\(\int \frac {(a+b x)^{2/3}}{(a-b x)^{4/3}} \, dx\) [218]

Optimal result

Integrand size = 20, antiderivative size = 648 \[ \int \frac {(a+b x)^{2/3}}{(a-b x)^{4/3}} \, dx=\frac {3 (a+b x)^{2/3}}{b \sqrt [3]{a-b x}}+\frac {6 x \sqrt [3]{1-\frac {b^2 x^2}{a^2}}}{\sqrt [3]{a-b x} \sqrt [3]{a+b x} \left (1-\sqrt {3}-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}\right )}+\frac {3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^2 \sqrt [3]{1-\frac {b^2 x^2}{a^2}} \left (1-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}\right ) \sqrt {\frac {1+\sqrt [3]{1-\frac {b^2 x^2}{a^2}}+\left (1-\frac {b^2 x^2}{a^2}\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}\right )^2}} E\left (\arcsin \left (\frac {1+\sqrt {3}-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}}{1-\sqrt {3}-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}}\right )|-7+4 \sqrt {3}\right )}{b^2 x \sqrt [3]{a-b x} \sqrt [3]{a+b x} \sqrt {-\frac {1-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}}{\left (1-\sqrt {3}-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}\right )^2}}}-\frac {2 \sqrt {2} 3^{3/4} a^2 \sqrt [3]{1-\frac {b^2 x^2}{a^2}} \left (1-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}\right ) \sqrt {\frac {1+\sqrt [3]{1-\frac {b^2 x^2}{a^2}}+\left (1-\frac {b^2 x^2}{a^2}\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}}{1-\sqrt {3}-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}}\right ),-7+4 \sqrt {3}\right )}{b^2 x \sqrt [3]{a-b x} \sqrt [3]{a+b x} \sqrt {-\frac {1-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}}{\left (1-\sqrt {3}-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}\right )^2}}} \] Output:

3*(b*x+a)^(2/3)/b/(-b*x+a)^(1/3)+6*x*(1-b^2*x^2/a^2)^(1/3)/(-b*x+a)^(1/3)/ 
(b*x+a)^(1/3)/(1-3^(1/2)-(1-b^2*x^2/a^2)^(1/3))+3*3^(1/4)*(1/2*6^(1/2)+1/2 
*2^(1/2))*a^2*(1-b^2*x^2/a^2)^(1/3)*(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)*EllipticE((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)^(1/3)/(b*x+a)^(1/3)/(-(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)-2*2^(1/2)*3^(3/4) 
*a^2*(1-b^2*x^2/a^2)^(1/3)*(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)*Ell 
ipticF((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)^(1/3)/(b*x+a)^(1/3)/(-(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 = 65, normalized size of antiderivative = 0.10 \[ \int \frac {(a+b x)^{2/3}}{(a-b x)^{4/3}} \, dx=\frac {3\ 2^{2/3} (a+b x)^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{3},\frac {2}{3},\frac {a-b x}{2 a}\right )}{b \sqrt [3]{a-b x} \left (\frac {a+b x}{a}\right )^{2/3}} \] Input:

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

Output:

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

Rubi [C] (verified)

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

Time = 0.16 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.10, 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.

\(\displaystyle \int \frac {(a+b x)^{2/3}}{(a-b x)^{4/3}} \, dx\)

\(\Big \downarrow \) 80

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 79

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

Input:

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

Output:

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

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

Input:

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

Output:

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

Fricas [F]

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

integrate((b*x+a)^(2/3)/(-b*x+a)^(4/3),x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

integrate((b*x+a)**(2/3)/(-b*x+a)**(4/3),x)
 

Output:

Integral((a + b*x)**(2/3)/(a - b*x)**(4/3), x)
 

Maxima [F]

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

integrate((b*x+a)^(2/3)/(-b*x+a)^(4/3),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((b*x+a)^(2/3)/(-b*x+a)^(4/3),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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