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

Optimal. Leaf size=715 \[ -\frac {3 (b+2 c x) \sqrt [3]{-\frac {c \left (b x+c x^2\right )}{b^2}}}{\sqrt [3]{2} c \sqrt [3]{b x+c x^2} \left (-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}-\sqrt {3}+1\right )}+\frac {\sqrt [6]{2} 3^{3/4} b^2 \sqrt [3]{-\frac {c \left (b x+c x^2\right )}{b^2}} \left (1-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right ) \sqrt {\frac {2 \sqrt [3]{2} \left (-\frac {c x (b+c x)}{b^2}\right )^{2/3}+2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}+1}{\left (-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}-\sqrt {3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac {-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}+\sqrt {3}+1}{-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{c (b+2 c x) \sqrt [3]{b x+c x^2} \sqrt {-\frac {1-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}}{\left (-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}-\sqrt {3}+1\right )^2}}}-\frac {3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} b^2 \sqrt [3]{-\frac {c \left (b x+c x^2\right )}{b^2}} \left (1-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right ) \sqrt {\frac {2 \sqrt [3]{2} \left (-\frac {c x (b+c x)}{b^2}\right )^{2/3}+2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}+1}{\left (-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}-\sqrt {3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac {-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}+\sqrt {3}+1}{-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{2 \sqrt [3]{2} c (b+2 c x) \sqrt [3]{b x+c x^2} \sqrt {-\frac {1-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}}{\left (-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}-\sqrt {3}+1\right )^2}}} \]

[Out]

-3/2*(2*c*x+b)*(-c*(c*x^2+b*x)/b^2)^(1/3)*2^(2/3)/c/(c*x^2+b*x)^(1/3)/(1-2^(2/3)*(-c*x*(c*x+b)/b^2)^(1/3)-3^(1
/2))+2^(1/6)*3^(3/4)*b^2*(-c*(c*x^2+b*x)/b^2)^(1/3)*(1-2^(2/3)*(-c*x*(c*x+b)/b^2)^(1/3))*EllipticF((1-2^(2/3)*
(-c*x*(c*x+b)/b^2)^(1/3)+3^(1/2))/(1-2^(2/3)*(-c*x*(c*x+b)/b^2)^(1/3)-3^(1/2)),2*I-I*3^(1/2))*((1+2^(2/3)*(-c*
x*(c*x+b)/b^2)^(1/3)+2*2^(1/3)*(-c*x*(c*x+b)/b^2)^(2/3))/(1-2^(2/3)*(-c*x*(c*x+b)/b^2)^(1/3)-3^(1/2))^2)^(1/2)
/c/(2*c*x+b)/(c*x^2+b*x)^(1/3)/((-1+2^(2/3)*(-c*x*(c*x+b)/b^2)^(1/3))/(1-2^(2/3)*(-c*x*(c*x+b)/b^2)^(1/3)-3^(1
/2))^2)^(1/2)-3/4*3^(1/4)*b^2*(-c*(c*x^2+b*x)/b^2)^(1/3)*(1-2^(2/3)*(-c*x*(c*x+b)/b^2)^(1/3))*EllipticE((1-2^(
2/3)*(-c*x*(c*x+b)/b^2)^(1/3)+3^(1/2))/(1-2^(2/3)*(-c*x*(c*x+b)/b^2)^(1/3)-3^(1/2)),2*I-I*3^(1/2))*((1+2^(2/3)
*(-c*x*(c*x+b)/b^2)^(1/3)+2*2^(1/3)*(-c*x*(c*x+b)/b^2)^(2/3))/(1-2^(2/3)*(-c*x*(c*x+b)/b^2)^(1/3)-3^(1/2))^2)^
(1/2)*(1/2*6^(1/2)+1/2*2^(1/2))*2^(2/3)/c/(2*c*x+b)/(c*x^2+b*x)^(1/3)/((-1+2^(2/3)*(-c*x*(c*x+b)/b^2)^(1/3))/(
1-2^(2/3)*(-c*x*(c*x+b)/b^2)^(1/3)-3^(1/2))^2)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.86, antiderivative size = 715, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {622, 619, 235, 304, 219, 1879} \[ -\frac {3 (b+2 c x) \sqrt [3]{-\frac {c \left (b x+c x^2\right )}{b^2}}}{\sqrt [3]{2} c \sqrt [3]{b x+c x^2} \left (-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}-\sqrt {3}+1\right )}+\frac {\sqrt [6]{2} 3^{3/4} b^2 \sqrt [3]{-\frac {c \left (b x+c x^2\right )}{b^2}} \left (1-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right ) \sqrt {\frac {2 \sqrt [3]{2} \left (-\frac {c x (b+c x)}{b^2}\right )^{2/3}+2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}+1}{\left (-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}-\sqrt {3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac {-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}+\sqrt {3}+1}{-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{c (b+2 c x) \sqrt [3]{b x+c x^2} \sqrt {-\frac {1-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}}{\left (-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}-\sqrt {3}+1\right )^2}}}-\frac {3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} b^2 \sqrt [3]{-\frac {c \left (b x+c x^2\right )}{b^2}} \left (1-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right ) \sqrt {\frac {2 \sqrt [3]{2} \left (-\frac {c x (b+c x)}{b^2}\right )^{2/3}+2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}+1}{\left (-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}-\sqrt {3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac {-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}+\sqrt {3}+1}{-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{2 \sqrt [3]{2} c (b+2 c x) \sqrt [3]{b x+c x^2} \sqrt {-\frac {1-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}}{\left (-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}-\sqrt {3}+1\right )^2}}} \]

