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

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Rubi [A]  time = 0.02, antiderivative size = 68, 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 = {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 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])

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])

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*}

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Mathematica [A]  time = 0.02, 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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fricas [F]  time = 1.09, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (e x\right )^{m}}{\sqrt {d x^{4} + c}}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x\right )^{m}}{\sqrt {d x^{4} + c}}\,{d x} \]

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)

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maple [F]  time = 0.42, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x \right )^{m}}{\sqrt {d \,x^{4}+c}}\, dx \]

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

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (e\,x\right )}^m}{\sqrt {d\,x^4+c}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m/(c + d*x^4)^(1/2),x)

[Out]

int((e*x)^m/(c + d*x^4)^(1/2), x)

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sympy [C]  time = 1.07, size = 56, normalized size = 0.82 \[ \frac {e^{m} x x^{m} \Gamma \left (\frac {m}{4} + \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {m}{4} + \frac {1}{4} \\ \frac {m}{4} + \frac {5}{4} \end {matrix}\middle | {\frac {d x^{4} e^{i \pi }}{c}} \right )}}{4 \sqrt {c} \Gamma \left (\frac {m}{4} + \frac {5}{4}\right )} \]

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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