Integrand size = 13, antiderivative size = 781 \[ \int \left (b x+c x^2\right )^{2/3} \, dx=\frac {3 \left (-\frac {c x (b+c x)}{b^2}\right )^{2/3} (b+2 c x) \left (b x+c x^2\right )^{2/3}}{14 c \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{2/3}}-\frac {3 (b+2 c x) \left (b x+c x^2\right )^{2/3}}{7 \sqrt [3]{2} c \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{2/3} \left (1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right )}-\frac {3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} b^2 \left (b x+c x^2\right )^{2/3} \left (1-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right ) \sqrt {\frac {1+2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}+2 \sqrt [3]{2} \left (-\frac {c x (b+c x)}{b^2}\right )^{2/3}}{\left (1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right )^2}} E\left (\arcsin \left (\frac {1+\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}}{1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}}\right )|-7+4 \sqrt {3}\right )}{14 \sqrt [3]{2} c (b+2 c x) \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{2/3} \sqrt {-\frac {1-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}}{\left (1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right )^2}}}+\frac {\sqrt [6]{2} 3^{3/4} b^2 \left (b x+c x^2\right )^{2/3} \left (1-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right ) \sqrt {\frac {1+2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}+2 \sqrt [3]{2} \left (-\frac {c x (b+c x)}{b^2}\right )^{2/3}}{\left (1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}}{1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}}\right ),-7+4 \sqrt {3}\right )}{7 c (b+2 c x) \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{2/3} \sqrt {-\frac {1-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}}{\left (1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right )^2}}} \]
3/14*(-c*x*(c*x+b)/b^2)^(2/3)*(2*c*x+b)*(c*x^2+b*x)^(2/3)/c/(-c*(c*x^2+b*x )/b^2)^(2/3)-3/14*(2*c*x+b)*(c*x^2+b*x)^(2/3)*2^(2/3)/c/(-c*(c*x^2+b*x)/b^ 2)^(2/3)/(1-2^(2/3)*(-c*x*(c*x+b)/b^2)^(1/3)-3^(1/2))+1/7*2^(1/6)*3^(3/4)* b^2*(c*x^2+b*x)^(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^(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*(c*x^2+b*x)/b^2)^(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)-3/28*3^(1/4 )*b^2*(c*x^2+b*x)^(2/3)*(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*(c*x^2+b*x)/b^2)^ (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 = 10.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.06 \[ \int \left (b x+c x^2\right )^{2/3} \, dx=\frac {3 x (x (b+c x))^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {5}{3},\frac {8}{3},-\frac {c x}{b}\right )}{5 \left (1+\frac {c x}{b}\right )^{2/3}} \]
(3*x*(x*(b + c*x))^(2/3)*Hypergeometric2F1[-2/3, 5/3, 8/3, -((c*x)/b)])/(5 *(1 + (c*x)/b)^(2/3))
Time = 0.48 (sec) , antiderivative size = 690, normalized size of antiderivative = 0.88, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {1087, 1093, 1090, 233, 833, 760, 2418}
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.
| \(\displaystyle \int \left (b x+c x^2\right )^{2/3} \, dx\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {3 (b+2 c x) \left (b x+c x^2\right )^{2/3}}{14 c}-\frac {b^2 \int \frac {1}{\sqrt [3]{c x^2+b x}}dx}{7 c}\) |
| \(\Big \downarrow \) 1093 |
| \(\displaystyle \frac {3 (b+2 c x) \left (b x+c x^2\right )^{2/3}}{14 c}-\frac {b^2 \sqrt [3]{-\frac {c \left (b x+c x^2\right )}{b^2}} \int \frac {1}{\sqrt [3]{-\frac {c^2 x^2}{b^2}-\frac {c x}{b}}}dx}{7 c \sqrt [3]{b x+c x^2}}\) |
| \(\Big \downarrow \) 1090 |
