3.1.35 \(\int (b x+c x^2)^{5/3} \, dx\) [35]

3.1.35.1 Optimal result

Integrand size = 13, antiderivative size = 842 \[ \int \left (b x+c x^2\right )^{5/3} \, dx=\frac {15 \left (-\frac {c x (b+c x)}{b^2}\right )^{2/3} (b+2 c x) \left (b x+c x^2\right )^{5/3}}{364 c \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{5/3}}+\frac {3 \left (-\frac {c x (b+c x)}{b^2}\right )^{5/3} (b+2 c x) \left (b x+c x^2\right )^{5/3}}{26 c \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{5/3}}-\frac {15 (b+2 c x) \left (b x+c x^2\right )^{5/3}}{182 \sqrt [3]{2} c \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{5/3} \left (1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right )}-\frac {15 \sqrt [4]{3} \sqrt {2+\sqrt {3}} b^2 \left (b x+c x^2\right )^{5/3} \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 )}{364 \sqrt [3]{2} c (b+2 c x) \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{5/3} \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 {5\ 3^{3/4} b^2 \left (b x+c x^2\right )^{5/3} \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 )}{91\ 2^{5/6} c (b+2 c x) \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{5/3} \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}}} \]

output

15/364*(-c*x*(c*x+b)/b^2)^(2/3)*(2*c*x+b)*(c*x^2+b*x)^(5/3)/c/(-c*(c*x^2+b 
*x)/b^2)^(5/3)+3/26*(-c*x*(c*x+b)/b^2)^(5/3)*(2*c*x+b)*(c*x^2+b*x)^(5/3)/c 
/(-c*(c*x^2+b*x)/b^2)^(5/3)-15/364*(2*c*x+b)*(c*x^2+b*x)^(5/3)*2^(2/3)/c/( 
-c*(c*x^2+b*x)/b^2)^(5/3)/(1-2^(2/3)*(-c*x*(c*x+b)/b^2)^(1/3)-3^(1/2))+5/1 
82*3^(3/4)*b^2*(c*x^2+b*x)^(5/3)*(1-2^(2/3)*(-c*x*(c*x+b)/b^2)^(1/3))*Elli 
pticF((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)*2^(1/6)/c/(2*c*x+b)/(-c*(c*x^2+b*x)/b^2)^(5/3)/((-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)-15/728*3^(1/4)*b^2*(c*x^2+b*x)^(5/3)*(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)^(5/3)/((-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)
 

3.1.35.2 Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.02 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.06 \[ \int \left (b x+c x^2\right )^{5/3} \, dx=\frac {3 b x^2 (x (b+c x))^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {5}{3},\frac {8}{3},\frac {11}{3},-\frac {c x}{b}\right )}{8 \left (1+\frac {c x}{b}\right )^{2/3}} \]

input

Integrate[(b*x + c*x^2)^(5/3),x]
 

output

(3*b*x^2*(x*(b + c*x))^(2/3)*Hypergeometric2F1[-5/3, 8/3, 11/3, -((c*x)/b) 
])/(8*(1 + (c*x)/b)^(2/3))
 

3.1.35.3 Rubi [A] (warning: unable to verify)

Time = 0.55 (sec) , antiderivative size = 727, normalized size of antiderivative = 0.86, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {1087, 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.

\(\displaystyle \int \left (b x+c x^2\right )^{5/3} \, dx\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {3 (b+2 c x) \left (b x+c x^2\right )^{5/3}}{26 c}-\frac {5 b^2 \int \left (c x^2+b x\right )^{2/3}dx}{26 c}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {3 (b+2 c x) \left (b x+c x^2\right )^{5/3}}{26 c}-\frac {5 b^2 \left (\frac {3 (b+2 c x) \left (b x+c x^2\right )^{2/3}}{14 c}-\frac {b^2 \int \frac {1}{\sqrt [3]{c x^2+b x}}dx}{7 c}\right )}{26 c}\)

