Optimal. Leaf size=100 \[ \frac {4 (a C+A c) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {x \left (C \left (b^2-2 a c\right )+2 A c^2\right )+b c \left (\frac {a C}{c}+A\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]
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Rubi [A] time = 0.07, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1660, 12, 618, 206} \[ \frac {4 (a C+A c) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {x \left (C \left (b^2-2 a c\right )+2 A c^2\right )+b c \left (\frac {a C}{c}+A\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 618
Rule 1660
Rubi steps
\begin {align*} \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^2} \, dx &=-\frac {b c \left (A+\frac {a C}{c}\right )+\left (2 A c^2+\left (b^2-2 a c\right ) C\right ) x}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\int \frac {2 (A c+a C)}{a+b x+c x^2} \, dx}{-b^2+4 a c}\\ &=-\frac {b c \left (A+\frac {a C}{c}\right )+\left (2 A c^2+\left (b^2-2 a c\right ) C\right ) x}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {(2 (A c+a C)) \int \frac {1}{a+b x+c x^2} \, dx}{b^2-4 a c}\\ &=-\frac {b c \left (A+\frac {a C}{c}\right )+\left (2 A c^2+\left (b^2-2 a c\right ) C\right ) x}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {(4 (A c+a C)) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{b^2-4 a c}\\ &=-\frac {b c \left (A+\frac {a C}{c}\right )+\left (2 A c^2+\left (b^2-2 a c\right ) C\right ) x}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {4 (A c+a C) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 98, normalized size = 0.98 \[ \frac {a C (b-2 c x)+A c (b+2 c x)+b^2 C x}{c \left (4 a c-b^2\right ) (a+x (b+c x))}+\frac {4 (a C+A c) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.26, size = 511, normalized size = 5.11 \[ \left [-\frac {C a b^{3} - 4 \, A a b c^{2} + 2 \, {\left (C a^{2} c + A a c^{2} + {\left (C a c^{2} + A c^{3}\right )} x^{2} + {\left (C a b c + A b c^{2}\right )} x\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c - \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) - {\left (4 \, C a^{2} b - A b^{3}\right )} c + {\left (C b^{4} - 6 \, C a b^{2} c - 8 \, A a c^{3} + 2 \, {\left (4 \, C a^{2} + A b^{2}\right )} c^{2}\right )} x}{a b^{4} c - 8 \, a^{2} b^{2} c^{2} + 16 \, a^{3} c^{3} + {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{2} + {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x}, -\frac {C a b^{3} - 4 \, A a b c^{2} - 4 \, {\left (C a^{2} c + A a c^{2} + {\left (C a c^{2} + A c^{3}\right )} x^{2} + {\left (C a b c + A b c^{2}\right )} x\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) - {\left (4 \, C a^{2} b - A b^{3}\right )} c + {\left (C b^{4} - 6 \, C a b^{2} c - 8 \, A a c^{3} + 2 \, {\left (4 \, C a^{2} + A b^{2}\right )} c^{2}\right )} x}{a b^{4} c - 8 \, a^{2} b^{2} c^{2} + 16 \, a^{3} c^{3} + {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{2} + {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 108, normalized size = 1.08 \[ -\frac {4 \, {\left (C a + A c\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} - 4 \, a c\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {C b^{2} x - 2 \, C a c x + 2 \, A c^{2} x + C a b + A b c}{{\left (b^{2} c - 4 \, a c^{2}\right )} {\left (c x^{2} + b x + a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 146, normalized size = 1.46 \[ \frac {4 A c \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}}}+\frac {4 C a \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}}}+\frac {\frac {\left (A c +a C \right ) b}{\left (4 a c -b^{2}\right ) c}+\frac {\left (2 A \,c^{2}-2 C a c +C \,b^{2}\right ) x}{\left (4 a c -b^{2}\right ) c}}{c \,x^{2}+b x +a} \]
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 = 4.53, size = 172, normalized size = 1.72 \[ \frac {\frac {A\,b\,c+C\,a\,b}{c\,\left (4\,a\,c-b^2\right )}+\frac {x\,\left (C\,b^2+2\,A\,c^2-2\,C\,a\,c\right )}{c\,\left (4\,a\,c-b^2\right )}}{c\,x^2+b\,x+a}-\frac {4\,\mathrm {atan}\left (\frac {\left (\frac {2\,\left (A\,c+C\,a\right )\,\left (b^3-4\,a\,b\,c\right )}{{\left (4\,a\,c-b^2\right )}^{5/2}}-\frac {4\,c\,x\,\left (A\,c+C\,a\right )}{{\left (4\,a\,c-b^2\right )}^{3/2}}\right )\,\left (4\,a\,c-b^2\right )}{2\,A\,c+2\,C\,a}\right )\,\left (A\,c+C\,a\right )}{{\left (4\,a\,c-b^2\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.21, size = 376, normalized size = 3.76 \[ - 2 \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (A c + C a\right ) \log {\left (x + \frac {2 A b c + 2 C a b - 32 a^{2} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (A c + C a\right ) + 16 a b^{2} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (A c + C a\right ) - 2 b^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (A c + C a\right )}{4 A c^{2} + 4 C a c} \right )} + 2 \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (A c + C a\right ) \log {\left (x + \frac {2 A b c + 2 C a b + 32 a^{2} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (A c + C a\right ) - 16 a b^{2} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (A c + C a\right ) + 2 b^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (A c + C a\right )}{4 A c^{2} + 4 C a c} \right )} + \frac {A b c + C a b + x \left (2 A c^{2} - 2 C a c + C b^{2}\right )}{4 a^{2} c^{2} - a b^{2} c + x^{2} \left (4 a c^{3} - b^{2} c^{2}\right ) + x \left (4 a b c^{2} - b^{3} c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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