Optimal. Leaf size=108 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) (b e (3 d e-c f)-a f (c f+d e))}{2 e^{3/2} f^{5/2}}-\frac {x \left (a+b x^2\right ) (d e-c f)}{2 e f \left (e+f x^2\right )}+\frac {b x (3 d e-c f)}{2 e f^2} \]
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Rubi [A] time = 0.08, 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 = {526, 388, 205} \begin {gather*} -\frac {\tan ^{-1}\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 205
Rule 388
Rule 526
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 = 0.06, size = 95, normalized size = 0.88 \begin {gather*} -\frac {\tan ^{-1}\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 (b e-a f) (d e-c f)}{2 e f^2 \left (e+f x^2\right )}+\frac {b d x}{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 {\left (a+b x^2\right ) \left (c+d x^2\right )}{\left (e+f x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.44, size = 318, normalized size = 2.94 \begin {gather*} \left [\frac {4 \, b d e^{2} f^{2} x^{3} + {\left (3 \, b d e^{3} - a c e f^{2} - {\left (b c + a d\right )} e^{2} f + {\left (3 \, b d e^{2} f - a c f^{3} - {\left (b c + a d\right )} e f^{2}\right )} x^{2}\right )} \sqrt {-e f} \log \left (\frac {f x^{2} - 2 \, \sqrt {-e f} x - e}{f x^{2} + e}\right ) + 2 \, {\left (3 \, b d e^{3} f + a c e f^{3} - {\left (b c + a d\right )} e^{2} f^{2}\right )} x}{4 \, {\left (e^{2} f^{4} x^{2} + e^{3} f^{3}\right )}}, \frac {2 \, b d e^{2} f^{2} x^{3} - {\left (3 \, b d e^{3} - a c e f^{2} - {\left (b c + a d\right )} e^{2} f + {\left (3 \, b d e^{2} f - a c f^{3} - {\left (b c + a d\right )} e f^{2}\right )} x^{2}\right )} \sqrt {e f} \arctan \left (\frac {\sqrt {e f} x}{e}\right ) + {\left (3 \, b d e^{3} f + a c e f^{3} - {\left (b c + a d\right )} e^{2} f^{2}\right )} x}{2 \, {\left (e^{2} f^{4} x^{2} + e^{3} f^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, 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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maple [A] time = 0.01, size = 163, normalized size = 1.51 \begin {gather*} \frac {a c x}{2 \left (f \,x^{2}+e \right ) e}+\frac {a c \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{2 \sqrt {e f}\, e}-\frac {a d x}{2 \left (f \,x^{2}+e \right ) f}+\frac {a d \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{2 \sqrt {e f}\, f}-\frac {b c x}{2 \left (f \,x^{2}+e \right ) f}+\frac {b c \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{2 \sqrt {e f}\, f}+\frac {b d e x}{2 \left (f \,x^{2}+e \right ) f^{2}}-\frac {3 b d e \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{2 \sqrt {e f}\, f^{2}}+\frac {b d x}{f^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.20, size = 101, normalized size = 0.94 \begin {gather*} \frac {{\left (b d e^{2} + a c f^{2} - {\left (b c + a d\right )} e f\right )} x}{2 \, {\left (e f^{3} x^{2} + e^{2} f^{2}\right )}} + \frac {b d x}{f^{2}} - \frac {{\left (3 \, b d e^{2} - a c f^{2} - {\left (b c + a d\right )} e f\right )} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{2 \, \sqrt {e f} e 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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sympy [A] time = 1.37, 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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