3.9.49 \(\int \frac {(e x)^m}{(c+d x^4)^{3/2}} \, dx\) [849]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 71, normalized size of antiderivative = 1.00, 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 = {372, 371} \begin {gather*} \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}} \end {gather*}

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 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 372

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])

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*}

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Mathematica [A]
time = 1.24, size = 69, normalized size = 0.97 \begin {gather*} \frac {x (e x)^m \sqrt {1+\frac {d x^4}{c}} \, _2F_1\left (\frac {3}{2},\frac {1+m}{4};1+\frac {1+m}{4};-\frac {d x^4}{c}\right )}{c (1+m) \sqrt {c+d x^4}} \end {gather*}

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 = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (e x \right )^{m}}{\left (d \,x^{4}+c \right )^{\frac {3}{2}}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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((x*e)^m/(d*x^4 + c)^(3/2), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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Sympy [C] Result contains complex when optimal does not.
time = 0.83, size = 56, normalized size = 0.79 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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((x*e)^m/(d*x^4 + c)^(3/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (e\,x\right )}^m}{{\left (d\,x^4+c\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m/(c + d*x^4)^(3/2),x)

[Out]

int((e*x)^m/(c + d*x^4)^(3/2), x)

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