3.9.4 \(\int \frac {\sqrt {c+d x^4}}{(e x)^{3/2} (a+b x^4)} \, dx\) [804]

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

[Out]

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

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Rubi [A]
time = 0.08, antiderivative size = 69, normalized size of antiderivative = 1.00, 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 = {477, 525, 524} \begin {gather*} -\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}} \end {gather*}

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 477

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 524

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)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], 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])

Rule 525

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPar
t[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])

Rubi steps

\begin {align*} \int \frac {\sqrt {c+d x^4}}{(e x)^{3/2} \left (a+b x^4\right )} \, dx &=\frac {2 \text {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 ) \text {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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(143\) vs. \(2(69)=138\).
time = 10.09, size = 143, normalized size = 2.07 \begin {gather*} \frac {x \left (-70 a \left (c+d x^4\right )-10 (b c-4 a d) x^4 \sqrt {1+\frac {d x^4}{c}} F_1\left (\frac {7}{8};\frac {1}{2},1;\frac {15}{8};-\frac {d x^4}{c},-\frac {b x^4}{a}\right )+14 b d x^8 \sqrt {1+\frac {d x^4}{c}} F_1\left (\frac {15}{8};\frac {1}{2},1;\frac {23}{8};-\frac {d x^4}{c},-\frac {b x^4}{a}\right )\right )}{35 a^2 (e x)^{3/2} \sqrt {c+d x^4}} \end {gather*}

Antiderivative was successfully verified.

[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 = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {d \,x^{4}+c}}{\left (e x \right )^{\frac {3}{2}} \left (b \,x^{4}+a \right )}\, dx\]

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 = 0.00, 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((d*x^4+c)^(1/2)/(e*x)^(3/2)/(b*x^4+a),x, algorithm="maxima")

[Out]

e^(-3/2)*integrate(sqrt(d*x^4 + c)/((b*x^4 + a)*x^(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((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(x)*e^(-3/2)/(b*x^6 + a*x^2), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {c + d x^{4}}}{\left (e x\right )^{\frac {3}{2}} \left (a + b x^{4}\right )}\, dx \end {gather*}

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 = 0.00, 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((d*x^4+c)^(1/2)/(e*x)^(3/2)/(b*x^4+a),x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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