Integrand size = 20, antiderivative size = 86 \[ \int \frac {x^4}{\sqrt {c x^2} (a+b x)^2} \, dx=-\frac {2 a x^2}{b^3 \sqrt {c x^2}}+\frac {x^3}{2 b^2 \sqrt {c x^2}}+\frac {a^3 x}{b^4 \sqrt {c x^2} (a+b x)}+\frac {3 a^2 x \log (a+b x)}{b^4 \sqrt {c x^2}} \]
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Time = 0.02 (sec) , antiderivative size = 86, 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} \[ \int \frac {x^4}{\sqrt {c x^2} (a+b x)^2} \, dx=\frac {a^3 x}{b^4 \sqrt {c x^2} (a+b x)}+\frac {3 a^2 x \log (a+b x)}{b^4 \sqrt {c x^2}}-\frac {2 a x^2}{b^3 \sqrt {c x^2}}+\frac {x^3}{2 b^2 \sqrt {c x^2}} \]
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Rule 15
Rule 45
Rubi steps \begin{align*} \text {integral}& = \frac {x \int \frac {x^3}{(a+b x)^2} \, dx}{\sqrt {c x^2}} \\ & = \frac {x \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}{\sqrt {c x^2}} \\ & = -\frac {2 a x^2}{b^3 \sqrt {c x^2}}+\frac {x^3}{2 b^2 \sqrt {c x^2}}+\frac {a^3 x}{b^4 \sqrt {c x^2} (a+b x)}+\frac {3 a^2 x \log (a+b x)}{b^4 \sqrt {c x^2}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.80 \[ \int \frac {x^4}{\sqrt {c x^2} (a+b x)^2} \, dx=\frac {x \left (2 a^3-4 a^2 b x-3 a b^2 x^2+b^3 x^3+6 a^2 (a+b x) \log (a+b x)\right )}{2 b^4 \sqrt {c x^2} (a+b x)} \]
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Time = 0.15 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.80
method | result | size |
risch | \(\frac {x \left (\frac {1}{2} b \,x^{2}-2 a x \right )}{\sqrt {c \,x^{2}}\, b^{3}}+\frac {a^{3} x}{b^{4} \left (b x +a \right ) \sqrt {c \,x^{2}}}+\frac {3 a^{2} x \ln \left (b x +a \right )}{b^{4} \sqrt {c \,x^{2}}}\) | \(69\) |
default | \(\frac {x \left (b^{3} x^{3}+6 \ln \left (b x +a \right ) a^{2} b x -3 a \,b^{2} x^{2}+6 a^{3} \ln \left (b x +a \right )-4 a^{2} b x +2 a^{3}\right )}{2 \sqrt {c \,x^{2}}\, b^{4} \left (b x +a \right )}\) | \(74\) |
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Time = 0.23 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.86 \[ \int \frac {x^4}{\sqrt {c x^2} (a+b x)^2} \, dx=\frac {{\left (b^{3} x^{3} - 3 \, a b^{2} x^{2} - 4 \, a^{2} b x + 2 \, a^{3} + 6 \, {\left (a^{2} b x + a^{3}\right )} \log \left (b x + a\right )\right )} \sqrt {c x^{2}}}{2 \, {\left (b^{5} c x^{2} + a b^{4} c x\right )}} \]
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\[ \int \frac {x^4}{\sqrt {c x^2} (a+b x)^2} \, dx=\int \frac {x^{4}}{\sqrt {c x^{2}} \left (a + b x\right )^{2}}\, dx \]
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Time = 0.22 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.50 \[ \int \frac {x^4}{\sqrt {c x^2} (a+b x)^2} \, dx=-\frac {\sqrt {c x^{2}} a^{2}}{b^{4} c x + a b^{3} c} + \frac {x^{2}}{2 \, b^{2} \sqrt {c}} + \frac {3 \, \left (-1\right )^{\frac {2 \, a c x}{b}} a^{2} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{b^{4} \sqrt {c}} + \frac {2 \, a x}{b^{3} \sqrt {c}} + \frac {3 \, a^{2} \log \left (b x\right )}{b^{4} \sqrt {c}} - \frac {4 \, \sqrt {c x^{2}} a}{b^{3} c} + \frac {3 \, a^{2}}{2 \, b^{4} \sqrt {c}} \]
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Time = 0.30 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.13 \[ \int \frac {x^4}{\sqrt {c x^2} (a+b x)^2} \, dx=\frac {3 \, a^{2} \log \left ({\left | b x + a \right |}\right )}{b^{4} \sqrt {c} \mathrm {sgn}\left (x\right )} - \frac {{\left (3 \, a^{2} \log \left ({\left | a \right |}\right ) + a^{2}\right )} \mathrm {sgn}\left (x\right )}{b^{4} \sqrt {c}} + \frac {a^{3}}{{\left (b x + a\right )} b^{4} \sqrt {c} \mathrm {sgn}\left (x\right )} + \frac {b^{2} \sqrt {c} x^{2} \mathrm {sgn}\left (x\right ) - 4 \, a b \sqrt {c} x \mathrm {sgn}\left (x\right )}{2 \, b^{4} c} \]
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Timed out. \[ \int \frac {x^4}{\sqrt {c x^2} (a+b x)^2} \, dx=\int \frac {x^4}{\sqrt {c\,x^2}\,{\left (a+b\,x\right )}^2} \,d x \]
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