3.9.84 \(\int \frac {1}{x^2 (a+b x^6)^2 \sqrt {c+d x^6}} \, dx\) [884]

Optimal. Leaf size=62 \[ -\frac {\sqrt {1+\frac {d x^6}{c}} F_1\left (-\frac {1}{6};2,\frac {1}{2};\frac {5}{6};-\frac {b x^6}{a},-\frac {d x^6}{c}\right )}{a^2 x \sqrt {c+d x^6}} \]

[Out]

-AppellF1(-1/6,2,1/2,5/6,-b*x^6/a,-d*x^6/c)*(1+d*x^6/c)^(1/2)/a^2/x/(d*x^6+c)^(1/2)

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Rubi [A]
time = 0.03, 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 = {525, 524} \begin {gather*} -\frac {\sqrt {\frac {d x^6}{c}+1} F_1\left (-\frac {1}{6};2,\frac {1}{2};\frac {5}{6};-\frac {b x^6}{a},-\frac {d x^6}{c}\right )}{a^2 x \sqrt {c+d x^6}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(a + b*x^6)^2*Sqrt[c + d*x^6]),x]

[Out]

-((Sqrt[1 + (d*x^6)/c]*AppellF1[-1/6, 2, 1/2, 5/6, -((b*x^6)/a), -((d*x^6)/c)])/(a^2*x*Sqrt[c + d*x^6]))

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^2 \left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx &=\frac {\sqrt {1+\frac {d x^6}{c}} \int \frac {1}{x^2 \left (a+b x^6\right )^2 \sqrt {1+\frac {d x^6}{c}}} \, dx}{\sqrt {c+d x^6}}\\ &=-\frac {\sqrt {1+\frac {d x^6}{c}} F_1\left (-\frac {1}{6};2,\frac {1}{2};\frac {5}{6};-\frac {b x^6}{a},-\frac {d x^6}{c}\right )}{a^2 x \sqrt {c+d x^6}}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(226\) vs. \(2(62)=124\).
time = 10.22, size = 226, normalized size = 3.65 \begin {gather*} \frac {55 a \left (c+d x^6\right ) \left (6 a^2 d-7 b^2 c x^6-6 a b \left (c-d x^6\right )\right )-11 \left (7 b^2 c^2-24 a b c d+12 a^2 d^2\right ) x^6 \left (a+b x^6\right ) \sqrt {1+\frac {d x^6}{c}} F_1\left (\frac {5}{6};\frac {1}{2},1;\frac {11}{6};-\frac {d x^6}{c},-\frac {b x^6}{a}\right )+10 b d (7 b c-6 a d) x^{12} \left (a+b x^6\right ) \sqrt {1+\frac {d x^6}{c}} F_1\left (\frac {11}{6};\frac {1}{2},1;\frac {17}{6};-\frac {d x^6}{c},-\frac {b x^6}{a}\right )}{330 a^3 c (b c-a d) x \left (a+b x^6\right ) \sqrt {c+d x^6}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(a + b*x^6)^2*Sqrt[c + d*x^6]),x]

[Out]

(55*a*(c + d*x^6)*(6*a^2*d - 7*b^2*c*x^6 - 6*a*b*(c - d*x^6)) - 11*(7*b^2*c^2 - 24*a*b*c*d + 12*a^2*d^2)*x^6*(
a + b*x^6)*Sqrt[1 + (d*x^6)/c]*AppellF1[5/6, 1/2, 1, 11/6, -((d*x^6)/c), -((b*x^6)/a)] + 10*b*d*(7*b*c - 6*a*d
)*x^12*(a + b*x^6)*Sqrt[1 + (d*x^6)/c]*AppellF1[11/6, 1/2, 1, 17/6, -((d*x^6)/c), -((b*x^6)/a)])/(330*a^3*c*(b
*c - a*d)*x*(a + b*x^6)*Sqrt[c + d*x^6])

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{2} \left (b \,x^{6}+a \right )^{2} \sqrt {d \,x^{6}+c}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(b*x^6+a)^2/(d*x^6+c)^(1/2),x)

[Out]

int(1/x^2/(b*x^6+a)^2/(d*x^6+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^2/(b*x^6+a)^2/(d*x^6+c)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/((b*x^6 + a)^2*sqrt(d*x^6 + c)*x^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(1/x^2/(b*x^6+a)^2/(d*x^6+c)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(d*x^6 + c)/(b^2*d*x^20 + (b^2*c + 2*a*b*d)*x^14 + (2*a*b*c + a^2*d)*x^8 + a^2*c*x^2), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{2} \left (a + b x^{6}\right )^{2} \sqrt {c + d x^{6}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(b*x**6+a)**2/(d*x**6+c)**(1/2),x)

[Out]

Integral(1/(x**2*(a + b*x**6)**2*sqrt(c + d*x**6)), 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^2/(b*x^6+a)^2/(d*x^6+c)^(1/2),x, algorithm="giac")

[Out]

integrate(1/((b*x^6 + a)^2*sqrt(d*x^6 + c)*x^2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{x^2\,{\left (b\,x^6+a\right )}^2\,\sqrt {d\,x^6+c}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(a + b*x^6)^2*(c + d*x^6)^(1/2)),x)

[Out]

int(1/(x^2*(a + b*x^6)^2*(c + d*x^6)^(1/2)), x)

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