Optimal. Leaf size=401 \[ -\frac {a^{3/4} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right )^2 \Pi \left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e};2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{4 \sqrt [4]{c} d \sqrt {a+b x^2+c x^4} \left (c d^2-a e^2\right )}+\frac {\sqrt {e} \tan ^{-1}\left (\frac {x \sqrt {a e^2-b d e+c d^2}}{\sqrt {d} \sqrt {e} \sqrt {a+b x^2+c x^4}}\right )}{2 \sqrt {d} \sqrt {a e^2-b d e+c d^2}}+\frac {\sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} \sqrt {a+b x^2+c x^4} \left (\sqrt {c} d-\sqrt {a} e\right )} \]
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Rubi [A] time = 0.35, antiderivative size = 401, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1216, 1103, 1706} \[ -\frac {a^{3/4} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right )^2 \Pi \left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e};2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{4 \sqrt [4]{c} d \sqrt {a+b x^2+c x^4} \left (c d^2-a e^2\right )}+\frac {\sqrt {e} \tan ^{-1}\left (\frac {x \sqrt {a e^2-b d e+c d^2}}{\sqrt {d} \sqrt {e} \sqrt {a+b x^2+c x^4}}\right )}{2 \sqrt {d} \sqrt {a e^2-b d e+c d^2}}+\frac {\sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} \sqrt {a+b x^2+c x^4} \left (\sqrt {c} d-\sqrt {a} e\right )} \]
Antiderivative was successfully verified.
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Rule 1103
Rule 1216
Rule 1706
Rubi steps
\begin {align*} \int \frac {1}{\left (d+e x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx &=\frac {\sqrt {c} \int \frac {1}{\sqrt {a+b x^2+c x^4}} \, dx}{\sqrt {c} d-\sqrt {a} e}-\frac {\left (\sqrt {a} e\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}{\sqrt {c} d-\sqrt {a} e}\\ &=\frac {\sqrt {e} \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 )}{2 \sqrt {d} \sqrt {c d^2-b d e+a e^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} \left (\sqrt {c} d-\sqrt {a} e\right ) \sqrt {a+b x^2+c x^4}}-\frac {\sqrt [4]{a} \left (\frac {\sqrt {c} d}{\sqrt {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 )}{4 \sqrt [4]{c} d \left (\sqrt {c} d-\sqrt {a} e\right ) \sqrt {a+b x^2+c x^4}}\\ \end {align*}
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Mathematica [C] time = 0.22, size = 214, normalized size = 0.53 \[ -\frac {i \sqrt {\frac {\sqrt {b^2-4 a c}+b+2 c x^2}{\sqrt {b^2-4 a c}+b}} \sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1} \Pi \left (\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c d};i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {2} d \sqrt {\frac {c}{\sqrt {b^2-4 a c}+b}} \sqrt {a+b x^2+c x^4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 45.98, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{4} + b x^{2} + a}}{c e x^{6} + {\left (c d + b e\right )} x^{4} + {\left (b d + a e\right )} x^{2} + a d}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {c x^{4} + b x^{2} + a} {\left (e x^{2} + d\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 200, normalized size = 0.50 \[ \frac {\sqrt {2}\, \sqrt {\frac {b \,x^{2}}{2 a}-\frac {\sqrt {-4 a c +b^{2}}\, x^{2}}{2 a}+1}\, \sqrt {\frac {b \,x^{2}}{2 a}+\frac {\sqrt {-4 a c +b^{2}}\, x^{2}}{2 a}+1}\, \EllipticPi \left (\frac {\sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, x}{2}, -\frac {2 a e}{\left (-b +\sqrt {-4 a c +b^{2}}\right ) d}, \frac {\sqrt {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}\, \sqrt {2}}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}\right )}{\sqrt {-\frac {b}{a}+\frac {\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, d} \]
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 \[ \int \frac {1}{\sqrt {c x^{4} + b x^{2} + a} {\left (e x^{2} + d\right )}}\,{d x} \]
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 \[ \int \frac {1}{\left (e\,x^2+d\right )\,\sqrt {c\,x^4+b\,x^2+a}} \,d x \]
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 \[ \int \frac {1}{\left (d + e x^{2}\right ) \sqrt {a + b x^{2} + c x^{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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