Antiderivative was successfully verified.

[In]

Int[(b*x + c*x^2)^(-1/3),x]

[Out]

(-3*(b + 2*c*x)*(-((c*(b*x + c*x^2))/b^2))^(1/3))/(2^(1/3)*c*(b*x + c*x^2)^(1/3)*(1 - Sqrt[3] - 2^(2/3)*(-((c*
x*(b + c*x))/b^2))^(1/3))) - (3*3^(1/4)*Sqrt[2 + Sqrt[3]]*b^2*(-((c*(b*x + c*x^2))/b^2))^(1/3)*(1 - 2^(2/3)*(-
((c*x*(b + c*x))/b^2))^(1/3))*Sqrt[(1 + 2^(2/3)*(-((c*x*(b + c*x))/b^2))^(1/3) + 2*2^(1/3)*(-((c*x*(b + c*x))/
b^2))^(2/3))/(1 - Sqrt[3] - 2^(2/3)*(-((c*x*(b + c*x))/b^2))^(1/3))^2]*EllipticE[ArcSin[(1 + Sqrt[3] - 2^(2/3)
*(-((c*x*(b + c*x))/b^2))^(1/3))/(1 - Sqrt[3] - 2^(2/3)*(-((c*x*(b + c*x))/b^2))^(1/3))], -7 + 4*Sqrt[3]])/(2*
2^(1/3)*c*(b + 2*c*x)*(b*x + c*x^2)^(1/3)*Sqrt[-((1 - 2^(2/3)*(-((c*x*(b + c*x))/b^2))^(1/3))/(1 - Sqrt[3] - 2
^(2/3)*(-((c*x*(b + c*x))/b^2))^(1/3))^2)]) + (2^(1/6)*3^(3/4)*b^2*(-((c*(b*x + c*x^2))/b^2))^(1/3)*(1 - 2^(2/
3)*(-((c*x*(b + c*x))/b^2))^(1/3))*Sqrt[(1 + 2^(2/3)*(-((c*x*(b + c*x))/b^2))^(1/3) + 2*2^(1/3)*(-((c*x*(b + c
*x))/b^2))^(2/3))/(1 - Sqrt[3] - 2^(2/3)*(-((c*x*(b + c*x))/b^2))^(1/3))^2]*EllipticF[ArcSin[(1 + Sqrt[3] - 2^
(2/3)*(-((c*x*(b + c*x))/b^2))^(1/3))/(1 - Sqrt[3] - 2^(2/3)*(-((c*x*(b + c*x))/b^2))^(1/3))], -7 + 4*Sqrt[3]]
)/(c*(b + 2*c*x)*(b*x + c*x^2)^(1/3)*Sqrt[-((1 - 2^(2/3)*(-((c*x*(b + c*x))/b^2))^(1/3))/(1 - Sqrt[3] - 2^(2/3
)*(-((c*x*(b + c*x))/b^2))^(1/3))^2)])

