Optimal. Leaf size=40 \[ \frac {b+\left (c+a d^2\right ) x}{d^2 \sqrt {1-d^2 x^2}}-\frac {c \sin ^{-1}(d x)}{d^3} \]
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Rubi [A]
time = 0.03, 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 = {913, 1828, 12,
222} \begin {gather*} \frac {x \left (a d^2+c\right )+b}{d^2 \sqrt {1-d^2 x^2}}-\frac {c \text {ArcSin}(d x)}{d^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 222
Rule 913
Rule 1828
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 = 0.37, size = 70, normalized size = 1.75 \begin {gather*} \frac {b+\left (c+a d^2\right ) x}{d^2 \sqrt {1-d^2 x^2}}-\frac {c \log \left (-\sqrt {-d^2} x+\sqrt {1-d^2 x^2}\right )}{\left (-d^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.12, size = 151, normalized size = 3.78
method | result | size |
default | \(\frac {\left (-\sqrt {-d^{2} x^{2}+1}\, \mathrm {csgn}\left (d \right ) d^{3} a x -\arctan \left (\frac {\mathrm {csgn}\left (d \right ) d x}{\sqrt {-\left (d x -1\right ) \left (d x +1\right )}}\right ) c \,d^{2} x^{2}-\mathrm {csgn}\left (d \right ) d \sqrt {-d^{2} x^{2}+1}\, c x -\mathrm {csgn}\left (d \right ) d \sqrt {-d^{2} x^{2}+1}\, b +\arctan \left (\frac {\mathrm {csgn}\left (d \right ) d x}{\sqrt {-\left (d x -1\right ) \left (d x +1\right )}}\right ) c \right ) \sqrt {-d x +1}\, \mathrm {csgn}\left (d \right )}{\left (d x -1\right ) \sqrt {-d^{2} x^{2}+1}\, d^{3} \sqrt {d x +1}}\) | \(151\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 61, normalized size = 1.52 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 101 vs.
\(2 (38) = 76\).
time = 5.95, size = 101, normalized size = 2.52 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 186 vs.
\(2 (38) = 76\).
time = 2.73, size = 186, normalized size = 4.65 \begin {gather*} -\frac {\frac {8 \, c \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {d x + 1}\right )}{d^{2}} - \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}}}{d^{2}} + \frac {{\left (a d^{2} - b d + c\right )} \sqrt {d x + 1}}{d^{2} {\left (\sqrt {2} - \sqrt {-d x + 1}\right )}} + \frac {2 \, {\left (a d^{4} + b d^{3} + c d^{2}\right )} \sqrt {d x + 1} \sqrt {-d x + 1}}{{\left (d x - 1\right )} d^{4}}}{4 \, d} \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 = -0.02 \begin {gather*} \int \frac {c\,x^2+b\,x+a}{{\left (1-d\,x\right )}^{3/2}\,{\left (d\,x+1\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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