Integrand size = 37, antiderivative size = 159 \[ \int \frac {1}{(c e+d e x)^{7/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=-\frac {2 \sqrt {1-c^2-2 c d x-d^2 x^2}}{5 d e (c e+d e x)^{5/2}}-\frac {6 \sqrt {1-c^2-2 c d x-d^2 x^2}}{5 d e^3 \sqrt {c e+d e x}}-\frac {6 E\left (\left .\arcsin \left (\frac {\sqrt {c e+d e x}}{\sqrt {e}}\right )\right |-1\right )}{5 d e^{7/2}}+\frac {6 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c e+d e x}}{\sqrt {e}}\right ),-1\right )}{5 d e^{7/2}} \]
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Time = 0.08 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {707, 704, 313, 227, 1213, 435} \[ \int \frac {1}{(c e+d e x)^{7/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=\frac {6 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c e+d x e}}{\sqrt {e}}\right ),-1\right )}{5 d e^{7/2}}-\frac {6 E\left (\left .\arcsin \left (\frac {\sqrt {c e+d x e}}{\sqrt {e}}\right )\right |-1\right )}{5 d e^{7/2}}-\frac {6 \sqrt {-c^2-2 c d x-d^2 x^2+1}}{5 d e^3 \sqrt {c e+d e x}}-\frac {2 \sqrt {-c^2-2 c d x-d^2 x^2+1}}{5 d e (c e+d e x)^{5/2}} \]
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Rule 227
Rule 313
Rule 435
Rule 704
Rule 707
Rule 1213
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {1-c^2-2 c d x-d^2 x^2}}{5 d e (c e+d e x)^{5/2}}+\frac {3 \int \frac {1}{(c e+d e x)^{3/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx}{5 e^2} \\ & = -\frac {2 \sqrt {1-c^2-2 c d x-d^2 x^2}}{5 d e (c e+d e x)^{5/2}}-\frac {6 \sqrt {1-c^2-2 c d x-d^2 x^2}}{5 d e^3 \sqrt {c e+d e x}}-\frac {3 \int \frac {\sqrt {c e+d e x}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx}{5 e^4} \\ & = -\frac {2 \sqrt {1-c^2-2 c d x-d^2 x^2}}{5 d e (c e+d e x)^{5/2}}-\frac {6 \sqrt {1-c^2-2 c d x-d^2 x^2}}{5 d e^3 \sqrt {c e+d e x}}-\frac {6 \text {Subst}\left (\int \frac {x^2}{\sqrt {1-\frac {x^4}{e^2}}} \, dx,x,\sqrt {c e+d e x}\right )}{5 d e^5} \\ & = -\frac {2 \sqrt {1-c^2-2 c d x-d^2 x^2}}{5 d e (c e+d e x)^{5/2}}-\frac {6 \sqrt {1-c^2-2 c d x-d^2 x^2}}{5 d e^3 \sqrt {c e+d e x}}+\frac {6 \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^4}{e^2}}} \, dx,x,\sqrt {c e+d e x}\right )}{5 d e^4}-\frac {6 \text {Subst}\left (\int \frac {1+\frac {x^2}{e}}{\sqrt {1-\frac {x^4}{e^2}}} \, dx,x,\sqrt {c e+d e x}\right )}{5 d e^4} \\ & = -\frac {2 \sqrt {1-c^2-2 c d x-d^2 x^2}}{5 d e (c e+d e x)^{5/2}}-\frac {6 \sqrt {1-c^2-2 c d x-d^2 x^2}}{5 d e^3 \sqrt {c e+d e x}}+\frac {6 F\left (\left .\sin ^{-1}\left (\frac {\sqrt {c e+d e x}}{\sqrt {e}}\right )\right |-1\right )}{5 d e^{7/2}}-\frac {6 \text {Subst}\left (\int \frac {\sqrt {1+\frac {x^2}{e}}}{\sqrt {1-\frac {x^2}{e}}} \, dx,x,\sqrt {c e+d e x}\right )}{5 d e^4} \\ & = -\frac {2 \sqrt {1-c^2-2 c d x-d^2 x^2}}{5 d e (c e+d e x)^{5/2}}-\frac {6 \sqrt {1-c^2-2 c d x-d^2 x^2}}{5 d e^3 \sqrt {c e+d e x}}-\frac {6 E\left (\left .\sin ^{-1}\left (\frac {\sqrt {c e+d e x}}{\sqrt {e}}\right )\right |-1\right )}{5 d e^{7/2}}+\frac {6 F\left (\left .\sin ^{-1}\left (\frac {\sqrt {c e+d e x}}{\sqrt {e}}\right )\right |-1\right )}{5 d e^{7/2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.25 \[ \int \frac {1}{(c e+d e x)^{7/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=-\frac {2 (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},\frac {1}{2},-\frac {1}{4},(c+d x)^2\right )}{5 d (e (c+d x))^{7/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(281\) vs. \(2(131)=262\).
