Optimal. Leaf size=425 \[ -\frac {e^2 x \sqrt {a-c x^4}}{4 d \left (c d^2-a e^2\right ) \left (d+e x^2\right )^2}-\frac {3 e^2 \left (3 c d^2-a e^2\right ) x \sqrt {a-c x^4}}{8 d^2 \left (c d^2-a e^2\right )^2 \left (d+e x^2\right )}-\frac {3 a^{3/4} \sqrt [4]{c} e \left (3 c d^2-a e^2\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 d^2 \left (c d^2-a e^2\right )^2 \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} \sqrt [4]{c} \left (7 c d^2-2 \sqrt {a} \sqrt {c} d e-3 a e^2\right ) \sqrt {1-\frac {c x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 d^2 \left (\sqrt {c} d+\sqrt {a} e\right ) \left (c d^2-a e^2\right ) \sqrt {a-c x^4}}+\frac {3 \sqrt [4]{a} \left (5 c^2 d^4-2 a c d^2 e^2+a^2 e^4\right ) \sqrt {1-\frac {c x^4}{a}} \Pi \left (-\frac {\sqrt {a} e}{\sqrt {c} d};\left .\sin ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{c} d^3 \left (c d^2-a e^2\right )^2 \sqrt {a-c x^4}} \]
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Rubi [A]
time = 0.46, antiderivative size = 425, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 11, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1238, 1711,
1731, 1215, 230, 227, 1214, 1213, 435, 1233, 1232} \begin {gather*} -\frac {3 a^{3/4} \sqrt [4]{c} e \sqrt {1-\frac {c x^4}{a}} \left (3 c d^2-a e^2\right ) E\left (\left .\text {ArcSin}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 d^2 \sqrt {a-c x^4} \left (c d^2-a e^2\right )^2}+\frac {3 \sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (a^2 e^4-2 a c d^2 e^2+5 c^2 d^4\right ) \Pi \left (-\frac {\sqrt {a} e}{\sqrt {c} d};\left .\text {ArcSin}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{c} d^3 \sqrt {a-c x^4} \left (c d^2-a e^2\right )^2}-\frac {\sqrt [4]{a} \sqrt [4]{c} \sqrt {1-\frac {c x^4}{a}} \left (-2 \sqrt {a} \sqrt {c} d e-3 a e^2+7 c d^2\right ) F\left (\left .\text {ArcSin}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 d^2 \sqrt {a-c x^4} \left (\sqrt {a} e+\sqrt {c} d\right ) \left (c d^2-a e^2\right )}-\frac {3 e^2 x \sqrt {a-c x^4} \left (3 c d^2-a e^2\right )}{8 d^2 \left (d+e x^2\right ) \left (c d^2-a e^2\right )^2}-\frac {e^2 x \sqrt {a-c x^4}}{4 d \left (d+e x^2\right )^2 \left (c d^2-a e^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 227
Rule 230
Rule 435
Rule 1213
Rule 1214
Rule 1215
Rule 1232
Rule 1233
Rule 1238
Rule 1711
Rule 1731
Rubi steps
\begin {align*} \int \frac {1}{\left (d+e x^2\right )^3 \sqrt {a-c x^4}} \, dx &=-\frac {e^2 x \sqrt {a-c x^4}}{4 d \left (c d^2-a e^2\right ) \left (d+e x^2\right )^2}+\frac {\int \frac {4 c d^2-3 a e^2-4 c d e x^2+c e^2 x^4}{\left (d+e x^2\right )^2 \sqrt {a-c x^4}} \, dx}{4 d \left (c d^2-a e^2\right )}\\ &=-\frac {e^2 x \sqrt {a-c x^4}}{4 d \left (c d^2-a e^2\right ) \left (d+e x^2\right )^2}-\frac {3 e^2 \left (3 c d^2-a e^2\right ) x \sqrt {a-c x^4}}{8 d^2 \left (c d^2-a e^2\right )^2 \left (d+e x^2\right )}+\frac {\int \frac {8 c^2 d^4-5 a c d^2 e^2+3 a^2 e^4-4 c d e \left (4 c d^2-a e^2\right ) x^2-3 c e^2 \left (3 c d^2-a e^2\right ) x^4}{\left (d+e x^2\right ) \sqrt {a-c x^4}} \, dx}{8 d^2 \left (c d^2-a e^2\right )^2}\\ &=-\frac {e^2 x \sqrt {a-c x^4}}{4 d \left (c d^2-a e^2\right ) \left (d+e x^2\right )^2}-\frac {3 e^2 \left (3 c d^2-a e^2\right ) x \sqrt {a-c x^4}}{8 d^2 \left (c d^2-a e^2\right )^2 \left (d+e x^2\right )}-\frac {\int \frac {-3 c d e^2 \left (3 c d^2-a e^2\right )+4 c d e^2 \left (4 c d^2-a e^2\right )+3 c e^3 \left (3 c d^2-a e^2\right ) x^2}{\sqrt {a-c x^4}} \, dx}{8 d^2 e^2 \left (c d^2-a e^2\right )^2}+\frac {\left (3 \left (5 c^2 d^4-2 a c d^2 e^2+a^2 e^4\right )\right ) \int \frac {1}{\left (d+e x^2\right ) \sqrt {a-c x^4}} \, dx}{8 d^2 \left (c d^2-a e^2\right )^2}\\ &=-\frac {e^2 x \sqrt {a-c x^4}}{4 d \left (c d^2-a e^2\right ) \left (d+e x^2\right )^2}-\frac {3 e^2 \left (3 c d^2-a e^2\right ) x \sqrt {a-c x^4}}{8 d^2 \left (c d^2-a e^2\right )^2 \left (d+e x^2\right )}-\frac {\left (\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (7 c d^2-2 \sqrt {a} \sqrt {c} d e-3 a e^2\right )\right ) \int \frac {1}{\sqrt {a-c x^4}} \, dx}{8 d^2 \left (c d^2-a e^2\right )^2}-\frac {\left (3 \sqrt {a} \sqrt {c} e \left (3 c d^2-a e^2\right )\right ) \int \frac {1+\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a-c x^4}} \, dx}{8 d^2 \left (c d^2-a e^2\right )^2}+\frac {\left (3 \left (5 c^2 d^4-2 a c d^2 e^2+a^2 e^4\right ) \sqrt {1-\frac {c x^4}{a}}\right ) \int \frac {1}{\left (d+e x^2\right ) \sqrt {1-\frac {c x^4}{a}}} \, dx}{8 d^2 \left (c d^2-a e^2\right )^2 \sqrt {a-c x^4}}\\ &=-\frac {e^2 x \sqrt {a-c x^4}}{4 d \left (c d^2-a e^2\right ) \left (d+e x^2\right )^2}-\frac {3 e^2 \left (3 c d^2-a e^2\right ) x \sqrt {a-c x^4}}{8 d^2 \left (c d^2-a e^2\right )^2 \left (d+e x^2\right )}+\frac {3 \sqrt [4]{a} \left (5 c^2 d^4-2 a c d^2 e^2+a^2 e^4\right ) \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 )}{8 \sqrt [4]{c} d^3 \left (c d^2-a e^2\right )^2 \sqrt {a-c x^4}}-\frac {\left (\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (7 c d^2-2 \sqrt {a} \sqrt {c} d e-3 a e^2\right ) \sqrt {1-\frac {c x^4}{a}}\right ) \int \frac {1}{\sqrt {1-\frac {c x^4}{a}}} \, dx}{8 d^2 \left (c d^2-a e^2\right )^2 \sqrt {a-c x^4}}-\frac {\left (3 \sqrt {a} \sqrt {c} e \left (3 c d^2-a e^2\right ) \sqrt {1-\frac {c x^4}{a}}\right ) \int \frac {1+\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {1-\frac {c x^4}{a}}} \, dx}{8 d^2 \left (c d^2-a e^2\right )^2 \sqrt {a-c x^4}}\\ &=-\frac {e^2 x \sqrt {a-c x^4}}{4 d \left (c d^2-a e^2\right ) \left (d+e x^2\right )^2}-\frac {3 e^2 \left (3 c d^2-a e^2\right ) x \sqrt {a-c x^4}}{8 d^2 \left (c d^2-a e^2\right )^2 \left (d+e x^2\right )}-\frac {\sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (7 c d^2-2 \sqrt {a} \sqrt {c} d e-3 a e^2\right ) \sqrt {1-\frac {c x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 d^2 \left (c d^2-a e^2\right )^2 \sqrt {a-c x^4}}+\frac {3 \sqrt [4]{a} \left (5 c^2 d^4-2 a c d^2 e^2+a^2 e^4\right ) \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 )}{8 \sqrt [4]{c} d^3 \left (c d^2-a e^2\right )^2 \sqrt {a-c x^4}}-\frac {\left (3 \sqrt {a} \sqrt {c} e \left (3 c d^2-a e^2\right ) \sqrt {1-\frac {c x^4}{a}}\right ) \int \frac {\sqrt {1+\frac {\sqrt {c} x^2}{\sqrt {a}}}}{\sqrt {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}} \, dx}{8 d^2 \left (c d^2-a e^2\right )^2 \sqrt {a-c x^4}}\\ &=-\frac {e^2 x \sqrt {a-c x^4}}{4 d \left (c d^2-a e^2\right ) \left (d+e x^2\right )^2}-\frac {3 e^2 \left (3 c d^2-a e^2\right ) x \sqrt {a-c x^4}}{8 d^2 \left (c d^2-a e^2\right )^2 \left (d+e x^2\right )}-\frac {3 a^{3/4} \sqrt [4]{c} e \left (3 c d^2-a e^2\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 d^2 \left (c d^2-a e^2\right )^2 \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (7 c d^2-2 \sqrt {a} \sqrt {c} d e-3 a e^2\right ) \sqrt {1-\frac {c x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 d^2 \left (c d^2-a e^2\right )^2 \sqrt {a-c x^4}}+\frac {3 \sqrt [4]{a} \left (5 c^2 d^4-2 a c d^2 e^2+a^2 e^4\right ) \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 )}{8 \sqrt [4]{c} d^3 \left (c d^2-a e^2\right )^2 \sqrt {a-c x^4}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 10.67, size = 321, normalized size = 0.76 \begin {gather*} \frac {\frac {d e^2 x \left (a-c x^4\right ) \left (a e^2 \left (5 d+3 e x^2\right )-c d^2 \left (11 d+9 e x^2\right )\right )}{\left (d+e x^2\right )^2}-\frac {i \sqrt {1-\frac {c x^4}{a}} \left (3 \sqrt {a} \sqrt {c} d e \left (-3 c d^2+a e^2\right ) E\left (\left .i \sinh ^{-1}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )+\left (-7 c^2 d^4+9 \sqrt {a} c^{3/2} d^3 e+a c d^2 e^2-3 a^{3/2} \sqrt {c} d e^3\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )+3 \left (5 c^2 d^4-2 a c d^2 e^2+a^2 e^4\right ) \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 )\right )}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}}{8 d^3 \left (c d^2-a e^2\right )^2 \sqrt {a-c x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 960 vs. \(2 (363 ) = 726\).
time = 0.12, size = 961, normalized size = 2.26 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {a - c x^{4}} \left (d + e x^{2}\right )^{3}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {a-c\,x^4}\,{\left (e\,x^2+d\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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