Rule 219

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(2*Sqr
t[2 - Sqrt[3]]*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]*EllipticF[ArcSin[((1 + Sqrt[3
])*s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]])/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[-((s*(s + r*x))/((1 - S
qrt[3])*s + r*x)^2)]), x]] /; FreeQ[{a, b}, x] && NegQ[a]

Rule 235

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

Rule 304

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, -Dist[(S
qrt[2]*s)/(Sqrt[2 - Sqrt[3]]*r), Int[1/Sqrt[a + b*x^3], x], x] + Dist[1/r, Int[((1 + Sqrt[3])*s + r*x)/Sqrt[a
+ b*x^3], x], x]] /; FreeQ[{a, b}, x] && NegQ[a]

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 622

Int[((b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(b*x + c*x^2)^p/(-((c*(b*x + c*x^2))/b^2))^p, Int[(-((
c*x)/b) - (c^2*x^2)/b^2)^p, x], x] /; FreeQ[{b, c}, x] && RationalQ[p] && 3 <= Denominator[p] <= 4

Rule 1879

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Simplify[((1 + Sqrt[3])*d)/c]]
, s = Denom[Simplify[((1 + Sqrt[3])*d)/c]]}, Simp[(2*d*s^3*Sqrt[a + b*x^3])/(a*r^2*((1 - Sqrt[3])*s + r*x)), x
] + Simp[(3^(1/4)*Sqrt[2 + Sqrt[3]]*d*s*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]*Elli
pticE[ArcSin[((1 + Sqrt[3])*s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]])/(r^2*Sqrt[a + b*x^3]*Sqrt[-((s
*(s + r*x))/((1 - Sqrt[3])*s + r*x)^2)]), x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && EqQ[b*c^3 - 2*(5 + 3*Sqr
t[3])*a*d^3, 0]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt [3]{b x+c x^2}} \, dx &=\frac {\sqrt [3]{-\frac {c \left (b x+c x^2\right )}{b^2}} \int \frac {1}{\sqrt [3]{-\frac {c x}{b}-\frac {c^2 x^2}{b^2}}} \, dx}{\sqrt [3]{b x+c x^2}}\\ &=-\frac {\left (b^2 \sqrt [3]{-\frac {c \left (b x+c x^2\right )}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1-\frac {b^2 x^2}{c^2}}} \, dx,x,-\frac {c}{b}-\frac {2 c^2 x}{b^2}\right )}{\sqrt [3]{2} c^2 \sqrt [3]{b x+c x^2}}\\ &=\frac {\left (3 \sqrt [3]{-\frac {c \left (b x+c x^2\right )}{b^2}} \sqrt {-1-\frac {4 c x}{b}-\frac {4 c^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {-1+x^3}} \, dx,x,2^{2/3} \sqrt [3]{-\frac {c x \left (1+\frac {c x}{b}\right )}{b}}\right )}{2 \sqrt [3]{2} \left (-\frac {c}{b}-\frac {2 c^2 x}{b^2}\right ) \sqrt [3]{b x+c x^2}}\\ &=-\frac {\left (3 \sqrt [3]{-\frac {c \left (b x+c x^2\right )}{b^2}} \sqrt {-1-\frac {4 c x}{b}-\frac {4 c^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1+\sqrt {3}-x}{\sqrt {-1+x^3}} \, dx,x,2^{2/3} \sqrt [3]{-\frac {c x \left (1+\frac {c x}{b}\right )}{b}}\right )}{2 \sqrt [3]{2} \left (-\frac {c}{b}-\frac {2 c^2 x}{b^2}\right ) \sqrt [3]{b x+c x^2}}+\frac {\left (3 \sqrt {2+\sqrt {3}} \sqrt [3]{-\frac {c \left (b x+c x^2\right )}{b^2}} \sqrt {-1-\frac {4 c x}{b}-\frac {4 c^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^3}} \, dx,x,2^{2/3} \sqrt [3]{-\frac {c x \left (1+\frac {c x}{b}\right )}{b}}\right )}{2^{5/6} \left (-\frac {c}{b}-\frac {2 c^2 x}{b^2}\right ) \sqrt [3]{b x+c x^2}}\\ &=\frac {3 b^2 \sqrt [3]{-\frac {c \left (b x+c x^2\right )}{b^2}} \sqrt {-1-\frac {4 c x}{b}-\frac {4 c^2 x^2}{b^2}} \sqrt {-1-\frac {4 c x (b+c x)}{b^2}}}{\sqrt [3]{2} c (b+2 c x) \sqrt [3]{b x+c x^2} \left (1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right )}-\frac {3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} b^2 \sqrt [3]{-\frac {c \left (b x+c x^2\right )}{b^2}} \sqrt {-1-\frac {4 c x}{b}-\frac {4 c^2 x^2}{b^2}} \left (1-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right ) \sqrt {\frac {1+2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}+2 \sqrt [3]{2} \left (-\frac {c x (b+c x)}{b^2}\right )^{2/3}}{\left (1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right )^2}} E\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}}{1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}}\right )|-7+4 \sqrt {3}\right )}{2 \sqrt [3]{2} c (b+2 c x) \sqrt [3]{b x+c x^2} \sqrt {-1-\frac {4 c x (b+c x)}{b^2}} \sqrt {-\frac {1-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}}{\left (1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right )^2}}}+\frac {\sqrt [6]{2} 3^{3/4} b^2 \sqrt [3]{-\frac {c \left (b x+c x^2\right )}{b^2}} \sqrt {-1-\frac {4 c x}{b}-\frac {4 c^2 x^2}{b^2}} \left (1-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right ) \sqrt {\frac {1+2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}+2 \sqrt [3]{2} \left (-\frac {c x (b+c x)}{b^2}\right )^{2/3}}{\left (1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}}{1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}}\right )|-7+4 \sqrt {3}\right )}{c (b+2 c x) \sqrt [3]{b x+c x^2} \sqrt {-1-\frac {4 c x (b+c x)}{b^2}} \sqrt {-\frac {1-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}}{\left (1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right )^2}}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.01, size = 45, normalized size = 0.06 \[ \frac {3 x \sqrt [3]{\frac {c x}{b}+1} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};-\frac {c x}{b}\right )}{2 \sqrt [3]{x (b+c x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x + c*x^2)^(-1/3),x]

