Integrand size = 39, antiderivative size = 221 \[ \int \frac {A+B x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx=-\frac {2 a B \sqrt {-b c+a d} \sqrt {\frac {b (c+d x)}{b c-a d}} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right )|-\frac {(b c-a d) (1-e)}{a d}\right )}{b^2 \sqrt {d} (1-e) \sqrt {c+d x}}+\frac {2 \sqrt {a} (a B e+A (b-b e)) \sqrt {\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-e} \sqrt {a+b x}}{\sqrt {a}}\right ),-\frac {a d}{(b c-a d) (1-e)}\right )}{b^2 (1-e)^{3/2} \sqrt {c+d x}} \]
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Time = 0.15 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {164, 115, 114, 122, 120} \[ \int \frac {A+B x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx=\frac {2 \sqrt {a} (a B e+A (b-b e)) \sqrt {\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-e} \sqrt {a+b x}}{\sqrt {a}}\right ),-\frac {a d}{(b c-a d) (1-e)}\right )}{b^2 (1-e)^{3/2} \sqrt {c+d x}}-\frac {2 a B \sqrt {a d-b c} \sqrt {\frac {b (c+d x)}{b c-a d}} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|-\frac {(b c-a d) (1-e)}{a d}\right )}{b^2 \sqrt {d} (1-e) \sqrt {c+d x}} \]
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Rule 114
Rule 115
Rule 120
Rule 122
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+d x}} \, dx}{b (1-e)}+\left (A+\frac {a B e}{b-b e}\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx \\ & = \frac {\left (\left (A+\frac {a B e}{b-b e}\right ) \sqrt {\frac {b (c+d x)}{b c-a d}}\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx}{\sqrt {c+d x}}-\frac {\left (a B \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+\frac {b (-1+e) x}{a}}\right ) \int \frac {\sqrt {\frac {b e}{-b (-1+e)+b e}+\frac {b^2 (-1+e) x}{a (-b (-1+e)+b e)}}}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}}} \, dx}{b (1-e) \sqrt {c+d x} \sqrt {\frac {b \left (e+\frac {b (-1+e) x}{a}\right )}{-b (-1+e)+b e}}} \\ & = -\frac {2 a B \sqrt {-b c+a d} \sqrt {\frac {b (c+d x)}{b c-a d}} E\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right )|-\frac {(b c-a d) (1-e)}{a d}\right )}{b^2 \sqrt {d} (1-e) \sqrt {c+d x}}+\frac {2 \sqrt {a} (a B e+A (b-b e)) \sqrt {\frac {b (c+d x)}{b c-a d}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-e} \sqrt {a+b x}}{\sqrt {a}}\right )|-\frac {a d}{(b c-a d) (1-e)}\right )}{b^2 (1-e)^{3/2} \sqrt {c+d x}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 19.24 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.41 \[ \int \frac {A+B x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx=-\frac {2 \sqrt {\frac {a}{-1+e}} (a+b x)^{3/2} \left (-\frac {b B \sqrt {\frac {a}{-1+e}} (c+d x) (a e+b (-1+e) x)}{(a+b x)^2}-\frac {i a B d \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {\frac {-1+e+\frac {a}{a+b x}}{-1+e}} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{-1+e}}}{\sqrt {a+b x}}\right )|\frac {(b c-a d) (-1+e)}{a d}\right )}{\sqrt {a+b x}}+\frac {i d (a B e+A (b-b e)) \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {\frac {-1+e+\frac {a}{a+b x}}{-1+e}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{-1+e}}}{\sqrt {a+b x}}\right ),\frac {(b c-a d) (-1+e)}{a d}\right )}{\sqrt {a+b x}}\right )}{a b^2 d \sqrt {c+d x} \sqrt {e+\frac {b (-1+e) x}{a}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(728\) vs. \(2(195)=390\).