\(\Big \downarrow \) 1093

\(\displaystyle \frac {3 (b+2 c x) \left (b x+c x^2\right )^{5/3}}{26 c}-\frac {5 b^2 \left (\frac {3 (b+2 c x) \left (b x+c x^2\right )^{2/3}}{14 c}-\frac {b^2 \sqrt [3]{-\frac {c \left (b x+c x^2\right )}{b^2}} \int \frac {1}{\sqrt [3]{-\frac {c^2 x^2}{b^2}-\frac {c x}{b}}}dx}{7 c \sqrt [3]{b x+c x^2}}\right )}{26 c}\)

\(\Big \downarrow \) 1090

\(\displaystyle \frac {3 (b+2 c x) \left (b x+c x^2\right )^{5/3}}{26 c}-\frac {5 b^2 \left (\frac {b^4 \sqrt [3]{-\frac {c \left (b x+c x^2\right )}{b^2}} \int \frac {1}{\sqrt [3]{1-\frac {b^2 \left (-\frac {2 x c^2}{b^2}-\frac {c}{b}\right )^2}{c^2}}}d\left (-\frac {2 x c^2}{b^2}-\frac {c}{b}\right )}{7 \sqrt [3]{2} c^3 \sqrt [3]{b x+c x^2}}+\frac {3 (b+2 c x) \left (b x+c x^2\right )^{2/3}}{14 c}\right )}{26 c}\)

\(\Big \downarrow \) 233

\(\displaystyle \frac {3 (b+2 c x) \left (b x+c x^2\right )^{5/3}}{26 c}-\frac {5 b^2 \left (\frac {3 (b+2 c x) \left (b x+c x^2\right )^{2/3}}{14 c}-\frac {3 b^2 \sqrt {-\frac {b^2 \left (-\frac {2 c^2 x}{b^2}-\frac {c}{b}\right )^2}{c^2}} \sqrt [3]{-\frac {c \left (b x+c x^2\right )}{b^2}} \int \frac {\sqrt [3]{1-\frac {b^2 \left (-\frac {2 x c^2}{b^2}-\frac {c}{b}\right )^2}{c^2}}}{\sqrt {-\frac {b^2 \left (-\frac {2 x c^2}{b^2}-\frac {c}{b}\right )^2}{c^2}}}d\sqrt [3]{1-\frac {b^2 \left (-\frac {2 x c^2}{b^2}-\frac {c}{b}\right )^2}{c^2}}}{14 \sqrt [3]{2} c \left (-\frac {2 c^2 x}{b^2}-\frac {c}{b}\right ) \sqrt [3]{b x+c x^2}}\right )}{26 c}\)

\(\Big \downarrow \) 833

\(\displaystyle \frac {3 (b+2 c x) \left (b x+c x^2\right )^{5/3}}{26 c}-\frac {5 b^2 \left (\frac {3 (b+2 c x) \left (b x+c x^2\right )^{2/3}}{14 c}-\frac {3 b^2 \sqrt {-\frac {b^2 \left (-\frac {2 c^2 x}{b^2}-\frac {c}{b}\right )^2}{c^2}} \sqrt [3]{-\frac {c \left (b x+c x^2\right )}{b^2}} \left (\left (1+\sqrt {3}\right ) \int \frac {1}{\sqrt {-\frac {b^2 \left (-\frac {2 x c^2}{b^2}-\frac {c}{b}\right )^2}{c^2}}}d\sqrt [3]{1-\frac {b^2 \left (-\frac {2 x c^2}{b^2}-\frac {c}{b}\right )^2}{c^2}}-\int \frac {\frac {2 x c^2}{b^2}+\frac {c}{b}+\sqrt {3}+1}{\sqrt {-\frac {b^2 \left (-\frac {2 x c^2}{b^2}-\frac {c}{b}\right )^2}{c^2}}}d\sqrt [3]{1-\frac {b^2 \left (-\frac {2 x c^2}{b^2}-\frac {c}{b}\right )^2}{c^2}}\right )}{14 \sqrt [3]{2} c \left (-\frac {2 c^2 x}{b^2}-\frac {c}{b}\right ) \sqrt [3]{b x+c x^2}}\right )}{26 c}\)

