3.27.34 \(\int \frac {1}{(-b x+a^2 x^2)^{3/2} (a x^2+x \sqrt {-b x+a^2 x^2})^{3/2}} \, dx\) [2634]

3.27.34.1 Optimal result

Integrand size = 45, antiderivative size = 232 \[ \int \frac {1}{\left (-b x+a^2 x^2\right )^{3/2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\frac {2 \sqrt {-b x+a^2 x^2} \left (210 b^3-200 a^2 b^2 x-456 a^4 b x^2+1601 a^6 x^3\right ) \sqrt {x \left (a x+\sqrt {-b x+a^2 x^2}\right )}}{1155 b^5 x^4 \left (b-a^2 x\right )}+\sqrt {x \left (a x+\sqrt {-b x+a^2 x^2}\right )} \left (-\frac {4 \left (245 a b^2+461 a^3 b x+2533 a^5 x^2\right )}{1155 b^5 x^3}-\frac {6 a^{11/2} \sqrt {-a x+\sqrt {-b x+a^2 x^2}} \arctan \left (\frac {\sqrt {a} \sqrt {-a x+\sqrt {-b x+a^2 x^2}}}{\sqrt {b}}\right )}{b^{11/2} x}\right ) \]

output

2/1155*(a^2*x^2-b*x)^(1/2)*(1601*a^6*x^3-456*a^4*b*x^2-200*a^2*b^2*x+210*b 
^3)*(x*(a*x+(a^2*x^2-b*x)^(1/2)))^(1/2)/b^5/x^4/(-a^2*x+b)+(x*(a*x+(a^2*x^ 
2-b*x)^(1/2)))^(1/2)*(-4/1155*(2533*a^5*x^2+461*a^3*b*x+245*a*b^2)/b^5/x^3 
-6*a^(11/2)*(-a*x+(a^2*x^2-b*x)^(1/2))^(1/2)*arctan(a^(1/2)*(-a*x+(a^2*x^2 
-b*x)^(1/2))^(1/2)/b^(1/2))/b^(11/2)/x)
 

3.27.34.2 Mathematica [A] (verified)

Time = 4.51 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.02 \[ \int \frac {1}{\left (-b x+a^2 x^2\right )^{3/2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=-\frac {2 \sqrt {x \left (a x+\sqrt {x \left (-b+a^2 x\right )}\right )} \left (\sqrt {b} \left (210 b^3+10 a b^2 \left (-20 a x+49 \sqrt {x \left (-b+a^2 x\right )}\right )+2 a^3 b x \left (-228 a x+461 \sqrt {x \left (-b+a^2 x\right )}\right )+a^5 x^2 \left (1601 a x+5066 \sqrt {x \left (-b+a^2 x\right )}\right )\right )+3465 a^{11/2} x^2 \sqrt {x \left (-b+a^2 x\right )} \sqrt {-a x+\sqrt {x \left (-b+a^2 x\right )}} \arctan \left (\frac {\sqrt {a} \sqrt {-a x+\sqrt {x \left (-b+a^2 x\right )}}}{\sqrt {b}}\right )\right )}{1155 b^{11/2} x^3 \sqrt {x \left (-b+a^2 x\right )}} \]

input

Integrate[1/((-(b*x) + a^2*x^2)^(3/2)*(a*x^2 + x*Sqrt[-(b*x) + a^2*x^2])^( 
3/2)),x]
 

output

(-2*Sqrt[x*(a*x + Sqrt[x*(-b + a^2*x)])]*(Sqrt[b]*(210*b^3 + 10*a*b^2*(-20 
*a*x + 49*Sqrt[x*(-b + a^2*x)]) + 2*a^3*b*x*(-228*a*x + 461*Sqrt[x*(-b + a 
^2*x)]) + a^5*x^2*(1601*a*x + 5066*Sqrt[x*(-b + a^2*x)])) + 3465*a^(11/2)* 
x^2*Sqrt[x*(-b + a^2*x)]*Sqrt[-(a*x) + Sqrt[x*(-b + a^2*x)]]*ArcTan[(Sqrt[ 
a]*Sqrt[-(a*x) + Sqrt[x*(-b + a^2*x)]])/Sqrt[b]]))/(1155*b^(11/2)*x^3*Sqrt 
[x*(-b + a^2*x)])
 

3.27.34.3 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.

