Optimal. Leaf size=718 \[ \frac {e^2 x \sqrt {a+b x^2+c x^4}}{2 d \left (d+e x^2\right ) \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {c} e x \sqrt {a+b x^2+c x^4}}{2 d \left (\sqrt {a}+\sqrt {c} x^2\right ) \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt [4]{a} \sqrt [4]{c} e \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 d \sqrt {a+b x^2+c x^4} \left (a e^2-b d e+c d^2\right )}-\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\sqrt {a} e+\sqrt {c} d\right ) \left (3 c d^2-e (2 b d-a e)\right ) \Pi \left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e};2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{8 \sqrt [4]{a} \sqrt [4]{c} d^2 \sqrt {a+b x^2+c x^4} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {e} \left (3 c d^2-e (2 b d-a e)\right ) \tan ^{-1}\left (\frac {x \sqrt {a e^2-b d e+c d^2}}{\sqrt {d} \sqrt {e} \sqrt {a+b x^2+c x^4}}\right )}{4 d^{3/2} \left (a e^2-b d e+c d^2\right )^{3/2}}+\frac {\sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} d \sqrt {a+b x^2+c x^4} \left (\sqrt {c} d-\sqrt {a} e\right )} \]
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Rubi [A] time = 1.08, antiderivative size = 718, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1223, 1714, 1195, 1708, 1103, 1706} \[ \frac {e^2 x \sqrt {a+b x^2+c x^4}}{2 d \left (d+e x^2\right ) \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {c} e x \sqrt {a+b x^2+c x^4}}{2 d \left (\sqrt {a}+\sqrt {c} x^2\right ) \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {e} \left (3 c d^2-e (2 b d-a e)\right ) \tan ^{-1}\left (\frac {x \sqrt {a e^2-b d e+c d^2}}{\sqrt {d} \sqrt {e} \sqrt {a+b x^2+c x^4}}\right )}{4 d^{3/2} \left (a e^2-b d e+c d^2\right )^{3/2}}+\frac {\sqrt [4]{a} \sqrt [4]{c} e \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 d \sqrt {a+b x^2+c x^4} \left (a e^2-b d e+c d^2\right )}-\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\sqrt {a} e+\sqrt {c} d\right ) \left (3 c d^2-e (2 b d-a e)\right ) \Pi \left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e};2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{8 \sqrt [4]{a} \sqrt [4]{c} d^2 \sqrt {a+b x^2+c x^4} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} d \sqrt {a+b x^2+c x^4} \left (\sqrt {c} d-\sqrt {a} e\right )} \]
Antiderivative was successfully verified.
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Rule 1103
Rule 1195
Rule 1223
Rule 1706
Rule 1708
Rule 1714
Rubi steps
\begin {align*} \int \frac {1}{\left (d+e x^2\right )^2 \sqrt {a+b x^2+c x^4}} \, dx &=\frac {e^2 x \sqrt {a+b x^2+c x^4}}{2 d \left (c d^2-b d e+a e^2\right ) \left (d+e x^2\right )}-\frac {\int \frac {-2 c d^2+e (2 b d-a e)+2 c d e x^2+c e^2 x^4}{\left (d+e x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx}{2 d \left (c d^2-b d e+a e^2\right )}\\ &=\frac {e^2 x \sqrt {a+b x^2+c x^4}}{2 d \left (c d^2-b d e+a e^2\right ) \left (d+e x^2\right )}-\frac {\int \frac {\sqrt {a} c^{3/2} d e^2+c e \left (-2 c d^2+e (2 b d-a e)\right )+\left (2 c^2 d e^2-c e^2 \left (c d-\sqrt {a} \sqrt {c} e\right )\right ) x^2}{\left (d+e x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx}{2 c d e \left (c d^2-b d e+a e^2\right )}+\frac {\left (\sqrt {a} \sqrt {c} e\right ) \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+b x^2+c x^4}} \, dx}{2 d \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {\sqrt {c} e x \sqrt {a+b x^2+c x^4}}{2 d \left (c d^2-b d e+a e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {e^2 x \sqrt {a+b x^2+c x^4}}{2 d \left (c d^2-b d e+a e^2\right ) \left (d+e x^2\right )}+\frac {\sqrt [4]{a} \sqrt [4]{c} e \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 d \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x^2+c x^4}}+\frac {\sqrt {c} \int \frac {1}{\sqrt {a+b x^2+c x^4}} \, dx}{d \left (\sqrt {c} d-\sqrt {a} e\right )}-\frac {\left (\sqrt {a} e \left (3 c d^2-e (2 b d-a e)\right )\right ) \int \frac {1+\frac {\sqrt {c} x^2}{\sqrt {a}}}{\left (d+e x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx}{2 d \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {\sqrt {c} e x \sqrt {a+b x^2+c x^4}}{2 d \left (c d^2-b d e+a e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {e^2 x \sqrt {a+b x^2+c x^4}}{2 d \left (c d^2-b d e+a e^2\right ) \left (d+e x^2\right )}+\frac {\sqrt {e} \left (3 c d^2-e (2 b d-a e)\right ) \tan ^{-1}\left (\frac {\sqrt {c d^2-b d e+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {a+b x^2+c x^4}}\right )}{4 d^{3/2} \left (c d^2-b d e+a e^2\right )^{3/2}}+\frac {\sqrt [4]{a} \sqrt [4]{c} e \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 d \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x^2+c x^4}}+\frac {\sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} d \left (\sqrt {c} d-\sqrt {a} e\right ) \sqrt {a+b x^2+c x^4}}-\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \left (3 c d^2-e (2 b d-a e)\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \Pi \left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e};2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{8 \sqrt [4]{a} \sqrt [4]{c} d^2 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x^2+c x^4}}\\ \end {align*}
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Mathematica [C] time = 1.86, size = 1069, normalized size = 1.49 \[ \frac {2 i \sqrt {2} c \sqrt {\frac {2 c x^2+b+\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1} \left (e x^2+d\right ) F\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right ) d^2-6 i \sqrt {2} c \sqrt {\frac {2 c x^2+b+\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1} \left (e x^2+d\right ) \Pi \left (\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c d};i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right ) d^2+4 \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} e^2 x \left (c x^4+b x^2+a\right ) d+i \sqrt {2} \left (b-\sqrt {b^2-4 a c}\right ) e \sqrt {\frac {2 c x^2+b+\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1} \left (e x^2+d\right ) \left (E\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-F\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )\right ) d+4 i \sqrt {2} b e \sqrt {\frac {2 c x^2+b+\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1} \left (e x^2+d\right ) \Pi \left (\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c d};i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right ) d-2 i \sqrt {2} a e^2 \sqrt {\frac {2 c x^2+b+\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1} \left (e x^2+d\right ) \Pi \left (\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c d};i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{8 \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} d \left (c d^3+e (a e-b d) d\right ) \left (e x^2+d\right ) \sqrt {c x^4+b x^2+a}} \]
Antiderivative was successfully verified.
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fricas [F] time = 125.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{4} + b x^{2} + a}}{c e^{2} x^{8} + {\left (2 \, c d e + b e^{2}\right )} x^{6} + {\left (c d^{2} + 2 \, b d e + a e^{2}\right )} x^{4} + a d^{2} + {\left (b d^{2} + 2 \, a d e\right )} x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {c x^{4} + b x^{2} + a} {\left (e x^{2} + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 1279, normalized size = 1.78 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {c x^{4} + b x^{2} + a} {\left (e x^{2} + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (e\,x^2+d\right )}^2\,\sqrt {c\,x^4+b\,x^2+a}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (d + e x^{2}\right )^{2} \sqrt {a + b x^{2} + c x^{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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