3.906 \(\int \frac {1}{x^2 (a+b x^8) \sqrt {c+d x^8}} \, dx\)

Optimal. Leaf size=62 \[ -\frac {\sqrt {\frac {d x^8}{c}+1} F_1\left (-\frac {1}{8};1,\frac {1}{2};\frac {7}{8};-\frac {b x^8}{a},-\frac {d x^8}{c}\right )}{a x \sqrt {c+d x^8}} \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 62, 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 = {511, 510} \[ -\frac {\sqrt {\frac {d x^8}{c}+1} F_1\left (-\frac {1}{8};1,\frac {1}{2};\frac {7}{8};-\frac {b x^8}{a},-\frac {d x^8}{c}\right )}{a x \sqrt {c+d x^8}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(a + b*x^8)*Sqrt[c + d*x^8]),x]

[Out]

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

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])

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])

Rubi steps

\begin {align*} \int \frac {1}{x^2 \left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx &=\frac {\sqrt {1+\frac {d x^8}{c}} \int \frac {1}{x^2 \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 {1}{8};1,\frac {1}{2};\frac {7}{8};-\frac {b x^8}{a},-\frac {d x^8}{c}\right )}{a x \sqrt {c+d x^8}}\\ \end {align*}

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Mathematica [B]  time = 0.15, size = 141, normalized size = 2.27 \[ \frac {-5 x^8 \sqrt {\frac {d x^8}{c}+1} (b c-3 a d) F_1\left (\frac {7}{8};\frac {1}{2},1;\frac {15}{8};-\frac {d x^8}{c},-\frac {b x^8}{a}\right )+7 b d x^{16} \sqrt {\frac {d x^8}{c}+1} F_1\left (\frac {15}{8};\frac {1}{2},1;\frac {23}{8};-\frac {d x^8}{c},-\frac {b x^8}{a}\right )-35 a \left (c+d x^8\right )}{35 a^2 c x \sqrt {c+d x^8}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(x^2*(a + b*x^8)*Sqrt[c + d*x^8]),x]

[Out]

(-35*a*(c + d*x^8) - 5*(b*c - 3*a*d)*x^8*Sqrt[1 + (d*x^8)/c]*AppellF1[7/8, 1/2, 1, 15/8, -((d*x^8)/c), -((b*x^
8)/a)] + 7*b*d*x^16*Sqrt[1 + (d*x^8)/c]*AppellF1[15/8, 1/2, 1, 23/8, -((d*x^8)/c), -((b*x^8)/a)])/(35*a^2*c*x*
Sqrt[c + d*x^8])

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fricas [F]  time = 1.06, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d x^{8} + c}}{b d x^{18} + {\left (b c + a d\right )} x^{10} + a c x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(d*x^8 + c)/(b*d*x^18 + (b*c + a*d)*x^10 + a*c*x^2), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{8} + a\right )} \sqrt {d x^{8} + c} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(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^2), x)

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maple [F]  time = 0.64, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{8}+a \right ) \sqrt {d \,x^{8}+c}\, x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/x^2/(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 \[ \int \frac {1}{{\left (b x^{8} + a\right )} \sqrt {d x^{8} + c} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(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^2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{x^2\,\left (b\,x^8+a\right )\,\sqrt {d\,x^8+c}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} \left (a + b x^{8}\right ) \sqrt {c + d x^{8}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(x**2*(a + b*x**8)*sqrt(c + d*x**8)), x)

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