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

3.21.59.1 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)
 

3.21.59.2 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]
 

3.21.59.3 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
 

3.21.59.3.1 Defintions of rubi rules used

rule 2035

Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k   Subst 
[Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti 
onQ[m] && AlgebraicFunctionQ[Fx, x]
 

rule 2467

Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F 
racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p])   Int[x^( 
p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P 
olyQ[Px, x] &&  !IntegerQ[p] &&  !MonomialQ[Px, x] &&  !PolyQ[Fx, x]
 

rule 7299

Int[u_, x_] :> CannotIntegrate[u, x]
 

3.21.59.4 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)
 

3.21.59.5 Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.28 (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
 

3.21.59.6 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)
 

3.21.59.7 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)
 

3.21.59.8 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
 

3.21.59.9 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)