Optimal. Leaf size=130 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) (a f (3 c f+d e)+b e (c f+3 d e))}{8 e^{5/2} f^{5/2}}-\frac {x (b e (c f+3 d e)-a f (3 c f+d e))}{8 e^2 f^2 \left (e+f x^2\right )}-\frac {x \left (a+b x^2\right ) (d e-c f)}{4 e f \left (e+f x^2\right )^2} \]
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 130, 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, 385, 205} \begin {gather*} -\frac {x (b e (c f+3 d e)-a f (3 c f+d e))}{8 e^2 f^2 \left (e+f x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) (a f (3 c f+d e)+b e (c f+3 d e))}{8 e^{5/2} f^{5/2}}-\frac {x \left (a+b x^2\right ) (d e-c f)}{4 e f \left (e+f x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 205
Rule 385
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 )^3} \, dx &=-\frac {(d e-c f) x \left (a+b x^2\right )}{4 e f \left (e+f x^2\right )^2}-\frac {\int \frac {-a (d e+3 c f)-b (3 d e+c f) x^2}{\left (e+f x^2\right )^2} \, dx}{4 e f}\\ &=-\frac {(d e-c f) x \left (a+b x^2\right )}{4 e f \left (e+f x^2\right )^2}-\frac {(b e (3 d e+c f)-a f (d e+3 c f)) x}{8 e^2 f^2 \left (e+f x^2\right )}+\frac {(b e (3 d e+c f)+a f (d e+3 c f)) \int \frac {1}{e+f x^2} \, dx}{8 e^2 f^2}\\ &=-\frac {(d e-c f) x \left (a+b x^2\right )}{4 e f \left (e+f x^2\right )^2}-\frac {(b e (3 d e+c f)-a f (d e+3 c f)) x}{8 e^2 f^2 \left (e+f x^2\right )}+\frac {(b e (3 d e+c f)+a f (d e+3 c f)) \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{8 e^{5/2} f^{5/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.08, size = 130, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) (a f (3 c f+d e)+b e (c f+3 d e))}{8 e^{5/2} f^{5/2}}+\frac {x (a f (3 c f+d e)+b e (c f-5 d e))}{8 e^2 f^2 \left (e+f x^2\right )}+\frac {x (b e-a f) (d e-c f)}{4 e f^2 \left (e+f x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
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 )^3} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.34, size = 471, normalized size = 3.62 \begin {gather*} \left [-\frac {2 \, {\left (5 \, b d e^{3} f^{2} - 3 \, a c e f^{4} - {\left (b c + a d\right )} e^{2} f^{3}\right )} x^{3} + {\left (3 \, b d e^{4} + 3 \, a c e^{2} f^{2} + {\left (b c + a d\right )} e^{3} f + {\left (3 \, b d e^{2} f^{2} + 3 \, a c f^{4} + {\left (b c + a d\right )} e f^{3}\right )} x^{4} + 2 \, {\left (3 \, b d e^{3} f + 3 \, a c e f^{3} + {\left (b c + a d\right )} e^{2} 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^{4} f - 5 \, a c e^{2} f^{3} + {\left (b c + a d\right )} e^{3} f^{2}\right )} x}{16 \, {\left (e^{3} f^{5} x^{4} + 2 \, e^{4} f^{4} x^{2} + e^{5} f^{3}\right )}}, -\frac {{\left (5 \, b d e^{3} f^{2} - 3 \, a c e f^{4} - {\left (b c + a d\right )} e^{2} f^{3}\right )} x^{3} - {\left (3 \, b d e^{4} + 3 \, a c e^{2} f^{2} + {\left (b c + a d\right )} e^{3} f + {\left (3 \, b d e^{2} f^{2} + 3 \, a c f^{4} + {\left (b c + a d\right )} e f^{3}\right )} x^{4} + 2 \, {\left (3 \, b d e^{3} f + 3 \, a c e f^{3} + {\left (b c + a d\right )} e^{2} f^{2}\right )} x^{2}\right )} \sqrt {e f} \arctan \left (\frac {\sqrt {e