Optimal. Leaf size=718 \[ -\frac {\sqrt {\sqrt {4 a c+b^2}+b} \sqrt {1-\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}} \sqrt {1-\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}} \left (e \left (b-\sqrt {4 a c+b^2}\right )+2 c d\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{4 \sqrt {2} \sqrt {c} d \sqrt {a+b x^2-c x^4} \left (e (b d-a e)+c d^2\right )}+\frac {e \left (b-\sqrt {4 a c+b^2}\right ) \sqrt {\sqrt {4 a c+b^2}+b} \sqrt {1-\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}} \sqrt {1-\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}} E\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{4 \sqrt {2} \sqrt {c} d \sqrt {a+b x^2-c x^4} \left (e (b d-a e)+c d^2\right )}+\frac {\sqrt {\sqrt {4 a c+b^2}+b} \sqrt {1-\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}} \sqrt {1-\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}} \left (e (2 b d-a e)+3 c d^2\right ) \Pi \left (-\frac {\left (b+\sqrt {b^2+4 a c}\right ) e}{2 c d};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{2 \sqrt {2} \sqrt {c} d^2 \sqrt {a+b x^2-c x^4} \left (e (b d-a e)+c d^2\right )}-\frac {e^2 x \sqrt {a+b x^2-c x^4}}{2 d \left (d+e x^2\right ) \left (-a e^2+b d e+c d^2\right )} \]
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Rubi [A] time = 1.02, antiderivative size = 718, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {1223, 1716, 1202, 524, 424, 419, 1220, 537} \[ -\frac {\sqrt {\sqrt {4 a c+b^2}+b} \sqrt {1-\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}} \sqrt {1-\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}} \left (e \left (b-\sqrt {4 a c+b^2}\right )+2 c d\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{4 \sqrt {2} \sqrt {c} d \sqrt {a+b x^2-c x^4} \left (e (b d-a e)+c d^2\right )}+\frac {e \left (b-\sqrt {4 a c+b^2}\right ) \sqrt {\sqrt {4 a c+b^2}+b} \sqrt {1-\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}} \sqrt {1-\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}} E\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{4 \sqrt {2} \sqrt {c} d \sqrt {a+b x^2-c x^4} \left (e (b d-a e)+c d^2\right )}+\frac {\sqrt {\sqrt {4 a c+b^2}+b} \sqrt {1-\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}} \sqrt {1-\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}} \left (e (2 b d-a e)+3 c d^2\right ) \Pi \left (-\frac {\left (b+\sqrt {b^2+4 a c}\right ) e}{2 c d};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{2 \sqrt {2} \sqrt {c} d^2 \sqrt {a+b x^2-c x^4} \left (e (b d-a e)+c d^2\right )}-\frac {e^2 x \sqrt {a+b x^2-c x^4}}{2 d \left (d+e x^2\right ) \left (e (b d-a e)+c d^2\right )} \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 524
Rule 537
Rule 1202
Rule 1220
Rule 1223
Rule 1716
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+e (b d-a e)\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+e (b d-a e)\right )}\\ &=-\frac {e^2 x \sqrt {a+b x^2-c x^4}}{2 d \left (c d^2+e (b d-a e)\right ) \left (d+e x^2\right )}-\frac {\int \frac {c d e^2+c e^3 x^2}{\sqrt {a+b x^2-c x^4}} \, dx}{2 d e^2 \left (c d^2+e (b d-a e)\right )}+\frac {\left (3 c d^2+e (2 b d-a e)\right ) \int \frac {1}{\left (d+e x^2\right ) \sqrt {a+b x^2-c x^4}} \, dx}{2 d \left (c d^2+e (b d-a e)\right )}\\ &=-\frac {e^2 x \sqrt {a+b x^2-c x^4}}{2 d \left (c d^2+e (b d-a e)\right ) \left (d+e x^2\right )}-\frac {\left (\sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}}\right ) \int \frac {c d e^2+c e^3 x^2}{\sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}}} \, dx}{2 d e^2 \left (c d^2+e (b d-a e)\right ) \sqrt {a+b x^2-c x^4}}+\frac {\left (\left (3 c d^2+e (2 b d-a e)\right ) \sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}}\right ) \int \frac {1}{\sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}} \left (d+e x^2\right )} \, dx}{2 d \left (c d^2+e (b d-a e)\right ) \sqrt {a+b x^2-c x^4}}\\ &=-\frac {e^2 x \sqrt {a+b x^2-c x^4}}{2 d \left (c d^2+e (b d-a e)\right ) \left (d+e x^2\right )}+\frac {\sqrt {b+\sqrt {b^2+4 a c}} \left (3 c d^2+e (2 b d-a e)\right ) \sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}} \Pi \left (-\frac {\left (b+\sqrt {b^2+4 a c}\right ) e}{2 c d};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{2 \sqrt {2} \sqrt {c} d^2 \left (c d^2+e (b d-a e)\right ) \sqrt {a+b x^2-c x^4}}+\frac {\left (\left (b-\sqrt {b^2+4 a c}\right ) e \sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}}\right ) \int \frac {\sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}}}{\sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}}} \, dx}{4 d \left (c d^2+e (b d-a e)\right ) \sqrt {a+b x^2-c x^4}}-\frac {\left (\left (2 c d+\left (b-\sqrt {b^2+4 a c}\right ) e\right ) \sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}}\right ) \int \frac {1}{\sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}}} \, dx}{4 d \left (c d^2+e (b d-a e)\right ) \sqrt {a+b x^2-c x^4}}\\ &=-\frac {e^2 x \sqrt {a+b x^2-c x^4}}{2 d \left (c d^2+e (b d-a e)\right ) \left (d+e x^2\right )}+\frac {\left (b-\sqrt {b^2+4 a c}\right ) \sqrt {b+\sqrt {b^2+4 a c}} e \sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}} E\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{4 \sqrt {2} \sqrt {c} d \left (c d^2+e (b d-a e)\right ) \sqrt {a+b x^2-c x^4}}-\frac {\sqrt {b+\sqrt {b^2+4 a c}} \left (2 c d+\left (b-\sqrt {b^2+4 a c}\right ) e\right ) \sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{4 \sqrt {2} \sqrt {c} d \left (c d^2+e (b d-a e)\right ) \sqrt {a+b x^2-c x^4}}+\frac {\sqrt {b+\sqrt {b^2+4 a c}} \left (3 c d^2+e (2 b d-a e)\right ) \sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}} \Pi \left (-\frac {\left (b+\sqrt {b^2+4 a c}\right ) e}{2 c d};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{2 \sqrt {2} \sqrt {c} d^2 \left (c d^2+e (b d-a e)\right ) \sqrt {a+b x^2-c x^4}}\\ \end {align*}
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Mathematica [C] time = 5.53, size = 464, normalized size = 0.65 \[ -\frac {\sqrt {a+b x^2-c x^4} \left (4 d e^2 x+\frac {i \left (d+e x^2\right ) \sqrt {\frac {4 c x^2}{\sqrt {4 a c+b^2}-b}+2} \sqrt {1-\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}} \left (2 \left (e (a e-2 b d)-3 c d^2\right ) \Pi \left (-\frac {\left (b+\sqrt {b^2+4 a c}\right ) e}{2 c d};i \sinh ^{-1}\left (\sqrt {2} \sqrt {-\frac {c}{b+\sqrt {b^2+4 a c}}} x\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )+d \left (e \left (b-\sqrt {4 a c+b^2}\right )+2 c d\right ) F\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {-\frac {c}{b+\sqrt {b^2+4 a c}}} x\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )+d e \left (\sqrt {4 a c+b^2}-b\right ) E\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {-\frac {c}{b+\sqrt {b^2+4 a c}}} x\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )\right )}{\sqrt {-\frac {c}{\sqrt {4 a c+b^2}+b}} \left (-a-b x^2+c x^4\right )}\right )}{8 d^2 \left (d+e x^2\right ) \left (e (b d-a e)+c d^2\right )} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 [B] time = 0.04, size = 1293, normalized size = 1.80 \[ \text {result too large to display} \]
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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