Integrand size = 22, antiderivative size = 76 \[ \int \frac {\sqrt {c+d x}}{x^2 \sqrt {a+b x}} \, dx=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{a x}+\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} \sqrt {c}} \]
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Time = 0.02 (sec) , antiderivative size = 76, 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 = {96, 95, 214} \[ \int \frac {\sqrt {c+d x}}{x^2 \sqrt {a+b x}} \, dx=\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} \sqrt {c}}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{a x} \]
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Rule 95
Rule 96
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {a+b x} \sqrt {c+d x}}{a x}-\frac {(b c-a d) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 a} \\ & = -\frac {\sqrt {a+b x} \sqrt {c+d x}}{a x}-\frac {(b c-a d) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{a} \\ & = -\frac {\sqrt {a+b x} \sqrt {c+d x}}{a x}+\frac {(b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} \sqrt {c}} \\ \end{align*}
Time = 10.05 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.01 \[ \int \frac {\sqrt {c+d x}}{x^2 \sqrt {a+b x}} \, dx=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{a x}-\frac {(-b c+a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} \sqrt {c}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(146\) vs. \(2(60)=120\).
Time = 1.62 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.93
method | result | size |
default | \(-\frac {\sqrt {d x +c}\, \sqrt {b x +a}\, \left (\ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a d x -\ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\right )}{2 a \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, x \sqrt {a c}}\) | \(147\) |
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none
Time = 0.25 (sec) , antiderivative size = 252, normalized size of antiderivative = 3.32 \[ \int \frac {\sqrt {c+d x}}{x^2 \sqrt {a+b x}} \, dx=\left [-\frac {\sqrt {a c} {\left (b c - a d\right )} x \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, \sqrt {b x + a} \sqrt {d x + c} a c}{4 \, a^{2} c x}, -\frac {\sqrt {-a c} {\left (b c - a d\right )} x \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) + 2 \, \sqrt {b x + a} \sqrt {d x + c} a c}{2 \, a^{2} c x}\right ] \]
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\[ \int \frac {\sqrt {c+d x}}{x^2 \sqrt {a+b x}} \, dx=\int \frac {\sqrt {c + d x}}{x^{2} \sqrt {a + b x}}\, dx \]
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Exception generated. \[ \int \frac {\sqrt {c+d x}}{x^2 \sqrt {a+b x}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 413 vs. \(2 (60) = 120\).
Time = 0.53 (sec) , antiderivative size = 413, normalized size of antiderivative = 5.43 \[ \int \frac {\sqrt {c+d x}}{x^2 \sqrt {a+b x}} \, dx=\frac {{\left (\frac {{\left (\sqrt {b d} b^{4} c - \sqrt {b d} a b^{3} d\right )} \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} a b} - \frac {2 \, {\left (\sqrt {b d} b^{6} c^{2} - 2 \, \sqrt {b d} a b^{5} c d + \sqrt {b d} a^{2} b^{4} d^{2} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{4} c - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{3} d\right )}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b d + {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4}\right )} a}\right )} {\left | b \right |}}{b^{3}} \]
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Time = 5.85 (sec) , antiderivative size = 439, normalized size of antiderivative = 5.78 \[ \int \frac {\sqrt {c+d x}}{x^2 \sqrt {a+b x}} \, dx=\frac {\frac {\left (\frac {c\,b^2}{4}+\frac {a\,d\,b}{4}\right )\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{a^{3/2}\,\sqrt {c}\,d\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}-\frac {b^2}{4\,a\,d}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2\,\left (\frac {a^2\,d^2}{4}-\frac {3\,a\,b\,c\,d}{4}+\frac {b^2\,c^2}{4}\right )}{a^2\,c\,d\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}}{\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3}{{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^3}+\frac {b\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{d\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}-\frac {\left (a\,d+b\,c\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{\sqrt {a}\,\sqrt {c}\,d\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}}-\frac {d\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{4\,a\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}-\frac {\ln \left (\frac {\left (\sqrt {c}\,\sqrt {a+b\,x}-\sqrt {a}\,\sqrt {c+d\,x}\right )\,\left (b\,\sqrt {c}-\frac {\sqrt {a}\,d\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {c+d\,x}-\sqrt {c}}\right )}{\sqrt {c+d\,x}-\sqrt {c}}\right )\,\left (\sqrt {a}\,b\,c^{3/2}-a^{3/2}\,\sqrt {c}\,d\right )}{2\,a^2\,c}+\frac {\ln \left (\frac {\sqrt {a+b\,x}-\sqrt {a}}{\sqrt {c+d\,x}-\sqrt {c}}\right )\,\left (\sqrt {a}\,b\,c^{3/2}-a^{3/2}\,\sqrt {c}\,d\right )}{2\,a^2\,c} \]
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