Optimal. Leaf size=349 \[ -\frac {\left (-2 \sqrt {a} \sqrt {c} d e+a e^2+3 c d^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} c^{5/4}}+\frac {\left (-2 \sqrt {a} \sqrt {c} d e+a e^2+3 c d^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} c^{5/4}}-\frac {\left (2 \sqrt {a} \sqrt {c} d e+a e^2+3 c d^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} c^{5/4}}+\frac {\left (2 \sqrt {a} \sqrt {c} d e+a e^2+3 c d^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt {2} a^{7/4} c^{5/4}}+\frac {x \left (a e^2+3 c d^2+6 c d e x^2\right )}{12 a c \left (a+c x^4\right )}-\frac {e^2 x}{3 c \left (a+c x^4\right )} \]
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Rubi [A] time = 0.31, antiderivative size = 349, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {1207, 1179, 1168, 1162, 617, 204, 1165, 628} \begin {gather*} -\frac {\left (-2 \sqrt {a} \sqrt {c} d e+a e^2+3 c d^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} c^{5/4}}+\frac {\left (-2 \sqrt {a} \sqrt {c} d e+a e^2+3 c d^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} c^{5/4}}-\frac {\left (2 \sqrt {a} \sqrt {c} d e+a e^2+3 c d^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} c^{5/4}}+\frac {\left (2 \sqrt {a} \sqrt {c} d e+a e^2+3 c d^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt {2} a^{7/4} c^{5/4}}+\frac {x \left (a e^2+3 c d^2+6 c d e x^2\right )}{12 a c \left (a+c x^4\right )}-\frac {e^2 x}{3 c \left (a+c x^4\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1168
Rule 1179
Rule 1207
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right )^2}{\left (a+c x^4\right )^2} \, dx &=-\frac {e^2 x}{3 c \left (a+c x^4\right )}-\frac {\int \frac {-3 c d^2-a e^2-6 c d e x^2}{\left (a+c x^4\right )^2} \, dx}{3 c}\\ &=-\frac {e^2 x}{3 c \left (a+c x^4\right )}+\frac {x \left (3 c d^2+a e^2+6 c d e x^2\right )}{12 a c \left (a+c x^4\right )}+\frac {\int \frac {3 \left (3 c d^2+a e^2\right )+6 c d e x^2}{a+c x^4} \, dx}{12 a c}\\ &=-\frac {e^2 x}{3 c \left (a+c x^4\right )}+\frac {x \left (3 c d^2+a e^2+6 c d e x^2\right )}{12 a c \left (a+c x^4\right )}+\frac {\left (3 c d^2-2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx}{8 a^{3/2} c^{3/2}}+\frac {\left (3 c d^2+2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx}{8 a^{3/2} c^{3/2}}\\ &=-\frac {e^2 x}{3 c \left (a+c x^4\right )}+\frac {x \left (3 c d^2+a e^2+6 c d e x^2\right )}{12 a c \left (a+c x^4\right )}-\frac {\left (3 c d^2-2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{-\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{16 \sqrt {2} a^{7/4} c^{5/4}}-\frac {\left (3 c d^2-2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}-2 x}{-\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{16 \sqrt {2} a^{7/4} c^{5/4}}+\frac {\left (3 c d^2+2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{16 a^{3/2} c^{3/2}}+\frac {\left (3 c d^2+2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{16 a^{3/2} c^{3/2}}\\ &=-\frac {e^2 x}{3 c \left (a+c x^4\right )}+\frac {x \left (3 c d^2+a e^2+6 c d e x^2\right )}{12 a c \left (a+c x^4\right )}-\frac {\left (3 c d^2-2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} c^{5/4}}+\frac {\left (3 c d^2-2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} c^{5/4}}+\frac {\left (3 c d^2+2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} c^{5/4}}-\frac {\left (3 c d^2+2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} c^{5/4}}\\ &=-\frac {e^2 x}{3 c \left (a+c x^4\right )}+\frac {x \left (3 c d^2+a e^2+6 c d e x^2\right )}{12 a c \left (a+c x^4\right )}-\frac {\left (3 c d^2+2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} c^{5/4}}+\frac {\left (3 c d^2+2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} c^{5/4}}-\frac {\left (3 c d^2-2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} c^{5/4}}+\frac {\left (3 c d^2-2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} c^{5/4}}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 295, normalized size = 0.85 \begin {gather*} \frac {-\frac {8 a^{3/4} \sqrt [4]{c} \left (a e^2 x-c d x \left (d+2 e x^2\right )\right )}{a+c x^4}-\sqrt {2} \left (-2 \sqrt {a} \sqrt {c} d e+a e^2+3 c d^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )+\sqrt {2} \left (-2 \sqrt {a} \sqrt {c} d e+a e^2+3 c d^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )-2 \sqrt {2} \left (2 \sqrt {a} \sqrt {c} d e+a e^2+3 c d^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+2 \sqrt {2} \left (2 \sqrt {a} \sqrt {c} d e+a e^2+3 c d^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{32 a^{7/4} c^{5/4}} \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 (d+e x^2\right )^2}{\left (a+c x^4\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 1.77, size = 1596, normalized size = 4.57
