\(\int \frac {1}{x^2 (a c+b c x^3+d \sqrt {a+b x^3})} \, dx\) [560]

Optimal result

Integrand size = 29, antiderivative size = 319 \[ \int \frac {1}{x^2 \left (a c+b c x^3+d \sqrt {a+b x^3}\right )} \, dx=-\frac {c}{\left (a c^2-d^2\right ) x}+\frac {d \sqrt {1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (-\frac {1}{3},\frac {1}{2},1,\frac {2}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{\left (a c^2-d^2\right ) x \sqrt {a+b x^3}}+\frac {\sqrt [3]{b} c^{5/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}}{\sqrt {3}}\right )}{\sqrt {3} \left (a c^2-d^2\right )^{4/3}}+\frac {\sqrt [3]{b} c^{5/3} \log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 \left (a c^2-d^2\right )^{4/3}}-\frac {\sqrt [3]{b} c^{5/3} \log \left (\left (a c^2-d^2\right )^{2/3}-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+b^{2/3} c^{4/3} x^2\right )}{6 \left (a c^2-d^2\right )^{4/3}} \]

[Out]

-c/(a*c^2-d^2)/x+1/3*b^(1/3)*c^(5/3)*ln((a*c^2-d^2)^(1/3)+b^(1/3)*c^(2/3)*x)/(a*c^2-d^2)^(4/3)-1/6*b^(1/3)*c^(
5/3)*ln((a*c^2-d^2)^(2/3)-b^(1/3)*c^(2/3)*(a*c^2-d^2)^(1/3)*x+b^(2/3)*c^(4/3)*x^2)/(a*c^2-d^2)^(4/3)+1/3*b^(1/
3)*c^(5/3)*arctan(1/3*(1-2*b^(1/3)*c^(2/3)*x/(a*c^2-d^2)^(1/3))*3^(1/2))/(a*c^2-d^2)^(4/3)*3^(1/2)+d*AppellF1(
-1/3,1/2,1,2/3,-b*x^3/a,-b*c^2*x^3/(a*c^2-d^2))*(1+b*x^3/a)^(1/2)/(a*c^2-d^2)/x/(b*x^3+a)^(1/2)

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {2187, 331, 298, 31, 648, 631, 210, 642, 525, 524} \[ \int \frac {1}{x^2 \left (a c+b c x^3+d \sqrt {a+b x^3}\right )} \, dx=\frac {d \sqrt {\frac {b x^3}{a}+1} \operatorname {AppellF1}\left (-\frac {1}{3},\frac {1}{2},1,\frac {2}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{x \sqrt {a+b x^3} \left (a c^2-d^2\right )}+\frac {\sqrt [3]{b} c^{5/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}}{\sqrt {3}}\right )}{\sqrt {3} \left (a c^2-d^2\right )^{4/3}}-\frac {\sqrt [3]{b} c^{5/3} \log \left (-\sqrt [3]{b} c^{2/3} x \sqrt [3]{a c^2-d^2}+\left (a c^2-d^2\right )^{2/3}+b^{2/3} c^{4/3} x^2\right )}{6 \left (a c^2-d^2\right )^{4/3}}+\frac {\sqrt [3]{b} c^{5/3} \log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 \left (a c^2-d^2\right )^{4/3}}-\frac {c}{x \left (a c^2-d^2\right )} \]

[In]

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

[Out]

-(c/((a*c^2 - d^2)*x)) + (d*Sqrt[1 + (b*x^3)/a]*AppellF1[-1/3, 1/2, 1, 2/3, -((b*x^3)/a), -((b*c^2*x^3)/(a*c^2
- d^2))])/((a*c^2 - d^2)*x*Sqrt[a + b*x^3]) + (b^(1/3)*c^(5/3)*ArcTan[(1 - (2*b^(1/3)*c^(2/3)*x)/(a*c^2 - d^2
)^(1/3))/Sqrt[3]])/(Sqrt[3]*(a*c^2 - d^2)^(4/3)) + (b^(1/3)*c^(5/3)*Log[(a*c^2 - d^2)^(1/3) + b^(1/3)*c^(2/3)*
x])/(3*(a*c^2 - d^2)^(4/3)) - (b^(1/3)*c^(5/3)*Log[(a*c^2 - d^2)^(2/3) - b^(1/3)*c^(2/3)*(a*c^2 - d^2)^(1/3)*x
+ b^(2/3)*c^(4/3)*x^2])/(6*(a*c^2 - d^2)^(4/3))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 298

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Dist[-(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3
]^2*x^2), x], x] /; FreeQ[{a, b}, x]

