Integrand size = 17, antiderivative size = 78 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x} \, dx=\frac {b (b+2 c x) \sqrt {b x+c x^2}}{8 c}+\frac {1}{3} \left (b x+c x^2\right )^{3/2}-\frac {b^3 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{8 c^{3/2}} \]
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Time = 0.02 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {678, 626, 634, 212} \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x} \, dx=-\frac {b^3 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{8 c^{3/2}}+\frac {b (b+2 c x) \sqrt {b x+c x^2}}{8 c}+\frac {1}{3} \left (b x+c x^2\right )^{3/2} \]
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Rule 212
Rule 626
Rule 634
Rule 678
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \left (b x+c x^2\right )^{3/2}+\frac {1}{2} b \int \sqrt {b x+c x^2} \, dx \\ & = \frac {b (b+2 c x) \sqrt {b x+c x^2}}{8 c}+\frac {1}{3} \left (b x+c x^2\right )^{3/2}-\frac {b^3 \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{16 c} \\ & = \frac {b (b+2 c x) \sqrt {b x+c x^2}}{8 c}+\frac {1}{3} \left (b x+c x^2\right )^{3/2}-\frac {b^3 \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{8 c} \\ & = \frac {b (b+2 c x) \sqrt {b x+c x^2}}{8 c}+\frac {1}{3} \left (b x+c x^2\right )^{3/2}-\frac {b^3 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{8 c^{3/2}} \\ \end{align*}
Time = 0.48 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.23 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x} \, dx=\frac {\sqrt {x (b+c x)} \left (\sqrt {c} \left (3 b^2+14 b c x+8 c^2 x^2\right )+\frac {6 b^3 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}-\sqrt {b+c x}}\right )}{\sqrt {x} \sqrt {b+c x}}\right )}{24 c^{3/2}} \]
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Time = 2.04 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.94
method | result | size |
default | \(\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{3}+\frac {b \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{2}\) | \(73\) |
risch | \(\frac {\left (8 c^{2} x^{2}+14 b c x +3 b^{2}\right ) x \left (c x +b \right )}{24 c \sqrt {x \left (c x +b \right )}}-\frac {b^{3} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{16 c^{\frac {3}{2}}}\) | \(73\) |
pseudoelliptic | \(\frac {8 c^{\frac {5}{2}} \sqrt {x \left (c x +b \right )}\, x^{2}+14 b \,c^{\frac {3}{2}} x \sqrt {x \left (c x +b \right )}+3 b^{2} \sqrt {c}\, \sqrt {x \left (c x +b \right )}-3 \,\operatorname {arctanh}\left (\frac {\sqrt {x \left (c x +b \right )}}{x \sqrt {c}}\right ) b^{3}}{24 c^{\frac {3}{2}}}\) | \(79\) |
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Time = 0.27 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.88 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x} \, dx=\left [\frac {3 \, b^{3} \sqrt {c} \log \left (2 \, c x + b - 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + 2 \, {\left (8 \, c^{3} x^{2} + 14 \, b c^{2} x + 3 \, b^{2} c\right )} \sqrt {c x^{2} + b x}}{48 \, c^{2}}, \frac {3 \, b^{3} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + {\left (8 \, c^{3} x^{2} + 14 \, b c^{2} x + 3 \, b^{2} c\right )} \sqrt {c x^{2} + b x}}{24 \, c^{2}}\right ] \]
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Time = 1.11 (sec) , antiderivative size = 224, normalized size of antiderivative = 2.87 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x} \, dx=b \left (\begin {cases} - \frac {b^{2} \left (\begin {cases} \frac {\log {\left (b + 2 \sqrt {c} \sqrt {b x + c x^{2}} + 2 c x \right )}}{\sqrt {c}} & \text {for}\: \frac {b^{2}}{c} \neq 0 \\\frac {\left (\frac {b}{2 c} + x\right ) \log {\left (\frac {b}{2 c} + x \right )}}{\sqrt {c \left (\frac {b}{2 c} + x\right )^{2}}} & \text {otherwise} \end {cases}\right )}{8 c} + \left (\frac {b}{4 c} + \frac {x}{2}\right ) \sqrt {b x + c x^{2}} & \text {for}\: c \neq 0 \\\frac {2 \left (b x\right )^{\frac {3}{2}}}{3 b} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases}\right ) + c \left (\begin {cases} \frac {b^{3} \left (\begin {cases} \frac {\log {\left (b + 2 \sqrt {c} \sqrt {b x + c x^{2}} + 2 c x \right )}}{\sqrt {c}} & \text {for}\: \frac {b^{2}}{c} \neq 0 \\\frac {\left (\frac {b}{2 c} + x\right ) \log {\left (\frac {b}{2 c} + x \right )}}{\sqrt {c \left (\frac {b}{2 c} + x\right )^{2}}} & \text {otherwise} \end {cases}\right )}{16 c^{2}} + \sqrt {b x + c x^{2}} \left (- \frac {b^{2}}{8 c^{2}} + \frac {b x}{12 c} + \frac {x^{2}}{3}\right ) & \text {for}\: c \neq 0 \\\frac {2 \left (b x\right )^{\frac {5}{2}}}{5 b^{2}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases}\right ) \]
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Time = 0.18 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.01 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x} \, dx=\frac {1}{4} \, \sqrt {c x^{2} + b x} b x - \frac {b^{3} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{16 \, c^{\frac {3}{2}}} + \frac {1}{3} \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} + \frac {\sqrt {c x^{2} + b x} b^{2}}{8 \, c} \]
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Time = 0.28 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.90 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x} \, dx=\frac {b^{3} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} + b \right |}\right )}{16 \, c^{\frac {3}{2}}} + \frac {1}{24} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, c x + 7 \, b\right )} x + \frac {3 \, b^{2}}{c}\right )} \]
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Timed out. \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x} \, dx=\int \frac {{\left (c\,x^2+b\,x\right )}^{3/2}}{x} \,d x \]
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