Optimal. Leaf size=240 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a f \left (5 c^2 f^2+2 c d e f+d^2 e^2\right )+b e \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )}{16 e^{7/2} f^{7/2}}-\frac {x \left (a f \left (-15 c^2 f^2+4 c d e f+3 d^2 e^2\right )+b e \left (-3 c^2 f^2-4 c d e f+15 d^2 e^2\right )\right )}{48 e^3 f^3 \left (e+f x^2\right )}-\frac {x \left (c+d x^2\right ) (d e (a f+5 b e)-c f (5 a f+b e))}{24 e^2 f^2 \left (e+f x^2\right )^2}-\frac {x \left (c+d x^2\right )^2 (b e-a f)}{6 e f \left (e+f x^2\right )^3} \]
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Rubi [A] time = 0.28, antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {526, 385, 205} \begin {gather*} -\frac {x \left (a f \left (-15 c^2 f^2+4 c d e f+3 d^2 e^2\right )+b e \left (-3 c^2 f^2-4 c d e f+15 d^2 e^2\right )\right )}{48 e^3 f^3 \left (e+f x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a f \left (5 c^2 f^2+2 c d e f+d^2 e^2\right )+b e \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )}{16 e^{7/2} f^{7/2}}-\frac {x \left (c+d x^2\right ) (d e (a f+5 b e)-c f (5 a f+b e))}{24 e^2 f^2 \left (e+f x^2\right )^2}-\frac {x \left (c+d x^2\right )^2 (b e-a f)}{6 e f \left (e+f x^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 385
Rule 526
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )^2}{\left (e+f x^2\right )^4} \, dx &=-\frac {(b e-a f) x \left (c+d x^2\right )^2}{6 e f \left (e+f x^2\right )^3}-\frac {\int \frac {\left (c+d x^2\right ) \left (-c (b e+5 a f)-d (5 b e+a f) x^2\right )}{\left (e+f x^2\right )^3} \, dx}{6 e f}\\ &=-\frac {(b e-a f) x \left (c+d x^2\right )^2}{6 e f \left (e+f x^2\right )^3}-\frac {(d e (5 b e+a f)-c f (b e+5 a f)) x \left (c+d x^2\right )}{24 e^2 f^2 \left (e+f x^2\right )^2}+\frac {\int \frac {c (d e (5 b e+a f)+3 c f (b e+5 a f))+d (b e (15 d e+c f)+a f (3 d e+5 c f)) x^2}{\left (e+f x^2\right )^2} \, dx}{24 e^2 f^2}\\ &=-\frac {(b e-a f) x \left (c+d x^2\right )^2}{6 e f \left (e+f x^2\right )^3}-\frac {(d e (5 b e+a f)-c f (b e+5 a f)) x \left (c+d x^2\right )}{24 e^2 f^2 \left (e+f x^2\right )^2}-\frac {\left (a f \left (3 d^2 e^2+4 c d e f-15 c^2 f^2\right )+b e \left (15 d^2 e^2-4 c d e f-3 c^2 f^2\right )\right ) x}{48 e^3 f^3 \left (e+f x^2\right )}+\frac {\left (b e \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )+a f \left (d^2 e^2+2 c d e f+5 c^2 f^2\right )\right ) \int \frac {1}{e+f x^2} \, dx}{16 e^3 f^3}\\ &=-\frac {(b e-a f) x \left (c+d x^2\right )^2}{6 e f \left (e+f x^2\right )^3}-\frac {(d e (5 b e+a f)-c f (b e+5 a f)) x \left (c+d x^2\right )}{24 e^2 f^2 \left (e+f x^2\right )^2}-\frac {\left (a f \left (3 d^2 e^2+4 c d e f-15 c^2 f^2\right )+b e \left (15 d^2 e^2-4 c d e f-3 c^2 f^2\right )\right ) x}{48 e^3 f^3 \left (e+f x^2\right )}+\frac {\left (b e \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )+a f \left (d^2 e^2+2 c d e f+5 c^2 f^2\right )\right ) \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{16 