Optimal. Leaf size=86 \[ -\frac{2}{d \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}-\frac{4 \sqrt{c} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{d \left (b^2-4 a c\right )^{3/2}} \]
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Rubi [A] time = 0.0582978, antiderivative size = 86, 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 = {687, 688, 205} \[ -\frac{2}{d \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}-\frac{4 \sqrt{c} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{d \left (b^2-4 a c\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 687
Rule 688
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{(b d+2 c d x) \left (a+b x+c x^2\right )^{3/2}} \, dx &=-\frac{2}{\left (b^2-4 a c\right ) d \sqrt{a+b x+c x^2}}-\frac{(4 c) \int \frac{1}{(b d+2 c d x) \sqrt{a+b x+c x^2}} \, dx}{b^2-4 a c}\\ &=-\frac{2}{\left (b^2-4 a c\right ) d \sqrt{a+b x+c x^2}}-\frac{\left (16 c^2\right ) \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 )}{b^2-4 a c}\\ &=-\frac{2}{\left (b^2-4 a c\right ) d \sqrt{a+b x+c x^2}}-\frac{4 \sqrt{c} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} d}\\ \end{align*}
Mathematica [C] time = 0.0284685, size = 60, normalized size = 0.7 \[ -\frac{2 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{4 c (a+x (b+c x))}{4 a c-b^2}\right )}{d \left (b^2-4 a c\right ) \sqrt{a+x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.196, size = 158, normalized size = 1.8 \begin{align*} 2\,{\frac{1}{d \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{ \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+1/4\,{\frac{4\,ac-{b}^{2}}{c}}}}}}-4\,{\frac{1}{d \left ( 4\,ac-{b}^{2} \right ) }\ln \left ({ \left ( 1/2\,{\frac{4\,ac-{b}^{2}}{c}}+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+1/2\,{\frac{b}{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 [A] time = 4.43152, size = 695, normalized size = 8.08 \begin{align*} \left [-\frac{2 \,{\left ({\left (c x^{2} + b x + a\right )} \sqrt{-\frac{c}{b^{2} - 4 \, a c}} \log \left (-\frac{4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c + 4 \, \sqrt{c x^{2} + b x + a}{\left (b^{2} - 4 \, a c\right )} \sqrt{-\frac{c}{b^{2} - 4 \, a c}}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}\right ) + \sqrt{c x^{2} + b x + a}\right )}}{{\left (b^{2} c - 4 \, a c^{2}\right )} d x^{2} +{\left (b^{3} - 4 \, a b c\right )} d x +{\left (a b^{2} - 4 \, a^{2} c\right )} d}, -\frac{2 \,{\left (2 \,{\left (c x^{2} + b x + a\right )} \sqrt{\frac{c}{b^{2} - 4 \, a c}} \arctan \left (-\frac{\sqrt{c x^{2} + b x + a}{\left (b^{2} - 4 \, a c\right )} \sqrt{\frac{c}{b^{2} - 4 \, a c}}}{2 \,{\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + \sqrt{c x^{2} + b x + a}\right )}}{{\left (b^{2} c - 4 \, a c^{2}\right )} d x^{2} +{\left (b^{3} - 4 \, a b c\right )} d x +{\left (a b^{2} - 4 \, a^{2} c\right )} d}\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}{a b \sqrt{a + b x + c x^{2}} + 2 a c x \sqrt{a + b x + c x^{2}} + b^{2} x \sqrt{a + b x + c x^{2}} + 3 b c x^{2} \sqrt{a + b x + c x^{2}} + 2 c^{2} x^{3} \sqrt{a + b x + c x^{2}}}\, dx}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15744, size = 213, normalized size = 2.48 \begin{align*} \frac{8 \, c \arctan \left (\frac{2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} c + b \sqrt{c}}{\sqrt{b^{2} c - 4 \, a c^{2}}}\right )}{\sqrt{b^{2} c - 4 \, a c^{2}}{\left (b^{2} d - 4 \, a c d\right )}} - \frac{2 \,{\left (b^{4} d - 8 \, a b^{2} c d + 16 \, a^{2} c^{2} d\right )}}{{\left (b^{6} d^{2} - 12 \, a b^{4} c d^{2} + 48 \, a^{2} b^{2} c^{2} d^{2} - 64 \, a^{3} c^{3} d^{2}\right )} \sqrt{c x^{2} + b x + a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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