Optimal. Leaf size=127 \[ -\frac {a^{7/4} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{21 c^{5/4} \sqrt {a+c x^4}}+\frac {2 a x \sqrt {a+c x^4}}{21 c}+\frac {1}{7} x^5 \sqrt {a+c x^4} \]
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Rubi [A] time = 0.04, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {279, 321, 220} \[ -\frac {a^{7/4} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{21 c^{5/4} \sqrt {a+c x^4}}+\frac {1}{7} x^5 \sqrt {a+c x^4}+\frac {2 a x \sqrt {a+c x^4}}{21 c} \]
Antiderivative was successfully verified.
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Rule 220
Rule 279
Rule 321
Rubi steps
\begin {align*} \int x^4 \sqrt {a+c x^4} \, dx &=\frac {1}{7} x^5 \sqrt {a+c x^4}+\frac {1}{7} (2 a) \int \frac {x^4}{\sqrt {a+c x^4}} \, dx\\ &=\frac {2 a x \sqrt {a+c x^4}}{21 c}+\frac {1}{7} x^5 \sqrt {a+c x^4}-\frac {\left (2 a^2\right ) \int \frac {1}{\sqrt {a+c x^4}} \, dx}{21 c}\\ &=\frac {2 a x \sqrt {a+c x^4}}{21 c}+\frac {1}{7} x^5 \sqrt {a+c x^4}-\frac {a^{7/4} \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 )}{21 c^{5/4} \sqrt {a+c x^4}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 62, normalized size = 0.49 \[ \frac {x \sqrt {a+c x^4} \left (-\frac {a \, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {5}{4};-\frac {c x^4}{a}\right )}{\sqrt {\frac {c x^4}{a}+1}}+a+c x^4\right )}{7 c} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {c x^{4} + a} x^{4}, x\right ) \]
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 \[ \int \sqrt {c x^{4} + a} x^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.05, size = 108, normalized size = 0.85 \[ \frac {\sqrt {c \,x^{4}+a}\, x^{5}}{7}-\frac {2 \sqrt {-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}+1}\, a^{2} \EllipticF \left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, x , i\right )}{21 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, c}+\frac {2 \sqrt {c \,x^{4}+a}\, a x}{21 c} \]
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 \[ \int \sqrt {c x^{4} + a} x^{4}\,{d x} \]
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 \[ \int x^4\,\sqrt {c\,x^4+a} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.03, size = 39, normalized size = 0.31 \[ \frac {\sqrt {a} x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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