e^{7/2} f^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 242, normalized size = 1.01 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a f \left (5 c^2 f^2+2 c d e f+d^2 e^2\right )+b e \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )}{16 e^{7/2} f^{7/2}}+\frac {x \left (a f \left (5 c^2 f^2+2 c d e f+d^2 e^2\right )+b e \left (c^2 f^2+2 c d e f-11 d^2 e^2\right )\right )}{16 e^3 f^3 \left (e+f x^2\right )}+\frac {x (d e-c f) (b e (13 d e-c f)-a f (5 c f+7 d e))}{24 e^2 f^3 \left (e+f x^2\right )^2}-\frac {x (b e-a f) (d e-c f)^2}{6 e f^3 \left (e+f x^2\right )^3} \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 )^2}{\left (e+f x^2\right )^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 1.12, size = 1024, normalized size = 4.27 \begin {gather*} \left [-\frac {6 \, {\left (11 \, b d^{2} e^{4} f^{3} - 5 \, a c^{2} e f^{6} - {\left (2 \, b c d + a d^{2}\right )} e^{3} f^{4} - {\left (b c^{2} + 2 \, a c d\right )} e^{2} f^{5}\right )} x^{5} + 16 \, {\left (5 \, b d^{2} e^{5} f^{2} - 5 \, a c^{2} e^{2} f^{5} + {\left (2 \, b c d + a d^{2}\right )} e^{4} f^{3} - {\left (b c^{2} + 2 \, a c d\right )} e^{3} f^{4}\right )} x^{3} + 3 \, {\left (5 \, b d^{2} e^{6} + 5 \, a c^{2} e^{3} f^{3} + {\left (2 \, b c d + a d^{2}\right )} e^{5} f + {\left (b c^{2} + 2 \, a c d\right )} e^{4} f^{2} + {\left (5 \, b d^{2} e^{3} f^{3} + 5 \, a c^{2} f^{6} + {\left (2 \, b c d + a d^{2}\right )} e^{2} f^{4} + {\left (b c^{2} + 2 \, a c d\right )} e f^{5}\right )} x^{6} + 3 \, {\left (5 \, b d^{2} e^{4} f^{2} + 5 \, a c^{2} e f^{5} + {\left (2 \, b c d + a d^{2}\right )} e^{3} f^{3} + {\left (b c^{2} + 2 \, a c d\right )} e^{2} f^{4}\right )} x^{4} + 3 \, {\left (5 \, b d^{2} e^{5} f + 5 \, a c^{2} e^{2} f^{4} + {\left (2 \, b c d + a d^{2}\right )} e^{4} f^{2} + {\left (b c^{2} + 2 \, a c d\right )} e^{3} f^{3}\right )} x^{2}\right )} \sqrt {-e f} \log \left (\frac {f x^{2} - 2 \, \sqrt {-e f} x - e}{f x^{2} + e}\right ) + 6 \, {\left (5 \, b d^{2} e^{6} f - 11 \, a c^{2} e^{3} f^{4} + {\left (2 \, b c d + a d^{2}\right )} e^{5} f^{2} + {\left (b c^{2} + 2 \, a c d\right )} e^{4} f^{3}\right )} x}{96 \, {\left (e^{4} f^{7} x^{6} + 3 \, e^{5} f^{6} x^{4} + 3 \, e^{6} f^{5} x^{2} + e^{7} f^{4}\right )}}, -\frac {3 \, {\left (11 \, b d^{2} e^{4} f^{3} - 5 \, a c^{2} e f^{6} - {\left (2 \, b c d + a d^{2}\right )} e^{3} f^{4} - {\left (b c^{2} + 2 \, a c d\right )} e^{2} f^{5}\right )} x^{5} + 8 \, {\left (5 \, b d^{2} e^{5} f^{2} - 5 \, a c^{2} e^{2} f^{5} + {\left (2 \, b c d + a d^{2}\right )} e^{4} f^{3} - {\left (b c^{2} + 2 \, a c d\right )} e^{3} f^{4}\right )} x^{3} - 3 \, {\left (5 \, b d^{2} e^{6} + 5 \, a c^{2} e^{3} f^{3} + {\left (2 \, b c d + a d^{2}\right )} e^{5} f + {\left (b c^{2} + 2 \, a c d\right )} e^{4} f^{2} + {\left (5 \, b d^{2} e^{3} f^{3} + 5 \, a c^{2} f^{6} + {\left (2 \, b c d + a d^{2}\right )} e^{2} f^{4} + {\left (b c^{2} + 2 \, a c d\right )} e f^{5}\right )} x^{6} + 3 \, {\left (5 \, b d^{2} e^{4} f^{2} + 5 \, a c^{2} e f^{5} + {\left (2 \, b c d + a d^{2}\right )} e^{3} f^{3} + {\left (b c^{2} + 2 \, a c d\right )} e^{2} f^{4}\right )} x^{4} + 3 \, {\left (5 \, b d^{2} e^{5} f + 5 \, a c^{2} e^{2} f^{4} + {\left (2 \, b c d + a d^{2}\right )} e^{4} f^{2} + {\left (b c^{2} + 2 \, a c d\right )} e^{3} f^{3}\right )} x^{2}\right )} \sqrt {e f} \arctan \left (\frac {\sqrt {e f} x}{e}\right ) + 3 \, {\left (5 \, b d^{2} e^{6} f - 11 \, a c^{2} e^{3} f^{4} + {\left (2 \, b c d + a d^{2}\right )} e^{5} f^{2} + {\left (b c^{2} + 2 \, a c d\right )} e^{4} f^{3}\right )} x}{48 \, {\left (e^{4} f^{7} x^{6} + 3 \, e^{5} f^{6} x^{4} + 3 \, e^{6} f^{5} x^{2} + e^{7} f^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 311, normalized size = 1.30 \begin {gather*} \frac {{\left (5 \, a c^{2} f^{3} + b c^{2} f^{2} e + 2 \, a c d f^{2} e + 2 \, b c d f e^{2} + a d^{2} f e^{2} + 5 \, b d^{2} e^{3}\right )} \arctan \left (\sqrt {f} x e^{\left (-\frac {1}{2}\right )}\right ) e^{\left (-\frac {7}{2}\right )}}{16 \, f^{\frac {7}{2}}} + \frac {{\left (15 \, a c^{2} f^{5} x^{5} + 3 \, b c^{2} f^{4} x^{5} e + 6 \, a c d f^{4} x^{5} e + 6 \, b c d f^{3} x^{5} e^{2} + 3 \, a d^{2} f^{3} x^{5} e^{2} - 33 \, b d^{2} f^{2} x^{5} e^{3} + 40 \, a c^{2} f^{4} x^{3} e + 8 \, b c^{2} f^{3} x^{3} e^{2} + 16 \, a c d f^{3} x^{3} e^{2} - 16 \, b c d f^{2} x^{3} e^{3} - 8 \, a d^{2} f^{2} x^{3} e^{3} - 40 \, b d^{2} f x^{3} e^{4} + 33 \, a c^{2} f^{3} x e^{2} - 3 \, b c^{2} f^{2} x e^{3} - 6 \, a c d f^{2} x e^{3} - 6 \, b c d f x e^{4} - 3 \, a d^{2} f x e^{4} - 15 \, b d^{2} x e^{5}\right )} e^{\left (-3\right )}}{48 \, {\left (f x^{2} + e\right )}^{3} f^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 360, normalized size = 1.50 \begin {gather*} \frac {5 a \,c^{2} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{16 \sqrt {e f}\, e^{3}}+\frac {a c d \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{8 \sqrt {e f}\, e^{2} f}+\frac {a \,d^{2} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{16 \sqrt {e f}\, e \,f^{2}}+\frac {b \,c^{2} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{16 \sqrt {e f}\, e^{2} f}+\frac {b c d \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{8 \sqrt {e f}\, e \,f^{2}}+\frac {5 b \,d^{2} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{16 \sqrt {e f}\, f^{3}}+\frac {\frac {\left (5 a \,c^{2} f^{3}+2 a c d e \,f^{2}+a \,d^{2} e^{2} f +b \,c^{2} e \,f^{2}+2 b c d \,e^{2} f -11 b \,d^{2} e^{3}\right ) x^{5}}{16 e^{3} f}+\frac {\left (5 a \,c^{2} f^{3}+2 a c d e \,f^{2}-a \,d^{2} e^{2} f +b \,c^{2} e \,f^{2}-2 b c d \,e^{2} f -5 b \,d^{2} e^{3}\right ) x^{3}}{6 e^{2} f^{2}}+\frac {\left (11 a \,c^{2} f^{3}-2 a c d e \,f^{2}-a \,d^{2} e^{2} f -b \,c^{2} e \,f^{2}-2 b c d \,e^{2} f -5 b \,d^{2} e^{3}\right ) x}{16 e \,f^{3}}}{\left (f \,x^{2}+e \right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.01, size = 304, normalized size = 1.27 \begin {gather*} -\frac {3 \, {\left (11 \, b d^{2} e^{3} f^{2} - 5 \, a c^{2} f^{5} - {\left (2 \, b c d + a d^{2}\right )} e^{2} f^{3} - {\left (b c^{2} + 2 \, a c d\right )} e f^{4}\right )} x^{5} + 8 \, {\left (5 \, b d^{2} e^{4} f - 5 \, a c^{2} e f^{4} + {\left (2 \, b c d + a d^{2}\right )} e^{3} f^{2} - {\left (b c^{2} + 2 \, a c d\right )} e^{2} f^{3}\right )} x^{3} + 3 \, {\left (5 \, b d^{2} e^{5} - 11 \, a c^{2} e^{2} f^{3} + {\left (2 \, b c d + a d^{2}\right )} e^{4} f + {\left (b c^{2} + 2 \, a c d\right )} e^{3} f^{2}\right )} x}{48 \, {\left (e^{3} f^{6} x^{6} + 3 \, e^{4} f^{5} x^{4} + 3 \, e^{5} f^{4} x^{2} + e^{6} f^{3}\right )}} + \frac {{\left (5 \, b d^{2} e^{3} + 5 \, a c^{2} f^{3} + {\left (2 \, b c d + a d^{2}\right )} e^{2} f + {\left (b c^{2} + 2 \, a c d\right )} e f^{2}\right )} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{16 \, \sqrt {e f} e^{3} f^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.98, size = 303, normalized size = 1.26 \begin {gather*} \frac {\frac {x^3\,\left (b\,c^2\,e\,f^2+5\,a\,c^2\,f^3-2\,b\,c\,d\,e^2\,f+2\,a\,c\,d\,e\,f^2-5\,b\,d^2\,e^3-a\,d^2\,e^2\,f\right )}{6\,e^2\,f^2}-\frac {x\,\left (b\,c^2\,e\,f^2-11\,a\,c^2\,f^3+2\,b\,c\,d\,e^2\,f+2\,a\,c\,d\,e\,f^2+5\,b\,d^2\,e^3+a\,d^2\,e^2\,f\right )}{16\,e\,f^3}+\frac {x^5\,\left (b\,c^2\,e\,f^2+5\,a\,c^2\,f^3+2\,b\,c\,d\,e^2\,f+2\,a\,c\,d\,e\,f^2-11\,b\,d^2\,e^3+a\,d^2\,e^2\,f\right )}{16\,e^3\,f}}{e^3+3\,e^2\,f\,x^2+3\,e\,f^2\,x^4+f^3\,x^6}+\frac {\mathrm {atan}\left (\frac {\sqrt {f}\,x}{\sqrt {e}}\right )\,\left (b\,c^2\,e\,f^2+5\,a\,c^2\,f^3+2\,b\,c\,d\,e^2\,f+2\,a\,c\,d\,e\,f^2+5\,b\,d^2\,e^3+a\,d^2\,e^2\,f\right )}{16\,e^{7/2}\,f^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 42.07, size = 486, normalized size = 2.02 \begin {gather*} - \frac {\sqrt {- \frac {1}{e^{7} f^{7}}} \left (5 a c^{2} f^{3} + 2 a c d e f^{2} + a d^{2} e^{2} f + b c^{2} e f^{2} + 2 b c d e^{2} f + 5 b d^{2} e^{3}\right ) \log {\left (- e^{4} f^{3} \sqrt {- \frac {1}{e^{7} f^{7}}} + x \right )}}{32} + \frac {\sqrt {- \frac {1}{e^{7} f^{7}}} \left (5 a c^{2} f^{3} + 2 a c d e f^{2} + a d^{2} e^{2} f + b c^{2} e f^{2} + 2 b c d e^{2} f + 5 b d^{2} e^{3}\right ) \log {\left (e^{4} f^{3} \sqrt {- \frac {1}{e^{7} f^{7}}} + x \right )}}{32} + \frac {x^{5} \left (15 a c^{2} f^{5} + 6 a c d e f^{4} + 3 a d^{2} e^{2} f^{3} + 3 b c^{2} e f^{4} + 6 b c d e^{2} f^{3} - 33 b d^{2} e^{3} f^{2}\right ) + x^{3} \left (40 a c^{2} e f^{4} + 16 a c d e^{2} f^{3} - 8 a d^{2} e^{3} f^{2} + 8 b c^{2} e^{2} f^{3} - 16 b c d e^{3} f^{2} - 40 b d^{2} e^{4} f\right ) + x \left (33 a c^{2} e^{2} f^{3} - 6 a c d e^{3} f^{2} - 3 a d^{2} e^{4} f - 3 b c^{2} e^{3} f^{2} - 6 b c d e^{4} f - 15 b d^{2} e^{5}\right )}{48 e^{6} f^{3} + 144 e^{5} f^{4} x^{2} + 144 e^{4} f^{5} x^{4} + 48 e^{3} f^{6} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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