Integrand size = 45, antiderivative size = 145 \[ \int \frac {A+B x}{\sqrt {a+b x} \sqrt {c+\frac {b (-1+c) x}{a}} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx=-\frac {2 a^{3/2} B E\left (\arcsin \left (\frac {\sqrt {1-c} \sqrt {a+b x}}{\sqrt {a}}\right )|\frac {1-e}{1-c}\right )}{b^2 \sqrt {1-c} (1-e)}+\frac {2 \sqrt {a} (a B e+A (b-b e)) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c} \sqrt {a+b x}}{\sqrt {a}}\right ),\frac {1-e}{1-c}\right )}{b^2 \sqrt {1-c} (1-e)} \]
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Time = 0.08 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {164, 114, 120} \[ \int \frac {A+B x}{\sqrt {a+b x} \sqrt {c+\frac {b (-1+c) x}{a}} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx=\frac {2 \sqrt {a} (a B e+A (b-b e)) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c} \sqrt {a+b x}}{\sqrt {a}}\right ),\frac {1-e}{1-c}\right )}{b^2 \sqrt {1-c} (1-e)}-\frac {2 a^{3/2} B E\left (\arcsin \left (\frac {\sqrt {1-c} \sqrt {a+b x}}{\sqrt {a}}\right )|\frac {1-e}{1-c}\right )}{b^2 \sqrt {1-c} (1-e)} \]
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Rule 114
Rule 120
Rule 164
Rubi steps \begin{align*} \text {integral}& = -\frac {(a B) \int \frac {\sqrt {e+\frac {b (-1+e) x}{a}}}{\sqrt {a+b x} \sqrt {c+\frac {b (-1+c) x}{a}}} \, dx}{b (1-e)}+\left (A+\frac {a B e}{b-b e}\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+\frac {b (-1+c) x}{a}} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx \\ & = -\frac {2 a^{3/2} B E\left (\sin ^{-1}\left (\frac {\sqrt {1-c} \sqrt {a+b x}}{\sqrt {a}}\right )|\frac {1-e}{1-c}\right )}{b^2 \sqrt {1-c} (1-e)}+\frac {2 \sqrt {a} \left (A+\frac {a B e}{b-b e}\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {1-c} \sqrt {a+b x}}{\sqrt {a}}\right )|\frac {1-e}{1-c}\right )}{b \sqrt {1-c}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 16.59 (sec) , antiderivative size = 309, normalized size of antiderivative = 2.13 \[ \int \frac {A+B x}{\sqrt {a+b x} \sqrt {c+\frac {b (-1+c) x}{a}} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx=-\frac {2 \sqrt {\frac {a}{-1+c}} (a+b x)^{3/2} \left (-B \sqrt {\frac {a}{-1+c}} \left (-1+c+\frac {a}{a+b x}\right ) \left (-1+e+\frac {a}{a+b x}\right )-\frac {i a B (-1+e) \sqrt {\frac {-1+c+\frac {a}{a+b x}}{-1+c}} \sqrt {\frac {-1+e+\frac {a}{a+b x}}{-1+e}} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{-1+c}}}{\sqrt {a+b x}}\right )|\frac {-1+c}{-1+e}\right )}{\sqrt {a+b x}}+\frac {i (a B c+A (b-b c)) (-1+e) \sqrt {\frac {-1+c+\frac {a}{a+b x}}{-1+c}} \sqrt {\frac {-1+e+\frac {a}{a+b x}}{-1+e}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{-1+c}}}{\sqrt {a+b x}}\right ),\frac {-1+c}{-1+e}\right )}{\sqrt {a+b x}}\right )}{a b^2 (-1+e) \sqrt {c+\frac {b (-1+c) x}{a}} \sqrt {e+\frac {b (-1+e) x}{a}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(603\) vs. \(2(127)=254\).
