3.10.7 \(\int \frac {1}{x^4 (a+b x^8) \sqrt {c+d x^8}} \, dx\) [907]

Optimal. Leaf size=64 \[ -\frac {\sqrt {1+\frac {d x^8}{c}} F_1\left (-\frac {3}{8};1,\frac {1}{2};\frac {5}{8};-\frac {b x^8}{a},-\frac {d x^8}{c}\right )}{3 a x^3 \sqrt {c+d x^8}} \]

[Out]

-1/3*AppellF1(-3/8,1,1/2,5/8,-b*x^8/a,-d*x^8/c)*(1+d*x^8/c)^(1/2)/a/x^3/(d*x^8+c)^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {525, 524} \begin {gather*} -\frac {\sqrt {\frac {d x^8}{c}+1} F_1\left (-\frac {3}{8};1,\frac {1}{2};\frac {5}{8};-\frac {b x^8}{a},-\frac {d x^8}{c}\right )}{3 a x^3 \sqrt {c+d x^8}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(a + b*x^8)*Sqrt[c + d*x^8]),x]

[Out]

-1/3*(Sqrt[1 + (d*x^8)/c]*AppellF1[-3/8, 1, 1/2, 5/8, -((b*x^8)/a), -((d*x^8)/c)])/(a*x^3*Sqrt[c + d*x^8])

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 {1}{x^4 \left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx &=\frac {\sqrt {1+\frac {d x^8}{c}} \int \frac {1}{x^4 \left (a+b x^8\right ) \sqrt {1+\frac {d x^8}{c}}} \, dx}{\sqrt {c+d x^8}}\\ &=-\frac {\sqrt {1+\frac {d x^8}{c}} F_1\left (-\frac {3}{8};1,\frac {1}{2};\frac {5}{8};-\frac {b x^8}{a},-\frac {d x^8}{c}\right )}{3 a x^3 \sqrt {c+d x^8}}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(141\) vs. \(2(64)=128\).
time = 10.12, size = 141, normalized size = 2.20 \begin {gather*} \frac {-65 a \left (c+d x^8\right )+13 (-3 b c+a d) x^8 \sqrt {1+\frac {d x^8}{c}} F_1\left (\frac {5}{8};\frac {1}{2},1;\frac {13}{8};-\frac {d x^8}{c},-\frac {b x^8}{a}\right )+5 b d x^{16} \sqrt {1+\frac {d x^8}{c}} F_1\left (\frac {13}{8};\frac {1}{2},1;\frac {21}{8};-\frac {d x^8}{c},-\frac {b x^8}{a}\right )}{195 a^2 c x^3 \sqrt {c+d x^8}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(a + b*x^8)*Sqrt[c + d*x^8]),x]

[Out]

(-65*a*(c + d*x^8) + 13*(-3*b*c + a*d)*x^8*Sqrt[1 + (d*x^8)/c]*AppellF1[5/8, 1/2, 1, 13/8, -((d*x^8)/c), -((b*
x^8)/a)] + 5*b*d*x^16*Sqrt[1 + (d*x^8)/c]*AppellF1[13/8, 1/2, 1, 21/8, -((d*x^8)/c), -((b*x^8)/a)])/(195*a^2*c
*x^3*Sqrt[c + d*x^8])

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{4} \left (b \,x^{8}+a \right ) \sqrt {d \,x^{8}+c}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(b*x^8+a)/(d*x^8+c)^(1/2),x)

[Out]

int(1/x^4/(b*x^8+a)/(d*x^8+c)^(1/2),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(1/x^4/(b*x^8+a)/(d*x^8+c)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/((b*x^8 + a)*sqrt(d*x^8 + c)*x^4), x)

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^8+a)/(d*x^8+c)^(1/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{4} \left (a + b x^{8}\right ) \sqrt {c + d x^{8}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**4/(b*x**8+a)/(d*x**8+c)**(1/2),x)

[Out]

Integral(1/(x**4*(a + b*x**8)*sqrt(c + d*x**8)), 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(1/x^4/(b*x^8+a)/(d*x^8+c)^(1/2),x, algorithm="giac")

[Out]

integrate(1/((b*x^8 + a)*sqrt(d*x^8 + c)*x^4), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{x^4\,\left (b\,x^8+a\right )\,\sqrt {d\,x^8+c}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*(a + b*x^8)*(c + d*x^8)^(1/2)),x)

[Out]

int(1/(x^4*(a + b*x^8)*(c + d*x^8)^(1/2)), x)

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