\(\int \frac {1}{(a-b x)^{2/3} (a+b x)^{8/3}} \, dx\) [209]

Optimal result

Integrand size = 20, antiderivative size = 340 \[ \int \frac {1}{(a-b x)^{2/3} (a+b x)^{8/3}} \, dx=-\frac {3 \sqrt [3]{a-b x}}{10 a b (a+b x)^{5/3}}-\frac {3 \sqrt [3]{a-b x}}{10 a^2 b (a+b x)^{2/3}}+\frac {3^{3/4} \sqrt {2-\sqrt {3}} \left (1-\frac {b^2 x^2}{a^2}\right )^{2/3} \left (1-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}\right ) \sqrt {\frac {1+\sqrt [3]{1-\frac {b^2 x^2}{a^2}}+\left (1-\frac {b^2 x^2}{a^2}\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}}{1-\sqrt {3}-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}}\right ),-7+4 \sqrt {3}\right )}{10 b^2 x (a-b x)^{2/3} (a+b x)^{2/3} \sqrt {-\frac {1-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}}{\left (1-\sqrt {3}-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}\right )^2}}} \] Output:

-3/10*(-b*x+a)^(1/3)/a/b/(b*x+a)^(5/3)-3/10*(-b*x+a)^(1/3)/a^2/b/(b*x+a)^( 
2/3)+1/10*3^(3/4)*(1/2*6^(1/2)-1/2*2^(1/2))*(1-b^2*x^2/a^2)^(2/3)*(1-(1-b^ 
2*x^2/a^2)^(1/3))*((1+(1-b^2*x^2/a^2)^(1/3)+(1-b^2*x^2/a^2)^(2/3))/(1-3^(1 
/2)-(1-b^2*x^2/a^2)^(1/3))^2)^(1/2)*EllipticF((1+3^(1/2)-(1-b^2*x^2/a^2)^( 
1/3))/(1-3^(1/2)-(1-b^2*x^2/a^2)^(1/3)),2*I-I*3^(1/2))/b^2/x/(-b*x+a)^(2/3 
)/(b*x+a)^(2/3)/(-(1-(1-b^2*x^2/a^2)^(1/3))/(1-3^(1/2)-(1-b^2*x^2/a^2)^(1/ 
3))^2)^(1/2)
 

Mathematica [C] (verified)

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

Time = 0.01 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.21 \[ \int \frac {1}{(a-b x)^{2/3} (a+b x)^{8/3}} \, dx=-\frac {3 \sqrt [3]{a-b x} \left (\frac {a+b x}{a}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {8}{3},\frac {4}{3},\frac {a-b x}{2 a}\right )}{4\ 2^{2/3} a^2 b (a+b x)^{2/3}} \] Input:

Integrate[1/((a - b*x)^(2/3)*(a + b*x)^(8/3)),x]
 

Output:

(-3*(a - b*x)^(1/3)*((a + b*x)/a)^(2/3)*Hypergeometric2F1[1/3, 8/3, 4/3, ( 
a - b*x)/(2*a)])/(4*2^(2/3)*a^2*b*(a + b*x)^(2/3))
 

Rubi [C] (verified)

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

Time = 0.17 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.21, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {80, 27, 79}

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.

\(\displaystyle \int \frac {1}{(a-b x)^{2/3} (a+b x)^{8/3}} \, dx\)

\(\Big \downarrow \) 80

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 79

\(\displaystyle -\frac {3 \sqrt [3]{a-b x} \left (\frac {a+b x}{a}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {8}{3},\frac {4}{3},\frac {a-b x}{2 a}\right )}{4\ 2^{2/3} a^2 b (a+b x)^{2/3}}\)

Input:

Int[1/((a - b*x)^(2/3)*(a + b*x)^(8/3)),x]
 

Output:

