Optimal. Leaf size=443 \[ \frac {c \left (-b d^2 \left (\sqrt {b^2-4 a c}+b\right )+2 a c d^2+2 c^2\right ) \tanh ^{-1}\left (\frac {d^2 x \left (b-\sqrt {b^2-4 a c}\right )+2 c}{\sqrt {2} \sqrt {1-d^2 x^2} \sqrt {-b d^2 \left (b-\sqrt {b^2-4 a c}\right )+2 a c d^2+2 c^2}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {-b d^2 \left (b-\sqrt {b^2-4 a c}\right )+2 a c d^2+2 c^2} \left (b^2 d^2-\left (a d^2+c\right )^2\right )}-\frac {c \left (-b d^2 \left (b-\sqrt {b^2-4 a c}\right )+2 a c d^2+2 c^2\right ) \tanh ^{-1}\left (\frac {d^2 x \left (\sqrt {b^2-4 a c}+b\right )+2 c}{\sqrt {2} \sqrt {1-d^2 x^2} \sqrt {-b d^2 \left (\sqrt {b^2-4 a c}+b\right )+2 a c d^2+2 c^2}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {-b d^2 \left (\sqrt {b^2-4 a c}+b\right )+2 a c d^2+2 c^2} \left (b^2 d^2-\left (a d^2+c\right )^2\right )}+\frac {d^2 \left (b-x \left (a d^2+c\right )\right )}{\sqrt {1-d^2 x^2} \left (b^2 d^2-\left (a d^2+c\right )^2\right )} \]
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Rubi [A] time = 1.44, antiderivative size = 443, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {899, 976, 1034, 725, 206} \[ \frac {c \left (-b d^2 \left (\sqrt {b^2-4 a c}+b\right )+2 a c d^2+2 c^2\right ) \tanh ^{-1}\left (\frac {d^2 x \left (b-\sqrt {b^2-4 a c}\right )+2 c}{\sqrt {2} \sqrt {1-d^2 x^2} \sqrt {-b d^2 \left (b-\sqrt {b^2-4 a c}\right )+2 a c d^2+2 c^2}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {-b d^2 \left (b-\sqrt {b^2-4 a c}\right )+2 a c d^2+2 c^2} \left (b^2 d^2-\left (a d^2+c\right )^2\right )}-\frac {c \left (-b d^2 \left (b-\sqrt {b^2-4 a c}\right )+2 a c d^2+2 c^2\right ) \tanh ^{-1}\left (\frac {d^2 x \left (\sqrt {b^2-4 a c}+b\right )+2 c}{\sqrt {2} \sqrt {1-d^2 x^2} \sqrt {-b d^2 \left (\sqrt {b^2-4 a c}+b\right )+2 a c d^2+2 c^2}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {-b d^2 \left (\sqrt {b^2-4 a c}+b\right )+2 a c d^2+2 c^2} \left (b^2 d^2-\left (a d^2+c\right )^2\right )}+\frac {d^2 \left (b-x \left (a d^2+c\right )\right )}{\sqrt {1-d^2 x^2} \left (b^2 d^2-\left (a d^2+c\right )^2\right )} \]
Antiderivative was successfully verified.
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Rule 206
Rule 725
Rule 899
Rule 976
Rule 1034
Rubi steps
\begin {align*} \int \frac {1}{(1-d x)^{3/2} (1+d x)^{3/2} \left (a+b x+c x^2\right )} \, dx &=\int \frac {1}{\left (a+b x+c x^2\right ) \left (1-d^2 x^2\right )^{3/2}} \, dx\\ &=\frac {d^2 \left (b-\left (c+a d^2\right ) x\right )}{\left (b^2 d^2-\left (c+a d^2\right )^2\right ) \sqrt {1-d^2 x^2}}-\frac {\int \frac {2 d^2 \left (c^2-b^2 d^2+a c d^2\right )-2 b c d^4 x}{\left (a+b x+c x^2\right ) \sqrt {1-d^2 x^2}} \, dx}{2 d^2 \left (b^2 d^2-\left (c+a d^2\right )^2\right )}\\ &=\frac {d^2 \left (b-\left (c+a d^2\right ) x\right )}{\left (b^2 d^2-\left (c+a d^2\right )^2\right ) \sqrt {1-d^2 x^2}}+\frac {\left (c \left (2 c^2+2 a c d^2-b \left (b-\sqrt {b^2-4 a c}\right ) d^2\right )\right ) \int \frac {1}{\left (b+\sqrt {b^2-4 a c}+2 c x\right ) \sqrt {1-d^2 x^2}} \, dx}{\sqrt {b^2-4 a c} \left (b^2 d^2-\left (c+a d^2\right )^2\right )}-\frac {\left (c \left (2 c^2+2 a c d^2-b \left (b+\sqrt {b^2-4 a c}\right ) d^2\right )\right ) \int \frac {1}{\left (b-\sqrt {b^2-4 a c}+2 c x\right ) \sqrt {1-d^2 x^2}} \, dx}{\sqrt {b^2-4 a c} \left (b^2 