Optimal. Leaf size=123 \[ \frac{b \sqrt{c+d x^4} (e x)^{m+1}}{d e (m+3)}-\frac{\sqrt{\frac{d x^4}{c}+1} (e x)^{m+1} (b c (m+1)-a d (m+3)) \, _2F_1\left (\frac{1}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right )}{d e (m+1) (m+3) \sqrt{c+d x^4}} \]
[Out]
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Rubi [A] time = 0.183351, antiderivative size = 115, normalized size of antiderivative = 0.93, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{\sqrt{\frac{d x^4}{c}+1} (e x)^{m+1} \left (\frac{a}{m+1}-\frac{b c}{d (m+3)}\right ) \, _2F_1\left (\frac{1}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right )}{e \sqrt{c+d x^4}}+\frac{b \sqrt{c+d x^4} (e x)^{m+1}}{d e (m+3)} \]
Antiderivative was successfully verified.
[In] Int[((e*x)^m*(a + b*x^4))/Sqrt[c + d*x^4],x]
[Out]
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Rubi in Sympy [A] time = 18.3996, size = 100, normalized size = 0.81 \[ \frac{b \left (e x\right )^{m + 1} \sqrt{c + d x^{4}}}{d e \left (m + 3\right )} + \frac{\left (e x\right )^{m + 1} \sqrt{c + d x^{4}} \left (a d \left (m + 3\right ) - b c \left (m + 1\right )\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m}{4} + \frac{1}{4} \\ \frac{m}{4} + \frac{5}{4} \end{matrix}\middle |{- \frac{d x^{4}}{c}} \right )}}{c d e \sqrt{1 + \frac{d x^{4}}{c}} \left (m + 1\right ) \left (m + 3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x)**m*(b*x**4+a)/(d*x**4+c)**(1/2),x)
[Out]
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Mathematica [A] time = 0.133493, size = 110, normalized size = 0.89 \[ \frac{x \sqrt{c+d x^4} (e x)^m \left ((a d-b c) \, _2F_1\left (\frac{1}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right )+b c \, _2F_1\left (-\frac{1}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right )\right )}{c d (m+1) \sqrt{\frac{d x^4}{c}+1}} \]
Antiderivative was successfully verified.
[In] Integrate[((e*x)^m*(a + b*x^4))/Sqrt[c + d*x^4],x]
[Out]
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Maple [F] time = 0.036, size = 0, normalized size = 0. \[ \int{ \left ( ex \right ) ^{m} \left ( b{x}^{4}+a \right ){\frac{1}{\sqrt{d{x}^{4}+c}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^m*(b*x^4+a)/(d*x^4+c)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )} \left (e x\right )^{m}}{\sqrt{d x^{4} + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)*(e*x)^m/sqrt(d*x^4 + c),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x^{4} + a\right )} \left (e x\right )^{m}}{\sqrt{d x^{4} + c}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)*(e*x)^m/sqrt(d*x^4 + c),x, algorithm="fricas")
[Out]
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Sympy [A] time = 11.5919, size = 119, normalized size = 0.97 \[ \frac{a e^{m} x x^{m} \Gamma \left (\frac{m}{4} + \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m}{4} + \frac{1}{4} \\ \frac{m}{4} + \frac{5}{4} \end{matrix}\middle |{\frac{d x^{4} e^{i \pi }}{c}} \right )}}{4 \sqrt{c} \Gamma \left (\frac{m}{4} + \frac{5}{4}\right )} + \frac{b e^{m} x^{5} x^{m} \Gamma \left (\frac{m}{4} + \frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m}{4} + \frac{5}{4} \\ \frac{m}{4} + \frac{9}{4} \end{matrix}\middle |{\frac{d x^{4} e^{i \pi }}{c}} \right )}}{4 \sqrt{c} \Gamma \left (\frac{m}{4} + \frac{9}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x)**m*(b*x**4+a)/(d*x**4+c)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )} \left (e x\right )^{m}}{\sqrt{d x^{4} + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)*(e*x)^m/sqrt(d*x^4 + c),x, algorithm="giac")
[Out]