Integrand size = 20, antiderivative size = 75 \[ \int \frac {x}{\sqrt {a+b x} \sqrt {c+d x}} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x}}{b d}-\frac {(b c+a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2} d^{3/2}} \]
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Time = 0.03 (sec) , antiderivative size = 75, 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 = {81, 65, 223, 212} \[ \int \frac {x}{\sqrt {a+b x} \sqrt {c+d x}} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x}}{b d}-\frac {(a d+b c) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2} d^{3/2}} \]
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Rule 65
Rule 81
Rule 212
Rule 223
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a+b x} \sqrt {c+d x}}{b d}-\frac {(b c+a d) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 b d} \\ & = \frac {\sqrt {a+b x} \sqrt {c+d x}}{b d}-\frac {(b c+a d) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{b^2 d} \\ & = \frac {\sqrt {a+b x} \sqrt {c+d x}}{b d}-\frac {(b c+a d) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{b^2 d} \\ & = \frac {\sqrt {a+b x} \sqrt {c+d x}}{b d}-\frac {(b c+a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2} d^{3/2}} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00 \[ \int \frac {x}{\sqrt {a+b x} \sqrt {c+d x}} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x}}{b d}-\frac {(b c+a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{b^{3/2} d^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(147\) vs. \(2(59)=118\).
Time = 1.66 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.97
method | result | size |
default | \(-\frac {\left (\ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a d +\ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b c -2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\right ) \sqrt {b x +a}\, \sqrt {d x +c}}{2 d \sqrt {b d}\, b \sqrt {\left (b x +a \right ) \left (d x +c \right )}}\) | \(148\) |
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none
Time = 0.24 (sec) , antiderivative size = 232, normalized size of antiderivative = 3.09 \[ \int \frac {x}{\sqrt {a+b x} \sqrt {c+d x}} \, dx=\left [\frac {4 \, \sqrt {b x + a} \sqrt {d x + c} b d + {\left (b c + a d\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} - 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right )}{4 \, b^{2} d^{2}}, \frac {2 \, \sqrt {b x + a} \sqrt {d x + c} b d + {\left (b c + a d\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right )}{2 \, b^{2} d^{2}}\right ] \]
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\[ \int \frac {x}{\sqrt {a+b x} \sqrt {c+d x}} \, dx=\int \frac {x}{\sqrt {a + b x} \sqrt {c + d x}}\, dx \]
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Exception generated. \[ \int \frac {x}{\sqrt {a+b x} \sqrt {c+d x}} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.29 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.27 \[ \int \frac {x}{\sqrt {a+b x} \sqrt {c+d x}} \, dx=\frac {\frac {{\left (b c + a d\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} d} + \frac {\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a}}{b d}}{{\left | b \right |}} \]
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Time = 4.43 (sec) , antiderivative size = 259, normalized size of antiderivative = 3.45 \[ \int \frac {x}{\sqrt {a+b x} \sqrt {c+d x}} \, dx=\frac {\frac {\left (2\,a\,d+2\,b\,c\right )\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{d^3\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}+\frac {\left (2\,a\,d+2\,b\,c\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3}{b\,d^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^3}-\frac {8\,\sqrt {a}\,\sqrt {c}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{d^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}}{\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}{{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^4}+\frac {b^2}{d^2}-\frac {2\,b\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{d\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}}-\frac {2\,\mathrm {atanh}\left (\frac {\sqrt {d}\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {b}\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}\right )\,\left (a\,d+b\,c\right )}{b^{3/2}\,d^{3/2}} \]
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