Integrand size = 13, antiderivative size = 322 \[ \int \frac {1}{\left (b x+c x^2\right )^{2/3}} \, dx=\frac {\sqrt [3]{2} 3^{3/4} \sqrt {2-\sqrt {3}} b^2 \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{2/3} \left (1-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right ) \sqrt {\frac {1+2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}+2 \sqrt [3]{2} \left (-\frac {c x (b+c x)}{b^2}\right )^{2/3}}{\left (1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}}{1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}}\right ),-7+4 \sqrt {3}\right )}{c (b+2 c x) \left (b x+c x^2\right )^{2/3} \sqrt {-\frac {1-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}}{\left (1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right )^2}}} \]
2^(1/3)*3^(3/4)*b^2*(-c*(c*x^2+b*x)/b^2)^(2/3)*(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*6^(1/2)-1/2*2^(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)^(2/3)/((-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)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.13 \[ \int \frac {1}{\left (b x+c x^2\right )^{2/3}} \, dx=\frac {3 x \left (1+\frac {c x}{b}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {c x}{b}\right )}{(x (b+c x))^{2/3}} \]
Time = 0.29 (sec) , antiderivative size = 278, normalized size of antiderivative = 0.86, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {1093, 1090, 234, 760}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{\left (b x+c x^2\right )^{2/3}} \, dx\) |
\(\Big \downarrow \) 1093 |
\(\displaystyle \frac {\left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{2/3} \int \frac {1}{\left (-\frac {c^2 x^2}{b^2}-\frac {c x}{b}\right )^{2/3}}dx}{\left (b x+c x^2\right )^{2/3}}\) |
\(\Big \downarrow \) 1090 |
\(\displaystyle -\frac {\sqrt [3]{2} b^2 \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{2/3} \int \frac {1}{\left (1-\frac {b^2 \left (-\frac {2 x c^2}{b^2}-\frac {c}{b}\right )^2}{c^2}\right )^{2/3}}d\left (-\frac {2 x c^2}{b^2}-\frac {c}{b}\right )}{c^2 \left (b x+c x^2\right )^{2/3}}\) |
\(\Big \downarrow \) 234 |
\(\displaystyle \frac {3 \sqrt {-\frac {b^2 \left (-\frac {2 c^2 x}{b^2}-\frac {c}{b}\right )^2}{c^2}} \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{2/3} \int \frac {1}{\sqrt {-\frac {b^2 \left (-\frac {2 x c^2}{b^2}-\frac {c}{b}\right )^2}{c^2}}}d\sqrt [3]{1-\frac {b^2 \left (-\frac {2 x c^2}{b^2}-\frac {c}{b}\right )^2}{c^2}}}{2^{2/3} \left (-\frac {2 c^2 x}{b^2}-\frac {c}{b}\right ) \left (b x+c x^2\right )^{2/3}}\) |
\(\Big \downarrow \) 760 |
\(\displaystyle -\frac {\sqrt [3]{2} 3^{3/4} \sqrt {2-\sqrt {3}} \left (\frac {2 c^2 x}{b^2}+\frac {c}{b}+1\right ) \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{2/3} \sqrt {\frac {\left (-\frac {2 c^2 x}{b^2}-\frac {c}{b}\right )^2+\sqrt [3]{1-\frac {b^2 \left (-\frac {2 c^2 x}{b^2}-\frac {c}{b}\right )^2}{c^2}}+1}{\left (\frac {2 c^2 x}{b^2}+\frac {c}{b}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\frac {2 x c^2}{b^2}+\frac {c}{b}+\sqrt {3}+1}{\frac {2 x c^2}{b^2}+\frac {c}{b}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\left (-\frac {2 c^2 x}{b^2}-\frac {c}{b}\right ) \sqrt {-\frac {\frac {2 c^2 x}{b^2}+\frac {c}{b}+1}{\left (\frac {2 c^2 x}{b^2}+\frac {c}{b}-\sqrt {3}+1\right )^2}} \left (b x+c x^2\right )^{2/3}}\) |
-((2^(1/3)*3^(3/4)*Sqrt[2 - Sqrt[3]]*(1 + c/b + (2*c^2*x)/b^2)*(-((c*(b*x + c*x^2))/b^2))^(2/3)*Sqrt[(1 + (-(c/b) - (2*c^2*x)/b^2)^2 + (1 - (b^2*(-( c/b) - (2*c^2*x)/b^2)^2)/c^2)^(1/3))/(1 - Sqrt[3] + c/b + (2*c^2*x)/b^2)^2 ]*EllipticF[ArcSin[(1 + Sqrt[3] + c/b + (2*c^2*x)/b^2)/(1 - Sqrt[3] + c/b + (2*c^2*x)/b^2)], -7 + 4*Sqrt[3]])/((-(c/b) - (2*c^2*x)/b^2)*Sqrt[-((1 + c/b + (2*c^2*x)/b^2)/(1 - Sqrt[3] + c/b + (2*c^2*x)/b^2)^2)]*(b*x + c*x^2) ^(2/3)))
3.1.32.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-2/3), x_Symbol] :> Simp[3*(Sqrt[b*x^2]/(2*b*x)) Subst[Int[1/Sqrt[-a + x^3], x], x, (a + b*x^2)^(1/3)], x] /; FreeQ[{a, b }, x]
Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 - Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s *x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(- s)*((s + r*x)/((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]], x]] /; FreeQ[{a, b}, x ] && NegQ[a]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/(2*c*(-4* (c/(b^2 - 4*a*c)))^p) Subst[Int[Simp[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]
Int[((b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(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] /; F reeQ[{b, c}, x] && (IntegerQ[4*p] || IntegerQ[3*p])
\[\int \frac {1}{\left (c \,x^{2}+b x \right )^{\frac {2}{3}}}d x\]
\[ \int \frac {1}{\left (b x+c x^2\right )^{2/3}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x\right )}^{\frac {2}{3}}} \,d x } \]
\[ \int \frac {1}{\left (b x+c x^2\right )^{2/3}} \, dx=\int \frac {1}{\left (b x + c x^{2}\right )^{\frac {2}{3}}}\, dx \]
\[ \int \frac {1}{\left (b x+c x^2\right )^{2/3}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x\right )}^{\frac {2}{3}}} \,d x } \]
\[ \int \frac {1}{\left (b x+c x^2\right )^{2/3}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x\right )}^{\frac {2}{3}}} \,d x } \]
Time = 9.04 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.11 \[ \int \frac {1}{\left (b x+c x^2\right )^{2/3}} \, dx=\frac {3\,x\,{\left (\frac {c\,x}{b}+1\right )}^{2/3}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{3},\frac {2}{3};\ \frac {4}{3};\ -\frac {c\,x}{b}\right )}{{\left (c\,x^2+b\,x\right )}^{2/3}} \]