Optimal. Leaf size=159 \[ -\frac {6 \sqrt {-c^2-2 c d x-d^2 x^2+1}}{5 d e^3 \sqrt {c e+d e x}}-\frac {2 \sqrt {-c^2-2 c d x-d^2 x^2+1}}{5 d e (c e+d e x)^{5/2}}+\frac {6 F\left (\left .\sin ^{-1}\left (\frac {\sqrt {c e+d x e}}{\sqrt {e}}\right )\right |-1\right )}{5 d e^{7/2}}-\frac {6 E\left (\left .\sin ^{-1}\left (\frac {\sqrt {c e+d x e}}{\sqrt {e}}\right )\right |-1\right )}{5 d e^{7/2}} \]
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Rubi [A] time = 0.12, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {693, 690, 307, 221, 1199, 424} \[ -\frac {6 \sqrt {-c^2-2 c d x-d^2 x^2+1}}{5 d e^3 \sqrt {c e+d e x}}-\frac {2 \sqrt {-c^2-2 c d x-d^2 x^2+1}}{5 d e (c e+d e x)^{5/2}}+\frac {6 F\left (\left .\sin ^{-1}\left (\frac {\sqrt {c e+d x e}}{\sqrt {e}}\right )\right |-1\right )}{5 d e^{7/2}}-\frac {6 E\left (\left .\sin ^{-1}\left (\frac {\sqrt {c e+d x e}}{\sqrt {e}}\right )\right |-1\right )}{5 d e^{7/2}} \]
Antiderivative was successfully verified.
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Rule 221
Rule 307
Rule 424
Rule 690
Rule 693
Rule 1199
Rubi steps
\begin {align*} \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 {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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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*}
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Mathematica [C] time = 0.02, size = 40, normalized size = 0.25 \[ -\frac {2 (c+d x) \, _2F_1\left (-\frac {5}{4},\frac {1}{2};-\frac {1}{4};(c+d x)^2\right )}{5 d (e (c+d x))^{7/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} \sqrt {d e x + c e}}{d^{6} e^{4} x^{6} + 6 \, c d^{5} e^{4} x^{5} + {\left (15 \, c^{2} - 1\right )} d^{4} e^{4} x^{4} + 4 \, {\left (5 \, c^{3} - c\right )} d^{3} e^{4} x^{3} + 3 \, {\left (5 \, c^{4} - 2 \, c^{2}\right )} d^{2} e^{4} x^{2} + 2 \, {\left (3 \, c^{5} - 2 \, c^{3}\right )} d e^{4} x + {\left (c^{6} - c^{4}\right )} e^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} {\left (d e x + c e\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.13, size = 447, normalized size = 2.81 \[ \frac {\left (-6 d^{4} x^{4}-24 c \,d^{3} x^{3}-36 c^{2} d^{2} x^{2}-24 c^{3} d x -10 \sqrt {d x +c}\, \sqrt {2 d x +2 c +2}\, \sqrt {-2 d x -2 c +2}\, d^{2} x^{2} \EllipticE \left (\frac {\sqrt {-2 d x -2 c +2}}{2}, \sqrt {2}\right )+7 \sqrt {2 d x +2 c +2}\, \sqrt {-d x -c}\, \sqrt {-2 d x -2 c +2}\, d^{2} x^{2} \EllipticE \left (\frac {\sqrt {2 d x +2 c +2}}{2}, \sqrt {2}\right )-6 c^{4}-20 \sqrt {d x +c}\, \sqrt {2 d x +2 c +2}\, \sqrt {-2 d x -2 c +2}\, c d x \EllipticE \left (\frac {\sqrt {-2 d x -2 c +2}}{2}, \sqrt {2}\right )+14 \sqrt {2 d x +2 c +2}\, \sqrt {-d x -c}\, \sqrt {-2 d x -2 c +2}\, c d x \EllipticE \left (\frac {\sqrt {2 d x +2 c +2}}{2}, \sqrt {2}\right )+4 d^{2} x^{2}-10 \sqrt {d x +c}\, \sqrt {2 d x +2 c +2}\, \sqrt {-2 d x -2 c +2}\, c^{2} \EllipticE \left (\frac {\sqrt {-2 d x -2 c +2}}{2}, \sqrt {2}\right )+7 \sqrt {2 d x +2 c +2}\, \sqrt {-d x -c}\, \sqrt {-2 d x -2 c +2}\, c^{2} \EllipticE \left (\frac {\sqrt {2 d x +2 c +2}}{2}, \sqrt {2}\right )+8 c d x +4 c^{2}+2\right ) \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}\, \sqrt {\left (d x +c \right ) e}}{5 \left (d x +c \right )^{3} \left (d^{2} x^{2}+2 c d x +c^{2}-1\right ) d \,e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} {\left (d e x + c e\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (c\,e+d\,e\,x\right )}^{7/2}\,\sqrt {-c^2-2\,c\,d\,x-d^2\,x^2+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (e \left (c + d x\right )\right )^{\frac {7}{2}} \sqrt {- \left (c + d x - 1\right ) \left (c + d x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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