f} x}{e}\right ) + {\left (3 \, b d e^{4} f - 5 \, a c e^{2} f^{3} + {\left (b c + a d\right )} e^{3} f^{2}\right )} x}{8 \, {\left (e^{3} f^{5} x^{4} + 2 \, e^{4} f^{4} x^{2} + e^{5} f^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.46, size = 135, normalized size = 1.04 \begin {gather*} \frac {{\left (3 \, 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 {5}{2}\right )}}{8 \, f^{\frac {5}{2}}} + \frac {{\left (3 \, a c f^{3} x^{3} + b c f^{2} x^{3} e + a d f^{2} x^{3} e - 5 \, b d f x^{3} e^{2} + 5 \, a c f^{2} x e - b c f x e^{2} - a d f x e^{2} - 3 \, b d x e^{3}\right )} e^{\left (-2\right )}}{8 \, {\left (f x^{2} + e\right )}^{2} f^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 175, normalized size = 1.35 \begin {gather*} \frac {3 a c \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{8 \sqrt {e f}\, e^{2}}+\frac {a d \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{8 \sqrt {e f}\, e f}+\frac {b c \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{8 \sqrt {e f}\, e f}+\frac {3 b d \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{8 \sqrt {e f}\, f^{2}}+\frac {\frac {\left (3 a c \,f^{2}+a d e f +b c e f -5 b d \,e^{2}\right ) x^{3}}{8 e^{2} f}+\frac {\left (5 a c \,f^{2}-a d e f -b c e f -3 b d \,e^{2}\right ) x}{8 e \,f^{2}}}{\left (f \,x^{2}+e \right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.43, size = 145, normalized size = 1.12 \begin {gather*} -\frac {{\left (5 \, b d e^{2} f - 3 \, a c f^{3} - {\left (b c + a d\right )} e f^{2}\right )} x^{3} + {\left (3 \, b d e^{3} - 5 \, a c e f^{2} + {\left (b c + a d\right )} e^{2} f\right )} x}{8 \, {\left (e^{2} f^{4} x^{4} + 2 \, e^{3} f^{3} x^{2} + e^{4} f^{2}\right )}} + \frac {{\left (3 \, b d e^{2} + 3 \, a c f^{2} + {\left (b c + a d\right )} e f\right )} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{8 \, \sqrt {e f} e^{2} f^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.97, size = 136, normalized size = 1.05 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {f}\,x}{\sqrt {e}}\right )\,\left (3\,a\,c\,f^2+3\,b\,d\,e^2+a\,d\,e\,f+b\,c\,e\,f\right )}{8\,e^{5/2}\,f^{5/2}}-\frac {\frac {x\,\left (3\,b\,d\,e^2-5\,a\,c\,f^2+a\,d\,e\,f+b\,c\,e\,f\right )}{8\,e\,f^2}-\frac {x^3\,\left (3\,a\,c\,f^2-5\,b\,d\,e^2+a\,d\,e\,f+b\,c\,e\,f\right )}{8\,e^2\,f}}{e^2+2\,e\,f\,x^2+f^2\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 3.13, size = 246, normalized size = 1.89 \begin {gather*} - \frac {\sqrt {- \frac {1}{e^{5} f^{5}}} \left (3 a c f^{2} + a d e f + b c e f + 3 b d e^{2}\right ) \log {\left (- e^{3} f^{2} \sqrt {- \frac {1}{e^{5} f^{5}}} + x \right )}}{16} + \frac {\sqrt {- \frac {1}{e^{5} f^{5}}} \left (3 a c f^{2} + a d e f + b c e f + 3 b d e^{2}\right ) \log {\left (e^{3} f^{2} \sqrt {- \frac {1}{e^{5} f^{5}}} + x \right )}}{16} + \frac {x^{3} \left (3 a c f^{3} + a d e f^{2} + b c e f^{2} - 5 b d e^{2} f\right ) + x \left (5 a c e f^{2} - a d e^{2} f - b c e^{2} f - 3 b d e^{3}\right )}{8 e^{4} f^{2} + 16 e^{3} f^{3} x^{2} + 8 e^{2} f^{4} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________