\(\int \frac {1}{x^2 \sqrt {a+b x^3+c x^6}} \, dx\) [232]

Optimal result

Integrand size = 20, antiderivative size = 138 \[ \int \frac {1}{x^2 \sqrt {a+b x^3+c x^6}} \, dx=-\frac {\sqrt {1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (-\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {2}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{x \sqrt {a+b x^3+c x^6}} \]

[Out]

-AppellF1(-1/3,1/2,1/2,2/3,-2*c*x^3/(b-(-4*a*c+b^2)^(1/2)),-2*c*x^3/(b+(-4*a*c+b^2)^(1/2)))*(1+2*c*x^3/(b-(-4*
a*c+b^2)^(1/2)))^(1/2)*(1+2*c*x^3/(b+(-4*a*c+b^2)^(1/2)))^(1/2)/x/(c*x^6+b*x^3+a)^(1/2)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1399, 524} \[ \int \frac {1}{x^2 \sqrt {a+b x^3+c x^6}} \, dx=-\frac {\sqrt {\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^3}{\sqrt {b^2-4 a c}+b}+1} \operatorname {AppellF1}\left (-\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {2}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{x \sqrt {a+b x^3+c x^6}} \]

[In]

Int[1/(x^2*Sqrt[a + b*x^3 + c*x^6]),x]

[Out]

-((Sqrt[1 + (2*c*x^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[-1/3, 1/2,
1/2, 2/3, (-2*c*x^3)/(b - Sqrt[b^2 - 4*a*c]), (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])])/(x*Sqrt[a + b*x^3 + c*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 1399

Int[((d_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a +
b*x^n + c*x^(2*n))^FracPart[p]/((1 + 2*c*(x^n/(b + Rt[b^2 - 4*a*c, 2])))^FracPart[p]*(1 + 2*c*(x^n/(b - Rt[b^
2 - 4*a*c, 2])))^FracPart[p])), Int[(d*x)^m*(1 + 2*c*(x^n/(b + Sqrt[b^2 - 4*a*c])))^p*(1 + 2*c*(x^n/(b - Sqrt[
b^2 - 4*a*c])))^p, x], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}}\right ) \int \frac {1}{x^2 \sqrt {1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}}} \, dx}{\sqrt {a+b x^3+c x^6}} \\ & = -\frac {\sqrt {1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}} F_1\left (-\frac {1}{3};\frac {1}{2},\frac {1}{2};\frac {2}{3};-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{x \sqrt {a+b x^3+c x^6}} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(343\) vs. \(2(138)=276\).

Time = 10.23 (sec) , antiderivative size = 343, normalized size of antiderivative = 2.49 \[ \int \frac {1}{x^2 \sqrt {a+b x^3+c x^6}} \, dx=\frac {-20 \left (a+b x^3+c x^6\right )+5 b x^3 \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},\frac {1}{2},\frac {5}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )+8 c x^6 \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{2},\frac {1}{2},\frac {8}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )}{20 a x \sqrt {a+b x^3+c x^6}} \]

[In]

Integrate[1/(x^2*Sqrt[a + b*x^3 + c*x^6]),x]

[Out]

(-20*(a + b*x^3 + c*x^6) + 5*b*x^3*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + S
qrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[2/3, 1/2, 1/2, 5/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4
*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])] + 8*c*x^6*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4
*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[5/3, 1/2, 1/2, 8/3, (-2*c*x^3
)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])])/(20*a*x*Sqrt[a + b*x^3 + c*x^6])

Maple [F]

\[\int \frac {1}{x^{2} \sqrt {c \,x^{6}+b \,x^{3}+a}}d x\]

[In]

int(1/x^2/(c*x^6+b*x^3+a)^(1/2),x)

[Out]

int(1/x^2/(c*x^6+b*x^3+a)^(1/2),x)

Fricas [F]

\[ \int \frac {1}{x^2 \sqrt {a+b x^3+c x^6}} \, dx=\int { \frac {1}{\sqrt {c x^{6} + b x^{3} + a} x^{2}} \,d x } \]

[In]

integrate(1/x^2/(c*x^6+b*x^3+a)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(c*x^6 + b*x^3 + a)/(c*x^8 + b*x^5 + a*x^2), x)

Sympy [F]

\[ \int \frac {1}{x^2 \sqrt {a+b x^3+c x^6}} \, dx=\int \frac {1}{x^{2} \sqrt {a + b x^{3} + c x^{6}}}\, dx \]

[In]

integrate(1/x**2/(c*x**6+b*x**3+a)**(1/2),x)

[Out]

Integral(1/(x**2*sqrt(a + b*x**3 + c*x**6)), x)

Maxima [F]

\[ \int \frac {1}{x^2 \sqrt {a+b x^3+c x^6}} \, dx=\int { \frac {1}{\sqrt {c x^{6} + b x^{3} + a} x^{2}} \,d x } \]

[In]

integrate(1/x^2/(c*x^6+b*x^3+a)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(c*x^6 + b*x^3 + a)*x^2), x)

Giac [F]

\[ \int \frac {1}{x^2 \sqrt {a+b x^3+c x^6}} \, dx=\int { \frac {1}{\sqrt {c x^{6} + b x^{3} + a} x^{2}} \,d x } \]

[In]

integrate(1/x^2/(c*x^6+b*x^3+a)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(c*x^6 + b*x^3 + a)*x^2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^2 \sqrt {a+b x^3+c x^6}} \, dx=\int \frac {1}{x^2\,\sqrt {c\,x^6+b\,x^3+a}} \,d x \]

[In]

int(1/(x^2*(a + b*x^3 + c*x^6)^(1/2)),x)

[Out]

int(1/(x^2*(a + b*x^3 + c*x^6)^(1/2)), x)