Time = 5.52 (sec) , antiderivative size = 604, normalized size of antiderivative = 4.17
method | result | size |
default | \(\frac {2 \left (A F\left (\sqrt {\frac {\left (c -1\right ) \left (b e x +a e -b x \right )}{a \left (c -e \right )}}, \sqrt {\frac {c -e}{c -1}}\right ) b c e -A F\left (\sqrt {\frac {\left (c -1\right ) \left (b e x +a e -b x \right )}{a \left (c -e \right )}}, \sqrt {\frac {c -e}{c -1}}\right ) b \,e^{2}-B F\left (\sqrt {\frac {\left (c -1\right ) \left (b e x +a e -b x \right )}{a \left (c -e \right )}}, \sqrt {\frac {c -e}{c -1}}\right ) a c e +B F\left (\sqrt {\frac {\left (c -1\right ) \left (b e x +a e -b x \right )}{a \left (c -e \right )}}, \sqrt {\frac {c -e}{c -1}}\right ) a \,e^{2}-A F\left (\sqrt {\frac {\left (c -1\right ) \left (b e x +a e -b x \right )}{a \left (c -e \right )}}, \sqrt {\frac {c -e}{c -1}}\right ) b c +A F\left (\sqrt {\frac {\left (c -1\right ) \left (b e x +a e -b x \right )}{a \left (c -e \right )}}, \sqrt {\frac {c -e}{c -1}}\right ) b e +B F\left (\sqrt {\frac {\left (c -1\right ) \left (b e x +a e -b x \right )}{a \left (c -e \right )}}, \sqrt {\frac {c -e}{c -1}}\right ) a c -B F\left (\sqrt {\frac {\left (c -1\right ) \left (b e x +a e -b x \right )}{a \left (c -e \right )}}, \sqrt {\frac {c -e}{c -1}}\right ) a e -B E\left (\sqrt {\frac {\left (c -1\right ) \left (b e x +a e -b x \right )}{a \left (c -e \right )}}, \sqrt {\frac {c -e}{c -1}}\right ) a c +B E\left (\sqrt {\frac {\left (c -1\right ) \left (b e x +a e -b x \right )}{a \left (c -e \right )}}, \sqrt {\frac {c -e}{c -1}}\right ) a e \right ) \sqrt {-\frac {\left (-1+e \right ) \left (b c x +a c -b x \right )}{a \left (c -e \right )}}\, \sqrt {-\frac {\left (b x +a \right ) \left (-1+e \right )}{a}}\, \sqrt {\frac {\left (c -1\right ) \left (b e x +a e -b x \right )}{a \left (c -e \right )}}\, a}{\sqrt {b x +a}\, \sqrt {\frac {b c x +a c -b x}{a}}\, \sqrt {\frac {b e x +a e -b x}{a}}\, \left (-1+e \right )^{2} \left (c -1\right ) b^{2}}\) | \(604\) |
elliptic | \(\frac {\sqrt {\frac {\left (b x +a \right ) \left (b c x +a c -b x \right ) \left (b e x +a e -b x \right )}{a^{2}}}\, \left (\frac {2 A \left (-\frac {a e}{b \left (-1+e \right )}+\frac {a c}{b \left (c -1\right )}\right ) \sqrt {\frac {x +\frac {a c}{b \left (c -1\right )}}{-\frac {a e}{b \left (-1+e \right )}+\frac {a c}{b \left (c -1\right )}}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {a c}{b \left (c -1\right )}+\frac {a}{b}}}\, \sqrt {\frac {x +\frac {a e}{b \left (-1+e \right )}}{\frac {a e}{b \left (-1+e \right )}-\frac {a c}{b \left (c -1\right )}}}\, F\left (\sqrt {\frac {x +\frac {a c}{b \left (c -1\right )}}{-\frac {a e}{b \left (-1+e \right )}+\frac {a c}{b \left (c -1\right )}}}, \sqrt {\frac {\frac {a e}{b \left (-1+e \right )}-\frac {a c}{b \left (c -1\right )}}{-\frac {a c}{b \left (c -1\right )}+\frac {a}{b}}}\right )}{\sqrt {\frac {b^{3} c e \,x^{3}}{a^{2}}+\frac {3 b^{2} c e \,x^{2}}{a}-\frac {b^{3} c \,x^{3}}{a^{2}}-\frac {b^{3} e \,x^{3}}{a^{2}}+3 b c e x -\frac {2 b^{2} c \,x^{2}}{a}-\frac {2 b^{2} e \,x^{2}}{a}+\frac {b^{3} x^{3}}{a^{2}}+a c e -b c x -b e x +\frac {b^{2} x^{2}}{a}}}+\frac {2 B \left (-\frac {a e}{b \left (-1+e \right )}+\frac {a c}{b \left (c -1\right )}\right ) \sqrt {\frac {x +\frac {a c}{b \left (c -1\right )}}{-\frac {a e}{b \left (-1+e \right )}+\frac {a c}{b \left (c -1\right )}}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {a c}{b \left (c -1\right )}+\frac {a}{b}}}\, \sqrt {\frac {x +\frac {a e}{b \left (-1+e \right )}}{\frac {a e}{b \left (-1+e \right )}-\frac {a c}{b \left (c -1\right )}}}\, \left (\left (-\frac {a c}{b \left (c -1\right )}+\frac {a}{b}\right ) E\left (\sqrt {\frac {x +\frac {a c}{b \left (c -1\right )}}{-\frac {a e}{b \left (-1+e \right )}+\frac {a c}{b \left (c -1\right )}}}, \sqrt {\frac {\frac {a e}{b \left (-1+e \right )}-\frac {a c}{b \left (c -1\right )}}{-\frac {a c}{b \left (c -1\right )}+\frac {a}{b}}}\right )-\frac {a F\left (\sqrt {\frac {x +\frac {a c}{b \left (c -1\right )}}{-\frac {a e}{b \left (-1+e \right )}+\frac {a c}{b \left (c -1\right )}}}, \sqrt {\frac {\frac {a e}{b \left (-1+e \right )}-\frac {a c}{b \left (c -1\right )}}{-\frac {a c}{b \left (c -1\right )}+\frac {a}{b}}}\right )}{b}\right )}{\sqrt {\frac {b^{3} c e \,x^{3}}{a^{2}}+\frac {3 b^{2} c e \,x^{2}}{a}-\frac {b^{3} c \,x^{3}}{a^{2}}-\frac {b^{3} e \,x^{3}}{a^{2}}+3 b c e x -\frac {2 b^{2} c \,x^{2}}{a}-\frac {2 b^{2} e \,x^{2}}{a}+\frac {b^{3} x^{3}}{a^{2}}+a c e -b c x -b e x +\frac {b^{2} x^{2}}{a}}}\right )}{\sqrt {b x +a}\, \sqrt {\frac {b c x +a c -b x}{a}}\, \sqrt {\frac {b e x +a e -b x}{a}}}\) | \(891\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 1228, normalized size of antiderivative = 8.47 \[ \int \frac {A+B x}{\sqrt {a+b x} \sqrt {c+\frac {b (-1+c) x}{a}} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx=\text {Too large to display} \]
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\[ \int \frac {A+B x}{\sqrt {a+b x} \sqrt {c+\frac {b (-1+c) x}{a}} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx=\int \frac {A + B x}{\sqrt {a + b x} \sqrt {c + \frac {b c x}{a} - \frac {b x}{a}} \sqrt {e + \frac {b e x}{a} - \frac {b x}{a}}}\, dx \]
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\[ \int \frac {A+B x}{\sqrt {a+b x} \sqrt {c+\frac {b (-1+c) x}{a}} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx=\int { \frac {B x + A}{\sqrt {b x + a} \sqrt {\frac {b {\left (c - 1\right )} x}{a} + c} \sqrt {\frac {b {\left (e - 1\right )} x}{a} + e}} \,d x } \]
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Exception generated. \[ \int \frac {A+B x}{\sqrt {a+b x} \sqrt {c+\frac {b (-1+c) x}{a}} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx=\text {Exception raised: RuntimeError} \]
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Timed out. \[ \int \frac {A+B x}{\sqrt {a+b x} \sqrt {c+\frac {b (-1+c) x}{a}} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx=\int \frac {A+B\,x}{\sqrt {c+\frac {b\,x\,\left (c-1\right )}{a}}\,\sqrt {e+\frac {b\,x\,\left (e-1\right )}{a}}\,\sqrt {a+b\,x}} \,d x \]
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