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

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])

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 2187

Int[(u_.)/((c_) + (d_.)*(x_)^(n_) + (e_.)*Sqrt[(a_) + (b_.)*(x_)^(n_)]), x_Symbol] :> Dist[c, Int[u/(c^2 - a*e
^2 + c*d*x^n), x], x] - Dist[a*e, Int[u/((c^2 - a*e^2 + c*d*x^n)*Sqrt[a + b*x^n]), x], x] /; FreeQ[{a, b, c, d
, e, n}, x] && EqQ[b*c - a*d, 0]

Rubi steps \begin{align*} \text {integral}& = (a c) \int \frac {1}{x^2 \left (a^2 c^2-a d^2+a b c^2 x^3\right )} \, dx-(a d) \int \frac {1}{x^2 \sqrt {a+b x^3} \left (a^2 c^2-a d^2+a b c^2 x^3\right )} \, dx \\ & = -\frac {c}{\left (a c^2-d^2\right ) x}-\frac {\left (a b c^3\right ) \int \frac {x}{a^2 c^2-a d^2+a b c^2 x^3} \, dx}{a c^2-d^2}-\frac {\left (a d \sqrt {1+\frac {b x^3}{a}}\right ) \int \frac {1}{x^2 \sqrt {1+\frac {b x^3}{a}} \left (a^2 c^2-a d^2+a b c^2 x^3\right )} \, dx}{\sqrt {a+b x^3}} \\ & = -\frac {c}{\left (a c^2-d^2\right ) x}+\frac {d \sqrt {1+\frac {b x^3}{a}} F_1\left (-\frac {1}{3};\frac {1}{2},1;\frac {2}{3};-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{\left (a c^2-d^2\right ) x \sqrt {a+b x^3}}+\frac {\left (\sqrt [3]{a} b^{2/3} c^{7/3}\right ) \int \frac {1}{\sqrt [3]{a} \sqrt [3]{a c^2-d^2}+\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x} \, dx}{3 \left (a c^2-d^2\right )^{4/3}}-\frac {\left (\sqrt [3]{a} b^{2/3} c^{7/3}\right ) \int \frac {\sqrt [3]{a} \sqrt [3]{a c^2-d^2}+\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x}{a^{2/3} \left (a c^2-d^2\right )^{2/3}-a^{2/3} \sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+a^{2/3} b^{2/3} c^{4/3} x^2} \, dx}{3 \left (a c^2-d^2\right )^{4/3}} \\ & = -\frac {c}{\left (a c^2-d^2\right ) x}+\frac {d \sqrt {1+\frac {b x^3}{a}} F_1\left (-\frac {1}{3};\frac {1}{2},1;\frac {2}{3};-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{\left (a c^2-d^2\right ) x \sqrt {a+b x^3}}+\frac {\sqrt [3]{b} c^{5/3} \log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 \left (a c^2-d^2\right )^{4/3}}-\frac {\left (\sqrt [3]{b} c^{5/3}\right ) \int \frac {-a^{2/3} \sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2}+2 a^{2/3} b^{2/3} c^{4/3} x}{a^{2/3} \left (a c^2-d^2\right )^{2/3}-a^{2/3} \sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+a^{2/3} b^{2/3} c^{4/3} x^2} \, dx}{6 \left (a c^2-d^2\right )^{4/3}}-\frac {\left (a^{2/3} b^{2/3} c^{7/3}\right ) \int \frac {1}{a^{2/3} \left (a c^2-d^2\right )^{2/3}-a^{2/3} \sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+a^{2/3} b^{2/3} c^{4/3} x^2} \, dx}{2 \left (a c^2-d^2\right )} \\ & = -\frac {c}{\left (a c^2-d^2\right ) x}+\frac {d \sqrt {1+\frac {b x^3}{a}} F_1\left (-\frac {1}{3};\frac {1}{2},1;\frac {2}{3};-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{\left (a c^2-d^2\right ) x \sqrt {a+b x^3}}+\frac {\sqrt [3]{b} c^{5/3} \log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 \left (a c^2-d^2\right )^{4/3}}-\frac {\sqrt [3]{b} c^{5/3} \log \left (\left (a c^2-d^2\right )^{2/3}-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+b^{2/3} c^{4/3} x^2\right )}{6 \left (a c^2-d^2\right )^{4/3}}-\frac {\left (\sqrt [3]{b} c^{5/3}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}\right )}{\left (a c^2-d^2\right )^{4/3}} \\ & = -\frac {c}{\left (a c^2-d^2\right ) x}+\frac {d \sqrt {1+\frac {b x^3}{a}} F_1\left (-\frac {1}{3};\frac {1}{2},1;\frac {2}{3};-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{\left (a c^2-d^2\right ) x \sqrt {a+b x^3}}+\frac {\sqrt [3]{b} c^{5/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}}{\sqrt {3}}\right )}{\sqrt {3} \left (a c^2-d^2\right )^{4/3}}+\frac {\sqrt [3]{b} c^{5/3} \log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 \left (a c^2-d^2\right )^{4/3}}-\frac {\sqrt [3]{b} c^{5/3} \log \left (\left (a c^2-d^2\right )^{2/3}-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+b^{2/3} c^{4/3} x^2\right )}{6 \left (a c^2-d^2\right )^{4/3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 10.52 (sec) , antiderivative size = 496, normalized size of antiderivative = 1.55 \[ \int \frac {1}{x^2 \left (a c+b c x^3+d \sqrt {a+b x^3}\right )} \, dx=\frac {15 b d \sqrt [3]{a c^2-d^2} \left (a c^2+d^2\right ) x^3 \sqrt {1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )-6 b^2 c^2 d \sqrt [3]{a c^2-d^2} x^6 \sqrt {1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{2},1,\frac {8}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )-10 \left (a c^2-d^2\right ) \left (-6 a d \sqrt [3]{a c^2-d^2}-6 b d \sqrt [3]{a c^2-d^2} x^3+6 a c \sqrt [3]{a c^2-d^2} \sqrt {a+b x^3}+2 \sqrt {3} a \sqrt [3]{b} c^{5/3} x \sqrt {a+b x^3} \arctan \left (\frac {-1+\frac {2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}}{\sqrt {3}}\right )-2 a \sqrt [3]{b} c^{5/3} x \sqrt {a+b x^3} \log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )+a \sqrt [3]{b} c^{5/3} x \sqrt {a+b x^3} \log \left (\left (a c^2-d^2\right )^{2/3}-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+b^{2/3} c^{4/3} x^2\right )\right )}{60 a \left (a c^2-d^2\right )^{7/3} x \sqrt {a+b x^3}} \]

