Optimal. Leaf size=571 \[ -\frac{c \left (-c d^2 \left (-8 a^2 d^2-b \sqrt{b^2-4 a c}+5 b^2\right )-a b d^4 \left (\sqrt{b^2-4 a c}+b\right )+12 a c^2 d^2+4 c^3\right ) \tanh ^{-1}\left (\frac{d^2 x \left (b-\sqrt{b^2-4 a c}\right )+2 c}{\sqrt{2} \sqrt{1-d^2 x^2} \sqrt{-b d^2 \left (b-\sqrt{b^2-4 a c}\right )+2 a c d^2+2 c^2}}\right )}{\sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{-b d^2 \left (b-\sqrt{b^2-4 a c}\right )+2 a c d^2+2 c^2} \left (b^2 d^2-\left (a d^2+c\right )^2\right )}+\frac{c \left (-4 c d^2 \left (b^2-2 a^2 d^2\right )-b d^2 \left (\sqrt{b^2-4 a c}+b\right ) \left (c-a d^2\right )-2 a b^2 d^4+12 a c^2 d^2+4 c^3\right ) \tanh ^{-1}\left (\frac{d^2 x \left (\sqrt{b^2-4 a c}+b\right )+2 c}{\sqrt{2} \sqrt{1-d^2 x^2} \sqrt{-b d^2 \left (\sqrt{b^2-4 a c}+b\right )+2 a c d^2+2 c^2}}\right )}{\sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{-b d^2 \left (\sqrt{b^2-4 a c}+b\right )+2 a c d^2+2 c^2} \left (b^2 d^2-\left (a d^2+c\right )^2\right )}-\frac{\sqrt{1-d^2 x^2} \left (b \left (b^2 d^2-c \left (3 a d^2+c\right )\right )-c x \left (2 a c d^2-b^2 d^2+2 c^2\right )\right )}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (a d^2+c\right )^2\right ) \left (a+b x+c x^2\right )} \]
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Rubi [A] time = 5.23475, antiderivative size = 571, normalized size of antiderivative = 1., 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, 975, 1034, 725, 206} \[ -\frac{c \left (-c d^2 \left (-8 a^2 d^2-b \sqrt{b^2-4 a c}+5 b^2\right )-a b d^4 \left (\sqrt{b^2-4 a c}+b\right )+12 a c^2 d^2+4 c^3\right ) \tanh ^{-1}\left (\frac{d^2 x \left (b-\sqrt{b^2-4 a c}\right )+2 c}{\sqrt{2} \sqrt{1-d^2 x^2} \sqrt{-b d^2 \left (b-\sqrt{b^2-4 a c}\right )+2 a c d^2+2 c^2}}\right )}{\sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{-b d^2 \left (b-\sqrt{b^2-4 a c}\right )+2 a c d^2+2 c^2} \left (b^2 d^2-\left (a d^2+c\right )^2\right )}+\frac{c \left (-4 c d^2 \left (b^2-2 a^2 d^2\right )-b d^2 \left (\sqrt{b^2-4 a c}+b\right ) \left (c-a d^2\right )-2 a b^2 d^4+12 a c^2 d^2+4 c^3\right ) \tanh ^{-1}\left (\frac{d^2 x \left (\sqrt{b^2-4 a c}+b\right )+2 c}{\sqrt{2} \sqrt{1-d^2 x^2} \sqrt{-b d^2 \left (\sqrt{b^2-4 a c}+b\right )+2 a c d^2+2 c^2}}\right )}{\sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{-b d^2 \left (\sqrt{b^2-4 a c}+b\right )+2 a c d^2+2 c^2} \left (b^2 d^2-\left (a d^2+c\right )^2\right )}-\frac{\sqrt{1-d^2 x^2} \left (b \left (b^2 d^2-c \left (3 a d^2+c\right )\right )-c x \left (2 a c d^2-b^2 d^2+2 c^2\right )\right )}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (a d^2+c\right )^2\right ) \left (a+b x+c x^2\right )} \]
Antiderivative was successfully verified.
