Optimal. Leaf size=105 \[ \frac {1}{3} x \sqrt {a+c x^4}+\frac {a^{3/4} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{3 \sqrt [4]{c} \sqrt {a+c x^4}} \]
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Rubi [A]
time = 0.01, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {201, 226}
\begin {gather*} \frac {a^{3/4} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{3 \sqrt [4]{c} \sqrt {a+c x^4}}+\frac {1}{3} x \sqrt {a+c x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 226
Rubi steps
\begin {align*} \int \sqrt {a+c x^4} \, dx &=\frac {1}{3} x \sqrt {a+c x^4}+\frac {1}{3} (2 a) \int \frac {1}{\sqrt {a+c x^4}} \, dx\\ &=\frac {1}{3} x \sqrt {a+c x^4}+\frac {a^{3/4} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{3 \sqrt [4]{c} \sqrt {a+c x^4}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 3.73, size = 89, normalized size = 0.85 \begin {gather*} \frac {x \left (a+c x^4\right )-\frac {2 i a \sqrt {1+\frac {c x^4}{a}} F\left (\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}}{3 \sqrt {a+c x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.21, size = 85, normalized size = 0.81
method | result | size |
default | \(\frac {x \sqrt {c \,x^{4}+a}}{3}+\frac {2 a \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{3 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\) | \(85\) |
risch | \(\frac {x \sqrt {c \,x^{4}+a}}{3}+\frac {2 a \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{3 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\) | \(85\) |
elliptic | \(\frac {x \sqrt {c \,x^{4}+a}}{3}+\frac {2 a \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{3 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\) | \(85\) |
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 [A]
time = 0.10, size = 40, normalized size = 0.38 \begin {gather*} \frac {2}{3} \, \sqrt {c} \left (-\frac {a}{c}\right )^{\frac {3}{4}} {\rm ellipticF}\left (\frac {\left (-\frac {a}{c}\right )^{\frac {1}{4}}}{x}, -1\right ) + \frac {1}{3} \, \sqrt {c x^{4} + a} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.39, size = 37, normalized size = 0.35 \begin {gather*} \frac {\sqrt {a} x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {5}{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 [B]
time = 2.64, size = 37, normalized size = 0.35 \begin {gather*} \frac {x\,\sqrt {c\,x^4+a}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{2},\frac {1}{4};\ \frac {5}{4};\ -\frac {c\,x^4}{a}\right )}{\sqrt {\frac {c\,x^4}{a}+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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