3.803 \(\int \frac{\sqrt{c+d x^4}}{\sqrt{e x} (a+b x^4)} \, dx\)

Optimal. Leaf size=69 \[ \frac{2 \sqrt{e x} \sqrt{c+d x^4} F_1\left (\frac{1}{8};1,-\frac{1}{2};\frac{9}{8};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{a e \sqrt{\frac{d x^4}{c}+1}} \]

[Out]

(2*Sqrt[e*x]*Sqrt[c + d*x^4]*AppellF1[1/8, 1, -1/2, 9/8, -((b*x^4)/a), -((d*x^4)/c)])/(a*e*Sqrt[1 + (d*x^4)/c]
)

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Rubi [A]  time = 0.0729746, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {466, 430, 429} \[ \frac{2 \sqrt{e x} \sqrt{c+d x^4} F_1\left (\frac{1}{8};1,-\frac{1}{2};\frac{9}{8};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{a e \sqrt{\frac{d x^4}{c}+1}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c + d*x^4]/(Sqrt[e*x]*(a + b*x^4)),x]

[Out]

(2*Sqrt[e*x]*Sqrt[c + d*x^4]*AppellF1[1/8, 1, -1/2, 9/8, -((b*x^4)/a), -((d*x^4)/c)])/(a*e*Sqrt[1 + (d*x^4)/c]
)

Rule 466

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno
minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/e^n)^p*(c + (d*x^(k*n))/e^n)^q, x], x, (e*
x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege
rQ[p]

Rule 430

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^F
racPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(1 + (b*x^n)/a)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n,
p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] &&  !(IntegerQ[p] || GtQ[a, 0])

Rule 429

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p,
-q, 1 + 1/n, -((b*x^n)/a), -((d*x^n)/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n
, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rubi steps

\begin{align*} \int \frac{\sqrt{c+d x^4}}{\sqrt{e x} \left (a+b x^4\right )} \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{\sqrt{c+\frac{d x^8}{e^4}}}{a+\frac{b x^8}{e^4}} \, dx,x,\sqrt{e x}\right )}{e}\\ &=\frac{\left (2 \sqrt{c+d x^4}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{d x^8}{c e^4}}}{a+\frac{b x^8}{e^4}} \, dx,x,\sqrt{e x}\right )}{e \sqrt{1+\frac{d x^4}{c}}}\\ &=\frac{2 \sqrt{e x} \sqrt{c+d x^4} F_1\left (\frac{1}{8};1,-\frac{1}{2};\frac{9}{8};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{a e \sqrt{1+\frac{d x^4}{c}}}\\ \end{align*}

Mathematica [A]  time = 0.0311815, size = 68, normalized size = 0.99 \[ \frac{2 x \sqrt{c+d x^4} F_1\left (\frac{1}{8};-\frac{1}{2},1;\frac{9}{8};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )}{a \sqrt{e x} \sqrt{\frac{c+d x^4}{c}}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sqrt[c + d*x^4]/(Sqrt[e*x]*(a + b*x^4)),x]

[Out]

(2*x*Sqrt[c + d*x^4]*AppellF1[1/8, -1/2, 1, 9/8, -((d*x^4)/c), -((b*x^4)/a)])/(a*Sqrt[e*x]*Sqrt[(c + d*x^4)/c]
)

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Maple [F]  time = 0.049, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{b{x}^{4}+a}\sqrt{d{x}^{4}+c}{\frac{1}{\sqrt{ex}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(d*x^4 + c)/((b*x^4 + a)*sqrt(e*x)), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(c + d*x**4)/(sqrt(e*x)*(a + b*x**4)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(d*x^4 + c)/((b*x^4 + a)*sqrt(e*x)), x)