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 = 0.198591, 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 \[ \frac{\sqrt{\frac{d x^4}{c}+1} (e x)^{m+1} F_1\left (\frac{m+1}{4};2,\frac{1}{2};\frac{m+5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{a^2 e (m+1) \sqrt{c+d x^4}} \]
Antiderivative was successfully verified.
[In] Int[(e*x)^m/((a + b*x^4)^2*Sqrt[c + d*x^4]),x]
[Out]
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Rubi in Sympy [A] time = 28.0564, size = 66, normalized size = 0.81 \[ \frac{\left (e x\right )^{m + 1} \sqrt{c + d x^{4}} \operatorname{appellf_{1}}{\left (\frac{m}{4} + \frac{1}{4},\frac{1}{2},2,\frac{m}{4} + \frac{5}{4},- \frac{d x^{4}}{c},- \frac{b x^{4}}{a} \right )}}{a^{2} c e \sqrt{1 + \frac{d x^{4}}{c}} \left (m + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x)**m/(b*x**4+a)**2/(d*x**4+c)**(1/2),x)
[Out]
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Mathematica [B] time = 1.36066, size = 488, normalized size = 6.02 \[ \frac{x (e x)^m \left (-\frac{a b c d (m+5) \left (c+d x^4\right ) F_1\left (\frac{m+1}{4};-\frac{1}{2},1;\frac{m+5}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )}{\left (a+b x^4\right ) (b c-a d)^2 \left (2 x^4 \left (a d F_1\left (\frac{m+5}{4};\frac{1}{2},1;\frac{m+9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )-2 b c F_1\left (\frac{m+5}{4};-\frac{1}{2},2;\frac{m+9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )+a c (m+5) F_1\left (\frac{m+1}{4};-\frac{1}{2},1;\frac{m+5}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )}-\frac{a b c (m+5) \left (c+d x^4\right ) F_1\left (\frac{m+1}{4};2,-\frac{1}{2};\frac{m+5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{\left (a+b x^4\right )^2 (a d-b c) \left (2 x^4 \left (a d F_1\left (\frac{m+5}{4};2,\frac{1}{2};\frac{m+9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )-4 b c F_1\left (\frac{m+5}{4};3,-\frac{1}{2};\frac{m+9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )+a c (m+5) F_1\left (\frac{m+1}{4};2,-\frac{1}{2};\frac{m+5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )}+\frac{d^2 \sqrt{\frac{d x^4}{c}+1} \, _2F_1\left (\frac{1}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right )}{(b c-a d)^2}\right )}{(m+1) \sqrt{c+d x^4}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(e*x)^m/((a + b*x^4)^2*Sqrt[c + d*x^4]),x]
[Out]
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Maple [F] time = 0.051, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex \right ) ^{m}}{ \left ( b{x}^{4}+a \right ) ^{2}}{\frac{1}{\sqrt{d{x}^{4}+c}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^m/(b*x^4+a)^2/(d*x^4+c)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (e x\right )^{m}}{{\left (b x^{4} + a\right )}^{2} \sqrt{d x^{4} + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x)^m/((b*x^4 + a)^2*sqrt(d*x^4 + c)),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\left (e x\right )^{m}}{{\left (b^{2} x^{8} + 2 \, a b x^{4} + a^{2}\right )} \sqrt{d x^{4} + c}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x)^m/((b*x^4 + a)^2*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)**2/(d*x**4+c)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (e x\right )^{m}}{{\left (b x^{4} + a\right )}^{2} \sqrt{d x^{4} + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x)^m/((b*x^4 + a)^2*sqrt(d*x^4 + c)),x, algorithm="giac")
[Out]