Integrand size = 17, antiderivative size = 142 \[ \int \frac {\left (c x^2\right )^{5/2}}{a+b x} \, dx=\frac {a^4 c^2 \sqrt {c x^2}}{b^5}-\frac {a^3 c^2 x \sqrt {c x^2}}{2 b^4}+\frac {a^2 c^2 x^2 \sqrt {c x^2}}{3 b^3}-\frac {a c^2 x^3 \sqrt {c x^2}}{4 b^2}+\frac {c^2 x^4 \sqrt {c x^2}}{5 b}-\frac {a^5 c^2 \sqrt {c x^2} \log (a+b x)}{b^6 x} \]
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Time = 0.03 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {15, 45} \[ \int \frac {\left (c x^2\right )^{5/2}}{a+b x} \, dx=-\frac {a^5 c^2 \sqrt {c x^2} \log (a+b x)}{b^6 x}+\frac {a^4 c^2 \sqrt {c x^2}}{b^5}-\frac {a^3 c^2 x \sqrt {c x^2}}{2 b^4}+\frac {a^2 c^2 x^2 \sqrt {c x^2}}{3 b^3}-\frac {a c^2 x^3 \sqrt {c x^2}}{4 b^2}+\frac {c^2 x^4 \sqrt {c x^2}}{5 b} \]
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Rule 15
Rule 45
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c^2 \sqrt {c x^2}\right ) \int \frac {x^5}{a+b x} \, dx}{x} \\ & = \frac {\left (c^2 \sqrt {c x^2}\right ) \int \left (\frac {a^4}{b^5}-\frac {a^3 x}{b^4}+\frac {a^2 x^2}{b^3}-\frac {a x^3}{b^2}+\frac {x^4}{b}-\frac {a^5}{b^5 (a+b x)}\right ) \, dx}{x} \\ & = \frac {a^4 c^2 \sqrt {c x^2}}{b^5}-\frac {a^3 c^2 x \sqrt {c x^2}}{2 b^4}+\frac {a^2 c^2 x^2 \sqrt {c x^2}}{3 b^3}-\frac {a c^2 x^3 \sqrt {c x^2}}{4 b^2}+\frac {c^2 x^4 \sqrt {c x^2}}{5 b}-\frac {a^5 c^2 \sqrt {c x^2} \log (a+b x)}{b^6 x} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.54 \[ \int \frac {\left (c x^2\right )^{5/2}}{a+b x} \, dx=\frac {c^3 x \left (b x \left (60 a^4-30 a^3 b x+20 a^2 b^2 x^2-15 a b^3 x^3+12 b^4 x^4\right )-60 a^5 \log (a+b x)\right )}{60 b^6 \sqrt {c x^2}} \]
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Time = 0.13 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.52
method | result | size |
default | \(-\frac {\left (c \,x^{2}\right )^{\frac {5}{2}} \left (-12 b^{5} x^{5}+15 a \,b^{4} x^{4}-20 a^{2} b^{3} x^{3}+30 a^{3} b^{2} x^{2}+60 a^{5} \ln \left (b x +a \right )-60 a^{4} b x \right )}{60 b^{6} x^{5}}\) | \(74\) |
risch | \(\frac {c^{2} \sqrt {c \,x^{2}}\, \left (\frac {1}{5} b^{4} x^{5}-\frac {1}{4} a \,b^{3} x^{4}+\frac {1}{3} a^{2} b^{2} x^{3}-\frac {1}{2} a^{3} b \,x^{2}+a^{4} x \right )}{x \,b^{5}}-\frac {a^{5} c^{2} \ln \left (b x +a \right ) \sqrt {c \,x^{2}}}{b^{6} x}\) | \(89\) |
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Time = 0.22 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.64 \[ \int \frac {\left (c x^2\right )^{5/2}}{a+b x} \, dx=\frac {{\left (12 \, b^{5} c^{2} x^{5} - 15 \, a b^{4} c^{2} x^{4} + 20 \, a^{2} b^{3} c^{2} x^{3} - 30 \, a^{3} b^{2} c^{2} x^{2} + 60 \, a^{4} b c^{2} x - 60 \, a^{5} c^{2} \log \left (b x + a\right )\right )} \sqrt {c x^{2}}}{60 \, b^{6} x} \]
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\[ \int \frac {\left (c x^2\right )^{5/2}}{a+b x} \, dx=\int \frac {\left (c x^{2}\right )^{\frac {5}{2}}}{a + b x}\, dx \]
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Time = 0.23 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.03 \[ \int \frac {\left (c x^2\right )^{5/2}}{a+b x} \, dx=-\frac {\left (-1\right )^{\frac {2 \, c x}{b}} a^{5} c^{\frac {5}{2}} \log \left (\frac {2 \, c x}{b}\right )}{b^{6}} - \frac {\left (-1\right )^{\frac {2 \, a c x}{b}} a^{5} c^{\frac {5}{2}} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{b^{6}} - \frac {\left (c x^{2}\right )^{\frac {3}{2}} a c x}{4 \, b^{2}} - \frac {\sqrt {c x^{2}} a^{3} c^{2} x}{2 \, b^{4}} + \frac {\left (c x^{2}\right )^{\frac {5}{2}}}{5 \, b} + \frac {\left (c x^{2}\right )^{\frac {3}{2}} a^{2} c}{3 \, b^{3}} + \frac {\sqrt {c x^{2}} a^{4} c^{2}}{b^{5}} \]
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Time = 0.29 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.82 \[ \int \frac {\left (c x^2\right )^{5/2}}{a+b x} \, dx=-\frac {1}{60} \, {\left (\frac {60 \, a^{5} c^{2} \log \left ({\left | b x + a \right |}\right ) \mathrm {sgn}\left (x\right )}{b^{6}} - \frac {60 \, a^{5} c^{2} \log \left ({\left | a \right |}\right ) \mathrm {sgn}\left (x\right )}{b^{6}} - \frac {12 \, b^{4} c^{2} x^{5} \mathrm {sgn}\left (x\right ) - 15 \, a b^{3} c^{2} x^{4} \mathrm {sgn}\left (x\right ) + 20 \, a^{2} b^{2} c^{2} x^{3} \mathrm {sgn}\left (x\right ) - 30 \, a^{3} b c^{2} x^{2} \mathrm {sgn}\left (x\right ) + 60 \, a^{4} c^{2} x \mathrm {sgn}\left (x\right )}{b^{5}}\right )} \sqrt {c} \]
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Timed out. \[ \int \frac {\left (c x^2\right )^{5/2}}{a+b x} \, dx=\int \frac {{\left (c\,x^2\right )}^{5/2}}{a+b\,x} \,d x \]
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