Integrand size = 25, antiderivative size = 128 \[ \int \frac {\sqrt {a+b x} (e+f x)}{x (c+d x)^2} \, dx=\frac {(d e-c f) \sqrt {a+b x}}{c d (c+d x)}-\frac {\left (2 a d^2 e-b c (d e+c f)\right ) \arctan \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{c^2 d^{3/2} \sqrt {b c-a d}}-\frac {2 \sqrt {a} e \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{c^2} \]
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Time = 0.08 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {154, 162, 65, 214, 211} \[ \int \frac {\sqrt {a+b x} (e+f x)}{x (c+d x)^2} \, dx=-\frac {\arctan \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right ) \left (2 a d^2 e-b c (c f+d e)\right )}{c^2 d^{3/2} \sqrt {b c-a d}}-\frac {2 \sqrt {a} e \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{c^2}+\frac {\sqrt {a+b x} (d e-c f)}{c d (c+d x)} \]
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Rule 65
Rule 154
Rule 162
Rule 211
Rule 214
Rubi steps \begin{align*} \text {integral}& = \frac {(d e-c f) \sqrt {a+b x}}{c d (c+d x)}-\frac {\int \frac {-a d e-\frac {1}{2} b (d e+c f) x}{x \sqrt {a+b x} (c+d x)} \, dx}{c d} \\ & = \frac {(d e-c f) \sqrt {a+b x}}{c d (c+d x)}+\frac {(a e) \int \frac {1}{x \sqrt {a+b x}} \, dx}{c^2}-\frac {\left (2 a d^2 e-b c (d e+c f)\right ) \int \frac {1}{\sqrt {a+b x} (c+d x)} \, dx}{2 c^2 d} \\ & = \frac {(d e-c f) \sqrt {a+b x}}{c d (c+d x)}+\frac {(2 a e) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{b c^2}-\frac {\left (2 a d^2 e-b c (d e+c f)\right ) \text {Subst}\left (\int \frac {1}{c-\frac {a d}{b}+\frac {d x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{b c^2 d} \\ & = \frac {(d e-c f) \sqrt {a+b x}}{c d (c+d x)}-\frac {\left (2 a d^2 e-b c (d e+c f)\right ) \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{c^2 d^{3/2} \sqrt {b c-a d}}-\frac {2 \sqrt {a} e \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{c^2} \\ \end{align*}
Time = 0.46 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {a+b x} (e+f x)}{x (c+d x)^2} \, dx=\frac {\frac {c (d e-c f) \sqrt {a+b x}}{d (c+d x)}+\frac {\left (-2 a d^2 e+b c (d e+c f)\right ) \arctan \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{d^{3/2} \sqrt {b c-a d}}-2 \sqrt {a} e \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{c^2} \]
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Time = 1.62 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.86
method | result | size |
pseudoelliptic | \(\frac {-2 e \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) \sqrt {a}+\frac {-\frac {c \left (c f -d e \right ) \sqrt {b x +a}}{d x +c}+\frac {\left (2 a e \,d^{2}-c^{2} b f -b c d e \right ) \operatorname {arctanh}\left (\frac {d \sqrt {b x +a}}{\sqrt {\left (a d -b c \right ) d}}\right )}{\sqrt {\left (a d -b c \right ) d}}}{d}}{c^{2}}\) | \(110\) |
derivativedivides | \(2 b \left (-\frac {e \sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{b \,c^{2}}+\frac {\frac {b c \left (c f -d e \right ) \sqrt {b x +a}}{2 d \left (-d \left (b x +a \right )+a d -b c \right )}+\frac {\left (2 a e \,d^{2}-c^{2} b f -b c d e \right ) \operatorname {arctanh}\left (\frac {d \sqrt {b x +a}}{\sqrt {\left (a d -b c \right ) d}}\right )}{2 d \sqrt {\left (a d -b c \right ) d}}}{c^{2} b}\right )\) | \(137\) |
default | \(2 b \left (-\frac {e \sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{b \,c^{2}}+\frac {\frac {b c \left (c f -d e \right ) \sqrt {b x +a}}{2 d \left (-d \left (b x +a \right )+a d -b c \right )}+\frac {\left (2 a e \,d^{2}-c^{2} b f -b c d e \right ) \operatorname {arctanh}\left (\frac {d \sqrt {b x +a}}{\sqrt {\left (a d -b c \right ) d}}\right )}{2 d \sqrt {\left (a d -b c \right ) d}}}{c^{2} b}\right )\) | \(137\) |
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Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (110) = 220\).