(-3*(a - b*x)^(1/3)*((a + b*x)/a)^(2/3)*Hypergeometric2F1[1/3, 8/3, 4/3, ( 
a - b*x)/(2*a)])/(4*2^(2/3)*a^2*b*(a + b*x)^(2/3))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 79

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( 
a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 
, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] 
&&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] 
 ||  !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
 

rule 80

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c 
 + d*x)^FracPart[n]/((b/(b*c - a*d))^IntPart[n]*(b*((c + d*x)/(b*c - a*d))) 
^FracPart[n])   Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d) 
), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] &&  !IntegerQ[m] &&  !Integ 
erQ[n] && (RationalQ[m] ||  !SimplerQ[n + 1, m + 1])
 

Maple [F]

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

Input:

int(1/(-b*x+a)^(2/3)/(b*x+a)^(8/3),x)
 

Output:

int(1/(-b*x+a)^(2/3)/(b*x+a)^(8/3),x)
 

Fricas [F]

\[ \int \frac {1}{(a-b x)^{2/3} (a+b x)^{8/3}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {8}{3}} {\left (-b x + a\right )}^{\frac {2}{3}}} \,d x } \] Input:

integrate(1/(-b*x+a)^(2/3)/(b*x+a)^(8/3),x, algorithm="fricas")
 

Output:

integral(-(b*x + a)^(1/3)*(-b*x + a)^(1/3)/(b^4*x^4 + 2*a*b^3*x^3 - 2*a^3* 
b*x - a^4), x)
 

Sympy [F]

\[ \int \frac {1}{(a-b x)^{2/3} (a+b x)^{8/3}} \, dx=\int \frac {1}{\left (a - b x\right )^{\frac {2}{3}} \left (a + b x\right )^{\frac {8}{3}}}\, dx \] Input:

integrate(1/(-b*x+a)**(2/3)/(b*x+a)**(8/3),x)
 

Output:

Integral(1/((a - b*x)**(2/3)*(a + b*x)**(8/3)), x)
 

Maxima [F]

\[ \int \frac {1}{(a-b x)^{2/3} (a+b x)^{8/3}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {8}{3}} {\left (-b x + a\right )}^{\frac {2}{3}}} \,d x } \] Input:

integrate(1/(-b*x+a)^(2/3)/(b*x+a)^(8/3),x, algorithm="maxima")
 

Output:

integrate(1/((b*x + a)^(8/3)*(-b*x + a)^(2/3)), x)
 

Giac [F]

\[ \int \frac {1}{(a-b x)^{2/3} (a+b x)^{8/3}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {8}{3}} {\left (-b x + a\right )}^{\frac {2}{3}}} \,d x } \] Input:

integrate(1/(-b*x+a)^(2/3)/(b*x+a)^(8/3),x, algorithm="giac")
 

Output:

integrate(1/((b*x + a)^(8/3)*(-b*x + a)^(2/3)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a-b x)^{2/3} (a+b x)^{8/3}} \, dx=\int \frac {1}{{\left (a+b\,x\right )}^{8/3}\,{\left (a-b\,x\right )}^{2/3}} \,d x \] Input:

int(1/((a + b*x)^(8/3)*(a - b*x)^(2/3)),x)
 

Output:

int(1/((a + b*x)^(8/3)*(a - b*x)^(2/3)), x)
 

Reduce [F]

\[ \int \frac {1}{(a-b x)^{2/3} (a+b x)^{8/3}} \, dx=\int \frac {1}{\left (b x +a \right )^{\frac {2}{3}} \left (-b x +a \right )^{\frac {2}{3}} a^{2}+2 \left (b x +a \right )^{\frac {2}{3}} \left (-b x +a \right )^{\frac {2}{3}} a b x +\left (b x +a \right )^{\frac {2}{3}} \left (-b x +a \right )^{\frac {2}{3}} b^{2} x^{2}}d x \] Input:

int(1/(-b*x+a)^(2/3)/(b*x+a)^(8/3),x)
 

Output:

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