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 350, normalized size = 1.00 \begin {gather*} \frac {2 \, c d x^{3} e + c d^{2} x - a x e^{2}}{4 \, {\left (c x^{4} + a\right )} a c} + \frac {\sqrt {2} {\left (3 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{2} + \left (a c^{3}\right )^{\frac {1}{4}} a c e^{2} + 2 \, \left (a c^{3}\right )^{\frac {3}{4}} d e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{16 \, a^{2} c^{3}} + \frac {\sqrt {2} {\left (3 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{2} + \left (a c^{3}\right )^{\frac {1}{4}} a c e^{2} + 2 \, \left (a c^{3}\right )^{\frac {3}{4}} d e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{16 \, a^{2} c^{3}} + \frac {\sqrt {2} {\left (3 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{2} + \left (a c^{3}\right )^{\frac {1}{4}} a c e^{2} - 2 \, \left (a c^{3}\right )^{\frac {3}{4}} d e\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{32 \, a^{2} c^{3}} - \frac {\sqrt {2} {\left (3 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{2} + \left (a c^{3}\right )^{\frac {1}{4}} a c e^{2} - 2 \, \left (a c^{3}\right )^{\frac {3}{4}} d e\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{32 \, a^{2} c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 464, normalized size = 1.33 \begin {gather*} \frac {\sqrt {2}\, d e \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )}{8 \left (\frac {a}{c}\right )^{\frac {1}{4}} a c}+\frac {\sqrt {2}\, d e \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )}{8 \left (\frac {a}{c}\right )^{\frac {1}{4}} a c}+\frac {\sqrt {2}\, d e \ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}\right )}{16 \left (\frac {a}{c}\right )^{\frac {1}{4}} a c}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, e^{2} \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )}{16 a c}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, e^{2} \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )}{16 a c}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, e^{2} \ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}\right )}{32 a c}+\frac {3 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, d^{2} \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )}{16 a^{2}}+\frac {3 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, d^{2} \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )}{16 a^{2}}+\frac {3 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, d^{2} \ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}\right )}{32 a^{2}}+\frac {\frac {d e \,x^{3}}{2 a}-\frac {\left (a \,e^{2}-c \,d^{2}\right ) x}{4 a c}}{c \,x^{4}+a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.59, size = 324, normalized size = 0.93 \begin {gather*} \frac {2 \, c d e x^{3} + {\left (c d^{2} - a e^{2}\right )} x}{4 \, {\left (a c^{2} x^{4} + a^{2} c\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (3 \, c^{\frac {3}{2}} d^{2} + 2 \, \sqrt {a} c d e + a \sqrt {c} e^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} {\left (3 \, c^{\frac {3}{2}} d^{2} + 2 \, \sqrt {a} c d e + a \sqrt {c} e^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {\sqrt {2} {\left (3 \, c^{\frac {3}{2}} d^{2} - 2 \, \sqrt {a} c d e + a \sqrt {c} e^{2}\right )} \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (3 \, c^{\frac {3}{2}} d^{2} - 2 \, \sqrt {a} c d e + a \sqrt {c} e^{2}\right )} \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}}}{32 \, a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.79, size = 1565, normalized size = 4.48
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.07, size = 275, normalized size = 0.79 \begin {gather*} \operatorname {RootSum} {\left (65536 t^{4} a^{7} c^{5} + t^{2} \left (2048 a^{5} c^{3} d e^{3} + 6144 a^{4} c^{4} d^{3} e\right ) + a^{4} e^{8} + 20 a^{3} c d^{2} e^{6} + 118 a^{2} c^{2} d^{4} e^{4} + 180 a c^{3} d^{6} e^{2} + 81 c^{4} d^{8}, \left (t \mapsto t \log {\left (x + \frac {- 8192 t^{3} a^{6} c^{4} d e + 16 t a^{5} c e^{6} - 48 t a^{4} c^{2} d^{2} e^{4} - 144 t a^{3} c^{3} d^{4} e^{2} + 432 t a^{2} c^{4} d^{6}}{a^{4} e^{8} + 12 a^{3} c d^{2} e^{6} + 38 a^{2} c^{2} d^{4} e^{4} + 108 a c^{3} d^{6} e^{2} + 81 c^{4} d^{8}} \right )} \right )\right )} + \frac {2 c d e x^{3} + x \left (- a e^{2} + c d^{2}\right )}{4 a^{2} c + 4 a c^{2} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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