\(\int (a+b x^2)^{5/3} \, dx\) [71]

Optimal result

Integrand size = 11, antiderivative size = 569 \[ \int \left (a+b x^2\right )^{5/3} \, dx=\frac {30}{91} a x \left (a+b x^2\right )^{2/3}+\frac {3}{13} x \left (a+b x^2\right )^{5/3}-\frac {120 a^2 x}{91 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}+\frac {60 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^{7/3} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}\right )|-7+4 \sqrt {3}\right )}{91 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}-\frac {40 \sqrt {2} 3^{3/4} a^{7/3} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}\right ),-7+4 \sqrt {3}\right )}{91 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}} \] Output:

30/91*a*x*(b*x^2+a)^(2/3)+3/13*x*(b*x^2+a)^(5/3)-120*a^2*x/(91*(1-3^(1/2)) 
*a^(1/3)-91*(b*x^2+a)^(1/3))+60/91*3^(1/4)*(1/2*6^(1/2)+1/2*2^(1/2))*a^(7/ 
3)*(a^(1/3)-(b*x^2+a)^(1/3))*((a^(2/3)+a^(1/3)*(b*x^2+a)^(1/3)+(b*x^2+a)^( 
2/3))/((1-3^(1/2))*a^(1/3)-(b*x^2+a)^(1/3))^2)^(1/2)*EllipticE(((1+3^(1/2) 
)*a^(1/3)-(b*x^2+a)^(1/3))/((1-3^(1/2))*a^(1/3)-(b*x^2+a)^(1/3)),2*I-I*3^( 
1/2))/b/x/(-a^(1/3)*(a^(1/3)-(b*x^2+a)^(1/3))/((1-3^(1/2))*a^(1/3)-(b*x^2+ 
a)^(1/3))^2)^(1/2)-40/91*2^(1/2)*3^(3/4)*a^(7/3)*(a^(1/3)-(b*x^2+a)^(1/3)) 
*((a^(2/3)+a^(1/3)*(b*x^2+a)^(1/3)+(b*x^2+a)^(2/3))/((1-3^(1/2))*a^(1/3)-( 
b*x^2+a)^(1/3))^2)^(1/2)*EllipticF(((1+3^(1/2))*a^(1/3)-(b*x^2+a)^(1/3))/( 
(1-3^(1/2))*a^(1/3)-(b*x^2+a)^(1/3)),2*I-I*3^(1/2))/b/x/(-a^(1/3)*(a^(1/3) 
-(b*x^2+a)^(1/3))/((1-3^(1/2))*a^(1/3)-(b*x^2+a)^(1/3))^2)^(1/2)
 

Mathematica [C] (verified)

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

Time = 5.50 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.08 \[ \int \left (a+b x^2\right )^{5/3} \, dx=\frac {a x \left (a+b x^2\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {5}{3},\frac {1}{2},\frac {3}{2},-\frac {b x^2}{a}\right )}{\left (1+\frac {b x^2}{a}\right )^{2/3}} \] Input:

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.45 (sec) , antiderivative size = 614, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {211, 211, 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.

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

\(\Big \downarrow \) 211

\(\displaystyle \frac {10}{13} a \int \left (b x^2+a\right )^{2/3}dx+\frac {3}{13} x \left (a+b x^2\right )^{5/3}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {10}{13} a \left (\frac {4}{7} a \int \frac {1}{\sqrt [3]{b x^2+a}}dx+\frac {3}{7} x \left (a+b x^2\right )^{2/3}\right )+\frac {3}{13} x \left (a+b x^2\right )^{5/3}\)

\(\Big \downarrow \) 233

\(\displaystyle \frac {10}{13} a \left (\frac {6 a \sqrt {b x^2} \int \frac {\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}}{7 b x}+\frac {3}{7} x \left (a+b x^2\right )^{2/3}\right )+\frac {3}{13} x \left (a+b x^2\right )^{5/3}\)

\(\Big \downarrow \) 833

\(\displaystyle \frac {10}{13} a \left (\frac {6 a \sqrt {b x^2} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} \int \frac {1}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}\right )}{7 b x}+\frac {3}{7} x \left (a+b x^2\right )^{2/3}\right )+\frac {3}{13} x \left (a+b x^2\right )^{5/3}\)

\(\Big \downarrow \) 760

\(\displaystyle \frac {10}{13} a \left (\frac {6 a \sqrt {b x^2} \left (-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}\right )}{7 b x}+\frac {3}{7} x \left (a+b x^2\right )^{2/3}\right )+\frac {3}{13} x \left (a+b x^2\right )^{5/3}\)

\(\Big \downarrow \) 2418

\(\displaystyle \frac {10}{13} a \left (\frac {6 a \sqrt {b x^2} \left (-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}+\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right )|-7+4 \sqrt {3}\right )}{\sqrt {b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}-\frac {2 \sqrt {b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}\right )}{7 b x}+\frac {3}{7} x \left (a+b x^2\right )^{2/3}\right )+\frac {3}{13} x \left (a+b x^2\right )^{5/3}\)

Input:

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

Output:

(3*x*(a + b*x^2)^(5/3))/13 + (10*a*((3*x*(a + b*x^2)^(2/3))/7 + (6*a*Sqrt[ 
b*x^2]*((-2*Sqrt[b*x^2])/((1 - Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3)) + (3^ 
(1/4)*Sqrt[2 + Sqrt[3]]*a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3))*Sqrt[(a^(2/3 
) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) 
- (a + b*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a + b*x 
^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/ 
(Sqrt[b*x^2]*Sqrt[-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - Sqrt[3]) 
*a^(1/3) - (a + b*x^2)^(1/3))^2)]) - (2*Sqrt[2 - Sqrt[3]]*(1 + Sqrt[3])*a^ 
(1/3)*(a^(1/3) - (a + b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a + b*x^2)^(1 
/3) + (a + b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3))^2]*El 
lipticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3))/((1 - Sqrt[3])* 
a^(1/3) - (a + b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(3^(1/4)*Sqrt[b*x^2]*Sqrt[ 
-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a + b* 
x^2)^(1/3))^2)])))/(7*b*x)))/13
 

Defintions of rubi rules used

rule 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

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 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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [A] (verification not implemented)

Time = 0.56 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.05 \[ \int \left (a+b x^2\right )^{5/3} \, dx=a^{\frac {5}{3}} x {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{3}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )} \] Input:

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

Output:

a**(5/3)*x*hyper((-5/3, 1/2), (3/2,), b*x**2*exp_polar(I*pi)/a)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.07 \[ \int \left (a+b x^2\right )^{5/3} \, dx=\frac {x\,{\left (b\,x^2+a\right )}^{5/3}\,{{}}_2{\mathrm {F}}_1\left (-\frac {5}{3},\frac {1}{2};\ \frac {3}{2};\ -\frac {b\,x^2}{a}\right )}{{\left (\frac {b\,x^2}{a}+1\right )}^{5/3}} \] Input:

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

Output:

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

Reduce [F]

\[ \int \left (a+b x^2\right )^{5/3} \, dx=\frac {51 \left (b \,x^{2}+a \right )^{\frac {2}{3}} a x}{91}+\frac {3 \left (b \,x^{2}+a \right )^{\frac {2}{3}} b \,x^{3}}{13}+\frac {40 \left (\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {1}{3}}}d x \right ) a^{2}}{91} \] Input:

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

Output:

(51*(a + b*x**2)**(2/3)*a*x + 21*(a + b*x**2)**(2/3)*b*x**3 + 40*int((a + 
b*x**2)**(2/3)/(a + b*x**2),x)*a**2)/91