| \(\displaystyle \frac {b^4 \sqrt [3]{-\frac {c \left (b x+c x^2\right )}{b^2}} \int \frac {1}{\sqrt [3]{1-\frac {b^2 \left (-\frac {2 x c^2}{b^2}-\frac {c}{b}\right )^2}{c^2}}}d\left (-\frac {2 x c^2}{b^2}-\frac {c}{b}\right )}{7 \sqrt [3]{2} c^3 \sqrt [3]{b x+c x^2}}+\frac {3 (b+2 c x) \left (b x+c x^2\right )^{2/3}}{14 c}\) |
| \(\Big \downarrow \) 233 |
| \(\displaystyle \frac {3 (b+2 c x) \left (b x+c x^2\right )^{2/3}}{14 c}-\frac {3 b^2 \sqrt {-\frac {b^2 \left (-\frac {2 c^2 x}{b^2}-\frac {c}{b}\right )^2}{c^2}} \sqrt [3]{-\frac {c \left (b x+c x^2\right )}{b^2}} \int \frac {\sqrt [3]{1-\frac {b^2 \left (-\frac {2 x c^2}{b^2}-\frac {c}{b}\right )^2}{c^2}}}{\sqrt {-\frac {b^2 \left (-\frac {2 x c^2}{b^2}-\frac {c}{b}\right )^2}{c^2}}}d\sqrt [3]{1-\frac {b^2 \left (-\frac {2 x c^2}{b^2}-\frac {c}{b}\right )^2}{c^2}}}{14 \sqrt [3]{2} c \left (-\frac {2 c^2 x}{b^2}-\frac {c}{b}\right ) \sqrt [3]{b x+c x^2}}\) |
| \(\Big \downarrow \) 833 |
| \(\displaystyle \frac {3 (b+2 c x) \left (b x+c x^2\right )^{2/3}}{14 c}-\frac {3 b^2 \sqrt {-\frac {b^2 \left (-\frac {2 c^2 x}{b^2}-\frac {c}{b}\right )^2}{c^2}} \sqrt [3]{-\frac {c \left (b x+c x^2\right )}{b^2}} \left (\left (1+\sqrt {3}\right ) \int \frac {1}{\sqrt {-\frac {b^2 \left (-\frac {2 x c^2}{b^2}-\frac {c}{b}\right )^2}{c^2}}}d\sqrt [3]{1-\frac {b^2 \left (-\frac {2 x c^2}{b^2}-\frac {c}{b}\right )^2}{c^2}}-\int \frac {\frac {2 x c^2}{b^2}+\frac {c}{b}+\sqrt {3}+1}{\sqrt {-\frac {b^2 \left (-\frac {2 x c^2}{b^2}-\frac {c}{b}\right )^2}{c^2}}}d\sqrt [3]{1-\frac {b^2 \left (-\frac {2 x c^2}{b^2}-\frac {c}{b}\right )^2}{c^2}}\right )}{14 \sqrt [3]{2} c \left (-\frac {2 c^2 x}{b^2}-\frac {c}{b}\right ) \sqrt [3]{b x+c x^2}}\) |
| \(\Big \downarrow \) 760 |
| \(\displaystyle \frac {3 (b+2 c x) \left (b x+c x^2\right )^{2/3}}{14 c}-\frac {3 b^2 \sqrt {-\frac {b^2 \left (-\frac {2 c^2 x}{b^2}-\frac {c}{b}\right )^2}{c^2}} \sqrt [3]{-\frac {c \left (b x+c x^2\right )}{b^2}} \left (-\int \frac {\frac {2 x c^2}{b^2}+\frac {c}{b}+\sqrt {3}+1}{\sqrt {-\frac {b^2 \left (-\frac {2 x c^2}{b^2}-\frac {c}{b}\right )^2}{c^2}}}d\sqrt [3]{1-\frac {b^2 \left (-\frac {2 x c^2}{b^2}-\frac {c}{b}\right )^2}{c^2}}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (\frac {2 c^2 x}{b^2}+\frac {c}{b}+1\right ) \sqrt {\frac {\left (-\frac {2 c^2 x}{b^2}-\frac {c}{b}\right )^2+\sqrt [3]{1-\frac {b^2 \left (-\frac {2 c^2 x}{b^2}-\frac {c}{b}\right )^2}{c^2}}+1}{\left (\frac {2 c^2 x}{b^2}+\frac {c}{b}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\frac {2 x c^2}{b^2}+\frac {c}{b}+\sqrt {3}+1}{\frac {2 x c^2}{b^2}+\frac {c}{b}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {b^2 \left (-\frac {2 c^2 x}{b^2}-\frac {c}{b}\right )^2}{c^2}} \sqrt {-\frac {\frac {2 c^2 x}{b^2}+\frac {c}{b}+1}{\left (\frac {2 c^2 x}{b^2}+\frac {c}{b}-\sqrt {3}+1\right )^2}}}\right )}{14 \sqrt [3]{2} c \left (-\frac {2 c^2 x}{b^2}-\frac {c}{b}\right ) \sqrt [3]{b x+c x^2}}\) |
| \(\Big \downarrow \) 2418 |
| \(\displaystyle \frac {3 (b+2 c x) \left (b x+c x^2\right )^{2/3}}{14 c}-\frac {3 b^2 \sqrt {-\frac {b^2 \left (-\frac {2 c^2 x}{b^2}-\frac {c}{b}\right )^2}{c^2}} \sqrt [3]{-\frac {c \left (b x+c x^2\right )}{b^2}} \left (-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (\frac {2 c^2 x}{b^2}+\frac {c}{b}+1\right ) \sqrt {\frac {\left (-\frac {2 c^2 x}{b^2}-\frac {c}{b}\right )^2+\sqrt [3]{1-\frac {b^2 \left (-\frac {2 c^2 x}{b^2}-\frac {c}{b}\right )^2}{c^2}}+1}{\left (\frac {2 c^2 x}{b^2}+\frac {c}{b}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\frac {2 x c^2}{b^2}+\frac {c}{b}+\sqrt {3}+1}{\frac {2 x c^2}{b^2}+\frac {c}{b}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {b^2 \left (-\frac {2 c^2 x}{b^2}-\frac {c}{b}\right )^2}{c^2}} \sqrt {-\frac {\frac {2 c^2 x}{b^2}+\frac {c}{b}+1}{\left (\frac {2 c^2 x}{b^2}+\frac {c}{b}-\sqrt {3}+1\right )^2}}}+\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (\frac {2 c^2 x}{b^2}+\frac {c}{b}+1\right ) \sqrt {\frac {\left (-\frac {2 c^2 x}{b^2}-\frac {c}{b}\right )^2+\sqrt [3]{1-\frac {b^2 \left (-\frac {2 c^2 x}{b^2}-\frac {c}{b}\right )^2}{c^2}}+1}{\left (\frac {2 c^2 x}{b^2}+\frac {c}{b}-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {\frac {2 x c^2}{b^2}+\frac {c}{b}+\sqrt {3}+1}{\frac {2 x c^2}{b^2}+\frac {c}{b}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {-\frac {b^2 \left (-\frac {2 c^2 x}{b^2}-\frac {c}{b}\right )^2}{c^2}} \sqrt {-\frac {\frac {2 c^2 x}{b^2}+\frac {c}{b}+1}{\left (\frac {2 c^2 x}{b^2}+\frac {c}{b}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {-\frac {b^2 \left (-\frac {2 c^2 x}{b^2}-\frac {c}{b}\right )^2}{c^2}}}{\frac {2 c^2 x}{b^2}+\frac {c}{b}-\sqrt {3}+1}\right )}{14 \sqrt [3]{2} c \left (-\frac {2 c^2 x}{b^2}-\frac {c}{b}\right ) \sqrt [3]{b x+c x^2}}\) |
(3*(b + 2*c*x)*(b*x + c*x^2)^(2/3))/(14*c) - (3*b^2*Sqrt[-((b^2*(-(c/b) - (2*c^2*x)/b^2)^2)/c^2)]*(-((c*(b*x + c*x^2))/b^2))^(1/3)*((-2*Sqrt[-((b^2* (-(c/b) - (2*c^2*x)/b^2)^2)/c^2)])/(1 - Sqrt[3] + c/b + (2*c^2*x)/b^2) + ( 3^(1/4)*Sqrt[2 + Sqrt[3]]*(1 + c/b + (2*c^2*x)/b^2)*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 - Sq rt[3] + c/b + (2*c^2*x)/b^2)^2]*EllipticE[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]])/(Sqrt[-( (b^2*(-(c/b) - (2*c^2*x)/b^2)^2)/c^2)]*Sqrt[-((1 + c/b + (2*c^2*x)/b^2)/(1 - Sqrt[3] + c/b + (2*c^2*x)/b^2)^2)]) - (2*Sqrt[2 - Sqrt[3]]*(1 + Sqrt[3] )*(1 + c/b + (2*c^2*x)/b^2)*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]])/(3^(1/4)*Sqrt[-((b^2*(-(c/b) - ( 2*c^2*x)/b^2)^2)/c^2)]*Sqrt[-((1 + c/b + (2*c^2*x)/b^2)/(1 - Sqrt[3] + c/b + (2*c^2*x)/b^2)^2)])))/(14*2^(1/3)*c*(-(c/b) - (2*c^2*x)/b^2)*(b*x + c*x ^2)^(1/3))
3.1.36.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1/3), x_Symbol] :> Simp[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]
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[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3] ], s = Denom[Rt[b/a, 3]]}, Simp[(-(1 + Sqrt[3]))*(s/r) Int[1/Sqrt[a + b*x ^3], x], x] + Simp[1/r Int[((1 + Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x], x ]] /; FreeQ[{a, b}, x] && NegQ[a]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* p + 1))) Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
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[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N umer[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] + S imp[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]/(r^2*Sqrt[a + b*x^3]*Sqrt[(-s)*((s + r*x)/((1 - S qrt[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 + Sqrt[3])*s + r*x)/((1 - Sqrt[ 3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && EqQ[b*c^3 - 2*(5 + 3*Sqrt[3])*a*d^3, 0]
\[\int \left (c \,x^{2}+b x \right )^{\frac {2}{3}}d x\]
\[ \int \left (b x+c x^2\right )^{2/3} \, dx=\int { {\left (c x^{2} + b x\right )}^{\frac {2}{3}} \,d x } \]
\[ \int \left (b x+c x^2\right )^{2/3} \, dx=\int \left (b x + c x^{2}\right )^{\frac {2}{3}}\, dx \]
\[ \int \left (b x+c x^2\right )^{2/3} \, dx=\int { {\left (c x^{2} + b x\right )}^{\frac {2}{3}} \,d x } \]
\[ \int \left (b x+c x^2\right )^{2/3} \, dx=\int { {\left (c x^{2} + b x\right )}^{\frac {2}{3}} \,d x } \]
Time = 9.06 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.05 \[ \int \left (b x+c x^2\right )^{2/3} \, dx=\frac {3\,x\,{\left (c\,x^2+b\,x\right )}^{2/3}\,{{}}_2{\mathrm {F}}_1\left (-\frac {2}{3},\frac {5}{3};\ \frac {8}{3};\ -\frac {c\,x}{b}\right )}{5\,{\left (\frac {c\,x}{b}+1\right )}^{2/3}} \]