Optimal. Leaf size=63 \[ \frac {\left (2 a d^2+c\right ) \sin ^{-1}(d x)}{2 d^3}-\frac {b \sqrt {1-d^2 x^2}}{d^2}-\frac {c x \sqrt {1-d^2 x^2}}{2 d^2} \]
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Rubi [A] time = 0.06, antiderivative size = 63, 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, 1815, 641, 216} \begin {gather*} \frac {\left (2 a d^2+c\right ) \sin ^{-1}(d x)}{2 d^3}-\frac {b \sqrt {1-d^2 x^2}}{d^2}-\frac {c x \sqrt {1-d^2 x^2}}{2 d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 216
Rule 641
Rule 899
Rule 1815
Rubi steps
\begin {align*} \int \frac {a+b x+c x^2}{\sqrt {1-d x} \sqrt {1+d x}} \, dx &=\int \frac {a+b x+c x^2}{\sqrt {1-d^2 x^2}} \, dx\\ &=-\frac {c x \sqrt {1-d^2 x^2}}{2 d^2}-\frac {\int \frac {-c-2 a d^2-2 b d^2 x}{\sqrt {1-d^2 x^2}} \, dx}{2 d^2}\\ &=-\frac {b \sqrt {1-d^2 x^2}}{d^2}-\frac {c x \sqrt {1-d^2 x^2}}{2 d^2}-\frac {\left (-c-2 a d^2\right ) \int \frac {1}{\sqrt {1-d^2 x^2}} \, dx}{2 d^2}\\ &=-\frac {b \sqrt {1-d^2 x^2}}{d^2}-\frac {c x \sqrt {1-d^2 x^2}}{2 d^2}+\frac {\left (c+2 a d^2\right ) \sin ^{-1}(d x)}{2 d^3}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 45, normalized size = 0.71 \begin {gather*} \frac {\left (2 a d^2+c\right ) \sin ^{-1}(d x)-d \sqrt {1-d^2 x^2} (2 b+c x)}{2 d^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.00, size = 117, normalized size = 1.86 \begin {gather*} \frac {\left (-2 a d^2-c\right ) \tan ^{-1}\left (\frac {\sqrt {1-d x}}{\sqrt {d x+1}}\right )}{d^3}-\frac {\sqrt {1-d x} \left (\frac {2 b d (1-d x)}{d x+1}+2 b d-\frac {c (1-d x)}{d x+1}+c\right )}{d^3 \sqrt {d x+1} \left (\frac {1-d x}{d x+1}+1\right )^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 67, normalized size = 1.06 \begin {gather*} -\frac {{\left (c d x + 2 \, b d\right )} \sqrt {d x + 1} \sqrt {-d x + 1} + 2 \, {\left (2 \, a d^{2} + c\right )} \arctan \left (\frac {\sqrt {d x + 1} \sqrt {-d x + 1} - 1}{d x}\right )}{2 \, d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 76, normalized size = 1.21 \begin {gather*} -\frac {\sqrt {d x + 1} \sqrt {-d x + 1} {\left (\frac {{\left (d x + 1\right )} c}{d^{2}} + \frac {2 \, b d^{5} - c d^{4}}{d^{6}}\right )} - \frac {2 \, {\left (2 \, a d^{2} + c\right )} \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {d x + 1}\right )}{d^{2}}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 117, normalized size = 1.86 \begin {gather*} -\frac {\sqrt {-d x +1}\, \sqrt {d x +1}\, \left (-2 a \,d^{2} \arctan \left (\frac {d x \,\mathrm {csgn}\relax (d )}{\sqrt {-d^{2} x^{2}+1}}\right )+\sqrt {-d^{2} x^{2}+1}\, c d x \,\mathrm {csgn}\relax (d )+2 \sqrt {-d^{2} x^{2}+1}\, b d \,\mathrm {csgn}\relax (d )-c \arctan \left (\frac {d x \,\mathrm {csgn}\relax (d )}{\sqrt {-d^{2} x^{2}+1}}\right )\right ) \mathrm {csgn}\relax (d )}{2 \sqrt {-d^{2} x^{2}+1}\, d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 57, normalized size = 0.90 \begin {gather*} \frac {a \arcsin \left (d x\right )}{d} - \frac {\sqrt {-d^{2} x^{2} + 1} c x}{2 \, d^{2}} - \frac {\sqrt {-d^{2} x^{2} + 1} b}{d^{2}} + \frac {c \arcsin \left (d x\right )}{2 \, d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.76, size = 232, normalized size = 3.68 \begin {gather*} -\frac {\sqrt {1-d\,x}\,\left (\frac {b}{d^2}+\frac {b\,x}{d}\right )}{\sqrt {d\,x+1}}-\frac {4\,a\,\mathrm {atan}\left (\frac {d\,\left (\sqrt {1-d\,x}-1\right )}{\left (\sqrt {d\,x+1}-1\right )\,\sqrt {d^2}}\right )}{\sqrt {d^2}}-\frac {2\,c\,\mathrm {atan}\left (\frac {\sqrt {1-d\,x}-1}{\sqrt {d\,x+1}-1}\right )}{d^3}-\frac {\frac {14\,c\,{\left (\sqrt {1-d\,x}-1\right )}^3}{{\left (\sqrt {d\,x+1}-1\right )}^3}-\frac {14\,c\,{\left (\sqrt {1-d\,x}-1\right )}^5}{{\left (\sqrt {d\,x+1}-1\right )}^5}+\frac {2\,c\,{\left (\sqrt {1-d\,x}-1\right )}^7}{{\left (\sqrt {d\,x+1}-1\right )}^7}-\frac {2\,c\,\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 49.71, size = 282, normalized size = 4.48 \begin {gather*} - \frac {i a {G_{6, 6}^{6, 2}\left (\begin {matrix} \frac {1}{4}, \frac {3}{4} & \frac {1}{2}, \frac {1}{2}, 1, 1 \\0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, 0 & \end {matrix} \middle | {\frac {1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d} + \frac {a {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, 1 & \\- \frac {1}{4}, \frac {1}{4} & - \frac {1}{2}, 0, 0, 0 \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d} - \frac {i b {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 {b {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 c {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 {c {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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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