\(\Big \downarrow \) 760

\(\displaystyle \frac {3 (b+2 c x) \left (b x+c x^2\right )^{5/3}}{26 c}-\frac {5 b^2 \left (\frac {3 (b+2 c x) \left (b x+c x^2\right )^{2/3}}{14 c}-\frac {3 b^2 \sqrt {-\frac {b^2 \left (-\frac {2 c^2 x}{b^2}-\frac {c}{b}\right )^2}{c^2}} \sqrt [3]{-\frac {c \left (b x+c x^2\right )}{b^2}} \left (-\int \frac {\frac {2 x c^2}{b^2}+\frac {c}{b}+\sqrt {3}+1}{\sqrt {-\frac {b^2 \left (-\frac {2 x c^2}{b^2}-\frac {c}{b}\right )^2}{c^2}}}d\sqrt [3]{1-\frac {b^2 \left (-\frac {2 x c^2}{b^2}-\frac {c}{b}\right )^2}{c^2}}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (\frac {2 c^2 x}{b^2}+\frac {c}{b}+1\right ) \sqrt {\frac {\left (-\frac {2 c^2 x}{b^2}-\frac {c}{b}\right )^2+\sqrt [3]{1-\frac {b^2 \left (-\frac {2 c^2 x}{b^2}-\frac {c}{b}\right )^2}{c^2}}+1}{\left (\frac {2 c^2 x}{b^2}+\frac {c}{b}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\frac {2 x c^2}{b^2}+\frac {c}{b}+\sqrt {3}+1}{\frac {2 x c^2}{b^2}+\frac {c}{b}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {b^2 \left (-\frac {2 c^2 x}{b^2}-\frac {c}{b}\right )^2}{c^2}} \sqrt {-\frac {\frac {2 c^2 x}{b^2}+\frac {c}{b}+1}{\left (\frac {2 c^2 x}{b^2}+\frac {c}{b}-\sqrt {3}+1\right )^2}}}\right )}{14 \sqrt [3]{2} c \left (-\frac {2 c^2 x}{b^2}-\frac {c}{b}\right ) \sqrt [3]{b x+c x^2}}\right )}{26 c}\)

\(\Big \downarrow \) 2418

\(\displaystyle \frac {3 (b+2 c x) \left (b x+c x^2\right )^{5/3}}{26 c}-\frac {5 b^2 \left (\frac {3 (b+2 c x) \left (b x+c x^2\right )^{2/3}}{14 c}-\frac {3 b^2 \sqrt {-\frac {b^2 \left (-\frac {2 c^2 x}{b^2}-\frac {c}{b}\right )^2}{c^2}} \sqrt [3]{-\frac {c \left (b x+c x^2\right )}{b^2}} \left (-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (\frac {2 c^2 x}{b^2}+\frac {c}{b}+1\right ) \sqrt {\frac {\left (-\frac {2 c^2 x}{b^2}-\frac {c}{b}\right )^2+\sqrt [3]{1-\frac {b^2 \left (-\frac {2 c^2 x}{b^2}-\frac {c}{b}\right )^2}{c^2}}+1}{\left (\frac {2 c^2 x}{b^2}+\frac {c}{b}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\frac {2 x c^2}{b^2}+\frac {c}{b}+\sqrt {3}+1}{\frac {2 x c^2}{b^2}+\frac {c}{b}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {b^2 \left (-\frac {2 c^2 x}{b^2}-\frac {c}{b}\right )^2}{c^2}} \sqrt {-\frac {\frac {2 c^2 x}{b^2}+\frac {c}{b}+1}{\left (\frac {2 c^2 x}{b^2}+\frac {c}{b}-\sqrt {3}+1\right )^2}}}+\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (\frac {2 c^2 x}{b^2}+\frac {c}{b}+1\right ) \sqrt {\frac {\left (-\frac {2 c^2 x}{b^2}-\frac {c}{b}\right )^2+\sqrt [3]{1-\frac {b^2 \left (-\frac {2 c^2 x}{b^2}-\frac {c}{b}\right )^2}{c^2}}+1}{\left (\frac {2 c^2 x}{b^2}+\frac {c}{b}-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {\frac {2 x c^2}{b^2}+\frac {c}{b}+\sqrt {3}+1}{\frac {2 x c^2}{b^2}+\frac {c}{b}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {-\frac {b^2 \left (-\frac {2 c^2 x}{b^2}-\frac {c}{b}\right )^2}{c^2}} \sqrt {-\frac {\frac {2 c^2 x}{b^2}+\frac {c}{b}+1}{\left (\frac {2 c^2 x}{b^2}+\frac {c}{b}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {-\frac {b^2 \left (-\frac {2 c^2 x}{b^2}-\frac {c}{b}\right )^2}{c^2}}}{\frac {2 c^2 x}{b^2}+\frac {c}{b}-\sqrt {3}+1}\right )}{14 \sqrt [3]{2} c \left (-\frac {2 c^2 x}{b^2}-\frac {c}{b}\right ) \sqrt [3]{b x+c x^2}}\right )}{26 c}\)