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Rule 899
Rule 975
Rule 1034
Rule 725
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{1-d x} \sqrt{1+d x} \left (a+b x+c x^2\right )^2} \, dx &=\int \frac{1}{\left (a+b x+c x^2\right )^2 \sqrt{1-d^2 x^2}} \, dx\\ &=-\frac{\left (b \left (b^2 d^2-c \left (c+3 a d^2\right )\right )-c \left (2 c^2-b^2 d^2+2 a c d^2\right ) x\right ) \sqrt{1-d^2 x^2}}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (c+a d^2\right )^2\right ) \left (a+b x+c x^2\right )}-\frac{\int \frac{-2 c^3-6 a c^2 d^2+a b^2 d^4+2 c d^2 \left (b^2-2 a^2 d^2\right )-b c d^2 \left (c-a d^2\right ) x}{\left (a+b x+c x^2\right ) \sqrt{1-d^2 x^2}} \, dx}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (c+a d^2\right )^2\right )}\\ &=-\frac{\left (b \left (b^2 d^2-c \left (c+3 a d^2\right )\right )-c \left (2 c^2-b^2 d^2+2 a c d^2\right ) x\right ) \sqrt{1-d^2 x^2}}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (c+a d^2\right )^2\right ) \left (a+b x+c x^2\right )}+\frac{\left (c \left (4 c^3+12 a c^2 d^2-a b \left (b+\sqrt{b^2-4 a c}\right ) d^4-c d^2 \left (5 b^2-b \sqrt{b^2-4 a c}-8 a^2 d^2\right )\right )\right ) \int \frac{1}{\left (b-\sqrt{b^2-4 a c}+2 c x\right ) \sqrt{1-d^2 x^2}} \, dx}{\left (b^2-4 a c\right )^{3/2} \left (b^2 d^2-\left (c+a d^2\right )^2\right )}+\frac{\left (b c \left (b+\sqrt{b^2-4 a c}\right ) d^2 \left (c-a d^2\right )+2 c \left (-2 c^3-6 a c^2 d^2+a b^2 d^4+2 c d^2 \left (b^2-2 a^2 d^2\right )\right )\right ) \int \frac{1}{\left (b+\sqrt{b^2-4 a c}+2 c x\right ) \sqrt{1-d^2 x^2}} \, dx}{\left (b^2-4 a c\right )^{3/2} \left (b^2 d^2-\left (c+a d^2\right )^2\right )}\\ &=-\frac{\left (b \left (b^2 d^2-c \left (c+3 a d^2\right )\right )-c \left (2 c^2-b^2 d^2+2 a c d^2\right ) x\right ) \sqrt{1-d^2 x^2}}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (c+a d^2\right )^2\right ) \left (a+b x+c x^2\right )}-\frac{\left (c \left (4 c^3+12 a c^2 d^2-a b \left (b+\sqrt{b^2-4 a c}\right ) d^4-c d^2 \left (5 b^2-b \sqrt{b^2-4 a c}-8 a^2 d^2\right )\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 )}{\left (b^2-4 a c\right )^{3/2} \left (b^2 d^2-\left (c+a d^2\right )^2\right )}-\frac{\left (b c \left (b+\sqrt{b^2-4 a c}\right ) d^2 \left (c-a d^2\right )+2 c \left (-2 c^3-6 a c^2 d^2+a b^2 d^4+2 c d^2 \left (b^2-2 a^2 d^2\right )\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 )}{\left (b^2-4 a c\right )^{3/2} \left (b^2 d^2-\left (c+a d^2\right )^2\right )}\\ &=-\frac{\left (b \left (b^2 d^2-c \left (c+3 a d^2\right )\right )-c \left (2 c^2-b^2 d^2+2 a c d^2\right ) x\right ) \sqrt{1-d^2 x^2}}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (c+a d^2\right )^2\right ) \left (a+b x+c x^2\right )}-\frac{c \left (4 c^3+12 a c^2 d^2-a b \left (b+\sqrt{b^2-4 a c}\right ) d^4-c d^2 \left (5 b^2-b \sqrt{b^2-4 a c}-8 a^2 d^2\right )\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} \left (b^2-4 a c\right )^{3/2} \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 (b \left (b+\sqrt{b^2-4 a c}\right ) d^2 \left (c-a d^2\right )-2 \left (2 c^3+6 a c^2 d^2-a b^2 d^4-2 c d^2 \left (b^2-2 a^2 d^2\right )\right )\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} \left (b^2-4 a c\right )^{3/2} \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*}
Mathematica [A] time = 1.40914, size = 508, normalized size = 0.89 \[ \frac{\frac{c \left (c d^2 \left (8 a^2 d^2+b \sqrt{b^2-4 a c}-5 b^2\right )-a b d^4 \left (\sqrt{b^2-4 a c}+b\right )+12 a c^2 d^2+4 c^3\right ) \tanh ^{-1}\left (\frac{d^2 x \left (b-\sqrt{b^2-4 a c}\right )+2 c}{\sqrt{1-d^2 x^2} \sqrt{2 b d^2 \left (\sqrt{b^2-4 a c}-b\right )+4 a c d^2+4 c^2}}\right )}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{b d^2 \left (\sqrt{b^2-4 a c}-b\right )+2 a c d^2+2 c^2}}+\frac{c \left (c d^2 \left (-8 a^2 d^2+b \sqrt{b^2-4 a c}+5 b^2\right )+a b d^4 \left (b-\sqrt{b^2-4 a c}\right )-12 a c^2 d^2-4 c^3\right ) \tanh ^{-1}\left (\frac{d^2 x \left (\sqrt{b^2-4 a c}+b\right )+2 c}{\sqrt{1-d^2 x^2} \sqrt{-2 b d^2 \left (\sqrt{b^2-4 a c}+b\right )+4 a c d^2+4 c^2}}\right )}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{-b d^2 \left (\sqrt{b^2-4 a c}+b\right )+2 a c d^2+2 c^2}}+\frac{\sqrt{1-d^2 x^2} \left (-b c \left (3 a d^2+c\right )-2 c^2 x \left (a d^2+c\right )+b^2 c d^2 x+b^3 d^2\right )}{a+x (b+c x)}}{\left (b^2-4 a c\right ) \left (\left (a d^2+c\right )^2-b^2 d^2\right )} \]
Antiderivative was successfully verified.
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Maple [C] time = 1.205, size = 41837, normalized size = 73.3 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (c x^{2} + b x + a\right )}^{2} \sqrt{d x + 1} \sqrt{-d x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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