3.27.1 \(\int \frac {x^2 \sqrt {-b x+a^2 x^2}}{(a x^2+x \sqrt {-b x+a^2 x^2})^{3/2}} \, dx\) [2601]

3.27.1.1 Optimal result

Integrand size = 48, antiderivative size = 225 \[ \int \frac {x^2 \sqrt {-b x+a^2 x^2}}{\left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\frac {\sqrt {-b x+a^2 x^2} \left (-15 b^2-104 a^2 b x+96 a^4 x^2\right ) \sqrt {x \left (a x+\sqrt {-b x+a^2 x^2}\right )}}{120 a^2 b^2 x}+\sqrt {x \left (a x+\sqrt {-b x+a^2 x^2}\right )} \left (\frac {5 b^2+152 a^2 b x-96 a^4 x^2}{120 a b^2}+\frac {\sqrt {b} \sqrt {-a x+\sqrt {-b x+a^2 x^2}} \arctan \left (\frac {\sqrt {2} \sqrt {a} \sqrt {-a x+\sqrt {-b x+a^2 x^2}}}{\sqrt {b}}\right )}{8 \sqrt {2} a^{5/2} x}\right ) \]

output

1/120*(a^2*x^2-b*x)^(1/2)*(96*a^4*x^2-104*a^2*b*x-15*b^2)*(x*(a*x+(a^2*x^2 
-b*x)^(1/2)))^(1/2)/a^2/b^2/x+(x*(a*x+(a^2*x^2-b*x)^(1/2)))^(1/2)*(1/120*( 
-96*a^4*x^2+152*a^2*b*x+5*b^2)/a/b^2+1/16*b^(1/2)*(-a*x+(a^2*x^2-b*x)^(1/2 
))^(1/2)*arctan(2^(1/2)*a^(1/2)*(-a*x+(a^2*x^2-b*x)^(1/2))^(1/2)/b^(1/2))* 
2^(1/2)/a^(5/2)/x)
 

3.27.1.2 Mathematica [A] (verified)

Time = 3.97 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.10 \[ \int \frac {x^2 \sqrt {-b x+a^2 x^2}}{\left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\frac {\sqrt {x \left (a x+\sqrt {x \left (-b+a^2 x\right )}\right )} \left (2 \sqrt {a} x \left (15 b^3+96 a^5 x^2 \left (a x-\sqrt {x \left (-b+a^2 x\right )}\right )+a b^2 \left (89 a x+5 \sqrt {x \left (-b+a^2 x\right )}\right )+8 a^3 b x \left (-25 a x+19 \sqrt {x \left (-b+a^2 x\right )}\right )\right )+15 \sqrt {2} b^{5/2} \sqrt {x \left (-b+a^2 x\right )} \sqrt {-a x+\sqrt {x \left (-b+a^2 x\right )}} \arctan \left (\frac {\sqrt {2} \sqrt {a} \sqrt {-a x+\sqrt {x \left (-b+a^2 x\right )}}}{\sqrt {b}}\right )\right )}{240 a^{5/2} b^2 x \sqrt {x \left (-b+a^2 x\right )}} \]

input

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

output

(Sqrt[x*(a*x + Sqrt[x*(-b + a^2*x)])]*(2*Sqrt[a]*x*(15*b^3 + 96*a^5*x^2*(a 
*x - Sqrt[x*(-b + a^2*x)]) + a*b^2*(89*a*x + 5*Sqrt[x*(-b + a^2*x)]) + 8*a 
^3*b*x*(-25*a*x + 19*Sqrt[x*(-b + a^2*x)])) + 15*Sqrt[2]*b^(5/2)*Sqrt[x*(- 
b + a^2*x)]*Sqrt[-(a*x) + Sqrt[x*(-b + a^2*x)]]*ArcTan[(Sqrt[2]*Sqrt[a]*Sq 
rt[-(a*x) + Sqrt[x*(-b + a^2*x)]])/Sqrt[b]]))/(240*a^(5/2)*b^2*x*Sqrt[x*(- 
b + a^2*x)])
 

3.27.1.3 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.

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

\(\Big \downarrow \) 2467

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

\(\Big \downarrow \) 2035

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

\(\Big \downarrow \) 7299

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

input

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

output

$Aborted
 

3.27.1.3.1 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]
 

3.27.1.4 Maple [F]

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

input

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

output

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

3.27.1.5 Fricas [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.63 \[ \int \frac {x^2 \sqrt {-b x+a^2 x^2}}{\left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\left [\frac {15 \, \sqrt {2} \sqrt {a} b^{3} x \log \left (-\frac {4 \, a^{2} x^{2} + 4 \, \sqrt {a^{2} x^{2} - b x} a x - b x - 2 \, {\left (\sqrt {2} a^{\frac {3}{2}} x + \sqrt {2} \sqrt {a^{2} x^{2} - b x} \sqrt {a}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x}}{x}\right ) - 4 \, {\left (96 \, a^{6} x^{3} - 152 \, a^{4} b x^{2} - 5 \, a^{2} b^{2} x - {\left (96 \, a^{5} x^{2} - 104 \, a^{3} b x - 15 \, a b^{2}\right )} \sqrt {a^{2} x^{2} - b x}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x}}{480 \, a^{3} b^{2} x}, \frac {15 \, \sqrt {2} \sqrt {-a} b^{3} x \arctan \left (\frac {\sqrt {2} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x} \sqrt {-a}}{2 \, a x}\right ) - 2 \, {\left (96 \, a^{6} x^{3} - 152 \, a^{4} b x^{2} - 5 \, a^{2} b^{2} x - {\left (96 \, a^{5} x^{2} - 104 \, a^{3} b x - 15 \, a b^{2}\right )} \sqrt {a^{2} x^{2} - b x}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x}}{240 \, a^{3} b^{2} x}\right ] \]

input

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

output

[1/480*(15*sqrt(2)*sqrt(a)*b^3*x*log(-(4*a^2*x^2 + 4*sqrt(a^2*x^2 - b*x)*a 
*x - b*x - 2*(sqrt(2)*a^(3/2)*x + sqrt(2)*sqrt(a^2*x^2 - b*x)*sqrt(a))*sqr 
t(a*x^2 + sqrt(a^2*x^2 - b*x)*x))/x) - 4*(96*a^6*x^3 - 152*a^4*b*x^2 - 5*a 
^2*b^2*x - (96*a^5*x^2 - 104*a^3*b*x - 15*a*b^2)*sqrt(a^2*x^2 - b*x))*sqrt 
(a*x^2 + sqrt(a^2*x^2 - b*x)*x))/(a^3*b^2*x), 1/240*(15*sqrt(2)*sqrt(-a)*b 
^3*x*arctan(1/2*sqrt(2)*sqrt(a*x^2 + sqrt(a^2*x^2 - b*x)*x)*sqrt(-a)/(a*x) 
) - 2*(96*a^6*x^3 - 152*a^4*b*x^2 - 5*a^2*b^2*x - (96*a^5*x^2 - 104*a^3*b* 
x - 15*a*b^2)*sqrt(a^2*x^2 - b*x))*sqrt(a*x^2 + sqrt(a^2*x^2 - b*x)*x))/(a 
^3*b^2*x)]
 

3.27.1.6 Sympy [F]

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

input

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

output

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

3.27.1.7 Maxima [F]

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

input

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

output

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

3.27.1.8 Giac [F]

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

input

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

output

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

3.27.1.9 Mupad [F(-1)]

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

input

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

output

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