Optimal. Leaf size=124 \[ -\frac {10 e^3 \sqrt {-c^2-2 c d x-d^2 x^2+1} \sqrt {c e+d e x}}{21 d}-\frac {2 e \sqrt {-c^2-2 c d x-d^2 x^2+1} (c e+d e x)^{5/2}}{7 d}+\frac {10 e^{7/2} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {c e+d x e}}{\sqrt {e}}\right )\right |-1\right )}{21 d} \]
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Rubi [A] time = 0.10, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {692, 689, 221} \[ -\frac {10 e^3 \sqrt {-c^2-2 c d x-d^2 x^2+1} \sqrt {c e+d e x}}{21 d}-\frac {2 e \sqrt {-c^2-2 c d x-d^2 x^2+1} (c e+d e x)^{5/2}}{7 d}+\frac {10 e^{7/2} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {c e+d x e}}{\sqrt {e}}\right )\right |-1\right )}{21 d} \]
Antiderivative was successfully verified.
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Rule 221
Rule 689
Rule 692
Rubi steps
\begin {align*} \int \frac {(c e+d e x)^{7/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx &=-\frac {2 e (c e+d e x)^{5/2} \sqrt {1-c^2-2 c d x-d^2 x^2}}{7 d}+\frac {1}{7} \left (5 e^2\right ) \int \frac {(c e+d e x)^{3/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx\\ &=-\frac {10 e^3 \sqrt {c e+d e x} \sqrt {1-c^2-2 c d x-d^2 x^2}}{21 d}-\frac {2 e (c e+d e x)^{5/2} \sqrt {1-c^2-2 c d x-d^2 x^2}}{7 d}+\frac {1}{21} \left (5 e^4\right ) \int \frac {1}{\sqrt {c e+d e x} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx\\ &=-\frac {10 e^3 \sqrt {c e+d e x} \sqrt {1-c^2-2 c d x-d^2 x^2}}{21 d}-\frac {2 e (c e+d e x)^{5/2} \sqrt {1-c^2-2 c d x-d^2 x^2}}{7 d}+\frac {\left (10 e^3\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^4}{e^2}}} \, dx,x,\sqrt {c e+d e x}\right )}{21 d}\\ &=-\frac {10 e^3 \sqrt {c e+d e x} \sqrt {1-c^2-2 c d x-d^2 x^2}}{21 d}-\frac {2 e (c e+d e x)^{5/2} \sqrt {1-c^2-2 c d x-d^2 x^2}}{7 d}+\frac {10 e^{7/2} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {c e+d e x}}{\sqrt {e}}\right )\right |-1\right )}{21 d}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 86, normalized size = 0.69 \[ -\frac {2 e^3 \sqrt {e (c+d x)} \left (\sqrt {-c^2-2 c d x-d^2 x^2+1} \left (3 c^2+6 c d x+3 d^2 x^2+5\right )-5 \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};(c+d x)^2\right )\right )}{21 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.87, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (d^{3} e^{3} x^{3} + 3 \, c d^{2} e^{3} x^{2} + 3 \, c^{2} d e^{3} x + c^{3} e^{3}\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} \sqrt {d e x + c e}}{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}, x\right ) \]
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 \[ \int \frac {{\left (d e x + c e\right )}^{\frac {7}{2}}}{\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 263, normalized size = 2.12 \[ -\frac {\sqrt {\left (d x +c \right ) e}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}\, \left (6 d^{5} x^{5}+30 c \,d^{4} x^{4}+60 c^{2} d^{3} x^{3}+60 c^{3} d^{2} x^{2}+30 c^{4} d x +4 d^{3} x^{3}+6 c^{5}+12 c \,d^{2} x^{2}+12 c^{2} d x +4 c^{3}-10 d x -10 c +2 \sqrt {-2 d x -2 c +2}\, \sqrt {d x +c}\, \sqrt {2 d x +2 c +2}\, \EllipticF \left (\frac {\sqrt {-2 d x -2 c +2}}{2}, \sqrt {2}\right )+7 \sqrt {-2 d x -2 c +2}\, \sqrt {2 d x +2 c +2}\, \sqrt {-d x -c}\, \EllipticF \left (\frac {\sqrt {2 d x +2 c +2}}{2}, \sqrt {2}\right )\right ) e^{3}}{21 \left (d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}-d x -c \right ) d} \]
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 \[ \int \frac {{\left (d e x + c e\right )}^{\frac {7}{2}}}{\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}\,{d x} \]
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 \[ \int \frac {{\left (c\,e+d\,e\,x\right )}^{7/2}}{\sqrt {-c^2-2\,c\,d\,x-d^2\,x^2+1}} \,d x \]
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 \[ \int \frac {\left (e \left (c + d x\right )\right )^{\frac {7}{2}}}{\sqrt {- \left (c + d x - 1\right ) \left (c + d x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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