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 = 1.44267, antiderivative size = 443, 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, 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 899
Rule 976
Rule 1034
Rule 725
Rule 206
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*}
Mathematica [A] time = 2.50816, 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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Maple [C] time = 0.477, size = 11142, normalized size = 25.2 \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 )}{\left (d x + 1\right )}^{\frac{3}{2}}{\left (-d x + 1\right )}^{\frac{3}{2}}}\,{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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