Optimal. Leaf size=84 \[ \frac {(e x)^{1+m} \sqrt {1+\frac {d x^4}{c}} F_1\left (\frac {1+m}{4};3,\frac {3}{2};\frac {5+m}{4};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{a^3 c e (1+m) \sqrt {c+d x^4}} \]
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Rubi [A]
time = 0.04, antiderivative size = 84, 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};3,\frac {3}{2};\frac {m+5}{4};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{a^3 c 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 )^3 \left (c+d x^4\right )^{3/2}} \, dx &=\frac {\sqrt {1+\frac {d x^4}{c}} \int \frac {(e x)^m}{\left (a+b x^4\right )^3 \left (1+\frac {d x^4}{c}\right )^{3/2}} \, dx}{c \sqrt {c+d x^4}}\\ &=\frac {(e x)^{1+m} \sqrt {1+\frac {d x^4}{c}} F_1\left (\frac {1+m}{4};3,\frac {3}{2};\frac {5+m}{4};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{a^3 c e (1+m) \sqrt {c+d x^4}}\\ \end {align*}
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Mathematica [A]
time = 10.08, size = 77, normalized size = 0.92 \begin {gather*} \frac {x (e x)^m \left (1+\frac {d x^4}{c}\right )^{3/2} F_1\left (\frac {1+m}{4};3,\frac {3}{2};\frac {5+m}{4};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{a^3 (1+m) \left (c+d x^4\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {\left (e x \right )^{m}}{\left (b \,x^{4}+a \right )^{3} \left (d \,x^{4}+c \right )^{\frac {3}{2}}}\, dx\]
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(-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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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 )}^3\,{\left (d\,x^4+c\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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