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

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

[Out]

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

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Rubi [A]  time = 0.0503759, 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} \[ -\frac{\sqrt{\frac{d x^6}{c}+1} F_1\left (-\frac{1}{6};1,\frac{1}{2};\frac{5}{6};-\frac{b x^6}{a},-\frac{d x^6}{c}\right )}{a x \sqrt{c+d x^6}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[1 + (d*x^6)/c]*AppellF1[-1/6, 1, 1/2, 5/6, -((b*x^6)/a), -((d*x^6)/c)])/(a*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 ) \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 ) \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};1,\frac{1}{2};\frac{5}{6};-\frac{b x^6}{a},-\frac{d x^6}{c}\right )}{a x \sqrt{c+d x^6}}\\ \end{align*}

Mathematica [B]  time = 0.113662, size = 141, normalized size = 2.27 \[ \frac{10 b d x^{12} \sqrt{\frac{d x^6}{c}+1} F_1\left (\frac{11}{6};\frac{1}{2},1;\frac{17}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )-11 x^6 \sqrt{\frac{d x^6}{c}+1} (b c-2 a d) F_1\left (\frac{5}{6};\frac{1}{2},1;\frac{11}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )-55 a \left (c+d x^6\right )}{55 a^2 c x \sqrt{c+d x^6}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-55*a*(c + d*x^6) - 11*(b*c - 2*a*d)*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*x^12*Sqrt[1 + (d*x^6)/c]*AppellF1[11/6, 1/2, 1, 17/6, -((d*x^6)/c), -((b*x^6)/a)])/(55*a^2*c*
x*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 ) }{\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)/(d*x^6+c)^(1/2),x)

[Out]

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

[Out]

integrate(1/((b*x^6 + a)*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 d x^{14} +{\left (b c + a d\right )} x^{8} + a 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)/(d*x^6+c)^(1/2),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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