Optimal. Leaf size=69 \[ \frac {2 \sqrt {e x} \sqrt {c+d x^4} F_1\left (\frac {1}{8};1,-\frac {1}{2};\frac {9}{8};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{a e \sqrt {1+\frac {d x^4}{c}}} \]
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Rubi [A]
time = 0.05, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {477, 441, 440}
\begin {gather*} \frac {2 \sqrt {e x} \sqrt {c+d x^4} F_1\left (\frac {1}{8};1,-\frac {1}{2};\frac {9}{8};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{a e \sqrt {\frac {d x^4}{c}+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 440
Rule 441
Rule 477
Rubi steps
\begin {align*} \int \frac {\sqrt {c+d x^4}}{\sqrt {e x} \left (a+b x^4\right )} \, dx &=\frac {2 \text {Subst}\left (\int \frac {\sqrt {c+\frac {d x^8}{e^4}}}{a+\frac {b x^8}{e^4}} \, dx,x,\sqrt {e x}\right )}{e}\\ &=\frac {\left (2 \sqrt {c+d x^4}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {d x^8}{c e^4}}}{a+\frac {b x^8}{e^4}} \, dx,x,\sqrt {e x}\right )}{e \sqrt {1+\frac {d x^4}{c}}}\\ &=\frac {2 \sqrt {e x} \sqrt {c+d x^4} F_1\left (\frac {1}{8};1,-\frac {1}{2};\frac {9}{8};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{a e \sqrt {1+\frac {d x^4}{c}}}\\ \end {align*}
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Mathematica [A]
time = 10.05, size = 68, normalized size = 0.99 \begin {gather*} \frac {2 x \sqrt {c+d x^4} F_1\left (\frac {1}{8};-\frac {1}{2},1;\frac {9}{8};-\frac {d x^4}{c},-\frac {b x^4}{a}\right )}{a \sqrt {e x} \sqrt {\frac {c+d x^4}{c}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {d \,x^{4}+c}}{\sqrt {e x}\, \left (b \,x^{4}+a \right )}\, dx\]
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 {\sqrt {c + d x^{4}}}{\sqrt {e x} \left (a + b x^{4}\right )}\, 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 = -0.01 \begin {gather*} \int \frac {\sqrt {d\,x^4+c}}{\sqrt {e\,x}\,\left (b\,x^4+a\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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