[In]

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

[Out]

(15*b*d*(a*c^2 - d^2)^(1/3)*(a*c^2 + d^2)*x^3*Sqrt[1 + (b*x^3)/a]*AppellF1[2/3, 1/2, 1, 5/3, -((b*x^3)/a), -((
b*c^2*x^3)/(a*c^2 - d^2))] - 6*b^2*c^2*d*(a*c^2 - d^2)^(1/3)*x^6*Sqrt[1 + (b*x^3)/a]*AppellF1[5/3, 1/2, 1, 8/3
, -((b*x^3)/a), -((b*c^2*x^3)/(a*c^2 - d^2))] - 10*(a*c^2 - d^2)*(-6*a*d*(a*c^2 - d^2)^(1/3) - 6*b*d*(a*c^2 -
d^2)^(1/3)*x^3 + 6*a*c*(a*c^2 - d^2)^(1/3)*Sqrt[a + b*x^3] + 2*Sqrt[3]*a*b^(1/3)*c^(5/3)*x*Sqrt[a + b*x^3]*Arc
Tan[(-1 + (2*b^(1/3)*c^(2/3)*x)/(a*c^2 - d^2)^(1/3))/Sqrt[3]] - 2*a*b^(1/3)*c^(5/3)*x*Sqrt[a + b*x^3]*Log[(a*c
^2 - d^2)^(1/3) + b^(1/3)*c^(2/3)*x] + a*b^(1/3)*c^(5/3)*x*Sqrt[a + b*x^3]*Log[(a*c^2 - d^2)^(2/3) - b^(1/3)*c
^(2/3)*(a*c^2 - d^2)^(1/3)*x + b^(2/3)*c^(4/3)*x^2]))/(60*a*(a*c^2 - d^2)^(7/3)*x*Sqrt[a + b*x^3])

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 6.

