Optimal. Leaf size=157 \[ -\frac {14 e^3 (c e+d e x)^{3/2} \sqrt {1-c^2-2 c d x-d^2 x^2}}{45 d}-\frac {2 e (c e+d e x)^{7/2} \sqrt {1-c^2-2 c d x-d^2 x^2}}{9 d}+\frac {14 e^{9/2} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {c e+d e x}}{\sqrt {e}}\right )\right |-1\right )}{15 d}-\frac {14 e^{9/2} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {c e+d e x}}{\sqrt {e}}\right )\right |-1\right )}{15 d} \]
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Rubi [A]
time = 0.09, antiderivative size = 157, 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 = {706, 704, 313,
227, 1213, 435} \begin {gather*} -\frac {14 e^{9/2} F\left (\left .\text {ArcSin}\left (\frac {\sqrt {c e+d x e}}{\sqrt {e}}\right )\right |-1\right )}{15 d}+\frac {14 e^{9/2} E\left (\left .\text {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} \end {gather*}
Antiderivative was successfully verified.
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Rule 227
Rule 313
Rule 435
Rule 704
Rule 706
Rule 1213
Rubi steps
\begin {align*} \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 (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*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.06, size = 86, normalized size = 0.55 \begin {gather*} -\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 \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};(c+d x)^2\right )\right )}{45 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.93, size = 245, normalized size = 1.56 Too large to display
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.71, size = 115, normalized size = 0.73 \begin {gather*} -\frac {2 \, {\left ({\left (5 \, d^{4} x^{3} + 15 \, c d^{3} x^{2} + {\left (15 \, c^{2} + 7\right )} d^{2} x + {\left (5 \, c^{3} + 7 \, c\right )} d\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} \sqrt {d x + c} e^{\frac {9}{2}} - 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 )\right )}}{45 \, d^{2}} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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