input

Int[(b*x + c*x^2)^(5/3),x]
 

output

(3*(b + 2*c*x)*(b*x + c*x^2)^(5/3))/(26*c) - (5*b^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 - Sqrt[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[Arc 
Sin[(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))))/(26*c)
 

3.1.35.3.1 Defintions of rubi rules used

rule 233

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]
 

rule 760

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]
 

rule 833

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]
 

rule 1087

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])
 

rule 1090

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]
 

rule 1093

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])
 

rule 2418

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]
 

3.1.35.4 Maple [F]

\[\int \left (c \,x^{2}+b x \right )^{\frac {5}{3}}d x\]

input

int((c*x^2+b*x)^(5/3),x)
 

output

int((c*x^2+b*x)^(5/3),x)
 

3.1.35.5 Fricas [F]

\[ \int \left (b x+c x^2\right )^{5/3} \, dx=\int { {\left (c x^{2} + b x\right )}^{\frac {5}{3}} \,d x } \]

input

integrate((c*x^2+b*x)^(5/3),x, algorithm="fricas")
 

output

integral((c*x^2 + b*x)^(5/3), x)
 

3.1.35.6 Sympy [F]

\[ \int \left (b x+c x^2\right )^{5/3} \, dx=\int \left (b x + c x^{2}\right )^{\frac {5}{3}}\, dx \]

input

integrate((c*x**2+b*x)**(5/3),x)
 

output

Integral((b*x + c*x**2)**(5/3), x)
 

3.1.35.7 Maxima [F]

\[ \int \left (b x+c x^2\right )^{5/3} \, dx=\int { {\left (c x^{2} + b x\right )}^{\frac {5}{3}} \,d x } \]

input

integrate((c*x^2+b*x)^(5/3),x, algorithm="maxima")
 

output

integrate((c*x^2 + b*x)^(5/3), x)
 

3.1.35.8 Giac [F]

\[ \int \left (b x+c x^2\right )^{5/3} \, dx=\int { {\left (c x^{2} + b x\right )}^{\frac {5}{3}} \,d x } \]

input

integrate((c*x^2+b*x)^(5/3),x, algorithm="giac")
 

output

integrate((c*x^2 + b*x)^(5/3), x)
 

3.1.35.9 Mupad [B] (verification not implemented)

Time = 9.09 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.04 \[ \int \left (b x+c x^2\right )^{5/3} \, dx=\frac {3\,x\,{\left (c\,x^2+b\,x\right )}^{5/3}\,{{}}_2{\mathrm {F}}_1\left (-\frac {5}{3},\frac {8}{3};\ \frac {11}{3};\ -\frac {c\,x}{b}\right )}{8\,{\left (\frac {c\,x}{b}+1\right )}^{5/3}} \]

input

int((b*x + c*x^2)^(5/3),x)
 

output

(3*x*(b*x + c*x^2)^(5/3)*hypergeom([-5/3, 8/3], 11/3, -(c*x)/b))/(8*((c*x) 
/b + 1)^(5/3))