Optimal. Leaf size=406 \[ \frac {\tanh ^{-1}\left (\frac {e+2 f x}{\sqrt {e^2-4 d f}}\right ) \left (B (a e f-2 b d f+c d e)-A \left (2 a f^2-b e f-2 c d f+c e^2\right )\right )}{\sqrt {e^2-4 d f} \left (f \left (a^2 f-a b e+b^2 d\right )-c \left (b d e-a \left (e^2-2 d f\right )\right )+c^2 d^2\right )}+\frac {\log \left (a+b x+c x^2\right ) (-a B f+A b f-A c e+B c d)}{2 \left (f \left (a^2 f-a b e+b^2 d\right )-c \left (b d e-a \left (e^2-2 d f\right )\right )+c^2 d^2\right )}-\frac {\log \left (d+e x+f x^2\right ) (-a B f+A b f-A c e+B c d)}{2 \left (f \left (a^2 f-a b e+b^2 d\right )-c \left (b d e-a \left (e^2-2 d f\right )\right )+c^2 d^2\right )}-\frac {\tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (-b (a B f+A c e+B c d)+2 c (-a A f+a B e+A c d)+A b^2 f\right )}{\sqrt {b^2-4 a c} \left (f \left (a^2 f-a b e+b^2 d\right )-c \left (b d e-a \left (e^2-2 d f\right )\right )+c^2 d^2\right )} \]
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Rubi [A] time = 0.48, antiderivative size = 398, normalized size of antiderivative = 0.98, number of steps used = 9, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1022, 634, 618, 206, 628} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {e+2 f x}{\sqrt {e^2-4 d f}}\right ) \left (B (a e f-2 b d f+c d e)-A \left (2 a f^2-b e f-2 c d f+c e^2\right )\right )}{\sqrt {e^2-4 d f} \left (f \left (a^2 f-a b e+b^2 d\right )+a c \left (e^2-2 d f\right )-b c d e+c^2 d^2\right )}+\frac {\log \left (a+b x+c x^2\right ) (-a B f+A b f-A c e+B c d)}{2 \left (f \left (a^2 f-a b e+b^2 d\right )+a c \left (e^2-2 d f\right )-b c d e+c^2 d^2\right )}-\frac {\log \left (d+e x+f x^2\right ) (-a B f+A b f-A c e+B c d)}{2 \left (f \left (a^2 f-a b e+b^2 d\right )+a c \left (e^2-2 d f\right )-b c d e+c^2 d^2\right )}-\frac {\tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (-b (a B f+A c e+B c d)+2 c (-a A f+a B e+A c d)+A b^2 f\right )}{\sqrt {b^2-4 a c} \left (f \left (a^2 f-a b e+b^2 d\right )+a c \left (e^2-2 d f\right )-b c d e+c^2 d^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 628
Rule 634
Rule 1022
Rubi steps
\begin {align*} \int \frac {A+B x}{\left (a+b x+c x^2\right ) \left (d+e x+f x^2\right )} \, dx &=\frac {\int \frac {a B (c e-b f)+A \left (c^2 d+b^2 f-c (b e+a f)\right )+c (B c d-A c e+A b f-a B f) x}{a+b x+c x^2} \, dx}{c^2 d^2-b c d e+f \left (b^2 d-a b e+a^2 f\right )+a c \left (e^2-2 d f\right )}+\frac {\int \frac {-A f (b e-a f)+A c \left (e^2-d f\right )-B (c d e-b d f)-f (B c d-A c e+A b f-a B f) x}{d+e x+f x^2} \, dx}{c^2 d^2-b c d e+f \left (b^2 d-a b e+a^2 f\right )+a c \left (e^2-2 d f\right )}\\ &=\frac {(B c d-A c e+A b f-a B f) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 \left (c^2 d^2-b c d e+f \left (b^2 d-a b e+a^2 f\right )+a c \left (e^2-2 d f\right )\right )}-\frac {(B c d-A c e+A b f-a B f) \int \frac {e+2 f x}{d+e x+f x^2} \, dx}{2 \left (c^2 d^2-b c d e+f \left (b^2 d-a b e+a^2 f\right )+a c \left (e^2-2 d f\right )\right )}+\frac {\left (A b^2 f+2 c (A c d+a B e-a A f)-b (B c d+A c e+a B f)\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 \left (c^2 d^2-b c d e+f \left (b^2 d-a b e+a^2 f\right )+a c \left (e^2-2 d f\right )\right )}+\frac {\left (e f (B c d-A c e+A b f-a B f)+2 f \left (-A f (b e-a f)+A c \left (e^2-d f\right )-B (c d e-b d f)\right )\right ) \int \frac {1}{d+e x+f x^2} \, dx}{2 f \left (c^2 d^2-b c d e+f \left (b^2 d-a b e+a^2 f\right )+a c \left (e^2-2 d f\right )\right )}\\ &=\frac {(B c d-A c e+A b f-a B f) \log \left (a+b x+c x^2\right )}{2 \left (c^2 d^2-b c d e+f \left (b^2 d-a b e+a^2 f\right )+a c \left (e^2-2 d f\right )\right )}-\frac {(B c d-A c e+A b f-a B f) \log \left (d+e x+f x^2\right )}{2 \left (c^2 d^2-b c d e+f \left (b^2 d-a b e+a^2 f\right )+a c \left (e^2-2 d f\right )\right )}-\frac {\left (A b^2 f+2 c (A c d+a B e-a A f)-b (B c d+A c e+a B f)\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{c^2 d^2-b c d e+f \left (b^2 d-a b e+a^2 f\right )+a c \left (e^2-2 d f\right )}-\frac {\left (e f (B c d-A c e+A b f-a B f)+2 f \left (-A f (b e-a f)+A c \left (e^2-d f\right )-B (c d e-b d f)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{e^2-4 d f-x^2} \, dx,x,e+2 f x\right )}{f \left (c^2 d^2-b c d e+f \left (b^2 d-a b e+a^2 f\right )+a c \left (e^2-2 d f\right )\right )}\\ &=-\frac {\left (A b^2 f+2 c (A c d+a B e-a A f)-b (B c d+A c e+a B f)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (c^2 d^2-b c d e+f \left (b^2 d-a b e+a^2 f\right )+a c \left (e^2-2 d f\right )\right )}+\frac {\left (B (c d e-2 b d f+a e f)-A \left (c e^2-2 c d f-b e f+2 a f^2\right )\right ) \tanh ^{-1}\left (\frac {e+2 f x}{\sqrt {e^2-4 d f}}\right )}{\sqrt {e^2-4 d f} \left (c^2 d^2-b c d e+f \left (b^2 d-a b e+a^2 f\right )+a c \left (e^2-2 d f\right )\right )}+\frac {(B c d-A c e+A b f-a B f) \log \left (a+b x+c x^2\right )}{2 \left (c^2 d^2-b c d e+f \left (b^2 d-a b e+a^2 f\right )+a c \left (e^2-2 d f\right )\right )}-\frac {(B c d-A c e+A b f-a B f) \log \left (d+e x+f x^2\right )}{2 \left (c^2 d^2-b c d e+f \left (b^2 d-a b e+a^2 f\right )+a c \left (e^2-2 d f\right )\right )}\\ \end {align*}
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Mathematica [A] time = 0.46, size = 267, normalized size = 0.66 \begin {gather*} \frac {\frac {2 \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right ) \left (-b (a B f+A c e+B c d)+2 c (-a A f+a B e+A c d)+A b^2 f\right )}{\sqrt {4 a c-b^2}}-\frac {2 \tan ^{-1}\left (\frac {e+2 f x}{\sqrt {4 d f-e^2}}\right ) \left (A \left (-2 a f^2+b e f+2 c d f-c e^2\right )+B (a e f-2 b d f+c d e)\right )}{\sqrt {4 d f-e^2}}+\log (a+x (b+c x)) (-a B f+A b f-A c e+B c d)+\log (d+x (e+f x)) (a B f-A b f+A c e-B c d)}{2 \left (f \left (a^2 f-a b e+b^2 d\right )+a c \left (e^2-2 d f\right )-b c d e+c^2 d^2\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A+B x}{\left (a+b x+c x^2\right ) \left (d+e x+f x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 416, normalized size = 1.02 \begin {gather*} \frac {{\left (B c d - B a f + A b f - A c e\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (c^{2} d^{2} + b^{2} d f - 2 \, a c d f + a^{2} f^{2} - b c d e - a b f e + a c e^{2}\right )}} - \frac {{\left (B c d - B a f + A b f - A c e\right )} \log \left (f x^{2} + x e + d\right )}{2 \, {\left (c^{2} d^{2} + b^{2} d f - 2 \, a c d f + a^{2} f^{2} - b c d e - a b f e + a c e^{2}\right )}} - \frac {{\left (B b c d - 2 \, A c^{2} d + B a b f - A b^{2} f + 2 \, A a c f - 2 \, B a c e + A b c e\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (c^{2} d^{2} + b^{2} d f - 2 \, a c d f + a^{2} f^{2} - b c d e - a b f e + a c e^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {{\left (2 \, B b d f - 2 \, A c d f + 2 \, A a f^{2} - B c d e - B a f e - A b f e + A c e^{2}\right )} \arctan \left (\frac {2 \, f x + e}{\sqrt {4 \, d f - e^{2}}}\right )}{{\left (c^{2} d^{2} + b^{2} d f - 2 \, a c d f + a^{2} f^{2} - b c d e - a b f e + a c e^{2}\right )} \sqrt {4 \, d f - e^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 1698, normalized size = 4.18
result too large to display
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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