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

\(\Big \downarrow \) 2467

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

\(\Big \downarrow \) 2035

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

\(\Big \downarrow \) 7299

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

input

Int[1/((-(b*x) + a^2*x^2)^(3/2)*(a*x^2 + x*Sqrt[-(b*x) + a^2*x^2])^(3/2)), 
x]
 

output

$Aborted
 

3.27.34.3.1 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]
 

3.27.34.4 Maple [F]

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

input

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

output

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

3.27.34.5 Fricas [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 438, normalized size of antiderivative = 1.89 \[ \int \frac {1}{\left (-b x+a^2 x^2\right )^{3/2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\left [\frac {3465 \, {\left (a^{7} x^{5} - a^{5} b x^{4}\right )} \sqrt {a} \log \left (\frac {a^{2} x^{2} + 2 \, \sqrt {a^{2} x^{2} - b x} a x - b x + 2 \, \sqrt {a^{2} x^{2} - b x} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x} \sqrt {a}}{a^{2} x^{2} - b x}\right ) - 2 \, {\left (5066 \, a^{7} x^{4} - 4144 \, a^{5} b x^{3} - 432 \, a^{3} b^{2} x^{2} - 490 \, a b^{3} x + {\left (1601 \, a^{6} x^{3} - 456 \, a^{4} b x^{2} - 200 \, a^{2} b^{2} x + 210 \, b^{3}\right )} \sqrt {a^{2} x^{2} - b x}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x}}{1155 \, {\left (a^{2} b^{5} x^{5} - b^{6} x^{4}\right )}}, -\frac {2 \, {\left (3465 \, {\left (a^{7} x^{5} - a^{5} b x^{4}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x} \sqrt {-a}}{a x}\right ) + {\left (5066 \, a^{7} x^{4} - 4144 \, a^{5} b x^{3} - 432 \, a^{3} b^{2} x^{2} - 490 \, a b^{3} x + {\left (1601 \, a^{6} x^{3} - 456 \, a^{4} b x^{2} - 200 \, a^{2} b^{2} x + 210 \, b^{3}\right )} \sqrt {a^{2} x^{2} - b x}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x}\right )}}{1155 \, {\left (a^{2} b^{5} x^{5} - b^{6} x^{4}\right )}}\right ] \]

input

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

output

[1/1155*(3465*(a^7*x^5 - a^5*b*x^4)*sqrt(a)*log((a^2*x^2 + 2*sqrt(a^2*x^2 
- b*x)*a*x - b*x + 2*sqrt(a^2*x^2 - b*x)*sqrt(a*x^2 + sqrt(a^2*x^2 - b*x)* 
x)*sqrt(a))/(a^2*x^2 - b*x)) - 2*(5066*a^7*x^4 - 4144*a^5*b*x^3 - 432*a^3* 
b^2*x^2 - 490*a*b^3*x + (1601*a^6*x^3 - 456*a^4*b*x^2 - 200*a^2*b^2*x + 21 
0*b^3)*sqrt(a^2*x^2 - b*x))*sqrt(a*x^2 + sqrt(a^2*x^2 - b*x)*x))/(a^2*b^5* 
x^5 - b^6*x^4), -2/1155*(3465*(a^7*x^5 - a^5*b*x^4)*sqrt(-a)*arctan(sqrt(a 
*x^2 + sqrt(a^2*x^2 - b*x)*x)*sqrt(-a)/(a*x)) + (5066*a^7*x^4 - 4144*a^5*b 
*x^3 - 432*a^3*b^2*x^2 - 490*a*b^3*x + (1601*a^6*x^3 - 456*a^4*b*x^2 - 200 
*a^2*b^2*x + 210*b^3)*sqrt(a^2*x^2 - b*x))*sqrt(a*x^2 + sqrt(a^2*x^2 - b*x 
)*x))/(a^2*b^5*x^5 - b^6*x^4)]
 

3.27.34.6 Sympy [F]

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

input

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

output

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

3.27.34.7 Maxima [F]

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

input

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

output

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

3.27.34.8 Giac [F]

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

input

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

output

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

3.27.34.9 Mupad [F(-1)]

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

input

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

output

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