\(\int \frac {(a x+\sqrt {-b x+a^2 x^2})^{3/4}}{\sqrt {-b x+a^2 x^2}} \, dx\) [2059]

Optimal result

Integrand size = 41, antiderivative size = 147 \[ \int \frac {\left (a x+\sqrt {-b x+a^2 x^2}\right )^{3/4}}{\sqrt {-b x+a^2 x^2}} \, dx=\frac {4 \left (a x+\sqrt {-b x+a^2 x^2}\right )^{3/4}}{3 a}+\frac {\sqrt [4]{2} b^{3/4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{a x+\sqrt {-b x+a^2 x^2}}}{\sqrt [4]{b}}\right )}{a^{7/4}}-\frac {\sqrt [4]{2} b^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{a x+\sqrt {-b x+a^2 x^2}}}{\sqrt [4]{b}}\right )}{a^{7/4}} \] Output:

4/3*(a*x+(a^2*x^2-b*x)^(1/2))^(3/4)/a+2^(1/4)*b^(3/4)*arctan(2^(1/4)*a^(1/ 
4)*(a*x+(a^2*x^2-b*x)^(1/2))^(1/4)/b^(1/4))/a^(7/4)-2^(1/4)*b^(3/4)*arctan 
h(2^(1/4)*a^(1/4)*(a*x+(a^2*x^2-b*x)^(1/2))^(1/4)/b^(1/4))/a^(7/4)
 

Mathematica [F]

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

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

Output:

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

Rubi [F]

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.

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

\(\Big \downarrow \) 2467

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

\(\Big \downarrow \) 2035

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

\(\Big \downarrow \) 7299

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

Input:

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

Output:

$Aborted
 

Maple [F]

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

Input:

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

Output:

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.09 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.77 \[ \int \frac {\left (a x+\sqrt {-b x+a^2 x^2}\right )^{3/4}}{\sqrt {-b x+a^2 x^2}} \, dx=-\frac {3 \, \left (\frac {1}{8}\right )^{\frac {1}{4}} a \left (\frac {b^{3}}{a^{7}}\right )^{\frac {1}{4}} \log \left (4 \, \left (\frac {1}{8}\right )^{\frac {3}{4}} a^{5} \left (\frac {b^{3}}{a^{7}}\right )^{\frac {3}{4}} + {\left (a x + \sqrt {a^{2} x^{2} - b x}\right )}^{\frac {1}{4}} b^{2}\right ) - 3 i \, \left (\frac {1}{8}\right )^{\frac {1}{4}} a \left (\frac {b^{3}}{a^{7}}\right )^{\frac {1}{4}} \log \left (4 i \, \left (\frac {1}{8}\right )^{\frac {3}{4}} a^{5} \left (\frac {b^{3}}{a^{7}}\right )^{\frac {3}{4}} + {\left (a x + \sqrt {a^{2} x^{2} - b x}\right )}^{\frac {1}{4}} b^{2}\right ) + 3 i \, \left (\frac {1}{8}\right )^{\frac {1}{4}} a \left (\frac {b^{3}}{a^{7}}\right )^{\frac {1}{4}} \log \left (-4 i \, \left (\frac {1}{8}\right )^{\frac {3}{4}} a^{5} \left (\frac {b^{3}}{a^{7}}\right )^{\frac {3}{4}} + {\left (a x + \sqrt {a^{2} x^{2} - b x}\right )}^{\frac {1}{4}} b^{2}\right ) - 3 \, \left (\frac {1}{8}\right )^{\frac {1}{4}} a \left (\frac {b^{3}}{a^{7}}\right )^{\frac {1}{4}} \log \left (-4 \, \left (\frac {1}{8}\right )^{\frac {3}{4}} a^{5} \left (\frac {b^{3}}{a^{7}}\right )^{\frac {3}{4}} + {\left (a x + \sqrt {a^{2} x^{2} - b x}\right )}^{\frac {1}{4}} b^{2}\right ) - 4 \, {\left (a x + \sqrt {a^{2} x^{2} - b x}\right )}^{\frac {3}{4}}}{3 \, a} \] Input:

integrate((a*x+(a^2*x^2-b*x)^(1/2))^(3/4)/(a^2*x^2-b*x)^(1/2),x, algorithm 
="fricas")
 

Output:

-1/3*(3*(1/8)^(1/4)*a*(b^3/a^7)^(1/4)*log(4*(1/8)^(3/4)*a^5*(b^3/a^7)^(3/4 
) + (a*x + sqrt(a^2*x^2 - b*x))^(1/4)*b^2) - 3*I*(1/8)^(1/4)*a*(b^3/a^7)^( 
1/4)*log(4*I*(1/8)^(3/4)*a^5*(b^3/a^7)^(3/4) + (a*x + sqrt(a^2*x^2 - b*x)) 
^(1/4)*b^2) + 3*I*(1/8)^(1/4)*a*(b^3/a^7)^(1/4)*log(-4*I*(1/8)^(3/4)*a^5*( 
b^3/a^7)^(3/4) + (a*x + sqrt(a^2*x^2 - b*x))^(1/4)*b^2) - 3*(1/8)^(1/4)*a* 
(b^3/a^7)^(1/4)*log(-4*(1/8)^(3/4)*a^5*(b^3/a^7)^(3/4) + (a*x + sqrt(a^2*x 
^2 - b*x))^(1/4)*b^2) - 4*(a*x + sqrt(a^2*x^2 - b*x))^(3/4))/a
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((a*x+(a^2*x^2-b*x)^(1/2))^(3/4)/(a^2*x^2-b*x)^(1/2),x, algorithm 
="maxima")
 

Output:

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

Giac [F(-1)]

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

integrate((a*x+(a^2*x^2-b*x)^(1/2))^(3/4)/(a^2*x^2-b*x)^(1/2),x, algorithm 
="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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