Integrand size = 24, antiderivative size = 188 \[ \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx=\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}} \]
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Time = 0.04 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {732, 435} \[ \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx=\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}} \]
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Rule 435
Rule 732
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 \sqrt {b^2-4 a c} e x^2}{2 c d-b e-\sqrt {b^2-4 a c} e}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{c \sqrt {\frac {c (d+e x)}{2 c d-b e-\sqrt {b^2-4 a c} e}} \sqrt {a+b x+c x^2}} \\ & = \frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 21.54 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.94 \[ \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx=\frac {i \left (2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e\right ) \sqrt {\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {1-\frac {2 c (d+e x)}{2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e}} \left (E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {d+e x}\right )|\frac {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {d+e x}\right ),\frac {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e}\right )\right )}{\sqrt {2} c e \sqrt {\frac {c}{-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+x (b+c x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(745\) vs. \(2(164)=328\).
Time = 0.44 (sec) , antiderivative size = 746, normalized size of antiderivative = 3.97
method | result | size |
elliptic | \(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+b x +a \right )}\, \left (\frac {2 d \left (\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, F\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )}{\sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+a e x +x d b +a d}}+\frac {2 e \left (\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \left (\left (-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) E\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )+\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) F\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )}{2 c}\right )}{\sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+a e x +x d b +a d}}\right )}{\sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}}\) | \(746\) |
default | \(\frac {\sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}\, \left (e \sqrt {-4 a c +b^{2}}+b e -2 c d \right ) \sqrt {2}\, \sqrt {-\frac {\left (e x +d \right ) c}{e \sqrt {-4 a c +b^{2}}+b e -2 c d}}\, \sqrt {\frac {\left (-b -2 c x +\sqrt {-4 a c +b^{2}}\right ) e}{2 c d -b e +e \sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {\left (b +2 c x +\sqrt {-4 a c +b^{2}}\right ) e}{e \sqrt {-4 a c +b^{2}}+b e -2 c d}}\, \left (F\left (\sqrt {2}\, \sqrt {-\frac {\left (e x +d \right ) c}{e \sqrt {-4 a c +b^{2}}+b e -2 c d}}, \sqrt {-\frac {e \sqrt {-4 a c +b^{2}}+b e -2 c d}{2 c d -b e +e \sqrt {-4 a c +b^{2}}}}\right ) e b -2 d F\left (\sqrt {2}\, \sqrt {-\frac {\left (e x +d \right ) c}{e \sqrt {-4 a c +b^{2}}+b e -2 c d}}, \sqrt {-\frac {e \sqrt {-4 a c +b^{2}}+b e -2 c d}{2 c d -b e +e \sqrt {-4 a c +b^{2}}}}\right ) c -F\left (\sqrt {2}\, \sqrt {-\frac {\left (e x +d \right ) c}{e \sqrt {-4 a c +b^{2}}+b e -2 c d}}, \sqrt {-\frac {e \sqrt {-4 a c +b^{2}}+b e -2 c d}{2 c d -b e +e \sqrt {-4 a c +b^{2}}}}\right ) e \sqrt {-4 a c +b^{2}}-E\left (\sqrt {2}\, \sqrt {-\frac {\left (e x +d \right ) c}{e \sqrt {-4 a c +b^{2}}+b e -2 c d}}, \sqrt {-\frac {e \sqrt {-4 a c +b^{2}}+b e -2 c d}{2 c d -b e +e \sqrt {-4 a c +b^{2}}}}\right ) b e +2 E\left (\sqrt {2}\, \sqrt {-\frac {\left (e x +d \right ) c}{e \sqrt {-4 a c +b^{2}}+b e -2 c d}}, \sqrt {-\frac {e \sqrt {-4 a c +b^{2}}+b e -2 c d}{2 c d -b e +e \sqrt {-4 a c +b^{2}}}}\right ) c d +\sqrt {-4 a c +b^{2}}\, E\left (\sqrt {2}\, \sqrt {-\frac {\left (e x +d \right ) c}{e \sqrt {-4 a c +b^{2}}+b e -2 c d}}, \sqrt {-\frac {e \sqrt {-4 a c +b^{2}}+b e -2 c d}{2 c d -b e +e \sqrt {-4 a c +b^{2}}}}\right ) e \right )}{2 e \left (c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+a e x +x d b +a d \right ) c^{2}}\) | \(747\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.91 \[ \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx=-\frac {2 \, {\left (3 \, \sqrt {c e} c e {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) - {\left (2 \, c d - b e\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right )}}{3 \, c^{2} e} \]
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\[ \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx=\int \frac {\sqrt {d + e x}}{\sqrt {a + b x + c x^{2}}}\, dx \]
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\[ \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx=\int { \frac {\sqrt {e x + d}}{\sqrt {c x^{2} + b x + a}} \,d x } \]
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\[ \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx=\int { \frac {\sqrt {e x + d}}{\sqrt {c x^{2} + b x + a}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx=\int \frac {\sqrt {d+e\,x}}{\sqrt {c\,x^2+b\,x+a}} \,d x \]
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