Optimal. Leaf size=81 \[ \frac {\sqrt {\frac {d x^4}{c}+1} (e x)^{m+1} F_1\left (\frac {m+1}{4};2,\frac {1}{2};\frac {m+5}{4};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{a^2 e (m+1) \sqrt {c+d x^4}} \]
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Rubi [A] time = 0.06, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {511, 510} \[ \frac {\sqrt {\frac {d x^4}{c}+1} (e x)^{m+1} F_1\left (\frac {m+1}{4};2,\frac {1}{2};\frac {m+5}{4};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{a^2 e (m+1) \sqrt {c+d x^4}} \]
Antiderivative was successfully verified.
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Rule 510
Rule 511
Rubi steps
\begin {align*} \int \frac {(e x)^m}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx &=\frac {\sqrt {1+\frac {d x^4}{c}} \int \frac {(e x)^m}{\left (a+b x^4\right )^2 \sqrt {1+\frac {d x^4}{c}}} \, dx}{\sqrt {c+d x^4}}\\ &=\frac {(e x)^{1+m} \sqrt {1+\frac {d x^4}{c}} F_1\left (\frac {1+m}{4};2,\frac {1}{2};\frac {5+m}{4};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{a^2 e (1+m) \sqrt {c+d x^4}}\\ \end {align*}
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Mathematica [B] time = 0.23, size = 179, normalized size = 2.21 \[ \frac {x \sqrt {c+d x^4} (e x)^m \left (a^2 d^2 \, _2F_1\left (\frac {1}{2},\frac {m+1}{4};\frac {m+5}{4};-\frac {d x^4}{c}\right )-a b c d F_1\left (\frac {m+1}{4};-\frac {1}{2},1;\frac {m+5}{4};-\frac {d x^4}{c},-\frac {b x^4}{a}\right )+b c (b c-a d) F_1\left (\frac {m+1}{4};2,-\frac {1}{2};\frac {m+5}{4};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )\right )}{a^2 c (m+1) \sqrt {\frac {d x^4}{c}+1} (b c-a d)^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.15, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d x^{4} + c} \left (e x\right )^{m}}{b^{2} d x^{12} + {\left (b^{2} c + 2 \, a b d\right )} x^{8} + {\left (2 \, a b c + a^{2} d\right )} x^{4} + a^{2} c}, x\right ) \]
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 \[ \int \frac {\left (e x\right )^{m}}{{\left (b x^{4} + a\right )}^{2} \sqrt {d x^{4} + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.55, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x \right )^{m}}{\left (b \,x^{4}+a \right )^{2} \sqrt {d \,x^{4}+c}}\, dx \]
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 \[ \int \frac {\left (e x\right )^{m}}{{\left (b x^{4} + a\right )}^{2} \sqrt {d x^{4} + c}}\,{d x} \]
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 \[ \int \frac {{\left (e\,x\right )}^m}{{\left (b\,x^4+a\right )}^2\,\sqrt {d\,x^4+c}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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