Integrand size = 23, antiderivative size = 94 \[ \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx=\frac {2 \sqrt {-b} \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}} \]
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Time = 0.03 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {729, 113, 111} \[ \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx=\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}} \]
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Rule 111
Rule 113
Rule 729
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt {b+c x}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}} \, dx}{\sqrt {b x+c x^2}} \\ & = \frac {\left (\sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x}\right ) \int \frac {\sqrt {1+\frac {e x}{d}}}{\sqrt {x} \sqrt {1+\frac {c x}{b}}} \, dx}{\sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}} \\ & = \frac {2 \sqrt {-b} \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}} \\ \end{align*}
Time = 5.59 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.29 \[ \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx=-\frac {2 \left (c+\frac {b}{x}\right ) \sqrt {x} \sqrt {d+e x} \left (-\sqrt {x}+\frac {d \sqrt {1+\frac {d}{e x}} E\left (\arcsin \left (\frac {\sqrt {-\frac {d}{e}}}{\sqrt {x}}\right )|\frac {b e}{c d}\right )}{\sqrt {-\frac {d}{e}} \sqrt {1+\frac {b}{c x}} \left (e+\frac {d}{x}\right )}\right )}{c \sqrt {x (b+c x)}} \]
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Time = 1.94 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.29
method | result | size |
default | \(-\frac {2 \sqrt {e x +d}\, \sqrt {x \left (c x +b \right )}\, \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b \left (b e -c d \right )}{c^{2} x \left (c e \,x^{2}+b e x +c d x +b d \right )}\) | \(121\) |
elliptic | \(\frac {\sqrt {x \left (e x +d \right ) \left (c x +b \right )}\, \left (\frac {2 d b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}+\frac {2 e b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, \left (\left (-\frac {b}{c}+\frac {d}{e}\right ) E\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )-\frac {d F\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{e}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}\right )}{\sqrt {x \left (c x +b \right )}\, \sqrt {e x +d}}\) | \(322\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 299, normalized size of antiderivative = 3.18 \[ \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx=-\frac {2 \, {\left (3 \, \sqrt {c e} c e {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) - {\left (2 \, c d - b e\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right )}}{3 \, c^{2} e} \]
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\[ \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx=\int \frac {\sqrt {d + e x}}{\sqrt {x \left (b + c x\right )}}\, dx \]
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\[ \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx=\int { \frac {\sqrt {e x + d}}{\sqrt {c x^{2} + b x}} \,d x } \]
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\[ \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx=\int { \frac {\sqrt {e x + d}}{\sqrt {c x^{2} + b x}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx=\int \frac {\sqrt {d+e\,x}}{\sqrt {c\,x^2+b\,x}} \,d x \]
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