[Out]

(3*x*(1 + (c*x)/b)^(1/3)*Hypergeometric2F1[1/3, 2/3, 5/3, -((c*x)/b)])/(2*(x*(b + c*x))^(1/3))

________________________________________________________________________________________

fricas [F]  time = 0.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{{\left (c x^{2} + b x\right )}^{\frac {1}{3}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((c*x^2 + b*x)^(-1/3), x)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (c x^{2} + b x\right )}^{\frac {1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c*x^2 + b*x)^(-1/3), x)

________________________________________________________________________________________

maple [F]  time = 0.70, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c \,x^{2}+b x \right )^{\frac {1}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (c x^{2} + b x\right )}^{\frac {1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c*x^2 + b*x)^(-1/3), x)

________________________________________________________________________________________

mupad [B]  time = 0.20, size = 36, normalized size = 0.05 \[ \frac {3\,x\,{\left (\frac {c\,x}{b}+1\right )}^{1/3}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{3},\frac {2}{3};\ \frac {5}{3};\ -\frac {c\,x}{b}\right )}{2\,{\left (c\,x^2+b\,x\right )}^{1/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x + c*x^2)^(1/3),x)

[Out]

(3*x*((c*x)/b + 1)^(1/3)*hypergeom([1/3, 2/3], 5/3, -(c*x)/b))/(2*(b*x + c*x^2)^(1/3))

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt [3]{b x + c x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((b*x + c*x**2)**(-1/3), x)

________________________________________________________________________________________