Optimal. Leaf size=135 \[ \frac {x \left (a^2 d^4+2 a c d^2+b^2 d^2+c^2\right )+2 b d^2 \left (a+\frac {c}{d^2}\right )}{d^4 \sqrt {1-d^2 x^2}}-\frac {\sin ^{-1}(d x) \left (c \left (4 a+\frac {3 c}{d^2}\right )+2 b^2\right )}{2 d^3}+\frac {2 b c \sqrt {1-d^2 x^2}}{d^4}+\frac {c^2 x \sqrt {1-d^2 x^2}}{2 d^4} \]
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Rubi [A] time = 0.19, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {899, 1814, 1815, 641, 216} \[ \frac {x \left (a^2 d^4+2 a c d^2+b^2 d^2+c^2\right )+2 b d^2 \left (a+\frac {c}{d^2}\right )}{d^4 \sqrt {1-d^2 x^2}}-\frac {\sin ^{-1}(d x) \left (c \left (4 a+\frac {3 c}{d^2}\right )+2 b^2\right )}{2 d^3}+\frac {2 b c \sqrt {1-d^2 x^2}}{d^4}+\frac {c^2 x \sqrt {1-d^2 x^2}}{2 d^4} \]
Antiderivative was successfully verified.
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Rule 216
Rule 641
Rule 899
Rule 1814
Rule 1815
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^2}{(1-d x)^{3/2} (1+d x)^{3/2}} \, dx &=\int \frac {\left (a+b x+c x^2\right )^2}{\left (1-d^2 x^2\right )^{3/2}} \, dx\\ &=\frac {2 b \left (a+\frac {c}{d^2}\right ) d^2+\left (c^2+b^2 d^2+2 a c d^2+a^2 d^4\right ) x}{d^4 \sqrt {1-d^2 x^2}}-\int \frac {\frac {c^2+b^2 d^2+2 a c d^2}{d^4}+\frac {2 b c x}{d^2}+\frac {c^2 x^2}{d^2}}{\sqrt {1-d^2 x^2}} \, dx\\ &=\frac {2 b \left (a+\frac {c}{d^2}\right ) d^2+\left (c^2+b^2 d^2+2 a c d^2+a^2 d^4\right ) x}{d^4 \sqrt {1-d^2 x^2}}+\frac {c^2 x \sqrt {1-d^2 x^2}}{2 d^4}+\frac {\int \frac {-2 b^2-c \left (4 a+\frac {3 c}{d^2}\right )-4 b c x}{\sqrt {1-d^2 x^2}} \, dx}{2 d^2}\\ &=\frac {2 b \left (a+\frac {c}{d^2}\right ) d^2+\left (c^2+b^2 d^2+2 a c d^2+a^2 d^4\right ) x}{d^4 \sqrt {1-d^2 x^2}}+\frac {2 b c \sqrt {1-d^2 x^2}}{d^4}+\frac {c^2 x \sqrt {1-d^2 x^2}}{2 d^4}-\frac {\left (2 b^2+c \left (4 a+\frac {3 c}{d^2}\right )\right ) \int \frac {1}{\sqrt {1-d^2 x^2}} \, dx}{2 d^2}\\ &=\frac {2 b \left (a+\frac {c}{d^2}\right ) d^2+\left (c^2+b^2 d^2+2 a c d^2+a^2 d^4\right ) x}{d^4 \sqrt {1-d^2 x^2}}+\frac {2 b c \sqrt {1-d^2 x^2}}{d^4}+\frac {c^2 x \sqrt {1-d^2 x^2}}{2 d^4}-\frac {\left (2 b^2+c \left (4 a+\frac {3 c}{d^2}\right )\right ) \sin ^{-1}(d x)}{2 d^3}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 127, normalized size = 0.94 \[ \frac {d x \left (2 a^2 d^4+4 a c d^2+c^2 \left (3-d^2 x^2\right )\right )-\sqrt {1-d^2 x^2} \sin ^{-1}(d x) \left (4 a c d^2+2 b^2 d^2+3 c^2\right )+4 b d \left (a d^2+c \left (2-d^2 x^2\right )\right )+2 b^2 d^3 x}{2 d^5 \sqrt {1-d^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 204, normalized size = 1.51 \[ -\frac {4 \, a b d^{3} + 8 \, b c d - 4 \, {\left (a b d^{5} + 2 \, b c d^{3}\right )} x^{2} - {\left (c^{2} d^{3} x^{3} + 4 \, b c d^{3} x^{2} - 4 \, a b d^{3} - 8 \, b c d - {\left (2 \, a^{2} d^{5} + 2 \, {\left (b^{2} + 2 \, a c\right )} d^{3} + 3 \, c^{2} d\right )} x\right )} \sqrt {d x + 1} \sqrt {-d x + 1} + 2 \, {\left (2 \, {\left (b^{2} + 2 \, a c\right )} d^{2} - {\left (2 \, {\left (b^{2} + 2 \, a c\right )} d^{4} + 3 \, c^{2} d^{2}\right )} x^{2} + 3 \, c^{2}\right )} \arctan \left (\frac {\sqrt {d x + 1} \sqrt {-d x + 1} - 1}{d x}\right )}{2 \, {\left (d^{7} x^{2} - d^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.39, size = 387, normalized size = 2.87 \[ \frac {\sqrt {d x + 1} \sqrt {-d x + 1} {\left ({\left (d x + 1\right )} {\left (\frac {{\left (d x + 1\right )} c^{2}}{d^{5}} + \frac {4 \, b c d^{16} - 3 \, c^{2} d^{15}}{d^{20}}\right )} - \frac {a^{2} d^{19} + 2 \, a b d^{18} + b^{2} d^{17} + 2 \, a c d^{17} + 10 \, b c d^{16} - c^{2} d^{15}}{d^{20}}\right )}}{2 \, {\left (d x - 1\right )}} - \frac {{\left (2 \, b^{2} d^{2} + 4 \, a c d^{2} + 3 \, c^{2}\right )} \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {d x + 1}\right )}{d^{5}} + \frac {\frac {a^{2} d^{4} {\left (\sqrt {2} - \sqrt {-d x + 1}\right )}}{\sqrt {d x + 1}} - \frac {2 \, a b d^{3} {\left (\sqrt {2} - \sqrt {-d x + 1}\right )}}{\sqrt {d x + 1}} + \frac {b^{2} d^{2} {\left (\sqrt {2} - \sqrt {-d x + 1}\right )}}{\sqrt {d x + 1}} + \frac {2 \, a c d^{2} {\left (\sqrt {2} - \sqrt {-d x + 1}\right )}}{\sqrt {d x + 1}} - \frac {2 \, b c d {\left (\sqrt {2} - \sqrt {-d x + 1}\right )}}{\sqrt {d x + 1}} + \frac {c^{2} {\left (\sqrt {2} - \sqrt {-d x + 1}\right )}}{\sqrt {d x + 1}}}{4 \, d^{5}} - \frac {{\left (a^{2} d^{4} - 2 \, a b d^{3} + b^{2} d^{2} + 2 \, a c d^{2} - 2 \, b c d + c^{2}\right )} \sqrt {d x + 1}}{4 \, d^{5} {\left (\sqrt {2} - \sqrt {-d x + 1}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 380, normalized size = 2.81 \[ \frac {\sqrt {-d x +1}\, \left (-2 \sqrt {-d^{2} x^{2}+1}\, a^{2} d^{5} x \,\mathrm {csgn}\relax (d )-4 a c \,d^{4} x^{2} \arctan \left (\frac {d x \,\mathrm {csgn}\relax (d )}{\sqrt {-d^{2} x^{2}+1}}\right )-2 b^{2} d^{4} x^{2} \arctan \left (\frac {d x \,\mathrm {csgn}\relax (d )}{\sqrt {-d^{2} x^{2}+1}}\right )+\sqrt {-d^{2} x^{2}+1}\, c^{2} d^{3} x^{3} \mathrm {csgn}\relax (d )+4 \sqrt {-d^{2} x^{2}+1}\, b c \,d^{3} x^{2} \mathrm {csgn}\relax (d )-4 \sqrt {-d^{2} x^{2}+1}\, a c \,d^{3} x \,\mathrm {csgn}\relax (d )-2 \sqrt {-d^{2} x^{2}+1}\, b^{2} d^{3} x \,\mathrm {csgn}\relax (d )-3 c^{2} d^{2} x^{2} \arctan \left (\frac {d x \,\mathrm {csgn}\relax (d )}{\sqrt {-d^{2} x^{2}+1}}\right )-4 \sqrt {-d^{2} x^{2}+1}\, a b \,d^{3} \mathrm {csgn}\relax (d )+4 a c \,d^{2} \arctan \left (\frac {d x \,\mathrm {csgn}\relax (d )}{\sqrt {-d^{2} x^{2}+1}}\right )+2 b^{2} d^{2} \arctan \left (\frac {d x \,\mathrm {csgn}\relax (d )}{\sqrt {-d^{2} x^{2}+1}}\right )-3 \sqrt {-d^{2} x^{2}+1}\, c^{2} d x \,\mathrm {csgn}\relax (d )-8 \sqrt {-d^{2} x^{2}+1}\, b c d \,\mathrm {csgn}\relax (d )+3 c^{2} \arctan \left (\frac {d x \,\mathrm {csgn}\relax (d )}{\sqrt {-d^{2} x^{2}+1}}\right )\right ) \mathrm {csgn}\relax (d )}{2 \left (d x -1\right ) \sqrt {-d^{2} x^{2}+1}\, \sqrt {d x +1}\, d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 176, normalized size = 1.30 \[ \frac {a^{2} x}{\sqrt {-d^{2} x^{2} + 1}} - \frac {c^{2} x^{3}}{2 \, \sqrt {-d^{2} x^{2} + 1} d^{2}} - \frac {2 \, b c x^{2}}{\sqrt {-d^{2} x^{2} + 1} d^{2}} + \frac {2 \, a b}{\sqrt {-d^{2} x^{2} + 1} d^{2}} + \frac {{\left (b^{2} + 2 \, a c\right )} x}{\sqrt {-d^{2} x^{2} + 1} d^{2}} - \frac {{\left (b^{2} + 2 \, a c\right )} \arcsin \left (d x\right )}{d^{3}} + \frac {3 \, c^{2} x}{2 \, \sqrt {-d^{2} x^{2} + 1} d^{4}} - \frac {3 \, c^{2} \arcsin \left (d x\right )}{2 \, d^{5}} + \frac {4 \, b c}{\sqrt {-d^{2} x^{2} + 1} d^{4}} \]
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 \[ \int \frac {{\left (c\,x^2+b\,x+a\right )}^2}{{\left (1-d\,x\right )}^{3/2}\,{\left (d\,x+1\right )}^{3/2}} \,d x \]
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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