Integrand size = 20, antiderivative size = 106 \[ \int \frac {x^3 \sqrt {c x^2}}{(a+b x)^2} \, dx=\frac {3 a^2 \sqrt {c x^2}}{b^4}-\frac {a x \sqrt {c x^2}}{b^3}+\frac {x^2 \sqrt {c x^2}}{3 b^2}-\frac {a^4 \sqrt {c x^2}}{b^5 x (a+b x)}-\frac {4 a^3 \sqrt {c x^2} \log (a+b x)}{b^5 x} \]
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Time = 0.03 (sec) , antiderivative size = 106, 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} \[ \int \frac {x^3 \sqrt {c x^2}}{(a+b x)^2} \, dx=-\frac {a^4 \sqrt {c x^2}}{b^5 x (a+b x)}-\frac {4 a^3 \sqrt {c x^2} \log (a+b x)}{b^5 x}+\frac {3 a^2 \sqrt {c x^2}}{b^4}-\frac {a x \sqrt {c x^2}}{b^3}+\frac {x^2 \sqrt {c x^2}}{3 b^2} \]
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Rule 15
Rule 45
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c x^2} \int \frac {x^4}{(a+b x)^2} \, dx}{x} \\ & = \frac {\sqrt {c x^2} \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}{x} \\ & = \frac {3 a^2 \sqrt {c x^2}}{b^4}-\frac {a x \sqrt {c x^2}}{b^3}+\frac {x^2 \sqrt {c x^2}}{3 b^2}-\frac {a^4 \sqrt {c x^2}}{b^5 x (a+b x)}-\frac {4 a^3 \sqrt {c x^2} \log (a+b x)}{b^5 x} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.76 \[ \int \frac {x^3 \sqrt {c x^2}}{(a+b x)^2} \, dx=\frac {c x \left (-3 a^4+9 a^3 b x+6 a^2 b^2 x^2-2 a b^3 x^3+b^4 x^4-12 a^3 (a+b x) \log (a+b x)\right )}{3 b^5 \sqrt {c x^2} (a+b x)} \]
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Time = 0.14 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.82
method | result | size |
risch | \(\frac {\sqrt {c \,x^{2}}\, \left (\frac {1}{3} b^{2} x^{3}-a b \,x^{2}+3 a^{2} x \right )}{x \,b^{4}}-\frac {4 a^{3} \ln \left (b x +a \right ) \sqrt {c \,x^{2}}}{b^{5} x}-\frac {a^{4} \sqrt {c \,x^{2}}}{b^{5} x \left (b x +a \right )}\) | \(87\) |
default | \(-\frac {\sqrt {c \,x^{2}}\, \left (-b^{4} x^{4}+2 a \,b^{3} x^{3}+12 \ln \left (b x +a \right ) a^{3} b x -6 a^{2} b^{2} x^{2}+12 a^{4} \ln \left (b x +a \right )-9 a^{3} b x +3 a^{4}\right )}{3 x \,b^{5} \left (b x +a \right )}\) | \(88\) |
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Time = 0.22 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.78 \[ \int \frac {x^3 \sqrt {c x^2}}{(a+b x)^2} \, dx=\frac {{\left (b^{4} x^{4} - 2 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 9 \, a^{3} b x - 3 \, a^{4} - 12 \, {\left (a^{3} b x + a^{4}\right )} \log \left (b x + a\right )\right )} \sqrt {c x^{2}}}{3 \, {\left (b^{6} x^{2} + a b^{5} x\right )}} \]
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\[ \int \frac {x^3 \sqrt {c x^2}}{(a+b x)^2} \, dx=\int \frac {x^{3} \sqrt {c x^{2}}}{\left (a + b x\right )^{2}}\, dx \]
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Time = 0.34 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.27 \[ \int \frac {x^3 \sqrt {c x^2}}{(a+b x)^2} \, dx=\frac {\sqrt {c x^{2}} a^{3}}{b^{5} x + a b^{4}} - \frac {4 \, \left (-1\right )^{\frac {2 \, c x}{b}} a^{3} \sqrt {c} \log \left (\frac {2 \, c x}{b}\right )}{b^{5}} - \frac {4 \, \left (-1\right )^{\frac {2 \, a c x}{b}} a^{3} \sqrt {c} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{b^{5}} - \frac {\sqrt {c x^{2}} a x}{b^{3}} + \frac {3 \, \sqrt {c x^{2}} a^{2}}{b^{4}} + \frac {\left (c x^{2}\right )^{\frac {3}{2}}}{3 \, b^{2} c} \]
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Time = 0.30 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.91 \[ \int \frac {x^3 \sqrt {c x^2}}{(a+b x)^2} \, dx=-\frac {1}{3} \, \sqrt {c} {\left (\frac {12 \, a^{3} \log \left ({\left | b x + a \right |}\right ) \mathrm {sgn}\left (x\right )}{b^{5}} + \frac {3 \, a^{4} \mathrm {sgn}\left (x\right )}{{\left (b x + a\right )} b^{5}} - \frac {3 \, {\left (4 \, a^{3} \log \left ({\left | a \right |}\right ) + a^{3}\right )} \mathrm {sgn}\left (x\right )}{b^{5}} - \frac {b^{4} x^{3} \mathrm {sgn}\left (x\right ) - 3 \, a b^{3} x^{2} \mathrm {sgn}\left (x\right ) + 9 \, a^{2} b^{2} x \mathrm {sgn}\left (x\right )}{b^{6}}\right )} \]
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Timed out. \[ \int \frac {x^3 \sqrt {c x^2}}{(a+b x)^2} \, dx=\int \frac {x^3\,\sqrt {c\,x^2}}{{\left (a+b\,x\right )}^2} \,d x \]
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