\(\int \frac {1}{(a+b x)^{5/3} (c+d x)^{2/3}} \, dx\) [552]

Optimal result

Integrand size = 19, antiderivative size = 58 \[ \int \frac {1}{(a+b x)^{5/3} (c+d x)^{2/3}} \, dx=-\frac {3 \sqrt [3]{c+d x} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},1,\frac {1}{3},-\frac {d (a+b x)}{b c-a d}\right )}{2 (b c-a d) (a+b x)^{2/3}} \] Output:

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.26 \[ \int \frac {1}{(a+b x)^{5/3} (c+d x)^{2/3}} \, dx=-\frac {3 \left (\frac {b (c+d x)}{b c-a d}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {2}{3},\frac {1}{3},\frac {d (a+b x)}{-b c+a d}\right )}{2 b (a+b x)^{2/3} (c+d x)^{2/3}} \] Input:

Integrate[1/((a + b*x)^(5/3)*(c + d*x)^(2/3)),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.28, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {80, 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.

\(\displaystyle \int \frac {1}{(a+b x)^{5/3} (c+d x)^{2/3}} \, dx\)

\(\Big \downarrow \) 80

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

\(\Big \downarrow \) 79

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

Input:

Int[1/((a + b*x)^(5/3)*(c + d*x)^(2/3)),x]
 

Output:

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

Defintions of rubi rules used

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]

\[\int \frac {1}{\left (b x +a \right )^{\frac {5}{3}} \left (x d +c \right )^{\frac {2}{3}}}d x\]

Input:

int(1/(b*x+a)^(5/3)/(d*x+c)^(2/3),x)
 

Output:

int(1/(b*x+a)^(5/3)/(d*x+c)^(2/3),x)
 

Fricas [F]

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

integrate(1/(b*x+a)^(5/3)/(d*x+c)^(2/3),x, algorithm="fricas")
 

Output:

integral((b*x + a)^(1/3)*(d*x + c)^(1/3)/(b^2*d*x^3 + a^2*c + (b^2*c + 2*a 
*b*d)*x^2 + (2*a*b*c + a^2*d)*x), x)
 

Sympy [F]

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

integrate(1/(b*x+a)**(5/3)/(d*x+c)**(2/3),x)
 

Output:

Integral(1/((a + b*x)**(5/3)*(c + d*x)**(2/3)), x)
 

Maxima [F]

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

integrate(1/(b*x+a)^(5/3)/(d*x+c)^(2/3),x, algorithm="maxima")
 

Output:

integrate(1/((b*x + a)^(5/3)*(d*x + c)^(2/3)), x)
 

Giac [F]

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

integrate(1/(b*x+a)^(5/3)/(d*x+c)^(2/3),x, algorithm="giac")
 

Output:

integrate(1/((b*x + a)^(5/3)*(d*x + c)^(2/3)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+b x)^{5/3} (c+d x)^{2/3}} \, dx=\int \frac {1}{{\left (a+b\,x\right )}^{5/3}\,{\left (c+d\,x\right )}^{2/3}} \,d x \] Input:

int(1/((a + b*x)^(5/3)*(c + d*x)^(2/3)),x)
 

Output:

int(1/((a + b*x)^(5/3)*(c + d*x)^(2/3)), x)
 

Reduce [F]

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

int(1/(b*x+a)^(5/3)/(d*x+c)^(2/3),x)
 

Output:

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