Optimal. Leaf size=79 \[ -\frac {\sqrt {1-d^2 x^2} \left (2 \left (3 a d^2+2 c\right )+3 b d^2 x\right )}{6 d^4}+\frac {b \sin ^{-1}(d x)}{2 d^3}-\frac {c x^2 \sqrt {1-d^2 x^2}}{3 d^2} \]
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Rubi [A] time = 0.14, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {1609, 1809, 780, 216} \[ -\frac {\sqrt {1-d^2 x^2} \left (2 \left (3 a d^2+2 c\right )+3 b d^2 x\right )}{6 d^4}+\frac {b \sin ^{-1}(d x)}{2 d^3}-\frac {c x^2 \sqrt {1-d^2 x^2}}{3 d^2} \]
Antiderivative was successfully verified.
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Rule 216
Rule 780
Rule 1609
Rule 1809
Rubi steps
\begin {align*} \int \frac {x \left (a+b x+c x^2\right )}{\sqrt {1-d x} \sqrt {1+d x}} \, dx &=\int \frac {x \left (a+b x+c x^2\right )}{\sqrt {1-d^2 x^2}} \, dx\\ &=-\frac {c x^2 \sqrt {1-d^2 x^2}}{3 d^2}-\frac {\int \frac {x \left (-2 c-3 a d^2-3 b d^2 x\right )}{\sqrt {1-d^2 x^2}} \, dx}{3 d^2}\\ &=-\frac {c x^2 \sqrt {1-d^2 x^2}}{3 d^2}-\frac {\left (2 \left (2 c+3 a d^2\right )+3 b d^2 x\right ) \sqrt {1-d^2 x^2}}{6 d^4}+\frac {b \int \frac {1}{\sqrt {1-d^2 x^2}} \, dx}{2 d^2}\\ &=-\frac {c x^2 \sqrt {1-d^2 x^2}}{3 d^2}-\frac {\left (2 \left (2 c+3 a d^2\right )+3 b d^2 x\right ) \sqrt {1-d^2 x^2}}{6 d^4}+\frac {b \sin ^{-1}(d x)}{2 d^3}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 57, normalized size = 0.72 \[ \frac {3 b d \sin ^{-1}(d x)-\sqrt {1-d^2 x^2} \left (3 d^2 (2 a+b x)+2 c \left (d^2 x^2+2\right )\right )}{6 d^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 78, normalized size = 0.99 \[ -\frac {6 \, b d \arctan \left (\frac {\sqrt {d x + 1} \sqrt {-d x + 1} - 1}{d x}\right ) + {\left (2 \, c d^{2} x^{2} + 3 \, b d^{2} x + 6 \, a d^{2} + 4 \, c\right )} \sqrt {d x + 1} \sqrt {-d x + 1}}{6 \, d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.34, size = 101, normalized size = 1.28 \[ -\frac {\sqrt {d x + 1} \sqrt {-d x + 1} {\left ({\left (d x + 1\right )} {\left (\frac {2 \, {\left (d x + 1\right )} c}{d^{3}} + \frac {3 \, b d^{10} - 4 \, c d^{9}}{d^{12}}\right )} + \frac {3 \, {\left (2 \, a d^{11} - b d^{10} + 2 \, c d^{9}\right )}}{d^{12}}\right )} - \frac {6 \, b \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {d x + 1}\right )}{d^{2}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.04, size = 139, normalized size = 1.76 \[ -\frac {\sqrt {-d x +1}\, \sqrt {d x +1}\, \left (2 \sqrt {-d^{2} x^{2}+1}\, c \,d^{2} x^{2} \mathrm {csgn}\relax (d )+3 \sqrt {-d^{2} x^{2}+1}\, b \,d^{2} x \,\mathrm {csgn}\relax (d )+6 \sqrt {-d^{2} x^{2}+1}\, a \,d^{2} \mathrm {csgn}\relax (d )-3 b d \arctan \left (\frac {d x \,\mathrm {csgn}\relax (d )}{\sqrt {-d^{2} x^{2}+1}}\right )+4 \sqrt {-d^{2} x^{2}+1}\, c \,\mathrm {csgn}\relax (d )\right ) \mathrm {csgn}\relax (d )}{6 \sqrt {-d^{2} x^{2}+1}\, d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 