3.843 \(\int \frac{(e x)^m}{\sqrt{c+d x^4}} \, dx\)

Optimal. Leaf size=68 \[ \frac{\sqrt{\frac{d x^4}{c}+1} (e x)^{m+1} \, _2F_1\left (\frac{1}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right )}{e (m+1) \sqrt{c+d x^4}} \]

[Out]

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

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Rubi [A]  time = 0.0194269, antiderivative size = 68, 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} \[ \frac{\sqrt{\frac{d x^4}{c}+1} (e x)^{m+1} \, _2F_1\left (\frac{1}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right )}{e (m+1) \sqrt{c+d x^4}} \]

Antiderivative was successfully verified.

[In]

Int[(e*x)^m/Sqrt[c + d*x^4],x]

[Out]

((e*x)^(1 + m)*Sqrt[1 + (d*x^4)/c]*Hypergeometric2F1[1/2, (1 + m)/4, (5 + m)/4, -((d*x^4)/c)])/(e*(1 + m)*Sqrt
[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}{\sqrt{c+d x^4}} \, dx &=\frac{\sqrt{1+\frac{d x^4}{c}} \int \frac{(e x)^m}{\sqrt{1+\frac{d x^4}{c}}} \, dx}{\sqrt{c+d x^4}}\\ &=\frac{(e x)^{1+m} \sqrt{1+\frac{d x^4}{c}} \, _2F_1\left (\frac{1}{2},\frac{1+m}{4};\frac{5+m}{4};-\frac{d x^4}{c}\right )}{e (1+m) \sqrt{c+d x^4}}\\ \end{align*}

Mathematica [A]  time = 0.0155782, size = 66, normalized size = 0.97 \[ \frac{x \sqrt{\frac{d x^4}{c}+1} (e x)^m \, _2F_1\left (\frac{1}{2},\frac{m+1}{4};\frac{m+1}{4}+1;-\frac{d x^4}{c}\right )}{(m+1) \sqrt{c+d x^4}} \]

Antiderivative was successfully verified.

[In]

Integrate[(e*x)^m/Sqrt[c + d*x^4],x]

[Out]

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

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Maple [F]  time = 0.023, size = 0, normalized size = 0. \begin{align*} \int{ \left ( ex \right ) ^{m}{\frac{1}{\sqrt{d{x}^{4}+c}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m/(d*x^4+c)^(1/2),x)

[Out]

int((e*x)^m/(d*x^4+c)^(1/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m/(d*x^4+c)^(1/2),x, algorithm="maxima")

[Out]

integrate((e*x)^m/sqrt(d*x^4 + c), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m/(d*x^4+c)^(1/2),x, algorithm="fricas")

[Out]

integral((e*x)^m/sqrt(d*x^4 + c), x)

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Sympy [C]  time = 0.95311, size = 56, normalized size = 0.82 \begin{align*} \frac{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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**m/(d*x**4+c)**(1/2),x)

[Out]

e**m*x*x**m*gamma(m/4 + 1/4)*hyper((1/2, m/4 + 1/4), (m/4 + 5/4,), d*x**4*exp_polar(I*pi)/c)/(4*sqrt(c)*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}}{\sqrt{d x^{4} + c}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m/(d*x^4+c)^(1/2),x, algorithm="giac")

[Out]

integrate((e*x)^m/sqrt(d*x^4 + c), x)