Optimal. Leaf size=84 \[ \frac{\sqrt{\frac{d x^4}{c}+1} (e x)^{m+1} F_1\left (\frac{m+1}{4};1,\frac{3}{2};\frac{m+5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{a c e (m+1) \sqrt{c+d x^4}} \]
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Rubi [A] time = 0.0601804, antiderivative size = 84, normalized size of antiderivative = 1., 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} \[ \frac{\sqrt{\frac{d x^4}{c}+1} (e x)^{m+1} F_1\left (\frac{m+1}{4};1,\frac{3}{2};\frac{m+5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{a c e (m+1) \sqrt{c+d x^4}} \]
Antiderivative was successfully verified.
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Rule 511
Rule 510
Rubi steps
\begin{align*} \int \frac{(e x)^m}{\left (a+b x^4\right ) \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 ) \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};1,\frac{3}{2};\frac{5+m}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{a c e (1+m) \sqrt{c+d x^4}}\\ \end{align*}
Mathematica [B] time = 0.136401, size = 169, normalized size = 2.01 \[ \frac{x \sqrt{c+d x^4} (e x)^m \left (b^2 c^2 F_1\left (\frac{m+1}{4};-\frac{1}{2},1;\frac{m+5}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )+a d \left ((a d-b c) \, _2F_1\left (\frac{3}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right )-b c \, _2F_1\left (\frac{1}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right )\right )\right )}{a c^2 (m+1) \sqrt{\frac{d x^4}{c}+1} (b c-a d)^2} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.03, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex \right ) ^{m}}{b{x}^{4}+a} \left ( d{x}^{4}+c \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e x\right )^{m}}{{\left (b x^{4} + a\right )}{\left (d x^{4} + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{d x^{4} + c} \left (e x\right )^{m}}{b d^{2} x^{12} +{\left (2 \, b c d + a d^{2}\right )} x^{8} +{\left (b c^{2} + 2 \, a c d\right )} x^{4} + a c^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e x\right )^{m}}{{\left (b x^{4} + a\right )}{\left (d x^{4} + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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