Optimal. Leaf size=71 \[ \frac{2 b x+c}{\left (4 a b-c^2\right ) \left (a+b x^2+c x\right )}+\frac{4 b \tan ^{-1}\left (\frac{2 b x+c}{\sqrt{4 a b-c^2}}\right )}{\left (4 a b-c^2\right )^{3/2}} \]
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Rubi [A] time = 0.0397878, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {614, 618, 204} \[ \frac{2 b x+c}{\left (4 a b-c^2\right ) \left (a+b x^2+c x\right )}+\frac{4 b \tan ^{-1}\left (\frac{2 b x+c}{\sqrt{4 a b-c^2}}\right )}{\left (4 a b-c^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 614
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{\left (a+c x+b x^2\right )^2} \, dx &=\frac{c+2 b x}{\left (4 a b-c^2\right ) \left (a+c x+b x^2\right )}+\frac{(2 b) \int \frac{1}{a+c x+b x^2} \, dx}{4 a b-c^2}\\ &=\frac{c+2 b x}{\left (4 a b-c^2\right ) \left (a+c x+b x^2\right )}-\frac{(4 b) \operatorname{Subst}\left (\int \frac{1}{-4 a b+c^2-x^2} \, dx,x,c+2 b x\right )}{4 a b-c^2}\\ &=\frac{c+2 b x}{\left (4 a b-c^2\right ) \left (a+c x+b x^2\right )}+\frac{4 b \tan ^{-1}\left (\frac{c+2 b x}{\sqrt{4 a b-c^2}}\right )}{\left (4 a b-c^2\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0651828, size = 70, normalized size = 0.99 \[ \frac{2 b x+c}{\left (4 a b-c^2\right ) (a+x (b x+c))}+\frac{4 b \tan ^{-1}\left (\frac{2 b x+c}{\sqrt{4 a b-c^2}}\right )}{\left (4 a b-c^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.191, size = 68, normalized size = 1. \begin{align*}{\frac{2\,bx+c}{ \left ( 4\,ab-{c}^{2} \right ) \left ( b{x}^{2}+cx+a \right ) }}+4\,{\frac{b}{ \left ( 4\,ab-{c}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,bx+c}{\sqrt{4\,ab-{c}^{2}}}} \right ) } \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 = 2.19871, size = 724, normalized size = 10.2 \begin{align*} \left [\frac{4 \, a b c - c^{3} + 2 \,{\left (b^{2} x^{2} + b c x + a b\right )} \sqrt{-4 \, a b + c^{2}} \log \left (\frac{2 \, b^{2} x^{2} + 2 \, b c x - 2 \, a b + c^{2} + \sqrt{-4 \, a b + c^{2}}{\left (2 \, b x + c\right )}}{b x^{2} + c x + a}\right ) + 2 \,{\left (4 \, a b^{2} - b c^{2}\right )} x}{16 \, a^{3} b^{2} - 8 \, a^{2} b c^{2} + a c^{4} +{\left (16 \, a^{2} b^{3} - 8 \, a b^{2} c^{2} + b c^{4}\right )} x^{2} +{\left (16 \, a^{2} b^{2} c - 8 \, a b c^{3} + c^{5}\right )} x}, \frac{4 \, a b c - c^{3} - 4 \,{\left (b^{2} x^{2} + b c x + a b\right )} \sqrt{4 \, a b - c^{2}} \arctan \left (-\frac{2 \, b x + c}{\sqrt{4 \, a b - c^{2}}}\right ) + 2 \,{\left (4 \, a b^{2} - b c^{2}\right )} x}{16 \, a^{3} b^{2} - 8 \, a^{2} b c^{2} + a c^{4} +{\left (16 \, a^{2} b^{3} - 8 \, a b^{2} c^{2} + b c^{4}\right )} x^{2} +{\left (16 \, a^{2} b^{2} c - 8 \, a b c^{3} + c^{5}\right )} x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.17108, size = 265, normalized size = 3.73 \begin{align*} - 2 b \sqrt{- \frac{1}{\left (4 a b - c^{2}\right )^{3}}} \log{\left (x + \frac{- 32 a^{2} b^{3} \sqrt{- \frac{1}{\left (4 a b - c^{2}\right )^{3}}} + 16 a b^{2} c^{2} \sqrt{- \frac{1}{\left (4 a b - c^{2}\right )^{3}}} - 2 b c^{4} \sqrt{- \frac{1}{\left (4 a b - c^{2}\right )^{3}}} + 2 b c}{4 b^{2}} \right )} + 2 b \sqrt{- \frac{1}{\left (4 a b - c^{2}\right )^{3}}} \log{\left (x + \frac{32 a^{2} b^{3} \sqrt{- \frac{1}{\left (4 a b - c^{2}\right )^{3}}} - 16 a b^{2} c^{2} \sqrt{- \frac{1}{\left (4 a b - c^{2}\right )^{3}}} + 2 b c^{4} \sqrt{- \frac{1}{\left (4 a b - c^{2}\right )^{3}}} + 2 b c}{4 b^{2}} \right )} + \frac{2 b x + c}{4 a^{2} b - a c^{2} + x^{2} \left (4 a b^{2} - b c^{2}\right ) + x \left (4 a b c - c^{3}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24021, size = 90, normalized size = 1.27 \begin{align*} \frac{4 \, b \arctan \left (\frac{2 \, b x + c}{\sqrt{4 \, a b - c^{2}}}\right )}{{\left (4 \, a b - c^{2}\right )}^{\frac{3}{2}}} + \frac{2 \, b x + c}{{\left (b x^{2} + c x + a\right )}{\left (4 \, a b - c^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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