Optimal. Leaf size=40 \[ \frac {x \left (a d^2+c\right )+b}{d^2 \sqrt {1-d^2 x^2}}-\frac {c \sin ^{-1}(d x)}{d^3} \]
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Rubi [A] time = 0.05, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {899, 1814, 12, 216} \[ \frac {x \left (a d^2+c\right )+b}{d^2 \sqrt {1-d^2 x^2}}-\frac {c \sin ^{-1}(d x)}{d^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 216
Rule 899
Rule 1814
Rubi steps
\begin {align*} \int \frac {a+b x+c x^2}{(1-d x)^{3/2} (1+d x)^{3/2}} \, dx &=\int \frac {a+b x+c x^2}{\left (1-d^2 x^2\right )^{3/2}} \, dx\\ &=\frac {b+\left (c+a d^2\right ) x}{d^2 \sqrt {1-d^2 x^2}}-\int \frac {c}{d^2 \sqrt {1-d^2 x^2}} \, dx\\ &=\frac {b+\left (c+a d^2\right ) x}{d^2 \sqrt {1-d^2 x^2}}-\frac {c \int \frac {1}{\sqrt {1-d^2 x^2}} \, dx}{d^2}\\ &=\frac {b+\left (c+a d^2\right ) x}{d^2 \sqrt {1-d^2 x^2}}-\frac {c \sin ^{-1}(d x)}{d^3}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 39, normalized size = 0.98 \[ \frac {\frac {d \left (x \left (a d^2+c\right )+b\right )}{\sqrt {1-d^2 x^2}}-c \sin ^{-1}(d x)}{d^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.97, size = 101, normalized size = 2.52 \[ \frac {b d^{3} x^{2} - {\left (b d + {\left (a d^{3} + c d\right )} x\right )} \sqrt {d x + 1} \sqrt {-d x + 1} - b d + 2 \, {\left (c d^{2} x^{2} - c\right )} \arctan \left (\frac {\sqrt {d x + 1} \sqrt {-d x + 1} - 1}{d x}\right )}{d^{5} x^{2} - d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.30, size = 182, normalized size = 4.55 \[ -\frac {2 \, c \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {d x + 1}\right )}{d^{3}} + \frac {\frac {a d^{2} {\left (\sqrt {2} - \sqrt {-d x + 1}\right )}}{\sqrt {d x + 1}} - \frac {b d {\left (\sqrt {2} - \sqrt {-d x + 1}\right )}}{\sqrt {d x + 1}} + \frac {c {\left (\sqrt {2} - \sqrt {-d x + 1}\right )}}{\sqrt {d x + 1}}}{4 \, d^{3}} - \frac {{\left (a d^{2} - b d + c\right )} \sqrt {d x + 1}}{4 \, d^{3} {\left (\sqrt {2} - \sqrt {-d x + 1}\right )}} - \frac {{\left (a d^{5} + b d^{4} + c d^{3}\right )} \sqrt {d x + 1} \sqrt {-d x + 1}}{2 \, {\left (d x - 1\right )} d^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 151, normalized size = 3.78 \[ \frac {\left (-\sqrt {-d^{2} x^{2}+1}\, a \,d^{3} x \,\mathrm {csgn}\relax (d )-c \,d^{2} x^{2} \arctan \left (\frac {d x \,\mathrm {csgn}\relax (d )}{\sqrt {-\left (d x -1\right ) \left (d x +1\right )}}\right )-\sqrt {-d^{2} x^{2}+1}\, c d x \,\mathrm {csgn}\relax (d )-\sqrt {-d^{2} x^{2}+1}\, b d \,\mathrm {csgn}\relax (d )+c \arctan \left (\frac {d x \,\mathrm {csgn}\relax (d )}{\sqrt {-\left (d x -1\right ) \left (d x +1\right )}}\right )\right ) \sqrt {-d x +1}\, \mathrm {csgn}\relax (d )}{\left (d x -1\right ) \sqrt {-d^{2} x^{2}+1}\, \sqrt {d x +1}\, d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 61, normalized size = 1.52 \[ \frac {a x}{\sqrt {-d^{2} x^{2} + 1}} + \frac {c x}{\sqrt {-d^{2} x^{2} + 1} d^{2}} - \frac {c \arcsin \left (d x\right )}{d^{3}} + \frac {b}{\sqrt {-d^{2} x^{2} + 1} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {c\,x^2+b\,x+a}{{\left (1-d\,x\right )}^{3/2}\,{\left (d\,x+1\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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