Optimal. Leaf size=81 \[ \frac {(e x)^{1+m} \sqrt {1+\frac {d x^4}{c}} F_1\left (\frac {1+m}{4};2,\frac {1}{2};\frac {5+m}{4};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{a^2 e (1+m) \sqrt {c+d x^4}} \]
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Rubi [A]
time = 0.04, 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 = {525, 524}
\begin {gather*} \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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 524
Rule 525
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] Leaf count is larger than twice the leaf count of optimal. \(179\) vs. \(2(81)=162\).
time = 7.91, size = 179, normalized size = 2.21 \begin {gather*} \frac {x (e x)^m \sqrt {c+d x^4} \left (-a b c d F_1\left (\frac {1+m}{4};-\frac {1}{2},1;\frac {5+m}{4};-\frac {d x^4}{c},-\frac {b x^4}{a}\right )+b c (b c-a d) 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 d^2 \, _2F_1\left (\frac {1}{2},\frac {1+m}{4};\frac {5+m}{4};-\frac {d x^4}{c}\right )\right )}{a^2 c (b c-a d)^2 (1+m) \sqrt {1+\frac {d x^4}{c}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.07, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (e x\right )^{m}}{\left (a + b x^{4}\right )^{2} \sqrt {c + d x^{4}}}\, 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 {{\left (e\,x\right )}^m}{{\left (b\,x^4+a\right )}^2\,\sqrt {d\,x^4+c}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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