Optimal. Leaf size=274 \[ \frac{\log \left (a+b x+c x^2\right ) (-a B f+A b f+B c d)}{2 \left (f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2\right )}-\frac{\log \left (d+f x^2\right ) (-a B f+A b f+B c d)}{2 \left (f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2\right )}+\frac{\sqrt{f} \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{d}}\right ) (a A f-A c d+b B d)}{\sqrt{d} \left (f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2\right )}-\frac{\tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \left (2 A c (c d-a f)-b B (a f+c d)+A b^2 f\right )}{\sqrt{b^2-4 a c} \left (f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2\right )} \]
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Rubi [A] time = 0.281907, antiderivative size = 274, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {1023, 634, 618, 206, 628, 635, 205, 260} \[ \frac{\log \left (a+b x+c x^2\right ) (-a B f+A b f+B c d)}{2 \left (f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2\right )}-\frac{\log \left (d+f x^2\right ) (-a B f+A b f+B c d)}{2 \left (f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2\right )}+\frac{\sqrt{f} \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{d}}\right ) (a A f-A c d+b B d)}{\sqrt{d} \left (f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2\right )}-\frac{\tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \left (2 A c (c d-a f)-b B (a f+c d)+A b^2 f\right )}{\sqrt{b^2-4 a c} \left (f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2\right )} \]
Antiderivative was successfully verified.
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Rule 1023
Rule 634
Rule 618
Rule 206
Rule 628
Rule 635
Rule 205
Rule 260
Rubi steps
\begin{align*} \int \frac{A+B x}{\left (a+b x+c x^2\right ) \left (d+f x^2\right )} \, dx &=\frac{\int \frac{-a b B f+A \left (c^2 d+b^2 f-a c f\right )+c (B c d+A b f-a B f) x}{a+b x+c x^2} \, dx}{c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )}+\frac{\int \frac{f (b B d-A c d+a A f)-f (B c d+A b f-a B f) x}{d+f x^2} \, dx}{c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )}\\ &=\frac{(f (b B d-A c d+a A f)) \int \frac{1}{d+f x^2} \, dx}{c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )}+\frac{(B c d+A b f-a B f) \int \frac{b+2 c x}{a+b x+c x^2} \, dx}{2 \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )}-\frac{(f (B c d+A b f-a B f)) \int \frac{x}{d+f x^2} \, dx}{c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )}+\frac{\left (A b^2 f+2 A c (c d-a f)-b B (c d+a f)\right ) \int \frac{1}{a+b x+c x^2} \, dx}{2 \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )}\\ &=\frac{\sqrt{f} (b B d-A c d+a A f) \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{d}}\right )}{\sqrt{d} \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )}+\frac{(B c d+A b f-a B f) \log \left (a+b x+c x^2\right )}{2 \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )}-\frac{(B c d+A b f-a B f) \log \left (d+f x^2\right )}{2 \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )}-\frac{\left (A b^2 f+2 A c (c d-a f)-b B (c d+a 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^2-2 a c d f+f \left (b^2 d+a^2 f\right )}\\ &=\frac{\sqrt{f} (b B d-A c d+a A f) \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{d}}\right )}{\sqrt{d} \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )}-\frac{\left (A b^2 f+2 A c (c d-a f)-b B (c d+a 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^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )}+\frac{(B c d+A b f-a B f) \log \left (a+b x+c x^2\right )}{2 \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )}-\frac{(B c d+A b f-a B f) \log \left (d+f x^2\right )}{2 \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )}\\ \end{align*}
