3.1.36 \(\int (b x+c x^2)^{2/3} \, dx\) [36]

Optimal. Leaf size=781 \[ \frac {3 \left (-\frac {c x (b+c x)}{b^2}\right )^{2/3} (b+2 c x) \left (b x+c x^2\right )^{2/3}}{14 c \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{2/3}}-\frac {3 (b+2 c x) \left (b x+c x^2\right )^{2/3}}{7 \sqrt [3]{2} c \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{2/3} \left (1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right )}-\frac {3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} b^2 \left (b x+c x^2\right )^{2/3} \left (1-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right ) \sqrt {\frac {1+2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}+2 \sqrt [3]{2} \left (-\frac {c x (b+c x)}{b^2}\right )^{2/3}}{\left (1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right )^2}} E\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}}{1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}}\right )|-7+4 \sqrt {3}\right )}{14 \sqrt [3]{2} c (b+2 c x) \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{2/3} \sqrt {-\frac {1-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}}{\left (1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right )^2}}}+\frac {\sqrt [6]{2} 3^{3/4} b^2 \left (b x+c x^2\right )^{2/3} \left (1-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right ) \sqrt {\frac {1+2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}+2 \sqrt [3]{2} \left (-\frac {c x (b+c x)}{b^2}\right )^{2/3}}{\left (1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}}{1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}}\right )|-7+4 \sqrt {3}\right )}{7 c (b+2 c x) \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{2/3} \sqrt {-\frac {1-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}}{\left (1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right )^2}}} \]

[Out]

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)

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Rubi [A]
time = 0.65, antiderivative size = 781, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {636, 633, 201, 241, 310, 225, 1893} \begin {gather*} \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 {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 (\text {ArcSin}\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 )}{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 (-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 \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 {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 (\text {ArcSin}\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 )}{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 (-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}-\sqrt {3}+1\right )^2}}}+\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 (-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}-\sqrt {3}+1\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x + c*x^2)^(2/3),x]

[Out]

(3*(-((c*x*(b + c*x))/b^2))^(2/3)*(b + 2*c*x)*(b*x + c*x^2)^(2/3))/(14*c*(-((c*(b*x + c*x^2))/b^2))^(2/3)) - (
3*(b + 2*c*x)*(b*x + c*x^2)^(2/3))/(7*2^(1/3)*c*(-((c*(b*x + c*x^2))/b^2))^(2/3)*(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*(b*x + c*x^2)^(2/3)*(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]])/(14*2^(1/3)*c*(b
+ 2*c*x)*(-((c*(b*x + c*x^2))/b^2))^(2/3)*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*(b*x + c*x^2)^(2/3)*(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]])/(7*c*(b +
2*c*x)*(-((c*(b*x + c*x^2))/b^2))^(2/3)*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 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 225

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]*Sq
rt[(-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]

Rule 241

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 310

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Dist[(-(
1 + Sqrt[3]))*(s/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 633

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 636

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 1893

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]/(r^2
*Sqrt[a + b*x^3]*Sqrt[(-s)*((s + r*x)/((1 - Sqrt[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*Sqr
t[3])*a*d^3, 0]

Rubi steps

\begin {align*} \int \left (b x+c x^2\right )^{2/3} \, dx &=\frac {\left (b x+c x^2\right )^{2/3} \int \left (-\frac {c x}{b}-\frac {c^2 x^2}{b^2}\right )^{2/3} \, dx}{\left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{2/3}}\\ &=-\frac {\left (b^2 \left (b x+c x^2\right )^{2/3}\right ) \text {Subst}\left (\int \left (1-\frac {b^2 x^2}{c^2}\right )^{2/3} \, dx,x,-\frac {c}{b}-\frac {2 c^2 x}{b^2}\right )}{4 \sqrt [3]{2} c^2 \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{2/3}}\\ &=\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 {\left (b^2 \left (b x+c x^2\right )^{2/3}\right ) \text {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 )}{7 \sqrt [3]{2} c^2 \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{2/3}}\\ &=\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 {\left (3 \left (b x+c x^2\right )^{2/3} \sqrt {-1-\frac {4 c x}{b}-\frac {4 c^2 x^2}{b^2}}\right ) \text {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 )}{14 \sqrt [3]{2} \left (-\frac {c}{b}-\frac {2 c^2 x}{b^2}\right ) \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{2/3}}\\ &=\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 {\left (3 \left (b x+c x^2\right )^{2/3} \sqrt {-1-\frac {4 c x}{b}-\frac {4 c^2 x^2}{b^2}}\right ) \text {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 )}{14 \sqrt [3]{2} \left (-\frac {c}{b}-\frac {2 c^2 x}{b^2}\right ) \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{2/3}}+\frac {\left (3 \sqrt {2+\sqrt {3}} \left (b x+c x^2\right )^{2/3} \sqrt {-1-\frac {4 c x}{b}-\frac {4 c^2 x^2}{b^2}}\right ) \text {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 )}{7\ 2^{5/6} \left (-\frac {c}{b}-\frac {2 c^2 x}{b^2}\right ) \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{2/3}}\\ &=\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 \left (b x+c x^2\right )^{2/3} \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}}}{7 \sqrt [3]{2} c (b+2 c x) \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} \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 )}{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 {-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 \left (b x+c x^2\right )^{2/3} \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 )}{7 c (b+2 c x) \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{2/3} \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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.02, size = 45, normalized size = 0.06 \begin {gather*} \frac {3 x (x (b+c x))^{2/3} \, _2F_1\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}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*x + c*x^2)^(2/3),x]

[Out]

(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))

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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \left (c \,x^{2}+b x \right )^{\frac {2}{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c*x^2 + b*x)^(2/3), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((c*x^2 + b*x)^(2/3), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (b x + c x^{2}\right )^{\frac {2}{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((b*x + c*x**2)**(2/3), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c*x^2 + b*x)^(2/3), x)

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Mupad [B]
time = 0.16, size = 36, normalized size = 0.05 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + c*x^2)^(2/3),x)

[Out]

(3*x*(b*x + c*x^2)^(2/3)*hypergeom([-2/3, 5/3], 8/3, -(c*x)/b))/(5*((c*x)/b + 1)^(2/3))

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