Optimal. Leaf size=184 \[ \frac {x \left (-2 c e g (-a e g+b d g+b e f)+b^2 e^2 g^2+c^2 \left (d^2 g^2+d e f g+e^2 f^2\right )\right )}{e^3 g^3}+\frac {\log (d+e x) \left (a e^2-b d e+c d^2\right )^2}{e^4 (e f-d g)}-\frac {\log (f+g x) \left (a g^2-b f g+c f^2\right )^2}{g^4 (e f-d g)}-\frac {c x^2 (-2 b e g+c d g+c e f)}{2 e^2 g^2}+\frac {c^2 x^3}{3 e g} \]
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Rubi [A] time = 0.31, antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {893} \[ \frac {x \left (-2 c e g (-a e g+b d g+b e f)+b^2 e^2 g^2+c^2 \left (d^2 g^2+d e f g+e^2 f^2\right )\right )}{e^3 g^3}+\frac {\log (d+e x) \left (a e^2-b d e+c d^2\right )^2}{e^4 (e f-d g)}-\frac {\log (f+g x) \left (a g^2-b f g+c f^2\right )^2}{g^4 (e f-d g)}-\frac {c x^2 (-2 b e g+c d g+c e f)}{2 e^2 g^2}+\frac {c^2 x^3}{3 e g} \]
Antiderivative was successfully verified.
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Rule 893
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x) (f+g x)} \, dx &=\int \left (\frac {b^2 e^2 g^2-2 c e g (b e f+b d g-a e g)+c^2 \left (e^2 f^2+d e f g+d^2 g^2\right )}{e^3 g^3}-\frac {c (c e f+c d g-2 b e g) x}{e^2 g^2}+\frac {c^2 x^2}{e g}+\frac {\left (c d^2-b d e+a e^2\right )^2}{e^3 (e f-d g) (d+e x)}+\frac {\left (c f^2-b f g+a g^2\right )^2}{g^3 (-e f+d g) (f+g x)}\right ) \, dx\\ &=\frac {\left (b^2 e^2 g^2-2 c e g (b e f+b d g-a e g)+c^2 \left (e^2 f^2+d e f g+d^2 g^2\right )\right ) x}{e^3 g^3}-\frac {c (c e f+c d g-2 b e g) x^2}{2 e^2 g^2}+\frac {c^2 x^3}{3 e g}+\frac {\left (c d^2-b d e+a e^2\right )^2 \log (d+e x)}{e^4 (e f-d g)}-\frac {\left (c f^2-b f g+a g^2\right )^2 \log (f+g x)}{g^4 (e f-d g)}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 177, normalized size = 0.96 \[ -\frac {e g x (d g-e f) \left (6 c e g (2 a e g+b (-2 d g-2 e f+e g x))+6 b^2 e^2 g^2+c^2 \left (6 d^2 g^2-3 d e g (g x-2 f)+e^2 \left (6 f^2-3 f g x+2 g^2 x^2\right )\right )\right )-6 g^4 \log (d+e x) \left (e (a e-b d)+c d^2\right )^2+6 e^4 \log (f+g x) \left (g (a g-b f)+c f^2\right )^2}{6 e^4 g^4 (e f-d g)} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.68, size = 313, normalized size = 1.70 \[ \frac {6 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + a^{2} e^{4} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}\right )} g^{4} \log \left (e x + d\right ) + 2 \, {\left (c^{2} e^{4} f g^{3} - c^{2} d e^{3} g^{4}\right )} x^{3} - 3 \, {\left (c^{2} e^{4} f^{2} g^{2} - 2 \, b c e^{4} f g^{3} - {\left (c^{2} d^{2} e^{2} - 2 \, b c d e^{3}\right )} g^{4}\right )} x^{2} + 6 \, {\left (c^{2} e^{4} f^{3} g - 2 \, b c e^{4} f^{2} g^{2} + {\left (b^{2} + 2 \, a c\right )} e^{4} f g^{3} - {\left (c^{2} d^{3} e - 2 \, b c d^{2} e^{2} + {\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} g^{4}\right )} x - 6 \, {\left (c^{2} e^{4} f^{4} - 2 \, b c e^{4} f^{3} g - 2 \, a b e^{4} f g^{3} + a^{2} e^{4} g^{4} + {\left (b^{2} + 2 \, a c\right )} e^{4} f^{2} g^{2}\right )} \log \left (g x + f\right )}{6 \, {\left (e^{5} f g^{4} - d e^{4} g^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 281, normalized size = 1.53 \[ \frac {{\left (c^{2} f^{4} - 2 \, b c f^{3} g + b^{2} f^{2} g^{2} + 2 \, a c f^{2} g^{2} - 2 \, a b f g^{3} + a^{2} g^{4}\right )} \log \left ({\left | g x + f \right |}\right )}{d g^{5} - f g^{4} e} - \frac {{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} \log \left ({\left | x e + d \right |}\right )}{d g e^{4} - f e^{5}} + \frac {{\left (2 \, c^{2} g^{2} x^{3} e^{2} - 3 \, c^{2} d g^{2} x^{2} e + 6 \, c^{2} d^{2} g^{2} x - 3 \, c^{2} f g x^{2} e^{2} + 6 \, b c g^{2} x^{2} e^{2} + 6 \, c^{2} d f g x e - 12 \, b c d g^{2} x e + 6 \, c^{2} f^{2} x e^{2} - 12 \, b c f g x e^{2} + 6 \, b^{2} g^{2} x e^{2} + 12 \, a c g^{2} x e^{2}\right )} e^{\left (-3\right )}}{6 \, g^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 444, normalized size = 2.41 \[ -\frac {a^{2} \ln \left (e x +d \right )}{d g -e f}+\frac {a^{2} \ln \left (g x +f \right )}{d g -e f}+\frac {2 a b d \ln \left (e x +d \right )}{\left (d g -e f \right ) e}-\frac {2 a b f \ln \left (g x +f \right )}{\left (d g -e f \right ) g}-\frac {2 a c \,d^{2} \ln \left (e x +d \right )}{\left (d g -e f \right ) e^{2}}+\frac {2 a c \,f^{2} \ln \left (g x +f \right )}{\left (d g -e f \right ) g^{2}}-\frac {b^{2} d^{2} \ln \left (e x +d \right )}{\left (d g -e f \right ) e^{2}}+\frac {b^{2} f^{2} \ln \left (g x +f \right )}{\left (d g -e f \right ) g^{2}}+\frac {2 b c \,d^{3} \ln \left (e x +d \right )}{\left (d g -e f \right ) e^{3}}-\frac {2 b c \,f^{3} \ln \left (g x +f \right )}{\left (d g -e f \right ) g^{3}}-\frac {c^{2} d^{4} \ln \left (e x +d \right )}{\left (d g -e f \right ) e^{4}}+\frac {c^{2} x^{3}}{3 e g}+\frac {c^{2} f^{4} \ln \left (g x +f \right )}{\left (d g -e f \right ) g^{4}}+\frac {b c \,x^{2}}{e g}-\frac {c^{2} d \,x^{2}}{2 e^{2} g}-\frac {c^{2} f \,x^{2}}{2 e \,g^{2}}+\frac {2 a c x}{e g}+\frac {b^{2} x}{e g}-\frac {2 b c d x}{e^{2} g}-\frac {2 b c f x}{e \,g^{2}}+\frac {c^{2} d^{2} x}{e^{3} g}+\frac {c^{2} d f x}{e^{2} g^{2}}+\frac {c^{2} f^{2} x}{e \,g^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 255, normalized size = 1.39 \[ \frac {{\left (c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + a^{2} e^{4} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}\right )} \log \left (e x + d\right )}{e^{5} f - d e^{4} g} - \frac {{\left (c^{2} f^{4} - 2 \, b c f^{3} g - 2 \, a b f g^{3} + a^{2} g^{4} + {\left (b^{2} + 2 \, a c\right )} f^{2} g^{2}\right )} \log \left (g x + f\right )}{e f g^{4} - d g^{5}} + \frac {2 \, c^{2} e^{2} g^{2} x^{3} - 3 \, {\left (c^{2} e^{2} f g + {\left (c^{2} d e - 2 \, b c e^{2}\right )} g^{2}\right )} x^{2} + 6 \, {\left (c^{2} e^{2} f^{2} + {\left (c^{2} d e - 2 \, b c e^{2}\right )} f g + {\left (c^{2} d^{2} - 2 \, b c d e + {\left (b^{2} + 2 \, a c\right )} e^{2}\right )} g^{2}\right )} x}{6 \, e^{3} g^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.51, size = 266, normalized size = 1.45 \[ x\,\left (\frac {b^2+2\,a\,c}{e\,g}+\frac {\left (\frac {c^2\,\left (d\,g+e\,f\right )}{e^2\,g^2}-\frac {2\,b\,c}{e\,g}\right )\,\left (d\,g+e\,f\right )}{e\,g}-\frac {c^2\,d\,f}{e^2\,g^2}\right )-x^2\,\left (\frac {c^2\,\left (d\,g+e\,f\right )}{2\,e^2\,g^2}-\frac {b\,c}{e\,g}\right )+\frac {\ln \left (d+e\,x\right )\,\left (e^2\,\left (b^2\,d^2+2\,a\,c\,d^2\right )+a^2\,e^4+c^2\,d^4-2\,a\,b\,d\,e^3-2\,b\,c\,d^3\,e\right )}{e^5\,f-d\,e^4\,g}+\frac {\ln \left (f+g\,x\right )\,\left (g^2\,\left (b^2\,f^2+2\,a\,c\,f^2\right )+a^2\,g^4+c^2\,f^4-2\,a\,b\,f\,g^3-2\,b\,c\,f^3\,g\right )}{d\,g^5-e\,f\,g^4}+\frac {c^2\,x^3}{3\,e\,g} \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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