Integrand size = 18, antiderivative size = 107 \[ \int \frac {x \left (c x^2\right )^{3/2}}{a+b x} \, dx=-\frac {a^3 c \sqrt {c x^2}}{b^4}+\frac {a^2 c x \sqrt {c x^2}}{2 b^3}-\frac {a c x^2 \sqrt {c x^2}}{3 b^2}+\frac {c x^3 \sqrt {c x^2}}{4 b}+\frac {a^4 c \sqrt {c x^2} \log (a+b x)}{b^5 x} \]
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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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {15, 45} \[ \int \frac {x \left (c x^2\right )^{3/2}}{a+b x} \, dx=\frac {a^4 c \sqrt {c x^2} \log (a+b x)}{b^5 x}-\frac {a^3 c \sqrt {c x^2}}{b^4}+\frac {a^2 c x \sqrt {c x^2}}{2 b^3}-\frac {a c x^2 \sqrt {c x^2}}{3 b^2}+\frac {c x^3 \sqrt {c x^2}}{4 b} \]
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Rule 15
Rule 45
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c \sqrt {c x^2}\right ) \int \frac {x^4}{a+b x} \, dx}{x} \\ & = \frac {\left (c \sqrt {c x^2}\right ) \int \left (-\frac {a^3}{b^4}+\frac {a^2 x}{b^3}-\frac {a x^2}{b^2}+\frac {x^3}{b}+\frac {a^4}{b^4 (a+b x)}\right ) \, dx}{x} \\ & = -\frac {a^3 c \sqrt {c x^2}}{b^4}+\frac {a^2 c x \sqrt {c x^2}}{2 b^3}-\frac {a c x^2 \sqrt {c x^2}}{3 b^2}+\frac {c x^3 \sqrt {c x^2}}{4 b}+\frac {a^4 c \sqrt {c x^2} \log (a+b x)}{b^5 x} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.60 \[ \int \frac {x \left (c x^2\right )^{3/2}}{a+b x} \, dx=\frac {\left (c x^2\right )^{3/2} \left (b x \left (-12 a^3+6 a^2 b x-4 a b^2 x^2+3 b^3 x^3\right )+12 a^4 \log (a+b x)\right )}{12 b^5 x^3} \]
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Time = 0.13 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.59
method | result | size |
default | \(\frac {\left (c \,x^{2}\right )^{\frac {3}{2}} \left (3 b^{4} x^{4}-4 a \,b^{3} x^{3}+6 a^{2} b^{2} x^{2}+12 a^{4} \ln \left (b x +a \right )-12 a^{3} b x \right )}{12 b^{5} x^{3}}\) | \(63\) |
risch | \(\frac {c \sqrt {c \,x^{2}}\, \left (\frac {1}{4} b^{3} x^{4}-\frac {1}{3} a \,b^{2} x^{3}+\frac {1}{2} a^{2} b \,x^{2}-a^{3} x \right )}{x \,b^{4}}+\frac {a^{4} c \ln \left (b x +a \right ) \sqrt {c \,x^{2}}}{b^{5} x}\) | \(74\) |
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Time = 0.24 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.63 \[ \int \frac {x \left (c x^2\right )^{3/2}}{a+b x} \, dx=\frac {{\left (3 \, b^{4} c x^{4} - 4 \, a b^{3} c x^{3} + 6 \, a^{2} b^{2} c x^{2} - 12 \, a^{3} b c x + 12 \, a^{4} c \log \left (b x + a\right )\right )} \sqrt {c x^{2}}}{12 \, b^{5} x} \]
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\[ \int \frac {x \left (c x^2\right )^{3/2}}{a+b x} \, dx=\int \frac {x \left (c x^{2}\right )^{\frac {3}{2}}}{a + b x}\, dx \]
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Time = 0.23 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.16 \[ \int \frac {x \left (c x^2\right )^{3/2}}{a+b x} \, dx=\frac {\left (-1\right )^{\frac {2 \, c x}{b}} a^{4} c^{\frac {3}{2}} \log \left (\frac {2 \, c x}{b}\right )}{b^{5}} + \frac {\left (-1\right )^{\frac {2 \, a c x}{b}} a^{4} c^{\frac {3}{2}} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{b^{5}} + \frac {\left (c x^{2}\right )^{\frac {3}{2}} x}{4 \, b} + \frac {\sqrt {c x^{2}} a^{2} c x}{2 \, b^{3}} - \frac {\left (c x^{2}\right )^{\frac {3}{2}} a}{3 \, b^{2}} - \frac {\sqrt {c x^{2}} a^{3} c}{b^{4}} \]
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Time = 0.37 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.76 \[ \int \frac {x \left (c x^2\right )^{3/2}}{a+b x} \, dx=\frac {1}{12} \, c^{\frac {3}{2}} {\left (\frac {12 \, a^{4} \log \left ({\left | b x + a \right |}\right ) \mathrm {sgn}\left (x\right )}{b^{5}} - \frac {12 \, a^{4} \log \left ({\left | a \right |}\right ) \mathrm {sgn}\left (x\right )}{b^{5}} + \frac {3 \, b^{3} x^{4} \mathrm {sgn}\left (x\right ) - 4 \, a b^{2} x^{3} \mathrm {sgn}\left (x\right ) + 6 \, a^{2} b x^{2} \mathrm {sgn}\left (x\right ) - 12 \, a^{3} x \mathrm {sgn}\left (x\right )}{b^{4}}\right )} \]
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Timed out. \[ \int \frac {x \left (c x^2\right )^{3/2}}{a+b x} \, dx=\int \frac {x\,{\left (c\,x^2\right )}^{3/2}}{a+b\,x} \,d x \]
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