Time = 0.83 (sec) , antiderivative size = 1200, normalized size of antiderivative = 3.76

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1200\)
default	\(\text {Expression too large to display}\)	\(2404\)

[In]

int(1/x^2/(a*c+b*c*x^3+d*(b*x^3+a)^(1/2)),x,method=_RETURNVERBOSE)

[Out]

(b*x^3+a)^(1/2)*(d+c*(b*x^3+a)^(1/2))/(a*c+b*c*x^3+d*(b*x^3+a)^(1/2))*(c*(-1/(a*c^2-d^2)/x-(-1/3/b/c^2/((a*c^2
-d^2)/b/c^2)^(1/3)*ln(x+((a*c^2-d^2)/b/c^2)^(1/3))+1/6/b/c^2/((a*c^2-d^2)/b/c^2)^(1/3)*ln(x^2-((a*c^2-d^2)/b/c
^2)^(1/3)*x+((a*c^2-d^2)/b/c^2)^(2/3))+1/3*3^(1/2)/b/c^2/((a*c^2-d^2)/b/c^2)^(1/3)*arctan(1/3*3^(1/2)*(2/((a*c
^2-d^2)/b/c^2)^(1/3)*x-1)))*b*c^2/(a*c^2-d^2))+d/a/(a*c^2-d^2)*(b*x^3+a)^(1/2)/x+1/3*I*d/a/(a*c^2-d^2)*3^(1/2)
*(-b^2*a)^(1/3)*(I*(x+1/2/b*(-b^2*a)^(1/3)-1/2*I*3^(1/2)/b*(-b^2*a)^(1/3))*3^(1/2)*b/(-b^2*a)^(1/3))^(1/2)*((x
-1/b*(-b^2*a)^(1/3))/(-3/2/b*(-b^2*a)^(1/3)+1/2*I*3^(1/2)/b*(-b^2*a)^(1/3)))^(1/2)*(-I*(x+1/2/b*(-b^2*a)^(1/3)
+1/2*I*3^(1/2)/b*(-b^2*a)^(1/3))*3^(1/2)*b/(-b^2*a)^(1/3))^(1/2)/(b*x^3+a)^(1/2)*((-3/2/b*(-b^2*a)^(1/3)+1/2*I
*3^(1/2)/b*(-b^2*a)^(1/3))*EllipticE(1/3*3^(1/2)*(I*(x+1/2/b*(-b^2*a)^(1/3)-1/2*I*3^(1/2)/b*(-b^2*a)^(1/3))*3^
(1/2)*b/(-b^2*a)^(1/3))^(1/2),(I*3^(1/2)/b*(-b^2*a)^(1/3)/(-3/2/b*(-b^2*a)^(1/3)+1/2*I*3^(1/2)/b*(-b^2*a)^(1/3
)))^(1/2))+1/b*(-b^2*a)^(1/3)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/b*(-b^2*a)^(1/3)-1/2*I*3^(1/2)/b*(-b^2*a)^(1/3))
*3^(1/2)*b/(-b^2*a)^(1/3))^(1/2),(I*3^(1/2)/b*(-b^2*a)^(1/3)/(-3/2/b*(-b^2*a)^(1/3)+1/2*I*3^(1/2)/b*(-b^2*a)^(
1/3)))^(1/2)))+1/3*I/d/b^2*c^2*2^(1/2)*sum(1/(a*c^2-d^2)/_alpha*(-b^2*a)^(1/3)*(1/2*I*b*(2*x+1/b*((-b^2*a)^(1/
3)-I*3^(1/2)*(-b^2*a)^(1/3)))/(-b^2*a)^(1/3))^(1/2)*(b*(x-1/b*(-b^2*a)^(1/3))/(-3*(-b^2*a)^(1/3)+I*3^(1/2)*(-b
^2*a)^(1/3)))^(1/2)*(-1/2*I*b*(2*x+1/b*((-b^2*a)^(1/3)+I*3^(1/2)*(-b^2*a)^(1/3)))/(-b^2*a)^(1/3))^(1/2)/(b*x^3
+a)^(1/2)*(I*(-b^2*a)^(1/3)*3^(1/2)*_alpha*b-I*(-b^2*a)^(2/3)*3^(1/2)+2*_alpha^2*b^2-(-b^2*a)^(1/3)*_alpha*b-(
-b^2*a)^(2/3))*EllipticPi(1/3*3^(1/2)*(I*(x+1/2/b*(-b^2*a)^(1/3)-1/2*I*3^(1/2)/b*(-b^2*a)^(1/3))*3^(1/2)*b/(-b
^2*a)^(1/3))^(1/2),-1/2/b*c^2*(2*I*(-b^2*a)^(1/3)*3^(1/2)*_alpha^2*b-I*(-b^2*a)^(2/3)*3^(1/2)*_alpha+I*3^(1/2)
*a*b-3*(-b^2*a)^(2/3)*_alpha-3*a*b)/d^2,(I*3^(1/2)/b*(-b^2*a)^(1/3)/(-3/2/b*(-b^2*a)^(1/3)+1/2*I*3^(1/2)/b*(-b
^2*a)^(1/3)))^(1/2)),_alpha=RootOf(_Z^3*b*c^2+a*c^2-d^2)))

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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