\(\int \frac {1}{(a+b x^2)^{7/3}} \, dx\) [75]

Optimal result

Integrand size = 11, antiderivative size = 574 \[ \int \frac {1}{\left (a+b x^2\right )^{7/3}} \, dx=\frac {3 x}{8 a \left (a+b x^2\right )^{4/3}}+\frac {15 x}{16 a^2 \sqrt [3]{a+b x^2}}+\frac {15 x}{16 a^2 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}-\frac {15 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}\right )|-7+4 \sqrt {3}\right )}{32 a^{5/3} b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}+\frac {5\ 3^{3/4} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}\right ),-7+4 \sqrt {3}\right )}{8 \sqrt {2} a^{5/3} b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}} \] Output:

3/8*x/a/(b*x^2+a)^(4/3)+15/16*x/a^2/(b*x^2+a)^(1/3)+15/16*x/a^2/((1-3^(1/2 
))*a^(1/3)-(b*x^2+a)^(1/3))-15/32*3^(1/4)*(1/2*6^(1/2)+1/2*2^(1/2))*(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))/a^( 
5/3)/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)+5/16*3^(3/4)*(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))*2^(1/2)/a^(5/3)/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 = 5.26 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.13 \[ \int \frac {1}{\left (a+b x^2\right )^{7/3}} \, dx=\frac {3 x \left (7 a+5 b x^2\right )-5 x \left (a+b x^2\right ) \sqrt [3]{1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},-\frac {b x^2}{a}\right )}{16 a^2 \left (a+b x^2\right )^{4/3}} \] Input:

Integrate[(a + b*x^2)^(-7/3),x]
 

Output:

(3*x*(7*a + 5*b*x^2) - 5*x*(a + b*x^2)*(1 + (b*x^2)/a)^(1/3)*Hypergeometri 
c2F1[1/3, 1/2, 3/2, -((b*x^2)/a)])/(16*a^2*(a + b*x^2)^(4/3))
 

Rubi [A] (warning: unable to verify)

Time = 0.46 (sec) , antiderivative size = 624, normalized size of antiderivative = 1.09, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {215, 215, 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.

\(\displaystyle \int \frac {1}{\left (a+b x^2\right )^{7/3}} \, dx\)

\(\Big \downarrow \) 215

\(\displaystyle \frac {5 \int \frac {1}{\left (b x^2+a\right )^{4/3}}dx}{8 a}+\frac {3 x}{8 a \left (a+b x^2\right )^{4/3}}\)

\(\Big \downarrow \) 215

\(\displaystyle \frac {5 \left (\frac {3 x}{2 a \sqrt [3]{a+b x^2}}-\frac {\int \frac {1}{\sqrt [3]{b x^2+a}}dx}{2 a}\right )}{8 a}+\frac {3 x}{8 a \left (a+b x^2\right )^{4/3}}\)

\(\Big \downarrow \) 233

\(\displaystyle \frac {5 \left (\frac {3 x}{2 a \sqrt [3]{a+b x^2}}-\frac {3 \sqrt {b x^2} \int \frac {\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}}{4 a b x}\right )}{8 a}+\frac {3 x}{8 a \left (a+b x^2\right )^{4/3}}\)

\(\Big \downarrow \) 833

\(\displaystyle \frac {5 \left (\frac {3 x}{2 a \sqrt [3]{a+b x^2}}-\frac {3 \sqrt {b x^2} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} \int \frac {1}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}\right )}{4 a b x}\right )}{8 a}+\frac {3 x}{8 a \left (a+b x^2\right )^{4/3}}\)

\(\Big \downarrow \) 760

\(\displaystyle \frac {5 \left (\frac {3 x}{2 a \sqrt [3]{a+b x^2}}-\frac {3 \sqrt {b x^2} \left (-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}\right )}{4 a b x}\right )}{8 a}+\frac {3 x}{8 a \left (a+b x^2\right )^{4/3}}\)

\(\Big \downarrow \) 2418

\(\displaystyle \frac {5 \left (\frac {3 x}{2 a \sqrt [3]{a+b x^2}}-\frac {3 \sqrt {b x^2} \left (-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}+\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right )|-7+4 \sqrt {3}\right )}{\sqrt {b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}-\frac {2 \sqrt {b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}\right )}{4 a b x}\right )}{8 a}+\frac {3 x}{8 a \left (a+b x^2\right )^{4/3}}\)

Input:

Int[(a + b*x^2)^(-7/3),x]
 

Output:

(3*x)/(8*a*(a + b*x^2)^(4/3)) + (5*((3*x)/(2*a*(a + b*x^2)^(1/3)) - (3*Sqr 
t[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]* 
EllipticF[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]*Sqr 
t[-((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)])))/(4*a*b*x)))/(8*a)
 

Defintions of rubi rules used

rule 215

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) 
/(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1))   Int[(a + b*x^2)^(p + 1 
), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (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]

\[\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {7}{3}}}d x\]

Input:

int(1/(b*x^2+a)^(7/3),x)
 

Output:

int(1/(b*x^2+a)^(7/3),x)
 

Fricas [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{7/3}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {7}{3}}} \,d x } \] Input:

integrate(1/(b*x^2+a)^(7/3),x, algorithm="fricas")
 

Output:

integral((b*x^2 + a)^(2/3)/(b^3*x^6 + 3*a*b^2*x^4 + 3*a^2*b*x^2 + a^3), x)
 

Sympy [A] (verification not implemented)

Time = 0.61 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.04 \[ \int \frac {1}{\left (a+b x^2\right )^{7/3}} \, dx=\frac {x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{3} \\ \frac {3}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{a^{\frac {7}{3}}} \] Input:

integrate(1/(b*x**2+a)**(7/3),x)
 

Output:

x*hyper((1/2, 7/3), (3/2,), b*x**2*exp_polar(I*pi)/a)/a**(7/3)
 

Maxima [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{7/3}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {7}{3}}} \,d x } \] Input:

integrate(1/(b*x^2+a)^(7/3),x, algorithm="maxima")
 

Output:

integrate((b*x^2 + a)^(-7/3), x)
 

Giac [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{7/3}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {7}{3}}} \,d x } \] Input:

integrate(1/(b*x^2+a)^(7/3),x, algorithm="giac")
 

Output:

integrate((b*x^2 + a)^(-7/3), x)
 

Mupad [B] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.06 \[ \int \frac {1}{\left (a+b x^2\right )^{7/3}} \, dx=\frac {x\,{\left (\frac {b\,x^2}{a}+1\right )}^{7/3}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{3};\ \frac {3}{2};\ -\frac {b\,x^2}{a}\right )}{{\left (b\,x^2+a\right )}^{7/3}} \] Input:

int(1/(a + b*x^2)^(7/3),x)
 

Output:

(x*((b*x^2)/a + 1)^(7/3)*hypergeom([1/2, 7/3], 3/2, -(b*x^2)/a))/(a + b*x^ 
2)^(7/3)
 

Reduce [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{7/3}} \, dx=\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {1}{3}} a^{2}+2 \left (b \,x^{2}+a \right )^{\frac {1}{3}} a b \,x^{2}+\left (b \,x^{2}+a \right )^{\frac {1}{3}} b^{2} x^{4}}d x \] Input:

int(1/(b*x^2+a)^(7/3),x)
 

Output:

int(1/((a + b*x**2)**(1/3)*a**2 + 2*(a + b*x**2)**(1/3)*a*b*x**2 + (a + b* 
x**2)**(1/3)*b**2*x**4),x)