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

Optimal. Leaf size=71 \[ \frac{2 (e x)^{3/2} \sqrt{c+d x^4} F_1\left (\frac{3}{8};1,-\frac{1}{2};\frac{11}{8};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{3 a e \sqrt{\frac{d x^4}{c}+1}} \]

[Out]

(2*(e*x)^(3/2)*Sqrt[c + d*x^4]*AppellF1[3/8, 1, -1/2, 11/8, -((b*x^4)/a), -((d*x^4)/c)])/(3*a*e*Sqrt[1 + (d*x^
4)/c])

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Rubi [A]  time = 0.114422, antiderivative size = 71, 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, 511, 510} \[ \frac{2 (e x)^{3/2} \sqrt{c+d x^4} F_1\left (\frac{3}{8};1,-\frac{1}{2};\frac{11}{8};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{3 a e \sqrt{\frac{d x^4}{c}+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(e*x)^(3/2)*Sqrt[c + d*x^4]*AppellF1[3/8, 1, -1/2, 11/8, -((b*x^4)/a), -((d*x^4)/c)])/(3*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 511

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

Rule 510

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

Rubi steps

\begin{align*} \int \frac{\sqrt{e x} \sqrt{c+d x^4}}{a+b x^4} \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{x^2 \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{x^2 \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 (e x)^{3/2} \sqrt{c+d x^4} F_1\left (\frac{3}{8};1,-\frac{1}{2};\frac{11}{8};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{3 a e \sqrt{1+\frac{d x^4}{c}}}\\ \end{align*}

Mathematica [A]  time = 0.0333433, size = 70, normalized size = 0.99 \[ \frac{2 x \sqrt{e x} \sqrt{c+d x^4} F_1\left (\frac{3}{8};-\frac{1}{2},1;\frac{11}{8};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )}{3 a \sqrt{\frac{c+d x^4}{c}}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*x*Sqrt[e*x]*Sqrt[c + d*x^4]*AppellF1[3/8, -1/2, 1, 11/8, -((d*x^4)/c), -((b*x^4)/a)])/(3*a*Sqrt[(c + d*x^4)
/c])

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{b{x}^{4}+a}\sqrt{ex}\sqrt{d{x}^{4}+c}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((e*x)^(1/2)*(d*x^4+c)^(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} \sqrt{e x}}{b x^{4} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(d*x^4 + c)*sqrt(e*x)/(b*x^4 + a), 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((e*x)^(1/2)*(d*x^4+c)^(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{e x} \sqrt{c + d x^{4}}}{a + b x^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(e*x)*sqrt(c + d*x**4)/(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} \sqrt{e x}}{b x^{4} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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