Optimal. Leaf size=69 \[ \frac {2 \sqrt {e x} \sqrt {c+d x^4} F_1\left (\frac {1}{8};1,-\frac {1}{2};\frac {9}{8};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{a e \sqrt {\frac {d x^4}{c}+1}} \]
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Rubi [A] time = 0.07, 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 = {466, 430, 429} \[ \frac {2 \sqrt {e x} \sqrt {c+d x^4} F_1\left (\frac {1}{8};1,-\frac {1}{2};\frac {9}{8};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{a e \sqrt {\frac {d x^4}{c}+1}} \]
Antiderivative was successfully verified.
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Rule 429
Rule 430
Rule 466
Rubi steps
\begin {align*} \int \frac {\sqrt {c+d x^4}}{\sqrt {e x} \left (a+b x^4\right )} \, dx &=\frac {2 \operatorname {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 ) \operatorname {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 = 0.05, size = 68, normalized size = 0.99 \[ \frac {2 x \sqrt {c+d x^4} F_1\left (\frac {1}{8};-\frac {1}{2},1;\frac {9}{8};-\frac {d x^4}{c},-\frac {b x^4}{a}\right )}{a \sqrt {e x} \sqrt {\frac {c+d x^4}{c}}} \]
Warning: Unable to verify antiderivative.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {\sqrt {d x^{4} + c}}{{\left (b x^{4} + a\right )} \sqrt {e x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.64, 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 \[ \int \frac {\sqrt {d x^{4} + c}}{{\left (b x^{4} + a\right )} \sqrt {e x}}\,{d x} \]
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 \[ \int \frac {\sqrt {d\,x^4+c}}{\sqrt {e\,x}\,\left (b\,x^4+a\right )} \,d x \]
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 \[ \int \frac {\sqrt {c + d x^{4}}}{\sqrt {e x} \left (a + b x^{4}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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