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 = 0.11, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {1629, 635, 205, 260} \begin {gather*} -\frac {\log \left (d+f x^2\right ) (-a B f-A b f+B c d)}{2 f^2}-\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 {x (A c+b B)}{f}+\frac {B c x^2}{2 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 260
Rule 635
Rule 1629
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+b x+c x^2\right )}{d+f x^2} \, dx &=\int \left (\frac {b B+A c}{f}+\frac {B c x}{f}-\frac {b B d+A c d-a A f+(B c d-A b f-a B f) x}{f \left (d+f x^2\right )}\right ) \, dx\\ &=\frac {(b B+A c) x}{f}+\frac {B c x^2}{2 f}-\frac {\int \frac {b B d+A c d-a A f+(B c d-A b f-a B f) x}{d+f x^2} \, dx}{f}\\ &=\frac {(b B+A c) x}{f}+\frac {B c x^2}{2 f}-\frac {(b B d+A c d-a A f) \int \frac {1}{d+f x^2} \, dx}{f}-\frac {(B c d-A b f-a B f) \int \frac {x}{d+f x^2} \, dx}{f}\\ &=\frac {(b B+A c) x}{f}+\frac {B c x^2}{2 f}-\frac {(b B d+A c d-a A f) \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {d}}\right )}{\sqrt {d} f^{3/2}}-\frac {(B c d-A b f-a B f) \log \left (d+f x^2\right )}{2 f^2}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 86, normalized size = 0.91 \begin {gather*} \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} \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 )}{d+f x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.40, size = 200, normalized size = 2.13 \begin {gather*} \left [\frac {B c d f x^{2} + 2 \, {\left (B b + A c\right )} d f x - {\left (A a f - {\left (B b + A c\right )} d\right )} \sqrt {-d f} \log \left (\frac {f x^{2} - 2 \, \sqrt {-d f} x - d}{f x^{2} + d}\right ) - {\left (B c d^{2} - {\left (B a + A b\right )} d f\right )} \log \left (f x^{2} + d\right )}{2 \, d f^{2}}, \frac {B c d f x^{2} + 2 \, {\left (B b + A c\right )} d f x + 2 \, {\left (A a f - {\left (B b + A c\right )} d\right )} \sqrt {d f} \arctan \left (\frac {\sqrt {d f} x}{d}\right ) - {\left (B c d^{2} - {\left (B a + A b\right )} d f\right )} \log \left (f x^{2} + d\right )}{2 \, d f^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 87, normalized size = 0.93 \begin {gather*} -\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 )} \log \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 133, normalized size = 1.41 \begin {gather*} \frac {A a \arctan \left (\frac {f x}{\sqrt {d f}}\right )}{\sqrt {d f}}-\frac {A c d \arctan \left (\frac {f x}{\sqrt {d f}}\right )}{\sqrt {d f}\, f}-\frac {B b d \arctan \left (\frac {f x}{\sqrt {d f}}\right )}{\sqrt {d f}\, f}+\frac {B c \,x^{2}}{2 f}+\frac {A b \ln \left (f \,x^{2}+d \right )}{2 f}+\frac {A c x}{f}+\frac {B a \ln \left (f \,x^{2}+d \right )}{2 f}+\frac {B b x}{f}-\frac {B c d \ln \left (f \,x^{2}+d \right )}{2 f^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 84, normalized size = 0.89 \begin {gather*} \frac {{\left (A a f - {\left (B b + A c\right )} d\right )} \arctan \left (\frac {f x}{\sqrt {d f}}\right )}{\sqrt {d f} f} + \frac {B c x^{2} + 2 \, {\left (B b + A c\right )} x}{2 \, f} - \frac {{\left (B c d - {\left (B a + A b\right )} f\right )} \log \left (f x^{2} + d\right )}{2 \, f^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.44, size = 97, normalized size = 1.03 \begin {gather*} \frac {x\,\left (A\,c+B\,b\right )}{f}-\frac {\mathrm {atan}\left (\frac {\sqrt {f}\,x}{\sqrt {d}}\right )\,\left (A\,c\,d-A\,a\,f+B\,b\,d\right )}{\sqrt {d}\,f^{3/2}}+\frac {B\,c\,x^2}{2\,f}+\frac {\ln \left (f\,x^2+d\right )\,\left (4\,A\,b\,d\,f^3+4\,B\,a\,d\,f^3-4\,B\,c\,d^2\,f^2\right )}{8\,d\,f^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.85, size = 333, normalized size = 3.54 \begin {gather*} \frac {B c x^{2}}{2 f} + x \left (\frac {A c}{f} + \frac {B b}{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 )} + \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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