Integrand size = 37, antiderivative size = 157 \[ \int \frac {(c e+d e x)^{9/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=-\frac {14 e^3 (c e+d e x)^{3/2} \sqrt {1-c^2-2 c d x-d^2 x^2}}{45 d}-\frac {2 e (c e+d e x)^{7/2} \sqrt {1-c^2-2 c d x-d^2 x^2}}{9 d}+\frac {14 e^{9/2} E\left (\left .\arcsin \left (\frac {\sqrt {c e+d e x}}{\sqrt {e}}\right )\right |-1\right )}{15 d}-\frac {14 e^{9/2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c e+d e x}}{\sqrt {e}}\right ),-1\right )}{15 d} \]
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Time = 0.09 (sec) , antiderivative size = 157, 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 = {706, 704, 313, 227, 1213, 435} \[ \int \frac {(c e+d e x)^{9/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=-\frac {14 e^{9/2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c e+d x e}}{\sqrt {e}}\right ),-1\right )}{15 d}+\frac {14 e^{9/2} E\left (\left .\arcsin \left (\frac {\sqrt {c e+d x e}}{\sqrt {e}}\right )\right |-1\right )}{15 d}-\frac {14 e^3 \sqrt {-c^2-2 c d x-d^2 x^2+1} (c e+d e x)^{3/2}}{45 d}-\frac {2 e \sqrt {-c^2-2 c d x-d^2 x^2+1} (c e+d e x)^{7/2}}{9 d} \]
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Rule 227
Rule 313
Rule 435
Rule 704
Rule 706
Rule 1213
Rubi steps \begin{align*} \text {integral}& = -\frac {2 e (c e+d e x)^{7/2} \sqrt {1-c^2-2 c d x-d^2 x^2}}{9 d}+\frac {1}{9} \left (7 e^2\right ) \int \frac {(c e+d e x)^{5/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx \\ & = -\frac {14 e^3 (c e+d e x)^{3/2} \sqrt {1-c^2-2 c d x-d^2 x^2}}{45 d}-\frac {2 e (c e+d e x)^{7/2} \sqrt {1-c^2-2 c d x-d^2 x^2}}{9 d}+\frac {1}{15} \left (7 e^4\right ) \int \frac {\sqrt {c e+d e x}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx \\ & = -\frac {14 e^3 (c e+d e x)^{3/2} \sqrt {1-c^2-2 c d x-d^2 x^2}}{45 d}-\frac {2 e (c e+d e x)^{7/2} \sqrt {1-c^2-2 c d x-d^2 x^2}}{9 d}+\frac {\left (14 e^3\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-\frac {x^4}{e^2}}} \, dx,x,\sqrt {c e+d e x}\right )}{15 d} \\ & = -\frac {14 e^3 (c e+d e x)^{3/2} \sqrt {1-c^2-2 c d x-d^2 x^2}}{45 d}-\frac {2 e (c e+d e x)^{7/2} \sqrt {1-c^2-2 c d x-d^2 x^2}}{9 d}-\frac {\left (14 e^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^4}{e^2}}} \, dx,x,\sqrt {c e+d e x}\right )}{15 d}+\frac {\left (14 e^4\right ) \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 )}{15 d} \\ & = -\frac {14 e^3 (c e+d e x)^{3/2} \sqrt {1-c^2-2 c d x-d^2 x^2}}{45 d}-\frac {2 e (c e+d e x)^{7/2} \sqrt {1-c^2-2 c d x-d^2 x^2}}{9 d}-\frac {14 e^{9/2} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {c e+d e x}}{\sqrt {e}}\right )\right |-1\right )}{15 d}+\frac {\left (14 e^4\right ) \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 )}{15 d} \\ & = -\frac {14 e^3 (c e+d e x)^{3/2} \sqrt {1-c^2-2 c d x-d^2 x^2}}{45 d}-\frac {2 e (c e+d e x)^{7/2} \sqrt {1-c^2-2 c d x-d^2 x^2}}{9 d}+\frac {14 e^{9/2} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {c e+d e x}}{\sqrt {e}}\right )\right |-1\right )}{15 d}-\frac {14 e^{9/2} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {c e+d e x}}{\sqrt {e}}\right )\right |-1\right )}{15 d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.06 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.55 \[ \int \frac {(c e+d e x)^{9/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=-\frac {2 e^3 (e (c+d x))^{3/2} \left (\sqrt {1-c^2-2 c d x-d^2 x^2} \left (7+5 c^2+10 c d x+5 d^2 x^2\right )-7 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},(c+d x)^2\right )\right )}{45 d} \]
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Time = 2.63 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.56
method | result | size |