Mathematica [A] time = 0.404446, size = 212, normalized size = 0.77 \[ \frac{\sqrt{d} \left (\sqrt{4 a c-b^2} (-a B f+A b f+B c d) \left (\log (a+x (b+c x))-\log \left (d+f x^2\right )\right )+2 \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right ) \left (2 A c (c d-a f)-b B (a f+c d)+A b^2 f\right )\right )+2 \sqrt{f} \sqrt{4 a c-b^2} \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{d}}\right ) (a A f-A c d+b B d)}{2 \sqrt{d} \sqrt{4 a c-b^2} \left (f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2\right )} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.199, size = 745, normalized size = 2.7 \begin{align*}{\frac{\ln \left ( c{x}^{2}+bx+a \right ) Abf}{2\,{a}^{2}{f}^{2}-4\,acdf+2\,{b}^{2}df+2\,{c}^{2}{d}^{2}}}-{\frac{\ln \left ( c{x}^{2}+bx+a \right ) Baf}{2\,{a}^{2}{f}^{2}-4\,acdf+2\,{b}^{2}df+2\,{c}^{2}{d}^{2}}}+{\frac{c\ln \left ( c{x}^{2}+bx+a \right ) Bd}{2\,{a}^{2}{f}^{2}-4\,acdf+2\,{b}^{2}df+2\,{c}^{2}{d}^{2}}}-2\,{\frac{Aacf}{ \left ({a}^{2}{f}^{2}-2\,acdf+{b}^{2}df+{c}^{2}{d}^{2} \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+{\frac{A{b}^{2}f}{{a}^{2}{f}^{2}-2\,acdf+{b}^{2}df+{c}^{2}{d}^{2}}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+2\,{\frac{A{c}^{2}d}{ \left ({a}^{2}{f}^{2}-2\,acdf+{b}^{2}df+{c}^{2}{d}^{2} \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{Babf}{{a}^{2}{f}^{2}-2\,acdf+{b}^{2}df+{c}^{2}{d}^{2}}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{Bbcd}{{a}^{2}{f}^{2}-2\,acdf+{b}^{2}df+{c}^{2}{d}^{2}}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{f\ln \left ( f{x}^{2}+d \right ) Ab}{2\,{a}^{2}{f}^{2}-4\,acdf+2\,{b}^{2}df+2\,{c}^{2}{d}^{2}}}+{\frac{f\ln \left ( f{x}^{2}+d \right ) aB}{2\,{a}^{2}{f}^{2}-4\,acdf+2\,{b}^{2}df+2\,{c}^{2}{d}^{2}}}-{\frac{\ln \left ( f{x}^{2}+d \right ) Bcd}{2\,{a}^{2}{f}^{2}-4\,acdf+2\,{b}^{2}df+2\,{c}^{2}{d}^{2}}}+{\frac{Aa{f}^{2}}{{a}^{2}{f}^{2}-2\,acdf+{b}^{2}df+{c}^{2}{d}^{2}}\arctan \left ({fx{\frac{1}{\sqrt{df}}}} \right ){\frac{1}{\sqrt{df}}}}-{\frac{Acdf}{{a}^{2}{f}^{2}-2\,acdf+{b}^{2}df+{c}^{2}{d}^{2}}\arctan \left ({fx{\frac{1}{\sqrt{df}}}} \right ){\frac{1}{\sqrt{df}}}}+{\frac{Bbdf}{{a}^{2}{f}^{2}-2\,acdf+{b}^{2}df+{c}^{2}{d}^{2}}\arctan \left ({fx{\frac{1}{\sqrt{df}}}} \right ){\frac{1}{\sqrt{df}}}} \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 [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23739, size = 359, normalized size = 1.31 \begin{align*} \frac{{\left (B c d - B a f + A b f\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}\right )}} - \frac{{\left (B c d - B a f + A b f\right )} \log \left (f x^{2} + d\right )}{2 \,{\left (c^{2} d^{2} + b^{2} d f - 2 \, a c d f + a^{2} f^{2}\right )}} + \frac{{\left (B b d f - A c d f + A a f^{2}\right )} \arctan \left (\frac{f x}{\sqrt{d f}}\right )}{{\left (c^{2} d^{2} + b^{2} d f - 2 \, a c d f + a^{2} f^{2}\right )} \sqrt{d f}} - \frac{{\left (B b c d - 2 \, A c^{2} d + B a b f - A b^{2} f + 2 \, A a c f\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}\right )} \sqrt{-b^{2} + 4 \, a c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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