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 = 0.04, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 204
Rule 614
Rule 618
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*}
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Mathematica [A] time = 0.06, 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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fricas [B] time = 1.08, size = 334, normalized size = 4.70 \[ \left [\frac {4 \, a b c - c^{3} + 2 \, {\left (b^{2} x^{2} + b c x + a b\right )} \sqrt {-4 \, a b + c^{2}} \log \left (\frac {2 \, b^{2} x^{2} + 2 \, b c x - 2 \, a b + c^{2} + \sqrt {-4 \, a b + c^{2}} {\left (2 \, b x + c\right )}}{b x^{2} + c x + a}\right ) + 2 \, {\left (4 \, a b^{2} - b c^{2}\right )} x}{16 \, a^{3} b^{2} - 8 \, a^{2} b c^{2} + a c^{4} + {\left (16 \, a^{2} b^{3} - 8 \, a b^{2} c^{2} + b c^{4}\right )} x^{2} + {\left (16 \, a^{2} b^{2} c - 8 \, a b c^{3} + c^{5}\right )} x}, \frac {4 \, a b c - c^{3} - 4 \, {\left (b^{2} x^{2} + b c x + a b\right )} \sqrt {4 \, a b - c^{2}} \arctan \left (-\frac {2 \, b x + c}{\sqrt {4 \, a b - c^{2}}}\right ) + 2 \, {\left (4 \, a b^{2} - b c^{2}\right )} x}{16 \, a^{3} b^{2} - 8 \, a^{2} b c^{2} + a c^{4} + {\left (16 \, a^{2} b^{3} - 8 \, a b^{2} c^{2} + b c^{4}\right )} x^{2} + {\left (16 \, a^{2} b^{2} c - 8 \, a b c^{3} + c^{5}\right )} x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 67, normalized size = 0.94 \[ \frac {4 \, b \arctan \left (\frac {2 \, b x + c}{\sqrt {4 \, a b - c^{2}}}\right )}{{\left (4 \, a b - c^{2}\right )}^{\frac {3}{2}}} + \frac {2 \, b x + c}{{\left (b x^{2} + c x + a\right )} {\left (4 \, a b - c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 68, normalized size = 0.96 \[ \frac {4 b \arctan \left (\frac {2 b x +c}{\sqrt {4 a b -c^{2}}}\right )}{\left (4 a b -c^{2}\right )^{\frac {3}{2}}}+\frac {2 b x +c}{\left (4 a b -c^{2}\right ) \left (b \,x^{2}+c x +a \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.17, size = 119, normalized size = 1.68 \[ \frac {\frac {c}{4\,a\,b-c^2}+\frac {2\,b\,x}{4\,a\,b-c^2}}{b\,x^2+c\,x+a}-\frac {4\,b\,\mathrm {atan}\left (\frac {\left (\frac {2\,b\,\left (c^3-4\,a\,b\,c\right )}{{\left (4\,a\,b-c^2\right )}^{5/2}}-\frac {4\,b^2\,x}{{\left (4\,a\,b-c^2\right )}^{3/2}}\right )\,\left (4\,a\,b-c^2\right )}{2\,b}\right )}{{\left (4\,a\,b-c^2\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.59, size = 265, normalized size = 3.73 \[ - 2 b \sqrt {- \frac {1}{\left (4 a b - c^{2}\right )^{3}}} \log {\left (x + \frac {- 32 a^{2} b^{3} \sqrt {- \frac {1}{\left (4 a b - c^{2}\right )^{3}}} + 16 a b^{2} c^{2} \sqrt {- \frac {1}{\left (4 a b - c^{2}\right )^{3}}} - 2 b c^{4} \sqrt {- \frac {1}{\left (4 a b - c^{2}\right )^{3}}} + 2 b c}{4 b^{2}} \right )} + 2 b \sqrt {- \frac {1}{\left (4 a b - c^{2}\right )^{3}}} \log {\left (x + \frac {32 a^{2} b^{3} \sqrt {- \frac {1}{\left (4 a b - c^{2}\right )^{3}}} - 16 a b^{2} c^{2} \sqrt {- \frac {1}{\left (4 a b - c^{2}\right )^{3}}} + 2 b c^{4} \sqrt {- \frac {1}{\left (4 a b - c^{2}\right )^{3}}} + 2 b c}{4 b^{2}} \right )} + \frac {2 b x + c}{4 a^{2} b - a c^{2} + x^{2} \left (4 a b^{2} - b c^{2}\right ) + x \left (4 a b c - c^{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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