Time = 2.96 (sec) , antiderivative size = 729, normalized size of antiderivative = 3.30
method | result | size |
elliptic | \(\frac {\sqrt {\frac {\left (b x +a \right ) \left (d x +c \right ) \left (b e x +a e -b x \right )}{a}}\, \left (\frac {2 A \left (\frac {a e}{b \left (-1+e \right )}-\frac {c}{d}\right ) \sqrt {\frac {x +\frac {a e}{b \left (-1+e \right )}}{\frac {a e}{b \left (-1+e \right )}-\frac {c}{d}}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {a e}{b \left (-1+e \right )}+\frac {a}{b}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {a e}{b \left (-1+e \right )}+\frac {c}{d}}}\, F\left (\sqrt {\frac {x +\frac {a e}{b \left (-1+e \right )}}{\frac {a e}{b \left (-1+e \right )}-\frac {c}{d}}}, \sqrt {\frac {-\frac {a e}{b \left (-1+e \right )}+\frac {c}{d}}{-\frac {a e}{b \left (-1+e \right )}+\frac {a}{b}}}\right )}{\sqrt {\frac {b^{2} d e \,x^{3}}{a}+2 b d e \,x^{2}+\frac {b^{2} c e \,x^{2}}{a}-\frac {d \,x^{3} b^{2}}{a}+a d e x +2 b c e x -b d \,x^{2}-\frac {b^{2} c \,x^{2}}{a}+a c e -b c x}}+\frac {2 B \left (\frac {a e}{b \left (-1+e \right )}-\frac {c}{d}\right ) \sqrt {\frac {x +\frac {a e}{b \left (-1+e \right )}}{\frac {a e}{b \left (-1+e \right )}-\frac {c}{d}}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {a e}{b \left (-1+e \right )}+\frac {a}{b}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {a e}{b \left (-1+e \right )}+\frac {c}{d}}}\, \left (\left (-\frac {a e}{b \left (-1+e \right )}+\frac {a}{b}\right ) E\left (\sqrt {\frac {x +\frac {a e}{b \left (-1+e \right )}}{\frac {a e}{b \left (-1+e \right )}-\frac {c}{d}}}, \sqrt {\frac {-\frac {a e}{b \left (-1+e \right )}+\frac {c}{d}}{-\frac {a e}{b \left (-1+e \right )}+\frac {a}{b}}}\right )-\frac {a F\left (\sqrt {\frac {x +\frac {a e}{b \left (-1+e \right )}}{\frac {a e}{b \left (-1+e \right )}-\frac {c}{d}}}, \sqrt {\frac {-\frac {a e}{b \left (-1+e \right )}+\frac {c}{d}}{-\frac {a e}{b \left (-1+e \right )}+\frac {a}{b}}}\right )}{b}\right )}{\sqrt {\frac {b^{2} d e \,x^{3}}{a}+2 b d e \,x^{2}+\frac {b^{2} c e \,x^{2}}{a}-\frac {d \,x^{3} b^{2}}{a}+a d e x +2 b c e x -b d \,x^{2}-\frac {b^{2} c \,x^{2}}{a}+a c e -b c x}}\right )}{\sqrt {b x +a}\, \sqrt {d x +c}\, \sqrt {\frac {b e x +a e -b x}{a}}}\) | \(729\) |
default | \(\frac {2 \sqrt {b x +a}\, \sqrt {d x +c}\, \sqrt {\frac {d \left (b e x +a e -b x \right )}{a d e -b c e +b c}}\, \sqrt {-\frac {\left (b x +a \right ) \left (-1+e \right )}{a}}\, \sqrt {-\frac {\left (d x +c \right ) b \left (-1+e \right )}{a d e -b c e +b c}}\, \left (A F\left (\sqrt {\frac {d \left (b e x +a e -b x \right )}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{d a}}\right ) a b d \,e^{2}-A F\left (\sqrt {\frac {d \left (b e x +a e -b x \right )}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{d a}}\right ) b^{2} c \,e^{2}-B F\left (\sqrt {\frac {d \left (b e x +a e -b x \right )}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{d a}}\right ) a^{2} d \,e^{2}+B F\left (\sqrt {\frac {d \left (b e x +a e -b x \right )}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{d a}}\right ) a b c \,e^{2}-A F\left (\sqrt {\frac {d \left (b e x +a e -b x \right )}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{d a}}\right ) a b d e +2 A F\left (\sqrt {\frac {d \left (b e x +a e -b x \right )}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{d a}}\right ) b^{2} c e +B F\left (\sqrt {\frac {d \left (b e x +a e -b x \right )}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{d a}}\right ) a^{2} d e -2 B F\left (\sqrt {\frac {d \left (b e x +a e -b x \right )}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{d a}}\right ) a b c e -B E\left (\sqrt {\frac {d \left (b e x +a e -b x \right )}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{d a}}\right ) a^{2} d e +B E\left (\sqrt {\frac {d \left (b e x +a e -b x \right )}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{d a}}\right ) a b c e -A F\left (\sqrt {\frac {d \left (b e x +a e -b x \right )}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{d a}}\right ) b^{2} c +B F\left (\sqrt {\frac {d \left (b e x +a e -b x \right )}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{d a}}\right ) a b c -B E\left (\sqrt {\frac {d \left (b e x +a e -b x \right )}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{d a}}\right ) a b c \right )}{\sqrt {\frac {b e x +a e -b x}{a}}\, \left (b d \,x^{2}+a d x +b c x +a c \right ) \left (-1+e \right )^{2} b^{2} d}\) | \(940\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 1126, normalized size of antiderivative = 5.10 \[ \int \frac {A+B x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx=\text {Too large to display} \]
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\[ \int \frac {A+B x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx=\int \frac {A + B x}{\sqrt {a + b x} \sqrt {c + d x} \sqrt {e + \frac {b e x}{a} - \frac {b x}{a}}}\, dx \]
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\[ \int \frac {A+B x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx=\int { \frac {B x + A}{\sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {b {\left (e - 1\right )} x}{a} + e}} \,d x } \]
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\[ \int \frac {A+B x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx=\int { \frac {B x + A}{\sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {b {\left (e - 1\right )} x}{a} + e}} \,d x } \]
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Timed out. \[ \int \frac {A+B x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx=\int \frac {A+B\,x}{\sqrt {e+\frac {b\,x\,\left (e-1\right )}{a}}\,\sqrt {a+b\,x}\,\sqrt {c+d\,x}} \,d x \]
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