Optimal. Leaf size=46 \[ -\frac {2 a x}{b^3}+\frac {x^2}{2 b^2}+\frac {a^3}{b^4 (a+b x)}+\frac {3 a^2 \log (a+b x)}{b^4} \]
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Rubi [A]
time = 0.02, antiderivative size = 46, 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^3}{b^4 (a+b x)}+\frac {3 a^2 \log (a+b x)}{b^4}-\frac {2 a x}{b^3}+\frac {x^2}{2 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 1598
Rubi steps
\begin {align*} \int \frac {x^7}{\left (a x^2+b x^3\right )^2} \, dx &=\int \frac {x^3}{(a+b x)^2} \, dx\\ &=\int \left (-\frac {2 a}{b^3}+\frac {x}{b^2}-\frac {a^3}{b^3 (a+b x)^2}+\frac {3 a^2}{b^3 (a+b x)}\right ) \, dx\\ &=-\frac {2 a x}{b^3}+\frac {x^2}{2 b^2}+\frac {a^3}{b^4 (a+b x)}+\frac {3 a^2 \log (a+b x)}{b^4}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 43, normalized size = 0.93 \begin {gather*} \frac {-4 a b x+b^2 x^2+\frac {2 a^3}{a+b x}+6 a^2 \log (a+b x)}{2 b^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.35, size = 46, normalized size = 1.00
method | result | size |
risch | \(-\frac {2 a x}{b^{3}}+\frac {x^{2}}{2 b^{2}}+\frac {a^{3}}{b^{4} \left (b x +a \right )}+\frac {3 a^{2} \ln \left (b x +a \right )}{b^{4}}\) | \(45\) |
default | \(-\frac {-\frac {1}{2} b \,x^{2}+2 a x}{b^{3}}+\frac {a^{3}}{b^{4} \left (b x +a \right )}+\frac {3 a^{2} \ln \left (b x +a \right )}{b^{4}}\) | \(46\) |
norman | \(\frac {\frac {3 a^{3} x^{3}}{b^{4}}+\frac {x^{6}}{2 b}-\frac {3 a \,x^{5}}{2 b^{2}}}{x^{3} \left (b x +a \right )}+\frac {3 a^{2} \ln \left (b x +a \right )}{b^{4}}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 47, normalized size = 1.02 \begin {gather*} \frac {a^{3}}{b^{5} x + a b^{4}} + \frac {3 \, a^{2} \log \left (b x + a\right )}{b^{4}} + \frac {b x^{2} - 4 \, a x}{2 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.43, size = 62, normalized size = 1.35 \begin {gather*} \frac {b^{3} x^{3} - 3 \, a b^{2} x^{2} - 4 \, a^{2} b x + 2 \, a^{3} + 6 \, {\left (a^{2} b x + a^{3}\right )} \log \left (b x + a\right )}{2 \, {\left (b^{5} x + a b^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 44, normalized size = 0.96 \begin {gather*} \frac {a^{3}}{a b^{4} + b^{5} x} + \frac {3 a^{2} \log {\left (a + b x \right )}}{b^{4}} - \frac {2 a x}{b^{3}} + \frac {x^{2}}{2 b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.62, size = 48, normalized size = 1.04 \begin {gather*} \frac {3 \, a^{2} \log \left ({\left | b x + a \right |}\right )}{b^{4}} + \frac {a^{3}}{{\left (b x + a\right )} b^{4}} + \frac {b^{2} x^{2} - 4 \, a b x}{2 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 50, normalized size = 1.09 \begin {gather*} \frac {x^2}{2\,b^2}+\frac {3\,a^2\,\ln \left (a+b\,x\right )}{b^4}+\frac {a^3}{b\,\left (x\,b^4+a\,b^3\right )}-\frac {2\,a\,x}{b^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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