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

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)*Sq
rt[c + d*x^4])

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Rubi [A]  time = 0.0231556, 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, Rules used = {365, 364} \[ \frac{\sqrt{\frac{d x^4}{c}+1} (e x)^{m+1} \, _2F_1\left (\frac{3}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right )}{c e (m+1) \sqrt{c+d x^4}} \]

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)*Sq
rt[c + d*x^4])

Rule 365

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])
/(1 + (b*x^n)/a)^FracPart[p], Int[(c*x)^m*(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[
p, 0] &&  !(ILtQ[p, 0] || GtQ[a, 0])

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int \frac{(e x)^m}{\left (c+d x^4\right )^{3/2}} \, dx &=\frac{\sqrt{1+\frac{d x^4}{c}} \int \frac{(e x)^m}{\left (1+\frac{d x^4}{c}\right )^{3/2}} \, dx}{c \sqrt{c+d x^4}}\\ &=\frac{(e x)^{1+m} \sqrt{1+\frac{d x^4}{c}} \, _2F_1\left (\frac{3}{2},\frac{1+m}{4};\frac{5+m}{4};-\frac{d x^4}{c}\right )}{c e (1+m) \sqrt{c+d x^4}}\\ \end{align*}

Mathematica [A]  time = 0.0175721, size = 69, normalized size = 0.97 \[ \frac{x \sqrt{\frac{d x^4}{c}+1} (e x)^m \, _2F_1\left (\frac{3}{2},\frac{m+1}{4};\frac{m+1}{4}+1;-\frac{d x^4}{c}\right )}{c (m+1) \sqrt{c+d x^4}} \]

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, 1 + (1 + m)/4, -((d*x^4)/c)])/(c*(1 + m)*Sqrt
[c + d*x^4])

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Maple [F]  time = 0.018, size = 0, normalized size = 0. \begin{align*} \int{ \left ( ex \right ) ^{m} \left ( d{x}^{4}+c \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

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. \begin{align*} \int \frac{\left (e x\right )^{m}}{{\left (d x^{4} + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

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. \begin{align*}{\rm integral}\left (\frac{\sqrt{d x^{4} + c} \left (e x\right )^{m}}{d^{2} x^{8} + 2 \, c d x^{4} + c^{2}}, x\right ) \end{align*}

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(sqrt(d*x^4 + c)*(e*x)^m/(d^2*x^8 + 2*c*d*x^4 + c^2), x)

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Sympy [C]  time = 1.6566, size = 56, normalized size = 0.79 \begin{align*} \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 )} \end{align*}

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_polar(I*pi)/c)/(4*c**(3/2)*gamma
(m/4 + 5/4))

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

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)