Time = 3.44 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.77
method | result | size |
default | \(-\frac {\left (3 E\left (\frac {\sqrt {-2 d x -2 c +2}}{2}, \sqrt {2}\right ) d^{2} x^{2} \sqrt {-2 d x -2 c +2}\, \sqrt {d x +c}\, \sqrt {2 d x +2 c +2}+6 d^{4} x^{4}+6 E\left (\frac {\sqrt {-2 d x -2 c +2}}{2}, \sqrt {2}\right ) c d x \sqrt {-2 d x -2 c +2}\, \sqrt {d x +c}\, \sqrt {2 d x +2 c +2}+24 c \,d^{3} x^{3}+3 E\left (\frac {\sqrt {-2 d x -2 c +2}}{2}, \sqrt {2}\right ) c^{2} \sqrt {-2 d x -2 c +2}\, \sqrt {d x +c}\, \sqrt {2 d x +2 c +2}+36 c^{2} d^{2} x^{2}+24 c^{3} d x +6 c^{4}-4 d^{2} x^{2}-8 c d x -4 c^{2}-2\right ) \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}\, \sqrt {e \left (d x +c \right )}}{5 e^{4} \left (d x +c \right )^{3} \left (d^{2} x^{2}+2 c d x +c^{2}-1\right ) d}\) | \(282\) |
elliptic | \(\frac {\sqrt {-e \left (d x +c \right ) \left (d^{2} x^{2}+2 c d x +c^{2}-1\right )}\, \left (-\frac {2 \sqrt {-d^{3} e \,x^{3}-3 c \,d^{2} e \,x^{2}-3 c^{2} d e x -e \,c^{3}+d e x +c e}}{5 d^{4} e^{4} \left (x +\frac {c}{d}\right )^{3}}-\frac {6 \left (-d^{3} e \,x^{2}-2 c \,d^{2} e x -d e \,c^{2}+d e \right )}{5 d^{2} e^{4} \sqrt {\left (x +\frac {c}{d}\right ) \left (-d^{3} e \,x^{2}-2 c \,d^{2} e x -d e \,c^{2}+d e \right )}}-\frac {6 c \left (-\frac {c +1}{d}+\frac {c -1}{d}\right ) \sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {c -1}{d}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c +1}{d}}}\, F\left (\sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}, \sqrt {\frac {-\frac {c -1}{d}+\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c}{d}}}\right )}{5 e^{3} \sqrt {-d^{3} e \,x^{3}-3 c \,d^{2} e \,x^{2}-3 c^{2} d e x -e \,c^{3}+d e x +c e}}-\frac {6 d \left (-\frac {c +1}{d}+\frac {c -1}{d}\right ) \sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {c -1}{d}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c +1}{d}}}\, \left (\left (-\frac {c -1}{d}+\frac {c}{d}\right ) E\left (\sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}, \sqrt {\frac {-\frac {c -1}{d}+\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c}{d}}}\right )-\frac {c F\left (\sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}, \sqrt {\frac {-\frac {c -1}{d}+\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c}{d}}}\right )}{d}\right )}{5 e^{3} \sqrt {-d^{3} e \,x^{3}-3 c \,d^{2} e \,x^{2}-3 c^{2} d e x -e \,c^{3}+d e x +c e}}\right )}{\sqrt {e \left (d x +c \right )}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}}\) | \(717\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.06 \[ \int \frac {1}{(c e+d e x)^{7/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=-\frac {2 \, {\left (3 \, {\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}\right )} \sqrt {-d^{3} e} {\rm weierstrassZeta}\left (\frac {4}{d^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right )\right ) + {\left (3 \, d^{3} x^{2} + 6 \, c d^{2} x + {\left (3 \, c^{2} + 1\right )} d\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} \sqrt {d e x + c e}\right )}}{5 \, {\left (d^{5} e^{4} x^{3} + 3 \, c d^{4} e^{4} x^{2} + 3 \, c^{2} d^{3} e^{4} x + c^{3} d^{2} e^{4}\right )}} \]
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\[ \int \frac {1}{(c e+d e x)^{7/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=\int \frac {1}{\left (e \left (c + d x\right )\right )^{\frac {7}{2}} \sqrt {- \left (c + d x - 1\right ) \left (c + d x + 1\right )}}\, dx \]
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\[ \int \frac {1}{(c e+d e x)^{7/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=\int { \frac {1}{\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} {\left (d e x + c e\right )}^{\frac {7}{2}}} \,d x } \]
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\[ \int \frac {1}{(c e+d e x)^{7/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=\int { \frac {1}{\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} {\left (d e x + c e\right )}^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(c e+d e x)^{7/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=\int \frac {1}{{\left (c\,e+d\,e\,x\right )}^{7/2}\,\sqrt {-c^2-2\,c\,d\,x-d^2\,x^2+1}} \,d x \]
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