Integrand size = 19, antiderivative size = 116 \[ \int \sqrt {a+b x} \sqrt {c+d x} \, dx=\frac {(b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b d}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 b}-\frac {(b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{3/2} d^{3/2}} \]
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Time = 0.04 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {52, 65, 223, 212} \[ \int \sqrt {a+b x} \sqrt {c+d x} \, dx=-\frac {(b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{3/2} d^{3/2}}+\frac {\sqrt {a+b x} \sqrt {c+d x} (b c-a d)}{4 b d}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 b} \]
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Rule 52
Rule 65
Rule 212
Rule 223
Rubi steps \begin{align*} \text {integral}& = \frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 b}+\frac {(b c-a d) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{4 b} \\ & = \frac {(b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b d}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 b}-\frac {(b c-a d)^2 \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 b d} \\ & = \frac {(b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b d}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 b}-\frac {(b c-a d)^2 \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{4 b^2 d} \\ & = \frac {(b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b d}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 b}-\frac {(b c-a d)^2 \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{4 b^2 d} \\ & = \frac {(b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b d}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 b}-\frac {(b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{3/2} d^{3/2}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.82 \[ \int \sqrt {a+b x} \sqrt {c+d x} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} (b c+a d+2 b d x)}{4 b d}-\frac {(b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{4 b^{3/2} d^{3/2}} \]
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Time = 1.50 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.21
method | result | size |
default | \(\frac {\sqrt {b x +a}\, \left (d x +c \right )^{\frac {3}{2}}}{2 d}-\frac {\left (-a d +b c \right ) \left (\frac {\sqrt {b x +a}\, \sqrt {d x +c}}{b}-\frac {\left (a d -b c \right ) \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d x}{\sqrt {b d}}+\sqrt {b d \,x^{2}+\left (a d +b c \right ) x +a c}\right )}{2 b \sqrt {d x +c}\, \sqrt {b x +a}\, \sqrt {b d}}\right )}{4 d}\) | \(140\) |
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Time = 0.25 (sec) , antiderivative size = 300, normalized size of antiderivative = 2.59 \[ \int \sqrt {a+b x} \sqrt {c+d x} \, dx=\left [\frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} - 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \, {\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, b^{2} d^{2}}, \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 2 \, {\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, b^{2} d^{2}}\right ] \]
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\[ \int \sqrt {a+b x} \sqrt {c+d x} \, dx=\int \sqrt {a + b x} \sqrt {c + d x}\, dx \]
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Exception generated. \[ \int \sqrt {a+b x} \sqrt {c+d x} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (90) = 180\).
Time = 0.34 (sec) , antiderivative size = 232, normalized size of antiderivative = 2.00 \[ \int \sqrt {a+b x} \sqrt {c+d x} \, dx=-\frac {\frac {4 \, {\left (\frac {{\left (b^{2} c - a b d\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d}} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a}\right )} a {\left | b \right |}}{b^{2}} - \frac {{\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, b x + 2 \, a + \frac {b c d - 5 \, a d^{2}}{d^{2}}\right )} \sqrt {b x + a} + \frac {{\left (b^{3} c^{2} + 2 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} d}\right )} {\left | b \right |}}{b^{2}}}{4 \, b} \]
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Time = 0.12 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.76 \[ \int \sqrt {a+b x} \sqrt {c+d x} \, dx=\left (\frac {x}{2}+\frac {a\,d+b\,c}{4\,b\,d}\right )\,\sqrt {a+b\,x}\,\sqrt {c+d\,x}-\frac {\ln \left (a\,d+b\,c+2\,b\,d\,x+2\,\sqrt {b}\,\sqrt {d}\,\sqrt {a+b\,x}\,\sqrt {c+d\,x}\right )\,{\left (a\,d-b\,c\right )}^2}{8\,b^{3/2}\,d^{3/2}} \]
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