Optimal. Leaf size=132 \[ \frac{\sqrt{\frac{d x^4}{c}+1} (e x)^{m+1} (a d (1-m)+b c (m+1)) \, _2F_1\left (\frac{1}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right )}{2 c d e (m+1) \sqrt{c+d x^4}}-\frac{(e x)^{m+1} (b c-a d)}{2 c d e \sqrt{c+d x^4}} \]
[Out]
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Rubi [A] time = 0.188099, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{\sqrt{\frac{d x^4}{c}+1} (e x)^{m+1} (a d (1-m)+b c (m+1)) \, _2F_1\left (\frac{1}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right )}{2 c d e (m+1) \sqrt{c+d x^4}}-\frac{(e x)^{m+1} (b c-a d)}{2 c d e \sqrt{c+d x^4}} \]
Antiderivative was successfully verified.
[In] Int[((e*x)^m*(a + b*x^4))/(c + d*x^4)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 18.1395, size = 105, normalized size = 0.8 \[ \frac{\left (e x\right )^{m + 1} \left (a d - b c\right )}{2 c d e \sqrt{c + d x^{4}}} + \frac{\left (e x\right )^{m + 1} \sqrt{c + d x^{4}} \left (a d \left (- m + 1\right ) + b c \left (m + 1\right )\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m}{4} + \frac{1}{4} \\ \frac{m}{4} + \frac{5}{4} \end{matrix}\middle |{- \frac{d x^{4}}{c}} \right )}}{2 c^{2} d e \sqrt{1 + \frac{d x^{4}}{c}} \left (m + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x)**m*(b*x**4+a)/(d*x**4+c)**(3/2),x)
[Out]
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Mathematica [A] time = 0.132366, size = 110, normalized size = 0.83 \[ \frac{x \sqrt{\frac{d x^4}{c}+1} (e x)^m \left ((a d-b c) \, _2F_1\left (\frac{3}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right )+b c \, _2F_1\left (\frac{1}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right )\right )}{c d (m+1) \sqrt{c+d x^4}} \]
Antiderivative was successfully verified.
[In] Integrate[((e*x)^m*(a + b*x^4))/(c + d*x^4)^(3/2),x]
[Out]
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Maple [F] time = 0.032, size = 0, normalized size = 0. \[ \int{ \left ( ex \right ) ^{m} \left ( b{x}^{4}+a \right ) \left ( d{x}^{4}+c \right ) ^{-{\frac{3}{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^m*(b*x^4+a)/(d*x^4+c)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )} \left (e x\right )^{m}}{{\left (d x^{4} + c\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)*(e*x)^m/(d*x^4 + c)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x^{4} + a\right )} \left (e x\right )^{m}}{{\left (d x^{4} + c\right )}^{\frac{3}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)*(e*x)^m/(d*x^4 + c)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 178.942, size = 119, normalized size = 0.9 \[ \frac{a e^{m} x x^{m} \Gamma \left (\frac{m}{4} + \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{m}{4} + \frac{1}{4} \\ \frac{m}{4} + \frac{5}{4} \end{matrix}\middle |{\frac{d x^{4} e^{i \pi }}{c}} \right )}}{4 c^{\frac{3}{2}} \Gamma \left (\frac{m}{4} + \frac{5}{4}\right )} + \frac{b e^{m} x^{5} x^{m} \Gamma \left (\frac{m}{4} + \frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{m}{4} + \frac{5}{4} \\ \frac{m}{4} + \frac{9}{4} \end{matrix}\middle |{\frac{d x^{4} e^{i \pi }}{c}} \right )}}{4 c^{\frac{3}{2}} \Gamma \left (\frac{m}{4} + \frac{9}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x)**m*(b*x**4+a)/(d*x**4+c)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )} \left (e x\right )^{m}}{{\left (d x^{4} + c\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)*(e*x)^m/(d*x^4 + c)^(3/2),x, algorithm="giac")
[Out]