Optimal. Leaf size=108 \[ \frac {b (3 d e-c f) x}{2 e f^2}-\frac {(d e-c f) x \left (a+b x^2\right )}{2 e f \left (e+f x^2\right )}-\frac {(b e (3 d e-c f)-a f (d e+c f)) \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{2 e^{3/2} f^{5/2}} \]
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Rubi [A]
time = 0.06, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {540, 396, 211}
\begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) (b e (3 d e-c f)-a f (c f+d e))}{2 e^{3/2} f^{5/2}}-\frac {x \left (a+b x^2\right ) (d e-c f)}{2 e f \left (e+f x^2\right )}+\frac {b x (3 d e-c f)}{2 e f^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 396
Rule 540
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )}{\left (e+f x^2\right )^2} \, dx &=-\frac {(d e-c f) x \left (a+b x^2\right )}{2 e f \left (e+f x^2\right )}-\frac {\int \frac {-a (d e+c f)-b (3 d e-c f) x^2}{e+f x^2} \, dx}{2 e f}\\ &=\frac {b (3 d e-c f) x}{2 e f^2}-\frac {(d e-c f) x \left (a+b x^2\right )}{2 e f \left (e+f x^2\right )}-\frac {(b e (3 d e-c f)-a f (d e+c f)) \int \frac {1}{e+f x^2} \, dx}{2 e f^2}\\ &=\frac {b (3 d e-c f) x}{2 e f^2}-\frac {(d e-c f) x \left (a+b x^2\right )}{2 e f \left (e+f x^2\right )}-\frac {(b e (3 d e-c f)-a f (d e+c f)) \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{2 e^{3/2} f^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 95, normalized size = 0.88 \begin {gather*} \frac {b d x}{f^2}+\frac {(b e-a f) (d e-c f) x}{2 e f^2 \left (e+f x^2\right )}-\frac {(b e (3 d e-c f)-a f (d e+c f)) \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{2 e^{3/2} f^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 97, normalized size = 0.90
method | result | size |
default | \(\frac {b d x}{f^{2}}+\frac {\frac {\left (a c \,f^{2}-a d e f -b c e f +b d \,e^{2}\right ) x}{2 e \left (f \,x^{2}+e \right )}+\frac {\left (a c \,f^{2}+a d e f +b c e f -3 b d \,e^{2}\right ) \arctan \left (\frac {f x}{\sqrt {f e}}\right )}{2 e \sqrt {f e}}}{f^{2}}\) | \(97\) |
risch | \(\frac {b d x}{f^{2}}+\frac {\left (a c \,f^{2}-a d e f -b c e f +b d \,e^{2}\right ) x}{2 e \,f^{2} \left (f \,x^{2}+e \right )}-\frac {\ln \left (f x +\sqrt {-f e}\right ) a c}{4 \sqrt {-f e}\, e}-\frac {\ln \left (f x +\sqrt {-f e}\right ) a d}{4 f \sqrt {-f e}}-\frac {\ln \left (f x +\sqrt {-f e}\right ) b c}{4 f \sqrt {-f e}}+\frac {3 e \ln \left (f x +\sqrt {-f e}\right ) b d}{4 f^{2} \sqrt {-f e}}+\frac {\ln \left (-f x +\sqrt {-f e}\right ) a c}{4 \sqrt {-f e}\, e}+\frac {\ln \left (-f x +\sqrt {-f e}\right ) a d}{4 f \sqrt {-f e}}+\frac {\ln \left (-f x +\sqrt {-f e}\right ) b c}{4 f \sqrt {-f e}}-\frac {3 e \ln \left (-f x +\sqrt {-f e}\right ) b d}{4 f^{2} \sqrt {-f e}}\) | \(250\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 96, normalized size = 0.89 \begin {gather*} \frac {{\left (a c f^{2} + b d e^{2} - {\left (b c e + a d e\right )} f\right )} x}{2 \, {\left (f^{3} x^{2} e + f^{2} e^{2}\right )}} + \frac {b d x}{f^{2}} + \frac {{\left (a c f^{2} - 3 \, b d e^{2} + {\left (b c e + a d e\right )} f\right )} \arctan \left (\sqrt {f} x e^{\left (-\frac {1}{2}\right )}\right ) e^{\left (-\frac {3}{2}\right )}}{2 \, f^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.67, size = 313, normalized size = 2.90 \begin {gather*} \left [\frac {2 \, a c f^{3} x e + 6 \, b d f x e^{3} + {\left (a c f^{3} x^{2} - 3 \, b d e^{3} - {\left (3 \, b d f x^{2} - {\left (b c + a d\right )} f\right )} e^{2} + {\left ({\left (b c + a d\right )} f^{2} x^{2} + a c f^{2}\right )} e\right )} \sqrt {-f e} \log \left (\frac {f x^{2} + 2 \, \sqrt {-f e} x - e}{f x^{2} + e}\right ) + 2 \, {\left (2 \, b d f^{2} x^{3} - {\left (b c + a d\right )} f^{2} x\right )} e^{2}}{4 \, {\left (f^{4} x^{2} e^{2} + f^{3} e^{3}\right )}}, \frac {a c f^{3} x e + 3 \, b d f x e^{3} + {\left (a c f^{3} x^{2} - 3 \, b d e^{3} - {\left (3 \, b d f x^{2} - {\left (b c + a d\right )} f\right )} e^{2} + {\left ({\left (b c + a d\right )} f^{2} x^{2} + a c f^{2}\right )} e\right )} \sqrt {f} \arctan \left (\sqrt {f} x e^{\left (-\frac {1}{2}\right )}\right ) e^{\frac {1}{2}} + {\left (2 \, b d f^{2} x^{3} - {\left (b c + a d\right )} f^{2} x\right )} e^{2}}{2 \, {\left (f^{4} x^{2} e^{2} + f^{3} e^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.73, size = 190, normalized size = 1.76 \begin {gather*} \frac {b d x}{f^{2}} + \frac {x \left (a c f^{2} - a d e f - b c e f + b d e^{2}\right )}{2 e^{2} f^{2} + 2 e f^{3} x^{2}} - \frac {\sqrt {- \frac {1}{e^{3} f^{5}}} \left (a c f^{2} + a d e f + b c e f - 3 b d e^{2}\right ) \log {\left (- e^{2} f^{2} \sqrt {- \frac {1}{e^{3} f^{5}}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{e^{3} f^{5}}} \left (a c f^{2} + a d e f + b c e f - 3 b d e^{2}\right ) \log {\left (e^{2} f^{2} \sqrt {- \frac {1}{e^{3} f^{5}}} + x \right )}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.28, size = 95, normalized size = 0.88 \begin {gather*} \frac {b d x}{f^{2}} + \frac {{\left (a c f^{2} + b c f e + a d f e - 3 \, b d e^{2}\right )} \arctan \left (\sqrt {f} x e^{\left (-\frac {1}{2}\right )}\right ) e^{\left (-\frac {3}{2}\right )}}{2 \, f^{\frac {5}{2}}} + \frac {{\left (a c f^{2} x - b c f x e - a d f x e + b d x e^{2}\right )} e^{\left (-1\right )}}{2 \, {\left (f x^{2} + e\right )} f^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.17, size = 95, normalized size = 0.88 \begin {gather*} \frac {b\,d\,x}{f^2}+\frac {\mathrm {atan}\left (\frac {\sqrt {f}\,x}{\sqrt {e}}\right )\,\left (a\,c\,f^2-3\,b\,d\,e^2+a\,d\,e\,f+b\,c\,e\,f\right )}{2\,e^{3/2}\,f^{5/2}}+\frac {x\,\left (a\,c\,f^2+b\,d\,e^2-a\,d\,e\,f-b\,c\,e\,f\right )}{2\,e\,\left (f^3\,x^2+e\,f^2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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