Integrand size = 20, antiderivative size = 107 \[ \int \frac {x^5}{\sqrt {c x^2} (a+b x)^2} \, dx=\frac {3 a^2 x^2}{b^4 \sqrt {c x^2}}-\frac {a x^3}{b^3 \sqrt {c x^2}}+\frac {x^4}{3 b^2 \sqrt {c x^2}}-\frac {a^4 x}{b^5 \sqrt {c x^2} (a+b x)}-\frac {4 a^3 x \log (a+b x)}{b^5 \sqrt {c x^2}} \]
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Time = 0.02 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 45} \[ \int \frac {x^5}{\sqrt {c x^2} (a+b x)^2} \, dx=-\frac {a^4 x}{b^5 \sqrt {c x^2} (a+b x)}-\frac {4 a^3 x \log (a+b x)}{b^5 \sqrt {c x^2}}+\frac {3 a^2 x^2}{b^4 \sqrt {c x^2}}-\frac {a x^3}{b^3 \sqrt {c x^2}}+\frac {x^4}{3 b^2 \sqrt {c x^2}} \]
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Rule 15
Rule 45
Rubi steps \begin{align*} \text {integral}& = \frac {x \int \frac {x^4}{(a+b x)^2} \, dx}{\sqrt {c x^2}} \\ & = \frac {x \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}{\sqrt {c x^2}} \\ & = \frac {3 a^2 x^2}{b^4 \sqrt {c x^2}}-\frac {a x^3}{b^3 \sqrt {c x^2}}+\frac {x^4}{3 b^2 \sqrt {c x^2}}-\frac {a^4 x}{b^5 \sqrt {c x^2} (a+b x)}-\frac {4 a^3 x \log (a+b x)}{b^5 \sqrt {c x^2}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.75 \[ \int \frac {x^5}{\sqrt {c x^2} (a+b x)^2} \, dx=\frac {x \left (-3 a^4+9 a^3 b x+6 a^2 b^2 x^2-2 a b^3 x^3+b^4 x^4-12 a^3 (a+b x) \log (a+b x)\right )}{3 b^5 \sqrt {c x^2} (a+b x)} \]
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Time = 0.14 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.76
method | result | size |
risch | \(\frac {x \left (\frac {1}{3} b^{2} x^{3}-a b \,x^{2}+3 a^{2} x \right )}{\sqrt {c \,x^{2}}\, b^{4}}-\frac {4 a^{3} x \ln \left (b x +a \right )}{b^{5} \sqrt {c \,x^{2}}}-\frac {a^{4} x}{b^{5} \left (b x +a \right ) \sqrt {c \,x^{2}}}\) | \(81\) |
default | \(-\frac {x \left (-b^{4} x^{4}+2 a \,b^{3} x^{3}+12 \ln \left (b x +a \right ) a^{3} b x -6 a^{2} b^{2} x^{2}+12 a^{4} \ln \left (b x +a \right )-9 a^{3} b x +3 a^{4}\right )}{3 \sqrt {c \,x^{2}}\, b^{5} \left (b x +a \right )}\) | \(86\) |
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Time = 0.22 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.79 \[ \int \frac {x^5}{\sqrt {c x^2} (a+b x)^2} \, dx=\frac {{\left (b^{4} x^{4} - 2 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 9 \, a^{3} b x - 3 \, a^{4} - 12 \, {\left (a^{3} b x + a^{4}\right )} \log \left (b x + a\right )\right )} \sqrt {c x^{2}}}{3 \, {\left (b^{6} c x^{2} + a b^{5} c x\right )}} \]
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\[ \int \frac {x^5}{\sqrt {c x^2} (a+b x)^2} \, dx=\int \frac {x^{5}}{\sqrt {c x^{2}} \left (a + b x\right )^{2}}\, dx \]
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Time = 0.22 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.57 \[ \int \frac {x^5}{\sqrt {c x^2} (a+b x)^2} \, dx=\frac {\sqrt {c x^{2}} a^{3}}{b^{5} c x + a b^{4} c} + \frac {\sqrt {c x^{2}} x^{2}}{3 \, b^{2} c} - \frac {5 \, a x^{2}}{3 \, b^{3} \sqrt {c}} - \frac {4 \, \left (-1\right )^{\frac {2 \, a c x}{b}} a^{3} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{b^{5} \sqrt {c}} + \frac {2 \, \sqrt {c x^{2}} a x}{3 \, b^{3} c} - \frac {20 \, a^{2} x}{3 \, b^{4} \sqrt {c}} - \frac {4 \, a^{3} \log \left (b x\right )}{b^{5} \sqrt {c}} + \frac {29 \, \sqrt {c x^{2}} a^{2}}{3 \, b^{4} c} - \frac {5 \, a^{3}}{b^{5} \sqrt {c}} \]
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Time = 0.29 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00 \[ \int \frac {x^5}{\sqrt {c x^2} (a+b x)^2} \, dx=-\frac {4 \, a^{3} \log \left ({\left | b x + a \right |}\right )}{b^{5} \sqrt {c} \mathrm {sgn}\left (x\right )} - \frac {a^{4}}{{\left (b x + a\right )} b^{5} \sqrt {c} \mathrm {sgn}\left (x\right )} + \frac {{\left (4 \, a^{3} \log \left ({\left | a \right |}\right ) + a^{3}\right )} \mathrm {sgn}\left (x\right )}{b^{5} \sqrt {c}} + \frac {b^{4} c x^{3} - 3 \, a b^{3} c x^{2} + 9 \, a^{2} b^{2} c x}{3 \, b^{6} c^{\frac {3}{2}} \mathrm {sgn}\left (x\right )} \]
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Timed out. \[ \int \frac {x^5}{\sqrt {c x^2} (a+b x)^2} \, dx=\int \frac {x^5}{\sqrt {c\,x^2}\,{\left (a+b\,x\right )}^2} \,d x \]
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