{3} \log \left ({\left | b x + a \right |}\right )}{b^{5}} - \frac {a^{4}}{{\left (b x + a\right )} b^{5}} + \frac {b^{4} x^{3} - 3 \, a b^{3} x^{2} + 9 \, a^{2} b^{2} x}{3 \, b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 62, normalized size = 1.07 \begin {gather*} \frac {x^3}{3\,b^2}-\frac {4\,a^3\,\ln \left (a+b\,x\right )}{b^5}-\frac {a\,x^2}{b^3}+\frac {3\,a^2\,x}{b^4}-\frac {a^4}{b\,\left (x\,b^5+a\,b^4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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