Optimal. Leaf size=95 \[ \frac{\sqrt{a+b x+c x^2}}{d^3 \left (b^2-4 a c\right ) (b+2 c x)^2}+\frac{\tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{2 \sqrt{c} d^3 \left (b^2-4 a c\right )^{3/2}} \]
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Rubi [A] time = 0.055632, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {693, 688, 205} \[ \frac{\sqrt{a+b x+c x^2}}{d^3 \left (b^2-4 a c\right ) (b+2 c x)^2}+\frac{\tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{2 \sqrt{c} d^3 \left (b^2-4 a c\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 693
Rule 688
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{(b d+2 c d x)^3 \sqrt{a+b x+c x^2}} \, dx &=\frac{\sqrt{a+b x+c x^2}}{\left (b^2-4 a c\right ) d^3 (b+2 c x)^2}+\frac{\int \frac{1}{(b d+2 c d x) \sqrt{a+b x+c x^2}} \, dx}{2 \left (b^2-4 a c\right ) d^2}\\ &=\frac{\sqrt{a+b x+c x^2}}{\left (b^2-4 a c\right ) d^3 (b+2 c x)^2}+\frac{(2 c) \operatorname{Subst}\left (\int \frac{1}{2 b^2 c d-8 a c^2 d+8 c^2 d x^2} \, dx,x,\sqrt{a+b x+c x^2}\right )}{\left (b^2-4 a c\right ) d^2}\\ &=\frac{\sqrt{a+b x+c x^2}}{\left (b^2-4 a c\right ) d^3 (b+2 c x)^2}+\frac{\tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{2 \sqrt{c} \left (b^2-4 a c\right )^{3/2} d^3}\\ \end{align*}
Mathematica [A] time = 0.284009, size = 107, normalized size = 1.13 \[ \frac{\sqrt{a+x (b+c x)} \left (\frac{2 \left (b^2-4 a c\right )}{(b+2 c x)^2}+\frac{\tanh ^{-1}\left (2 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}\right )}{\sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}\right )}{2 d^3 \left (b^2-4 a c\right )^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.192, size = 174, normalized size = 1.8 \begin{align*} -{\frac{1}{4\,{c}^{2}{d}^{3} \left ( 4\,ac-{b}^{2} \right ) }\sqrt{ \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}}} \left ( x+{\frac{b}{2\,c}} \right ) ^{-2}}+{\frac{1}{2\,c{d}^{3} \left ( 4\,ac-{b}^{2} \right ) }\ln \left ({ \left ({\frac{4\,ac-{b}^{2}}{2\,c}}+{\frac{1}{2}\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}} \right ) \left ( x+{\frac{b}{2\,c}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 4.18764, size = 936, normalized size = 9.85 \begin{align*} \left [\frac{{\left (4 \, c^{2} x^{2} + 4 \, b c x + b^{2}\right )} \sqrt{-b^{2} c + 4 \, a c^{2}} \log \left (-\frac{4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c + 4 \, \sqrt{-b^{2} c + 4 \, a c^{2}} \sqrt{c x^{2} + b x + a}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}\right ) + 4 \,{\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt{c x^{2} + b x + a}}{4 \,{\left (4 \,{\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} d^{3} x^{2} + 4 \,{\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} d^{3} x +{\left (b^{6} c - 8 \, a b^{4} c^{2} + 16 \, a^{2} b^{2} c^{3}\right )} d^{3}\right )}}, -\frac{{\left (4 \, c^{2} x^{2} + 4 \, b c x + b^{2}\right )} \sqrt{b^{2} c - 4 \, a c^{2}} \arctan \left (\frac{\sqrt{b^{2} c - 4 \, a c^{2}} \sqrt{c x^{2} + b x + a}}{2 \,{\left (c^{2} x^{2} + b c x + a c\right )}}\right ) - 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt{c x^{2} + b x + a}}{2 \,{\left (4 \,{\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} d^{3} x^{2} + 4 \,{\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} d^{3} x +{\left (b^{6} c - 8 \, a b^{4} c^{2} + 16 \, a^{2} b^{2} c^{3}\right )} d^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{b^{3} \sqrt{a + b x + c x^{2}} + 6 b^{2} c x \sqrt{a + b x + c x^{2}} + 12 b c^{2} x^{2} \sqrt{a + b x + c x^{2}} + 8 c^{3} x^{3} \sqrt{a + b x + c x^{2}}}\, dx}{d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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