87, normalized size = 1.10 \[ -\frac {\sqrt {-d^{2} x^{2} + 1} c x^{2}}{3 \, d^{2}} - \frac {\sqrt {-d^{2} x^{2} + 1} b x}{2 \, d^{2}} - \frac {\sqrt {-d^{2} x^{2} + 1} a}{d^{2}} + \frac {b \arcsin \left (d x\right )}{2 \, d^{3}} - \frac {2 \, \sqrt {-d^{2} x^{2} + 1} c}{3 \, d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.44, size = 244, normalized size = 3.09 \[ -\frac {\sqrt {1-d\,x}\,\left (\frac {a}{d^2}+\frac {a\,x}{d}\right )}{\sqrt {d\,x+1}}-\frac {2\,b\,\mathrm {atan}\left (\frac {\sqrt {1-d\,x}-1}{\sqrt {d\,x+1}-1}\right )}{d^3}-\frac {\frac {14\,b\,{\left (\sqrt {1-d\,x}-1\right )}^3}{{\left (\sqrt {d\,x+1}-1\right )}^3}-\frac {14\,b\,{\left (\sqrt {1-d\,x}-1\right )}^5}{{\left (\sqrt {d\,x+1}-1\right )}^5}+\frac {2\,b\,{\left (\sqrt {1-d\,x}-1\right )}^7}{{\left (\sqrt {d\,x+1}-1\right )}^7}-\frac {2\,b\,\left (\sqrt {1-d\,x}-1\right )}{\sqrt {d\,x+1}-1}}{d^3\,{\left (\frac {{\left (\sqrt {1-d\,x}-1\right )}^2}{{\left (\sqrt {d\,x+1}-1\right )}^2}+1\right )}^4}-\frac {\sqrt {1-d\,x}\,\left (\frac {2\,c}{3\,d^4}+\frac {c\,x^3}{3\,d}+\frac {c\,x^2}{3\,d^2}+\frac {2\,c\,x}{3\,d^3}\right )}{\sqrt {d\,x+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 82.40, size = 313, normalized size = 3.96 \[ - \frac {i a {G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{4} & 0, 0, \frac {1}{2}, 1 \\- \frac {1}{2}, - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, 0 & \end {matrix} \middle | {\frac {1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d^{2}} - \frac {a {G_{6, 6}^{2, 6}\left (\begin {matrix} -1, - \frac {3}{4}, - \frac {1}{2}, - \frac {1}{4}, 0, 1 & \\- \frac {3}{4}, - \frac {1}{4} & -1, - \frac {1}{2}, - \frac {1}{2}, 0 \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d^{2}} - \frac {i b {G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{4} & - \frac {1}{2}, - \frac {1}{2}, 0, 1 \\-1, - \frac {3}{4}, - \frac {1}{2}, - \frac {1}{4}, 0, 0 & \end {matrix} \middle | {\frac {1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d^{3}} + \frac {b {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {3}{2}, - \frac {5}{4}, -1, - \frac {3}{4}, - \frac {1}{2}, 1 & \\- \frac {5}{4}, - \frac {3}{4} & - \frac {3}{2}, -1, -1, 0 \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d^{3}} - \frac {i c {G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {5}{4}, - \frac {3}{4} & -1, -1, - \frac {1}{2}, 1 \\- \frac {3}{2}, - \frac {5}{4}, -1, - \frac {3}{4}, - \frac {1}{2}, 0 & \end {matrix} \middle | {\frac {1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d^{4}} - \frac {c {G_{6, 6}^{2, 6}\left (\begin {matrix} -2, - \frac {7}{4}, - \frac {3}{2}, - \frac {5}{4}, -1, 1 & \\- \frac {7}{4}, - \frac {5}{4} & -2, - \frac {3}{2}, - \frac {3}{2}, 0 \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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