3.804 \(\int \frac{\sqrt{c+d x^4}}{(e x)^{3/2} (a+b x^4)} \, dx\)

Optimal. Leaf size=69 \[ -\frac{2 \sqrt{c+d x^4} F_1\left (-\frac{1}{8};1,-\frac{1}{2};\frac{7}{8};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{a e \sqrt{e x} \sqrt{\frac{d x^4}{c}+1}} \]

[Out]

(-2*Sqrt[c + d*x^4]*AppellF1[-1/8, 1, -1/2, 7/8, -((b*x^4)/a), -((d*x^4)/c)])/(a*e*Sqrt[e*x]*Sqrt[1 + (d*x^4)/
c])

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Rubi [A]  time = 0.118679, 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, 511, 510} \[ -\frac{2 \sqrt{c+d x^4} F_1\left (-\frac{1}{8};1,-\frac{1}{2};\frac{7}{8};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{a e \sqrt{e x} \sqrt{\frac{d x^4}{c}+1}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c + d*x^4]/((e*x)^(3/2)*(a + b*x^4)),x]

[Out]

(-2*Sqrt[c + d*x^4]*AppellF1[-1/8, 1, -1/2, 7/8, -((b*x^4)/a), -((d*x^4)/c)])/(a*e*Sqrt[e*x]*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{c+d x^4}}{(e x)^{3/2} \left (a+b x^4\right )} \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{\sqrt{c+\frac{d x^8}{e^4}}}{x^2 \left (a+\frac{b x^8}{e^4}\right )} \, 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}}}{x^2 \left (a+\frac{b x^8}{e^4}\right )} \, dx,x,\sqrt{e x}\right )}{e \sqrt{1+\frac{d x^4}{c}}}\\ &=-\frac{2 \sqrt{c+d x^4} F_1\left (-\frac{1}{8};1,-\frac{1}{2};\frac{7}{8};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{a e \sqrt{e x} \sqrt{1+\frac{d x^4}{c}}}\\ \end{align*}

Mathematica [B]  time = 0.113718, size = 143, normalized size = 2.07 \[ \frac{x \left (14 b d x^8 \sqrt{\frac{d x^4}{c}+1} F_1\left (\frac{15}{8};\frac{1}{2},1;\frac{23}{8};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )-10 x^4 \sqrt{\frac{d x^4}{c}+1} (b c-4 a d) F_1\left (\frac{7}{8};\frac{1}{2},1;\frac{15}{8};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )-70 a \left (c+d x^4\right )\right )}{35 a^2 (e x)^{3/2} \sqrt{c+d x^4}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sqrt[c + d*x^4]/((e*x)^(3/2)*(a + b*x^4)),x]

[Out]

(x*(-70*a*(c + d*x^4) - 10*(b*c - 4*a*d)*x^4*Sqrt[1 + (d*x^4)/c]*AppellF1[7/8, 1/2, 1, 15/8, -((d*x^4)/c), -((
b*x^4)/a)] + 14*b*d*x^8*Sqrt[1 + (d*x^4)/c]*AppellF1[15/8, 1/2, 1, 23/8, -((d*x^4)/c), -((b*x^4)/a)]))/(35*a^2
*(e*x)^(3/2)*Sqrt[c + d*x^4])

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Maple [F]  time = 0.079, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{b{x}^{4}+a}\sqrt{d{x}^{4}+c} \left ( ex \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^4+c)^(1/2)/(e*x)^(3/2)/(b*x^4+a),x)

[Out]

int((d*x^4+c)^(1/2)/(e*x)^(3/2)/(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 )} \left (e x\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^4+c)^(1/2)/(e*x)^(3/2)/(b*x^4+a),x, algorithm="maxima")

[Out]

integrate(sqrt(d*x^4 + c)/((b*x^4 + a)*(e*x)^(3/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{d x^{4} + c} \sqrt{e x}}{b e^{2} x^{6} + a e^{2} x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^4+c)^(1/2)/(e*x)^(3/2)/(b*x^4+a),x, algorithm="fricas")

[Out]

integral(sqrt(d*x^4 + c)*sqrt(e*x)/(b*e^2*x^6 + a*e^2*x^2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c + d x^{4}}}{\left (e x\right )^{\frac{3}{2}} \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)**(3/2)/(b*x**4+a),x)

[Out]

Integral(sqrt(c + d*x**4)/((e*x)**(3/2)*(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 )} \left (e x\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^4+c)^(1/2)/(e*x)^(3/2)/(b*x^4+a),x, algorithm="giac")

[Out]

integrate(sqrt(d*x^4 + c)/((b*x^4 + a)*(e*x)^(3/2)), x)