d^2-\left (c+a d^2\right )^2\right )}\\ &=\frac {d^2 \left (b-\left (c+a d^2\right ) x\right )}{\left (b^2 d^2-\left (c+a d^2\right )^2\right ) \sqrt {1-d^2 x^2}}-\frac {\left (c \left (2 c^2+2 a c d^2-b \left (b-\sqrt {b^2-4 a c}\right ) d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 c^2-\left (b+\sqrt {b^2-4 a c}\right )^2 d^2-x^2} \, dx,x,\frac {2 c+\left (b+\sqrt {b^2-4 a c}\right ) d^2 x}{\sqrt {1-d^2 x^2}}\right )}{\sqrt {b^2-4 a c} \left (b^2 d^2-\left (c+a d^2\right )^2\right )}+\frac {\left (c \left (2 c^2+2 a c d^2-b \left (b+\sqrt {b^2-4 a c}\right ) d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 c^2-\left (b-\sqrt {b^2-4 a c}\right )^2 d^2-x^2} \, dx,x,\frac {2 c+\left (b-\sqrt {b^2-4 a c}\right ) d^2 x}{\sqrt {1-d^2 x^2}}\right )}{\sqrt {b^2-4 a c} \left (b^2 d^2-\left (c+a d^2\right )^2\right )}\\ &=\frac {d^2 \left (b-\left (c+a d^2\right ) x\right )}{\left (b^2 d^2-\left (c+a d^2\right )^2\right ) \sqrt {1-d^2 x^2}}+\frac {c \left (2 c^2+2 a c d^2-b \left (b+\sqrt {b^2-4 a c}\right ) d^2\right ) \tanh ^{-1}\left (\frac {2 c+\left (b-\sqrt {b^2-4 a c}\right ) d^2 x}{\sqrt {2} \sqrt {2 c^2+2 a c d^2-b \left (b-\sqrt {b^2-4 a c}\right ) d^2} \sqrt {1-d^2 x^2}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {2 c^2+2 a c d^2-b \left (b-\sqrt {b^2-4 a c}\right ) d^2} \left (b^2 d^2-\left (c+a d^2\right )^2\right )}-\frac {c \left (2 c^2+2 a c d^2-b \left (b-\sqrt {b^2-4 a c}\right ) d^2\right ) \tanh ^{-1}\left (\frac {2 c+\left (b+\sqrt {b^2-4 a c}\right ) d^2 x}{\sqrt {2} \sqrt {2 c^2+2 a c d^2-b \left (b+\sqrt {b^2-4 a c}\right ) d^2} \sqrt {1-d^2 x^2}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {2 c^2+2 a c d^2-b \left (b+\sqrt {b^2-4 a c}\right ) d^2} \left (b^2 d^2-\left (c+a d^2\right )^2\right )}\\ \end {align*}
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Mathematica [A] time = 2.39, size = 335, normalized size = 0.76 \[ \frac {d^2 \left (x \left (a d^2+c\right )-b\right )}{\sqrt {1-d^2 x^2} \left (a^2 d^4+2 a c d^2-b^2 d^2+c^2\right )}-\frac {2 \sqrt {2} c^3 \tanh ^{-1}\left (\frac {d^2 x \left (b-\sqrt {b^2-4 a c}\right )+2 c}{\sqrt {1-d^2 x^2} \sqrt {2 b d^2 \left (\sqrt {b^2-4 a c}-b\right )+4 a c d^2+4 c^2}}\right )}{\sqrt {b^2-4 a c} \left (b d^2 \left (\sqrt {b^2-4 a c}-b\right )+2 a c d^2+2 c^2\right )^{3/2}}+\frac {2 \sqrt {2} c^3 \tanh ^{-1}\left (\frac {d^2 x \left (\sqrt {b^2-4 a c}+b\right )+2 c}{\sqrt {1-d^2 x^2} \sqrt {-2 b d^2 \left (\sqrt {b^2-4 a c}+b\right )+4 a c d^2+4 c^2}}\right )}{\sqrt {b^2-4 a c} \left (-b d^2 \left (\sqrt {b^2-4 a c}+b\right )+2 a c d^2+2 c^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [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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giac [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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maple [C] time = 0.12, size = 11141, normalized size = 25.15 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (c x^{2} + b x + a\right )} {\left (d x + 1\right )}^{\frac {3}{2}} {\left (-d x + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (1-d\,x\right )}^{3/2}\,{\left (d\,x+1\right )}^{3/2}\,\left (c\,x^2+b\,x+a\right )} \,d x \]
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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