Optimal. Leaf size=126 \[ \frac {a^{3/4} e \sqrt {1-\frac {c x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} \sqrt {-a+c x^4}}+\frac {a^{3/4} \left (\frac {\sqrt {c} d}{\sqrt {a}}-e\right ) \sqrt {1-\frac {c x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} \sqrt {-a+c x^4}} \]
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Rubi [A]
time = 0.05, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1215, 230, 227,
1214, 1213, 435} \begin {gather*} \frac {a^{3/4} \sqrt {1-\frac {c x^4}{a}} \left (\frac {\sqrt {c} d}{\sqrt {a}}-e\right ) F\left (\left .\text {ArcSin}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} \sqrt {c x^4-a}}+\frac {a^{3/4} e \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 )}{c^{3/4} \sqrt {c x^4-a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 227
Rule 230
Rule 435
Rule 1213
Rule 1214
Rule 1215
Rubi steps
\begin {align*} \int \frac {d+e x^2}{\sqrt {-a+c x^4}} \, dx &=\frac {\left (\sqrt {a} e\right ) \int \frac {1+\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {-a+c x^4}} \, dx}{\sqrt {c}}+\left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {-a+c x^4}} \, dx\\ &=\frac {\left (\sqrt {a} e \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}{\sqrt {c} \sqrt {-a+c x^4}}+\frac {\left (\left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \sqrt {1-\frac {c x^4}{a}}\right ) \int \frac {1}{\sqrt {1-\frac {c x^4}{a}}} \, dx}{\sqrt {-a+c x^4}}\\ &=\frac {\sqrt [4]{a} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\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 )}{\sqrt [4]{c} \sqrt {-a+c x^4}}+\frac {\left (\sqrt {a} e \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 {c} \sqrt {-a+c x^4}}\\ &=\frac {a^{3/4} e \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 )}{c^{3/4} \sqrt {-a+c x^4}}+\frac {\sqrt [4]{a} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\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 )}{\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 = 10.03, size = 78, normalized size = 0.62 \begin {gather*} \frac {\sqrt {1-\frac {c x^4}{a}} \left (3 d x \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\frac {c x^4}{a}\right )+e 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 [A]
time = 0.12, size = 160, normalized size = 1.27
method | result | size |
default | \(\frac {e \sqrt {a}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\EllipticF \left (x \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\EllipticE \left (x \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}-a}\, \sqrt {c}}+\frac {d \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}-a}}\) | \(160\) |
elliptic | \(\frac {e \sqrt {a}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\EllipticF \left (x \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\EllipticE \left (x \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}-a}\, \sqrt {c}}+\frac {d \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}-a}}\) | \(160\) |
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 = 1.05, size = 73, normalized size = 0.58 \begin {gather*} - \frac {i d 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 \sqrt {a} \Gamma \left (\frac {5}{4}\right )} - \frac {i e 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 = -0.01 \begin {gather*} \int \frac {e\,x^2+d}{\sqrt {c\,x^4-a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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