Integrand size = 20, antiderivative size = 71 \[ \int \frac {(a+b x)^2}{x^2 \sqrt {c+d x}} \, dx=\frac {2 b^2 \sqrt {c+d x}}{d}-\frac {a^2 \sqrt {c+d x}}{c x}-\frac {a (4 b c-a d) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{c^{3/2}} \]
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Time = 0.03 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {91, 81, 65, 214} \[ \int \frac {(a+b x)^2}{x^2 \sqrt {c+d x}} \, dx=-\frac {a^2 \sqrt {c+d x}}{c x}-\frac {a (4 b c-a d) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{c^{3/2}}+\frac {2 b^2 \sqrt {c+d x}}{d} \]
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Rule 65
Rule 81
Rule 91
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {a^2 \sqrt {c+d x}}{c x}+\frac {\int \frac {\frac {1}{2} a (4 b c-a d)+b^2 c x}{x \sqrt {c+d x}} \, dx}{c} \\ & = \frac {2 b^2 \sqrt {c+d x}}{d}-\frac {a^2 \sqrt {c+d x}}{c x}+\frac {(a (4 b c-a d)) \int \frac {1}{x \sqrt {c+d x}} \, dx}{2 c} \\ & = \frac {2 b^2 \sqrt {c+d x}}{d}-\frac {a^2 \sqrt {c+d x}}{c x}+\frac {(a (4 b c-a d)) \text {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{c d} \\ & = \frac {2 b^2 \sqrt {c+d x}}{d}-\frac {a^2 \sqrt {c+d x}}{c x}-\frac {a (4 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{c^{3/2}} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b x)^2}{x^2 \sqrt {c+d x}} \, dx=\frac {\left (-a^2 d+2 b^2 c x\right ) \sqrt {c+d x}}{c d x}+\frac {a (-4 b c+a d) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{c^{3/2}} \]
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Time = 0.53 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\frac {2 b^{2} \sqrt {d x +c}+2 a d \left (-\frac {a \sqrt {d x +c}}{2 c x}+\frac {\left (a d -4 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 c^{\frac {3}{2}}}\right )}{d}\) | \(63\) |
default | \(\frac {2 b^{2} \sqrt {d x +c}+2 a d \left (-\frac {a \sqrt {d x +c}}{2 c x}+\frac {\left (a d -4 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 c^{\frac {3}{2}}}\right )}{d}\) | \(63\) |
risch | \(-\frac {a^{2} \sqrt {d x +c}}{c x}+\frac {2 b^{2} c \sqrt {d x +c}+\frac {a d \left (a d -4 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{\sqrt {c}}}{c d}\) | \(67\) |
pseudoelliptic | \(\frac {\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right ) a^{2} d^{2} x -4 \,\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right ) a b c d x +2 b^{2} c^{\frac {3}{2}} x \sqrt {d x +c}-a^{2} d \sqrt {d x +c}\, \sqrt {c}}{c^{\frac {3}{2}} d x}\) | \(83\) |
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Time = 0.24 (sec) , antiderivative size = 158, normalized size of antiderivative = 2.23 \[ \int \frac {(a+b x)^2}{x^2 \sqrt {c+d x}} \, dx=\left [-\frac {{\left (4 \, a b c d - a^{2} d^{2}\right )} \sqrt {c} x \log \left (\frac {d x + 2 \, \sqrt {d x + c} \sqrt {c} + 2 \, c}{x}\right ) - 2 \, {\left (2 \, b^{2} c^{2} x - a^{2} c d\right )} \sqrt {d x + c}}{2 \, c^{2} d x}, \frac {{\left (4 \, a b c d - a^{2} d^{2}\right )} \sqrt {-c} x \arctan \left (\frac {\sqrt {d x + c} \sqrt {-c}}{c}\right ) + {\left (2 \, b^{2} c^{2} x - a^{2} c d\right )} \sqrt {d x + c}}{c^{2} d x}\right ] \]
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Time = 13.18 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.58 \[ \int \frac {(a+b x)^2}{x^2 \sqrt {c+d x}} \, dx=- \frac {a^{2} \sqrt {d} \sqrt {\frac {c}{d x} + 1}}{c \sqrt {x}} + \frac {a^{2} d \operatorname {asinh}{\left (\frac {\sqrt {c}}{\sqrt {d} \sqrt {x}} \right )}}{c^{\frac {3}{2}}} + 2 a b \left (\begin {cases} \frac {2 \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {- c}} \right )}}{\sqrt {- c}} & \text {for}\: d \neq 0 \\\frac {\log {\left (x \right )}}{\sqrt {c}} & \text {otherwise} \end {cases}\right ) + b^{2} \left (\begin {cases} \frac {2 \sqrt {c + d x}}{d} & \text {for}\: d \neq 0 \\\frac {x}{\sqrt {c}} & \text {otherwise} \end {cases}\right ) \]
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Time = 0.27 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.30 \[ \int \frac {(a+b x)^2}{x^2 \sqrt {c+d x}} \, dx=-\frac {1}{2} \, {\left (\frac {2 \, \sqrt {d x + c} a^{2}}{{\left (d x + c\right )} c - c^{2}} - \frac {4 \, \sqrt {d x + c} b^{2}}{d^{2}} - \frac {{\left (4 \, b c - a d\right )} a \log \left (\frac {\sqrt {d x + c} - \sqrt {c}}{\sqrt {d x + c} + \sqrt {c}}\right )}{c^{\frac {3}{2}} d}\right )} d \]
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Time = 0.30 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.04 \[ \int \frac {(a+b x)^2}{x^2 \sqrt {c+d x}} \, dx=\frac {2 \, \sqrt {d x + c} b^{2} - \frac {\sqrt {d x + c} a^{2} d}{c x} + \frac {{\left (4 \, a b c d - a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{\sqrt {-c} c}}{d} \]
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Time = 0.16 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.83 \[ \int \frac {(a+b x)^2}{x^2 \sqrt {c+d x}} \, dx=\frac {2\,b^2\,\sqrt {c+d\,x}}{d}+\frac {a\,\mathrm {atanh}\left (\frac {\sqrt {c+d\,x}}{\sqrt {c}}\right )\,\left (a\,d-4\,b\,c\right )}{c^{3/2}}-\frac {a^2\,\sqrt {c+d\,x}}{c\,x} \]
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