Optimal. Leaf size=73 \[ \frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \Pi \left (-\frac {\sqrt {a} e}{\sqrt {c} d};\left .\sin ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{c} d \sqrt {c x^4-a}} \]
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Rubi [A] time = 0.04, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {1219, 1218} \[ \frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \Pi \left (-\frac {\sqrt {a} e}{\sqrt {c} d};\left .\sin ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{c} d \sqrt {c x^4-a}} \]
Antiderivative was successfully verified.
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Rule 1218
Rule 1219
Rubi steps
\begin {align*} \int \frac {1}{\left (d+e x^2\right ) \sqrt {-a+c x^4}} \, dx &=\frac {\sqrt {1-\frac {c x^4}{a}} \int \frac {1}{\left (d+e x^2\right ) \sqrt {1-\frac {c x^4}{a}}} \, dx}{\sqrt {-a+c x^4}}\\ &=\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \Pi \left (-\frac {\sqrt {a} e}{\sqrt {c} d};\left .\sin ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{c} d \sqrt {-a+c x^4}}\\ \end {align*}
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Mathematica [C] time = 0.15, size = 92, normalized size = 1.26 \[ -\frac {i \sqrt {1-\frac {c x^4}{a}} \Pi \left (-\frac {\sqrt {a} e}{\sqrt {c} d};\left .i \sinh ^{-1}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )}{d \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} \sqrt {c x^4-a}} \]
Antiderivative was successfully verified.
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fricas [F] time = 13.11, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{4} - a}}{c e x^{6} + c d x^{4} - a e x^{2} - a d}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {c x^{4} - a} {\left (e x^{2} + d\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 99, normalized size = 1.36 \[ \frac {\sqrt {\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}+1}\, \EllipticPi \left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}\, x , \frac {\sqrt {a}\, e}{\sqrt {c}\, d}, \frac {\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}-a}\, d} \]
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 \[ \int \frac {1}{\sqrt {c x^{4} - a} {\left (e x^{2} + d\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {c\,x^4-a}\,\left (e\,x^2+d\right )} \,d x \]
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 \[ \int \frac {1}{\sqrt {- a + c x^{4}} \left (d + e x^{2}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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