Integrand size = 19, antiderivative size = 72 \[ \int \frac {\sqrt {c+d x}}{\sqrt {a+b x}} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x}}{b}+\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2} \sqrt {d}} \]
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Time = 0.03 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {52, 65, 223, 212} \[ \int \frac {\sqrt {c+d x}}{\sqrt {a+b x}} \, dx=\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2} \sqrt {d}}+\frac {\sqrt {a+b x} \sqrt {c+d x}}{b} \]
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Rule 52
Rule 65
Rule 212
Rule 223
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a+b x} \sqrt {c+d x}}{b}+\frac {(b c-a d) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 b} \\ & = \frac {\sqrt {a+b x} \sqrt {c+d x}}{b}+\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} \\ & = \frac {\sqrt {a+b x} \sqrt {c+d x}}{b}+\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} \\ & = \frac {\sqrt {a+b x} \sqrt {c+d x}}{b}+\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} \sqrt {d}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {c+d x}}{\sqrt {a+b x}} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x}}{b}+\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{b^{3/2} \sqrt {d}} \]
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Time = 1.63 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.49
method | result | size |
default | \(\frac {\sqrt {b x +a}\, \sqrt {d x +c}}{b}-\frac {\left (a d -b c \right ) \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d x}{\sqrt {b d}}+\sqrt {b d \,x^{2}+\left (a d +b c \right ) x +a c}\right )}{2 b \sqrt {d x +c}\, \sqrt {b x +a}\, \sqrt {b d}}\) | \(107\) |
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none
Time = 0.24 (sec) , antiderivative size = 236, normalized size of antiderivative = 3.28 \[ \int \frac {\sqrt {c+d x}}{\sqrt {a+b x}} \, dx=\left [\frac {4 \, \sqrt {b x + a} \sqrt {d x + c} b d - {\left (b c - a d\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} - 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right )}{4 \, b^{2} d}, \frac {2 \, \sqrt {b x + a} \sqrt {d x + c} b d - {\left (b c - a d\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right )}{2 \, b^{2} d}\right ] \]
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\[ \int \frac {\sqrt {c+d x}}{\sqrt {a+b x}} \, dx=\int \frac {\sqrt {c + d x}}{\sqrt {a + b x}}\, dx \]
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Exception generated. \[ \int \frac {\sqrt {c+d x}}{\sqrt {a+b x}} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.29 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.29 \[ \int \frac {\sqrt {c+d x}}{\sqrt {a+b x}} \, dx=-\frac {{\left (\frac {{\left (b^{2} c - a b d\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d}} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a}\right )} {\left | b \right |}}{b^{3}} \]
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Time = 4.07 (sec) , antiderivative size = 260, normalized size of antiderivative = 3.61 \[ \int \frac {\sqrt {c+d x}}{\sqrt {a+b x}} \, dx=\frac {\frac {\left (2\,a\,d+2\,b\,c\right )\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{d^2\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}+\frac {\left (2\,a\,d+2\,b\,c\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3}{b\,d\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^3}-\frac {8\,\sqrt {a}\,\sqrt {c}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{d\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}}{\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}{{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^4}+\frac {b^2}{d^2}-\frac {2\,b\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{d\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}}-\frac {2\,\mathrm {atanh}\left (\frac {\sqrt {d}\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {b}\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}\right )\,\left (a\,d-b\,c\right )}{b^{3/2}\,\sqrt {d}} \]
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