Optimal. Leaf size=94 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{d}}\right ) (-a A f+A c d+b B d)}{\sqrt{d} f^{3/2}}-\frac{\log \left (d+f x^2\right ) (-a B f-A b f+B c d)}{2 f^2}+\frac{x (A c+b B)}{f}+\frac{B c x^2}{2 f} \]
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Rubi [A] time = 0.233197, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{d}}\right ) (-a A f+A c d+b B d)}{\sqrt{d} f^{3/2}}-\frac{\log \left (d+f x^2\right ) (-a B f-A b f+B c d)}{2 f^2}+\frac{x (A c+b B)}{f}+\frac{B c x^2}{2 f} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a + b*x + c*x^2))/(d + f*x^2),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{B c \int x\, dx}{f} + \left (A c + B b\right ) \int \frac{1}{f}\, dx + \frac{\left (A b f + B a f - B c d\right ) \log{\left (d + f x^{2} \right )}}{2 f^{2}} + \frac{\left (A a f - A c d - B b d\right ) \operatorname{atan}{\left (\frac{\sqrt{f} x}{\sqrt{d}} \right )}}{\sqrt{d} f^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+b*x+a)/(f*x**2+d),x)
[Out]
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Mathematica [A] time = 0.143179, size = 86, normalized size = 0.91 \[ \frac{\log \left (d+f x^2\right ) (a B f+A b f-B c d)-\frac{2 \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{d}}\right ) (-a A f+A c d+b B d)}{\sqrt{d}}+f x (2 A c+2 b B+B c x)}{2 f^2} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a + b*x + c*x^2))/(d + f*x^2),x]
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Maple [A] time = 0.01, size = 133, normalized size = 1.4 \[{\frac{Bc{x}^{2}}{2\,f}}+{\frac{Acx}{f}}+{\frac{bBx}{f}}+{\frac{\ln \left ( f{x}^{2}+d \right ) Ab}{2\,f}}+{\frac{\ln \left ( f{x}^{2}+d \right ) aB}{2\,f}}-{\frac{\ln \left ( f{x}^{2}+d \right ) Bcd}{2\,{f}^{2}}}+{Aa\arctan \left ({fx{\frac{1}{\sqrt{df}}}} \right ){\frac{1}{\sqrt{df}}}}-{\frac{Acd}{f}\arctan \left ({fx{\frac{1}{\sqrt{df}}}} \right ){\frac{1}{\sqrt{df}}}}-{\frac{Bbd}{f}\arctan \left ({fx{\frac{1}{\sqrt{df}}}} \right ){\frac{1}{\sqrt{df}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+b*x+a)/(f*x^2+d),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)*(B*x + A)/(f*x^2 + d),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.270519, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (A a f^{2} -{\left (B b + A c\right )} d f\right )} \log \left (\frac{2 \, d f x +{\left (f x^{2} - d\right )} \sqrt{-d f}}{f x^{2} + d}\right ) +{\left (B c f x^{2} + 2 \,{\left (B b + A c\right )} f x -{\left (B c d -{\left (B a + A b\right )} f\right )} \log \left (f x^{2} + d\right )\right )} \sqrt{-d f}}{2 \, \sqrt{-d f} f^{2}}, \frac{2 \,{\left (A a f^{2} -{\left (B b + A c\right )} d f\right )} \arctan \left (\frac{\sqrt{d f} x}{d}\right ) +{\left (B c f x^{2} + 2 \,{\left (B b + A c\right )} f x -{\left (B c d -{\left (B a + A b\right )} f\right )} \log \left (f x^{2} + d\right )\right )} \sqrt{d f}}{2 \, \sqrt{d f} f^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)*(B*x + A)/(f*x^2 + d),x, algorithm="fricas")
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Sympy [A] time = 3.84803, size = 332, normalized size = 3.53 \[ \frac{B c x^{2}}{2 f} + \left (\frac{A b f + B a f - B c d}{2 f^{2}} - \frac{\sqrt{- d f^{5}} \left (A a f - A c d - B b d\right )}{2 d f^{4}}\right ) \log{\left (x + \frac{- A b d f - B a d f + B c d^{2} + 2 d f^{2} \left (\frac{A b f + B a f - B c d}{2 f^{2}} - \frac{\sqrt{- d f^{5}} \left (A a f - A c d - B b d\right )}{2 d f^{4}}\right )}{A a f^{2} - A c d f - B b d f} \right )} + \left (\frac{A b f + B a f - B c d}{2 f^{2}} + \frac{\sqrt{- d f^{5}} \left (A a f - A c d - B b d\right )}{2 d f^{4}}\right ) \log{\left (x + \frac{- A b d f - B a d f + B c d^{2} + 2 d f^{2} \left (\frac{A b f + B a f - B c d}{2 f^{2}} + \frac{\sqrt{- d f^{5}} \left (A a f - A c d - B b d\right )}{2 d f^{4}}\right )}{A a f^{2} - A c d f - B b d f} \right )} + \frac{x \left (A c + B b\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+b*x+a)/(f*x**2+d),x)
[Out]
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GIAC/XCAS [A] time = 0.26355, size = 117, normalized size = 1.24 \[ -\frac{{\left (B b d + A c d - A a f\right )} \arctan \left (\frac{f x}{\sqrt{d f}}\right )}{\sqrt{d f} f} - \frac{{\left (B c d - B a f - A b f\right )}{\rm ln}\left (f x^{2} + d\right )}{2 \, f^{2}} + \frac{B c f x^{2} + 2 \, B b f x + 2 \, A c f x}{2 \, f^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)*(B*x + A)/(f*x^2 + d),x, algorithm="giac")
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