Integrand size = 11, antiderivative size = 64 \[ \int \frac {x^4}{(a+b x)^3} \, dx=-\frac {3 a x}{b^4}+\frac {x^2}{2 b^3}-\frac {a^4}{2 b^5 (a+b x)^2}+\frac {4 a^3}{b^5 (a+b x)}+\frac {6 a^2 \log (a+b x)}{b^5} \]
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Time = 0.02 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \[ \int \frac {x^4}{(a+b x)^3} \, dx=-\frac {a^4}{2 b^5 (a+b x)^2}+\frac {4 a^3}{b^5 (a+b x)}+\frac {6 a^2 \log (a+b x)}{b^5}-\frac {3 a x}{b^4}+\frac {x^2}{2 b^3} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {3 a}{b^4}+\frac {x}{b^3}+\frac {a^4}{b^4 (a+b x)^3}-\frac {4 a^3}{b^4 (a+b x)^2}+\frac {6 a^2}{b^4 (a+b x)}\right ) \, dx \\ & = -\frac {3 a x}{b^4}+\frac {x^2}{2 b^3}-\frac {a^4}{2 b^5 (a+b x)^2}+\frac {4 a^3}{b^5 (a+b x)}+\frac {6 a^2 \log (a+b x)}{b^5} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.86 \[ \int \frac {x^4}{(a+b x)^3} \, dx=\frac {-6 a b x+b^2 x^2-\frac {a^4}{(a+b x)^2}+\frac {8 a^3}{a+b x}+12 a^2 \log (a+b x)}{2 b^5} \]
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Time = 0.18 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.89
method | result | size |
risch | \(\frac {x^{2}}{2 b^{3}}-\frac {3 a x}{b^{4}}+\frac {4 a^{3} x +\frac {7 a^{4}}{2 b}}{b^{4} \left (b x +a \right )^{2}}+\frac {6 a^{2} \ln \left (b x +a \right )}{b^{5}}\) | \(57\) |
norman | \(\frac {\frac {9 a^{4}}{b^{5}}+\frac {x^{4}}{2 b}-\frac {2 a \,x^{3}}{b^{2}}+\frac {12 a^{3} x}{b^{4}}}{\left (b x +a \right )^{2}}+\frac {6 a^{2} \ln \left (b x +a \right )}{b^{5}}\) | \(59\) |
default | \(-\frac {-\frac {1}{2} b \,x^{2}+3 a x}{b^{4}}+\frac {6 a^{2} \ln \left (b x +a \right )}{b^{5}}-\frac {a^{4}}{2 b^{5} \left (b x +a \right )^{2}}+\frac {4 a^{3}}{b^{5} \left (b x +a \right )}\) | \(62\) |
parallelrisch | \(\frac {b^{4} x^{4}+12 \ln \left (b x +a \right ) x^{2} a^{2} b^{2}-4 a \,b^{3} x^{3}+24 \ln \left (b x +a \right ) x \,a^{3} b +12 a^{4} \ln \left (b x +a \right )+24 a^{3} b x +18 a^{4}}{2 b^{5} \left (b x +a \right )^{2}}\) | \(83\) |
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none
Time = 0.22 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.48 \[ \int \frac {x^4}{(a+b x)^3} \, dx=\frac {b^{4} x^{4} - 4 \, a b^{3} x^{3} - 11 \, a^{2} b^{2} x^{2} + 2 \, a^{3} b x + 7 \, a^{4} + 12 \, {\left (a^{2} b^{2} x^{2} + 2 \, a^{3} b x + a^{4}\right )} \log \left (b x + a\right )}{2 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} \]
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Time = 0.15 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.09 \[ \int \frac {x^4}{(a+b x)^3} \, dx=\frac {6 a^{2} \log {\left (a + b x \right )}}{b^{5}} - \frac {3 a x}{b^{4}} + \frac {7 a^{4} + 8 a^{3} b x}{2 a^{2} b^{5} + 4 a b^{6} x + 2 b^{7} x^{2}} + \frac {x^{2}}{2 b^{3}} \]
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none
Time = 0.21 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.08 \[ \int \frac {x^4}{(a+b x)^3} \, dx=\frac {8 \, a^{3} b x + 7 \, a^{4}}{2 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} + \frac {6 \, a^{2} \log \left (b x + a\right )}{b^{5}} + \frac {b x^{2} - 6 \, a x}{2 \, b^{4}} \]
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none
Time = 0.30 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.95 \[ \int \frac {x^4}{(a+b x)^3} \, dx=\frac {6 \, a^{2} \log \left ({\left | b x + a \right |}\right )}{b^{5}} + \frac {b^{3} x^{2} - 6 \, a b^{2} x}{2 \, b^{6}} + \frac {8 \, a^{3} b x + 7 \, a^{4}}{2 \, {\left (b x + a\right )}^{2} b^{5}} \]
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Time = 0.09 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.84 \[ \int \frac {x^4}{(a+b x)^3} \, dx=\frac {\frac {{\left (a+b\,x\right )}^2}{2}+\frac {4\,a^3}{a+b\,x}-\frac {a^4}{2\,{\left (a+b\,x\right )}^2}+6\,a^2\,\ln \left (a+b\,x\right )-4\,a\,b\,x}{b^5} \]
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