default | \(\frac {\sqrt {e \left (d x +c \right )}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}\, e^{4} \left (-10 d^{6} x^{6}-60 c \,d^{5} x^{5}-150 c^{2} d^{4} x^{4}-200 c^{3} d^{3} x^{3}-150 c^{4} d^{2} x^{2}-4 d^{4} x^{4}-60 c^{5} d x -16 c \,d^{3} x^{3}-10 c^{6}-24 c^{2} d^{2} x^{2}-16 c^{3} d x +21 \sqrt {-2 d x -2 c +2}\, \sqrt {d x +c}\, \sqrt {2 d x +2 c +2}\, E\left (\frac {\sqrt {-2 d x -2 c +2}}{2}, \sqrt {2}\right )-4 c^{4}+14 d^{2} x^{2}+28 c d x +14 c^{2}\right )}{45 d \left (d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}-d x -c \right )}\) | \(245\) |
risch | \(\frac {2 \left (5 d^{3} x^{3}+15 c \,d^{2} x^{2}+15 c^{2} d x +5 c^{3}+7 d x +7 c \right ) \left (d x +c \right ) \left (d^{2} x^{2}+2 c d x +c^{2}-1\right ) \sqrt {e \left (d x +c \right ) \left (-d^{2} x^{2}-2 c d x -c^{2}+1\right )}\, e^{5}}{45 d \sqrt {-e \left (d x +c \right ) \left (d^{2} x^{2}+2 c d x +c^{2}-1\right )}\, \sqrt {e \left (d x +c \right )}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}}+\frac {\left (\frac {14 c \left (-\frac {c -1}{d}+\frac {c +1}{d}\right ) \sqrt {\frac {x +\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c +1}{d}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {c +1}{d}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}\, F\left (\sqrt {\frac {x +\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c +1}{d}}}, \sqrt {\frac {-\frac {c +1}{d}+\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c}{d}}}\right )}{15 \sqrt {-d^{3} e \,x^{3}-3 c \,d^{2} e \,x^{2}-3 c^{2} d e x -e \,c^{3}+d e x +c e}}+\frac {14 d \left (-\frac {c +1}{d}+\frac {c -1}{d}\right ) \sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {c -1}{d}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c +1}{d}}}\, \left (\left (-\frac {c -1}{d}+\frac {c}{d}\right ) E\left (\sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}, \sqrt {\frac {-\frac {c -1}{d}+\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c}{d}}}\right )-\frac {c F\left (\sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}, \sqrt {\frac {-\frac {c -1}{d}+\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c}{d}}}\right )}{d}\right )}{15 \sqrt {-d^{3} e \,x^{3}-3 c \,d^{2} e \,x^{2}-3 c^{2} d e x -e \,c^{3}+d e x +c e}}\right ) \sqrt {e \left (d x +c \right ) \left (-d^{2} x^{2}-2 c d x -c^{2}+1\right )}\, e^{5}}{\sqrt {e \left (d x +c \right )}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}}\) | \(737\) |
elliptic | \(\text {Expression too large to display}\) | \(1255\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.17 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.82 \[ \int \frac {(c e+d e x)^{9/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=\frac {2 \, {\left (21 \, \sqrt {-d^{3} e} e^{4} {\rm weierstrassZeta}\left (\frac {4}{d^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right )\right ) - {\left (5 \, d^{4} e^{4} x^{3} + 15 \, c d^{3} e^{4} x^{2} + {\left (15 \, c^{2} + 7\right )} d^{2} e^{4} x + {\left (5 \, c^{3} + 7 \, c\right )} d e^{4}\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} \sqrt {d e x + c e}\right )}}{45 \, d^{2}} \]
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\[ \int \frac {(c e+d e x)^{9/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=\int \frac {\left (e \left (c + d x\right )\right )^{\frac {9}{2}}}{\sqrt {- \left (c + d x - 1\right ) \left (c + d x + 1\right )}}\, dx \]
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\[ \int \frac {(c e+d e x)^{9/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{\frac {9}{2}}}{\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}} \,d x } \]
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\[ \int \frac {(c e+d e x)^{9/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{\frac {9}{2}}}{\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}} \,d x } \]
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Timed out. \[ \int \frac {(c e+d e x)^{9/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^{9/2}}{\sqrt {-c^2-2\,c\,d\,x-d^2\,x^2+1}} \,d x \]
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