\(\int \frac {\sqrt [5]{a-b x}}{(a+b x)^{9/5}} \, dx\) [285]

Optimal result

Integrand size = 20, antiderivative size = 66 \[ \int \frac {\sqrt [5]{a-b x}}{(a+b x)^{9/5}} \, dx=-\frac {5 \sqrt [5]{a-b x} \operatorname {Hypergeometric2F1}\left (-\frac {4}{5},-\frac {1}{5},\frac {1}{5},\frac {a+b x}{2 a}\right )}{2\ 2^{4/5} b (a+b x)^{4/5} \sqrt [5]{1-\frac {b x}{a}}} \] Output:

-5/4*(-b*x+a)^(1/5)*hypergeom([-4/5, -1/5],[1/5],1/2*(b*x+a)/a)*2^(1/5)/b/ 
(b*x+a)^(4/5)/(1-b*x/a)^(1/5)
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.06 \[ \int \frac {\sqrt [5]{a-b x}}{(a+b x)^{9/5}} \, dx=-\frac {5 (a-b x)^{6/5} \left (\frac {a+b x}{a}\right )^{4/5} \operatorname {Hypergeometric2F1}\left (\frac {6}{5},\frac {9}{5},\frac {11}{5},\frac {a-b x}{2 a}\right )}{12\ 2^{4/5} a b (a+b x)^{4/5}} \] Input:

Integrate[(a - b*x)^(1/5)/(a + b*x)^(9/5),x]
 

Output:

(-5*(a - b*x)^(6/5)*((a + b*x)/a)^(4/5)*Hypergeometric2F1[6/5, 9/5, 11/5, 
(a - b*x)/(2*a)])/(12*2^(4/5)*a*b*(a + b*x)^(4/5))
 

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.06, 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.

\(\displaystyle \int \frac {\sqrt [5]{a-b x}}{(a+b x)^{9/5}} \, dx\)

\(\Big \downarrow \) 80

\(\displaystyle \frac {\left (\frac {a+b x}{a}\right )^{4/5} \int \frac {2\ 2^{4/5} \sqrt [5]{a-b x}}{\left (\frac {b x}{a}+1\right )^{9/5}}dx}{2\ 2^{4/5} a (a+b x)^{4/5}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (\frac {a+b x}{a}\right )^{4/5} \int \frac {\sqrt [5]{a-b x}}{\left (\frac {b x}{a}+1\right )^{9/5}}dx}{a (a+b x)^{4/5}}\)

\(\Big \downarrow \) 79

\(\displaystyle -\frac {5 (a-b x)^{6/5} \left (\frac {a+b x}{a}\right )^{4/5} \operatorname {Hypergeometric2F1}\left (\frac {6}{5},\frac {9}{5},\frac {11}{5},\frac {a-b x}{2 a}\right )}{12\ 2^{4/5} a b (a+b x)^{4/5}}\)

Input:

Int[(a - b*x)^(1/5)/(a + b*x)^(9/5),x]
 

Output:

(-5*(a - b*x)^(6/5)*((a + b*x)/a)^(4/5)*Hypergeometric2F1[6/5, 9/5, 11/5, 
(a - b*x)/(2*a)])/(12*2^(4/5)*a*b*(a + b*x)^(4/5))
 

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]

\[\int \frac {\left (-b x +a \right )^{\frac {1}{5}}}{\left (b x +a \right )^{\frac {9}{5}}}d x\]

Input:

int((-b*x+a)^(1/5)/(b*x+a)^(9/5),x)
 

Output:

int((-b*x+a)^(1/5)/(b*x+a)^(9/5),x)
 

Fricas [F]

\[ \int \frac {\sqrt [5]{a-b x}}{(a+b x)^{9/5}} \, dx=\int { \frac {{\left (-b x + a\right )}^{\frac {1}{5}}}{{\left (b x + a\right )}^{\frac {9}{5}}} \,d x } \] Input:

integrate((-b*x+a)^(1/5)/(b*x+a)^(9/5),x, algorithm="fricas")
 

Output:

integral((b*x + a)^(1/5)*(-b*x + a)^(1/5)/(b^2*x^2 + 2*a*b*x + a^2), x)
 

Sympy [F]

\[ \int \frac {\sqrt [5]{a-b x}}{(a+b x)^{9/5}} \, dx=\int \frac {\sqrt [5]{a - b x}}{\left (a + b x\right )^{\frac {9}{5}}}\, dx \] Input:

integrate((-b*x+a)**(1/5)/(b*x+a)**(9/5),x)
 

Output:

Integral((a - b*x)**(1/5)/(a + b*x)**(9/5), x)
 

Maxima [F]

\[ \int \frac {\sqrt [5]{a-b x}}{(a+b x)^{9/5}} \, dx=\int { \frac {{\left (-b x + a\right )}^{\frac {1}{5}}}{{\left (b x + a\right )}^{\frac {9}{5}}} \,d x } \] Input:

integrate((-b*x+a)^(1/5)/(b*x+a)^(9/5),x, algorithm="maxima")
 

Output:

integrate((-b*x + a)^(1/5)/(b*x + a)^(9/5), x)
 

Giac [F]

\[ \int \frac {\sqrt [5]{a-b x}}{(a+b x)^{9/5}} \, dx=\int { \frac {{\left (-b x + a\right )}^{\frac {1}{5}}}{{\left (b x + a\right )}^{\frac {9}{5}}} \,d x } \] Input:

integrate((-b*x+a)^(1/5)/(b*x+a)^(9/5),x, algorithm="giac")
 

Output:

integrate((-b*x + a)^(1/5)/(b*x + a)^(9/5), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt [5]{a-b x}}{(a+b x)^{9/5}} \, dx=\int \frac {{\left (a-b\,x\right )}^{1/5}}{{\left (a+b\,x\right )}^{9/5}} \,d x \] Input:

int((a - b*x)^(1/5)/(a + b*x)^(9/5),x)
 

Output:

int((a - b*x)^(1/5)/(a + b*x)^(9/5), x)
 

Reduce [F]

\[ \int \frac {\sqrt [5]{a-b x}}{(a+b x)^{9/5}} \, dx=\frac {-5 \left (b x +a \right )^{\frac {1}{5}} \left (-b x +a \right )^{\frac {1}{5}}-2 \left (\int \frac {\left (b x +a \right )^{\frac {1}{5}} \left (-b x +a \right )^{\frac {1}{5}}}{-b^{3} x^{3}-a \,b^{2} x^{2}+a^{2} b x +a^{3}}d x \right ) a^{2} b -2 \left (\int \frac {\left (b x +a \right )^{\frac {1}{5}} \left (-b x +a \right )^{\frac {1}{5}}}{-b^{3} x^{3}-a \,b^{2} x^{2}+a^{2} b x +a^{3}}d x \right ) a \,b^{2} x}{3 b \left (b x +a \right )} \] Input:

int((-b*x+a)^(1/5)/(b*x+a)^(9/5),x)
 

Output:

( - 5*(a + b*x)**(1/5)*(a - b*x)**(1/5) - 2*int(((a + b*x)**(1/5)*(a - b*x 
)**(1/5))/(a**3 + a**2*b*x - a*b**2*x**2 - b**3*x**3),x)*a**2*b - 2*int((( 
a + b*x)**(1/5)*(a - b*x)**(1/5))/(a**3 + a**2*b*x - a*b**2*x**2 - b**3*x* 
*3),x)*a*b**2*x)/(3*b*(a + b*x))