3.680 \(\int \frac{(e x)^m}{\left (c+d x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=71 \[ \frac{\sqrt{\frac{d x^4}{c}+1} (e x)^{m+1} \, _2F_1\left (\frac{3}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right )}{c e (m+1) \sqrt{c+d x^4}} \]

[Out]

((e*x)^(1 + m)*Sqrt[1 + (d*x^4)/c]*Hypergeometric2F1[3/2, (1 + m)/4, (5 + m)/4,
-((d*x^4)/c)])/(c*e*(1 + m)*Sqrt[c + d*x^4])

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Rubi [A]  time = 0.0695755, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{\sqrt{\frac{d x^4}{c}+1} (e x)^{m+1} \, _2F_1\left (\frac{3}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right )}{c e (m+1) \sqrt{c+d x^4}} \]

Antiderivative was successfully verified.

[In]  Int[(e*x)^m/(c + d*x^4)^(3/2),x]

[Out]

((e*x)^(1 + m)*Sqrt[1 + (d*x^4)/c]*Hypergeometric2F1[3/2, (1 + m)/4, (5 + m)/4,
-((d*x^4)/c)])/(c*e*(1 + m)*Sqrt[c + d*x^4])

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Rubi in Sympy [A]  time = 8.23275, size = 58, normalized size = 0.82 \[ \frac{\left (e x\right )^{m + 1} \sqrt{c + d x^{4}}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{m}{4} + \frac{1}{4} \\ \frac{m}{4} + \frac{5}{4} \end{matrix}\middle |{- \frac{d x^{4}}{c}} \right )}}{c^{2} e \sqrt{1 + \frac{d x^{4}}{c}} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**m/(d*x**4+c)**(3/2),x)

[Out]

(e*x)**(m + 1)*sqrt(c + d*x**4)*hyper((3/2, m/4 + 1/4), (m/4 + 5/4,), -d*x**4/c)
/(c**2*e*sqrt(1 + d*x**4/c)*(m + 1))

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Mathematica [A]  time = 0.0354781, size = 67, normalized size = 0.94 \[ \frac{x \sqrt{\frac{d x^4}{c}+1} (e x)^m \, _2F_1\left (\frac{3}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right )}{c (m+1) \sqrt{c+d x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(e*x)^m/(c + d*x^4)^(3/2),x]

[Out]

(x*(e*x)^m*Sqrt[1 + (d*x^4)/c]*Hypergeometric2F1[3/2, (1 + m)/4, (5 + m)/4, -((d
*x^4)/c)])/(c*(1 + m)*Sqrt[c + d*x^4])

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Maple [F]  time = 0.031, size = 0, normalized size = 0. \[ \int{ \left ( ex \right ) ^{m} \left ( d{x}^{4}+c \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^m/(d*x^4+c)^(3/2),x)

[Out]

int((e*x)^m/(d*x^4+c)^(3/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (e x\right )^{m}}{{\left (d x^{4} + c\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)^m/(d*x^4 + c)^(3/2),x, algorithm="maxima")

[Out]

integrate((e*x)^m/(d*x^4 + c)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\left (e x\right )^{m}}{{\left (d x^{4} + c\right )}^{\frac{3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)^m/(d*x^4 + c)^(3/2),x, algorithm="fricas")

[Out]

integral((e*x)^m/(d*x^4 + c)^(3/2), x)

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Sympy [A]  time = 2.79688, size = 56, normalized size = 0.79 \[ \frac{e^{m} x x^{m} \Gamma \left (\frac{m}{4} + \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{m}{4} + \frac{1}{4} \\ \frac{m}{4} + \frac{5}{4} \end{matrix}\middle |{\frac{d x^{4} e^{i \pi }}{c}} \right )}}{4 c^{\frac{3}{2}} \Gamma \left (\frac{m}{4} + \frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)**m/(d*x**4+c)**(3/2),x)

[Out]

e**m*x*x**m*gamma(m/4 + 1/4)*hyper((3/2, m/4 + 1/4), (m/4 + 5/4,), d*x**4*exp_po
lar(I*pi)/c)/(4*c**(3/2)*gamma(m/4 + 5/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (e x\right )^{m}}{{\left (d x^{4} + c\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)^m/(d*x^4 + c)^(3/2),x, algorithm="giac")

[Out]

integrate((e*x)^m/(d*x^4 + c)^(3/2), x)