Optimal. Leaf size=54 \[ \frac {a^{3/4} \sqrt {1-\frac {c x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{c} \sqrt {-a+c x^4}} \]
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Rubi [A]
time = 0.03, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {1214, 1213,
435} \begin {gather*} \frac {a^{3/4} \sqrt {1-\frac {c x^4}{a}} E\left (\left .\text {ArcSin}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{c} \sqrt {c x^4-a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 435
Rule 1213
Rule 1214
Rubi steps
\begin {align*} \int \frac {\sqrt {a}+\sqrt {c} x^2}{\sqrt {-a+c x^4}} \, dx &=\frac {\sqrt {1-\frac {c x^4}{a}} \int \frac {\sqrt {a}+\sqrt {c} x^2}{\sqrt {1-\frac {c x^4}{a}}} \, dx}{\sqrt {-a+c x^4}}\\ &=\frac {\left (\sqrt {a} \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}{\sqrt {-a+c x^4}}\\ &=\frac {a^{3/4} \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 )}{\sqrt [4]{c} \sqrt {-a+c x^4}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.04, size = 86, normalized size = 1.59 \begin {gather*} \frac {\sqrt {1-\frac {c x^4}{a}} \left (3 \sqrt {a} x \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\frac {c x^4}{a}\right )+\sqrt {c} x^3 \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};\frac {c x^4}{a}\right )\right )}{3 \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. 157 vs. \(2 (42 ) = 84\).
time = 0.14, size = 158, normalized size = 2.93
method | result | size |
default | \(\frac {\sqrt {a}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}-a}}+\frac {\sqrt {a}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\EllipticF \left (x \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\EllipticE \left (x \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}-a}}\) | \(158\) |
elliptic | \(\frac {\left (\sqrt {a}+x^{2} \sqrt {c}\right ) \sqrt {-\left (-c \,x^{4}+a \right ) c}\, \sqrt {-\left (-c \,x^{4}+a \right ) a}\, \left (\frac {\sqrt {c}\, \sqrt {a}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\EllipticF \left (x \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\EllipticE \left (x \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {c^{2} x^{4}-a c}}+\frac {a \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {a c \,x^{4}-a^{2}}}\right )}{\sqrt {c \,x^{4}-a}\, \left (c \,x^{2} \sqrt {-\left (-c \,x^{4}+a \right ) a}+a \sqrt {-\left (-c \,x^{4}+a \right ) c}\right )}\) | \(250\) |
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.01, size = 70, normalized size = 1.30 \begin {gather*} - \frac {i x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{4}}{a}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} - \frac {i \sqrt {c} x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{4}}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {7}{4}\right )} \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 = -0.02 \begin {gather*} \int \frac {\sqrt {a}+\sqrt {c}\,x^2}{\sqrt {c\,x^4-a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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