Integrand size = 22, antiderivative size = 88 \[ \int \frac {a+b x}{(c+d x) (e+f x)^{3/2}} \, dx=-\frac {2 (b e-a f)}{f (d e-c f) \sqrt {e+f x}}+\frac {2 (b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{\sqrt {d} (d e-c f)^{3/2}} \]
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Time = 0.05 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {79, 65, 214} \[ \int \frac {a+b x}{(c+d x) (e+f x)^{3/2}} \, dx=\frac {2 (b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{\sqrt {d} (d e-c f)^{3/2}}-\frac {2 (b e-a f)}{f \sqrt {e+f x} (d e-c f)} \]
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Rule 65
Rule 79
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (b e-a f)}{f (d e-c f) \sqrt {e+f x}}-\frac {(b c-a d) \int \frac {1}{(c+d x) \sqrt {e+f x}} \, dx}{d e-c f} \\ & = -\frac {2 (b e-a f)}{f (d e-c f) \sqrt {e+f x}}-\frac {(2 (b c-a d)) \text {Subst}\left (\int \frac {1}{c-\frac {d e}{f}+\frac {d x^2}{f}} \, dx,x,\sqrt {e+f x}\right )}{f (d e-c f)} \\ & = -\frac {2 (b e-a f)}{f (d e-c f) \sqrt {e+f x}}+\frac {2 (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{\sqrt {d} (d e-c f)^{3/2}} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00 \[ \int \frac {a+b x}{(c+d x) (e+f x)^{3/2}} \, dx=\frac {2 b e-2 a f}{f (-d e+c f) \sqrt {e+f x}}+\frac {2 (b c-a d) \arctan \left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{\sqrt {d} (-d e+c f)^{3/2}} \]
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Time = 2.92 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.90
method | result | size |
pseudoelliptic | \(\frac {-\frac {2 \left (a f -b e \right )}{\sqrt {f x +e}}-\frac {2 f \left (a d -b c \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{\sqrt {\left (c f -d e \right ) d}}}{\left (c f -d e \right ) f}\) | \(79\) |
derivativedivides | \(\frac {-\frac {2 \left (a f -b e \right )}{\left (c f -d e \right ) \sqrt {f x +e}}-\frac {2 f \left (a d -b c \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{\left (c f -d e \right ) \sqrt {\left (c f -d e \right ) d}}}{f}\) | \(89\) |
default | \(\frac {-\frac {2 \left (a f -b e \right )}{\left (c f -d e \right ) \sqrt {f x +e}}-\frac {2 f \left (a d -b c \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{\left (c f -d e \right ) \sqrt {\left (c f -d e \right ) d}}}{f}\) | \(89\) |
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Leaf count of result is larger than twice the leaf count of optimal. 175 vs. \(2 (76) = 152\).
Time = 0.24 (sec) , antiderivative size = 363, normalized size of antiderivative = 4.12 \[ \int \frac {a+b x}{(c+d x) (e+f x)^{3/2}} \, dx=\left [\frac {{\left ({\left (b c - a d\right )} f^{2} x + {\left (b c - a d\right )} e f\right )} \sqrt {d^{2} e - c d f} \log \left (\frac {d f x + 2 \, d e - c f + 2 \, \sqrt {d^{2} e - c d f} \sqrt {f x + e}}{d x + c}\right ) - 2 \, {\left (b d^{2} e^{2} + a c d f^{2} - {\left (b c d + a d^{2}\right )} e f\right )} \sqrt {f x + e}}{d^{3} e^{3} f - 2 \, c d^{2} e^{2} f^{2} + c^{2} d e f^{3} + {\left (d^{3} e^{2} f^{2} - 2 \, c d^{2} e f^{3} + c^{2} d f^{4}\right )} x}, -\frac {2 \, {\left ({\left ({\left (b c - a d\right )} f^{2} x + {\left (b c - a d\right )} e f\right )} \sqrt {-d^{2} e + c d f} \arctan \left (\frac {\sqrt {-d^{2} e + c d f} \sqrt {f x + e}}{d f x + d e}\right ) + {\left (b d^{2} e^{2} + a c d f^{2} - {\left (b c d + a d^{2}\right )} e f\right )} \sqrt {f x + e}\right )}}{d^{3} e^{3} f - 2 \, c d^{2} e^{2} f^{2} + c^{2} d e f^{3} + {\left (d^{3} e^{2} f^{2} - 2 \, c d^{2} e f^{3} + c^{2} d f^{4}\right )} x}\right ] \]
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Time = 3.13 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.25 \[ \int \frac {a+b x}{(c+d x) (e+f x)^{3/2}} \, dx=\begin {cases} \frac {2 \left (- \frac {a f - b e}{\sqrt {e + f x} \left (c f - d e\right )} - \frac {f \left (a d - b c\right ) \operatorname {atan}{\left (\frac {\sqrt {e + f x}}{\sqrt {\frac {c f - d e}{d}}} \right )}}{d \sqrt {\frac {c f - d e}{d}} \left (c f - d e\right )}\right )}{f} & \text {for}\: f \neq 0 \\\frac {\frac {b x}{d} + \frac {\left (a d - b c\right ) \left (\begin {cases} \frac {x}{c} & \text {for}\: d = 0 \\\frac {\log {\left (c + d x \right )}}{d} & \text {otherwise} \end {cases}\right )}{d}}{e^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
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Exception generated. \[ \int \frac {a+b x}{(c+d x) (e+f x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.28 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.99 \[ \int \frac {a+b x}{(c+d x) (e+f x)^{3/2}} \, dx=-\frac {2 \, {\left (b c - a d\right )} \arctan \left (\frac {\sqrt {f x + e} d}{\sqrt {-d^{2} e + c d f}}\right )}{\sqrt {-d^{2} e + c d f} {\left (d e - c f\right )}} - \frac {2 \, {\left (b e - a f\right )}}{{\left (d e f - c f^{2}\right )} \sqrt {f x + e}} \]
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Time = 1.47 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.09 \[ \int \frac {a+b x}{(c+d x) (e+f x)^{3/2}} \, dx=-\frac {2\,\left (a\,f-b\,e\right )}{f\,\sqrt {e+f\,x}\,\left (c\,f-d\,e\right )}-\frac {2\,\mathrm {atan}\left (\frac {2\,\sqrt {d}\,\sqrt {e+f\,x}\,\left (a\,d-b\,c\right )}{\left (2\,a\,d-2\,b\,c\right )\,\sqrt {c\,f-d\,e}}\right )\,\left (a\,d-b\,c\right )}{\sqrt {d}\,{\left (c\,f-d\,e\right )}^{3/2}} \]
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