Time = 0.32 (sec) , antiderivative size = 1008, normalized size of antiderivative = 7.88 \[ \int \frac {\sqrt {a+b x} (e+f x)}{x (c+d x)^2} \, dx=\left [-\frac {{\left (b c^{3} f + {\left (b c^{2} d - 2 \, a c d^{2}\right )} e + {\left (b c^{2} d f + {\left (b c d^{2} - 2 \, a d^{3}\right )} e\right )} x\right )} \sqrt {-b c d + a d^{2}} \log \left (\frac {b d x - b c + 2 \, a d - 2 \, \sqrt {-b c d + a d^{2}} \sqrt {b x + a}}{d x + c}\right ) - 2 \, {\left ({\left (b c d^{3} - a d^{4}\right )} e x + {\left (b c^{2} d^{2} - a c d^{3}\right )} e\right )} \sqrt {a} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) - 2 \, {\left ({\left (b c^{2} d^{2} - a c d^{3}\right )} e - {\left (b c^{3} d - a c^{2} d^{2}\right )} f\right )} \sqrt {b x + a}}{2 \, {\left (b c^{4} d^{2} - a c^{3} d^{3} + {\left (b c^{3} d^{3} - a c^{2} d^{4}\right )} x\right )}}, \frac {4 \, {\left ({\left (b c d^{3} - a d^{4}\right )} e x + {\left (b c^{2} d^{2} - a c d^{3}\right )} e\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) - {\left (b c^{3} f + {\left (b c^{2} d - 2 \, a c d^{2}\right )} e + {\left (b c^{2} d f + {\left (b c d^{2} - 2 \, a d^{3}\right )} e\right )} x\right )} \sqrt {-b c d + a d^{2}} \log \left (\frac {b d x - b c + 2 \, a d - 2 \, \sqrt {-b c d + a d^{2}} \sqrt {b x + a}}{d x + c}\right ) + 2 \, {\left ({\left (b c^{2} d^{2} - a c d^{3}\right )} e - {\left (b c^{3} d - a c^{2} d^{2}\right )} f\right )} \sqrt {b x + a}}{2 \, {\left (b c^{4} d^{2} - a c^{3} d^{3} + {\left (b c^{3} d^{3} - a c^{2} d^{4}\right )} x\right )}}, -\frac {{\left (b c^{3} f + {\left (b c^{2} d - 2 \, a c d^{2}\right )} e + {\left (b c^{2} d f + {\left (b c d^{2} - 2 \, a d^{3}\right )} e\right )} x\right )} \sqrt {b c d - a d^{2}} \arctan \left (\frac {\sqrt {b c d - a d^{2}} \sqrt {b x + a}}{b d x + a d}\right ) - {\left ({\left (b c d^{3} - a d^{4}\right )} e x + {\left (b c^{2} d^{2} - a c d^{3}\right )} e\right )} \sqrt {a} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) - {\left ({\left (b c^{2} d^{2} - a c d^{3}\right )} e - {\left (b c^{3} d - a c^{2} d^{2}\right )} f\right )} \sqrt {b x + a}}{b c^{4} d^{2} - a c^{3} d^{3} + {\left (b c^{3} d^{3} - a c^{2} d^{4}\right )} x}, -\frac {{\left (b c^{3} f + {\left (b c^{2} d - 2 \, a c d^{2}\right )} e + {\left (b c^{2} d f + {\left (b c d^{2} - 2 \, a d^{3}\right )} e\right )} x\right )} \sqrt {b c d - a d^{2}} \arctan \left (\frac {\sqrt {b c d - a d^{2}} \sqrt {b x + a}}{b d x + a d}\right ) - 2 \, {\left ({\left (b c d^{3} - a d^{4}\right )} e x + {\left (b c^{2} d^{2} - a c d^{3}\right )} e\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) - {\left ({\left (b c^{2} d^{2} - a c d^{3}\right )} e - {\left (b c^{3} d - a c^{2} d^{2}\right )} f\right )} \sqrt {b x + a}}{b c^{4} d^{2} - a c^{3} d^{3} + {\left (b c^{3} d^{3} - a c^{2} d^{4}\right )} x}\right ] \]
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Timed out. \[ \int \frac {\sqrt {a+b x} (e+f x)}{x (c+d x)^2} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {\sqrt {a+b x} (e+f x)}{x (c+d x)^2} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.28 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt {a+b x} (e+f x)}{x (c+d x)^2} \, dx=\frac {2 \, a e \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} c^{2}} + \frac {{\left (b c d e - 2 \, a d^{2} e + b c^{2} f\right )} \arctan \left (\frac {\sqrt {b x + a} d}{\sqrt {b c d - a d^{2}}}\right )}{\sqrt {b c d - a d^{2}} c^{2} d} + \frac {\sqrt {b x + a} b d e - \sqrt {b x + a} b c f}{{\left (b c + {\left (b x + a\right )} d - a d\right )} c d} \]
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Time = 3.39 (sec) , antiderivative size = 1814, normalized size of antiderivative = 14.17 \[ \int \frac {\sqrt {a+b x} (e+f x)}{x (c+d x)^2} \, dx=\text {Too large to display} \]
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