Optimal. Leaf size=81 \[ \frac{\sqrt{\frac{d x^4}{c}+1} (e x)^{m+1} F_1\left (\frac{m+1}{4};3,\frac{1}{2};\frac{m+5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{a^3 e (m+1) \sqrt{c+d x^4}} \]
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Rubi [A] time = 0.19977, antiderivative size = 81, 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 \[ \frac{\sqrt{\frac{d x^4}{c}+1} (e x)^{m+1} F_1\left (\frac{m+1}{4};3,\frac{1}{2};\frac{m+5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{a^3 e (m+1) \sqrt{c+d x^4}} \]
Antiderivative was successfully verified.
[In] Int[(e*x)^m/((a + b*x^4)^3*Sqrt[c + d*x^4]),x]
[Out]
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Rubi in Sympy [A] time = 28.1405, size = 66, normalized size = 0.81 \[ \frac{\left (e x\right )^{m + 1} \sqrt{c + d x^{4}} \operatorname{appellf_{1}}{\left (\frac{m}{4} + \frac{1}{4},\frac{1}{2},3,\frac{m}{4} + \frac{5}{4},- \frac{d x^{4}}{c},- \frac{b x^{4}}{a} \right )}}{a^{3} c e \sqrt{1 + \frac{d x^{4}}{c}} \left (m + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x)**m/(b*x**4+a)**3/(d*x**4+c)**(1/2),x)
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Mathematica [B] time = 0.590451, size = 209, normalized size = 2.58 \[ -\frac{a c (m+5) x (e x)^m F_1\left (\frac{m+1}{4};3,\frac{1}{2};\frac{m+5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{(m+1) \left (a+b x^4\right )^3 \sqrt{c+d x^4} \left (2 x^4 \left (a d F_1\left (\frac{m+5}{4};3,\frac{3}{2};\frac{m+9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )+6 b c F_1\left (\frac{m+5}{4};4,\frac{1}{2};\frac{m+9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )-a c (m+5) F_1\left (\frac{m+1}{4};3,\frac{1}{2};\frac{m+5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(e*x)^m/((a + b*x^4)^3*Sqrt[c + d*x^4]),x]
[Out]
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Maple [F] time = 0.05, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex \right ) ^{m}}{ \left ( b{x}^{4}+a \right ) ^{3}}{\frac{1}{\sqrt{d{x}^{4}+c}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^m/(b*x^4+a)^3/(d*x^4+c)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (e x\right )^{m}}{{\left (b x^{4} + a\right )}^{3} \sqrt{d x^{4} + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x)^m/((b*x^4 + a)^3*sqrt(d*x^4 + c)),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\left (e x\right )^{m}}{{\left (b^{3} x^{12} + 3 \, a b^{2} x^{8} + 3 \, a^{2} b x^{4} + a^{3}\right )} \sqrt{d x^{4} + c}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x)^m/((b*x^4 + a)^3*sqrt(d*x^4 + c)),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x)**m/(b*x**4+a)**3/(d*x**4+c)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (e x\right )^{m}}{{\left (b x^{4} + a\right )}^{3} \sqrt{d x^{4} + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x)^m/((b*x^4 + a)^3*sqrt(d*x^4 + c)),x, algorithm="giac")
[Out]