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

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

[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]))

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Rubi [A]  time = 0.0524712, antiderivative size = 62, normalized size of antiderivative = 1., 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} \[ -\frac{\sqrt{\frac{d x^6}{c}+1} F_1\left (-\frac{1}{6};2,\frac{1}{2};\frac{5}{6};-\frac{b x^6}{a},-\frac{d x^6}{c}\right )}{a^2 x \sqrt{c+d x^6}} \]

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 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{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*}

Mathematica [B]  time = 0.261364, size = 226, normalized size = 3.65 \[ \frac{-11 x^6 \left (a+b x^6\right ) \sqrt{\frac{d x^6}{c}+1} \left (12 a^2 d^2-24 a b c d+7 b^2 c^2\right ) F_1\left (\frac{5}{6};\frac{1}{2},1;\frac{11}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )+55 a \left (c+d x^6\right ) \left (6 a^2 d-6 a b \left (c-d x^6\right )-7 b^2 c x^6\right )+10 b d x^{12} \left (a+b x^6\right ) \sqrt{\frac{d x^6}{c}+1} (7 b c-6 a d) F_1\left (\frac{11}{6};\frac{1}{2},1;\frac{17}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )}{330 a^3 c x \left (a+b x^6\right ) \sqrt{c+d x^6} (b c-a d)} \]

Warning: Unable to verify antiderivative.

[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 = 0.063, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2} \left ( b{x}^{6}+a \right ) ^{2}}{\frac{1}{\sqrt{d{x}^{6}+c}}}}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{6} + a\right )}^{2} \sqrt{d x^{6} + c} x^{2}}\,{d x} \end{align*}

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., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{d x^{6} + c}}{b^{2} d x^{20} +{\left (b^{2} c + 2 \, a b d\right )} x^{14} +{\left (2 \, a b c + a^{2} d\right )} x^{8} + a^{2} c x^{2}}, x\right ) \end{align*}

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(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out

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

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)