\(\int \frac {-b^3+a^3 x^3}{\sqrt {b^2 x+a^2 x^3} (b^3+a^3 x^3)} \, dx\) [1810]

Optimal result

Integrand size = 44, antiderivative size = 123 \[ \int \frac {-b^3+a^3 x^3}{\sqrt {b^2 x+a^2 x^3} \left (b^3+a^3 x^3\right )} \, dx=-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right )}{3 \sqrt {a} \sqrt {b}}-\frac {4 \text {arctanh}\left (\frac {\sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right )}{3 \sqrt {a} \sqrt {b}} \] Output:

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

Mathematica [A] (verified)

Time = 0.92 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.05 \[ \int \frac {-b^3+a^3 x^3}{\sqrt {b^2 x+a^2 x^3} \left (b^3+a^3 x^3\right )} \, dx=-\frac {\sqrt {x} \sqrt {b^2+a^2 x^2} \left (\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )+4 \text {arctanh}\left (\frac {\sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )\right )}{3 \sqrt {a} \sqrt {b} \sqrt {x \left (b^2+a^2 x^2\right )}} \] Input:

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

Output:

-1/3*(Sqrt[x]*Sqrt[b^2 + a^2*x^2]*(Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[a]*Sqrt[b] 
*Sqrt[x])/Sqrt[b^2 + a^2*x^2]] + 4*ArcTanh[(Sqrt[a]*Sqrt[b]*Sqrt[x])/Sqrt[ 
b^2 + a^2*x^2]]))/(Sqrt[a]*Sqrt[b]*Sqrt[x*(b^2 + a^2*x^2)])
 

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 6.87 (sec) , antiderivative size = 1261, normalized size of antiderivative = 10.25, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {2467, 25, 2035, 7276, 2009}

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.

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

\(\Big \downarrow \) 2467

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2035

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 \sqrt {x} \sqrt {b^2+a^2 x^2} \left (-\frac {(-1)^{5/6} \sqrt {a} \arctan \left (\frac {\sqrt [6]{-1} \sqrt [6]{-a^3} \sqrt {a+\sqrt [3]{-1} \sqrt [3]{-a^3}} \sqrt {b} \sqrt {x}}{\sqrt {a} \sqrt {b^2+a^2 x^2}}\right )}{3 \sqrt [6]{-a^3} \sqrt {a+\sqrt [3]{-1} \sqrt [3]{-a^3}} \sqrt {b}}-\frac {\sqrt [6]{-1} \sqrt [6]{-a^3} \left ((-1)^{2/3} a^2-\left (-a^3\right )^{2/3}\right ) \arctan \left (\frac {\sqrt [6]{-1} \sqrt {a^2-\sqrt [3]{-1} \left (-a^3\right )^{2/3}} \sqrt {b} \sqrt {x}}{\sqrt [6]{-a^3} \sqrt {b^2+a^2 x^2}}\right )}{3 \sqrt {a^2-\sqrt [3]{-1} \left (-a^3\right )^{2/3}} \left (a^2+\sqrt [3]{-1} \left (-a^3\right )^{2/3}\right ) \sqrt {b}}+\frac {\sqrt [6]{-a^3} \text {arctanh}\left (\frac {\sqrt {a^2+\left (-a^3\right )^{2/3}} \sqrt {b} \sqrt {x}}{\sqrt [6]{-a^3} \sqrt {b^2+a^2 x^2}}\right )}{3 \sqrt {a^2+\left (-a^3\right )^{2/3}} \sqrt {b}}-\frac {(b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{2 \sqrt {a} \sqrt {b} \sqrt {b^2+a^2 x^2}}+\frac {\sqrt {a} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{3 \left (a+\sqrt [3]{-a^3}\right ) \sqrt {b} \sqrt {b^2+a^2 x^2}}+\frac {\sqrt {a} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{3 \left (a-\sqrt [3]{-1} \sqrt [3]{-a^3}\right ) \sqrt {b} \sqrt {b^2+a^2 x^2}}+\frac {\sqrt {a} \left (a-(-1)^{2/3} \sqrt [3]{-a^3}\right ) (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{3 \left (a^2+\sqrt [3]{-1} \left (-a^3\right )^{2/3}\right ) \sqrt {b} \sqrt {b^2+a^2 x^2}}-\frac {\left (a-\sqrt [3]{-a^3}\right ) (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticPi}\left (\frac {\left (a+\sqrt [3]{-a^3}\right )^2}{4 a \sqrt [3]{-a^3}},2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{6 \sqrt {a} \left (a+\sqrt [3]{-a^3}\right ) \sqrt {b} \sqrt {b^2+a^2 x^2}}-\frac {\left (a+\sqrt [3]{-1} \sqrt [3]{-a^3}\right ) (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticPi}\left (\frac {(-1)^{2/3} \left (a-\sqrt [3]{-1} \sqrt [3]{-a^3}\right )^2}{4 a \sqrt [3]{-a^3}},2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{6 \sqrt {a} \left (a-\sqrt [3]{-1} \sqrt [3]{-a^3}\right ) \sqrt {b} \sqrt {b^2+a^2 x^2}}-\frac {\left (a^2-2 (-1)^{2/3} \sqrt [3]{-a^3} a-\sqrt [3]{-1} \left (-a^3\right )^{2/3}\right ) (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticPi}\left (\frac {\sqrt [3]{-1} a^2 \left (a+(-1)^{2/3} \sqrt [3]{-a^3}\right )^2}{4 \left (-a^3\right )^{4/3}},2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{6 \sqrt {a} \left (a^2+\sqrt [3]{-1} \left (-a^3\right )^{2/3}\right ) \sqrt {b} \sqrt {b^2+a^2 x^2}}\right )}{\sqrt {a^2 x^3+b^2 x}}\)

Input:

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

Output:

(-2*Sqrt[x]*Sqrt[b^2 + a^2*x^2]*(-1/3*((-1)^(5/6)*Sqrt[a]*ArcTan[((-1)^(1/ 
6)*(-a^3)^(1/6)*Sqrt[a + (-1)^(1/3)*(-a^3)^(1/3)]*Sqrt[b]*Sqrt[x])/(Sqrt[a 
]*Sqrt[b^2 + a^2*x^2])])/((-a^3)^(1/6)*Sqrt[a + (-1)^(1/3)*(-a^3)^(1/3)]*S 
qrt[b]) - ((-1)^(1/6)*(-a^3)^(1/6)*((-1)^(2/3)*a^2 - (-a^3)^(2/3))*ArcTan[ 
((-1)^(1/6)*Sqrt[a^2 - (-1)^(1/3)*(-a^3)^(2/3)]*Sqrt[b]*Sqrt[x])/((-a^3)^( 
1/6)*Sqrt[b^2 + a^2*x^2])])/(3*Sqrt[a^2 - (-1)^(1/3)*(-a^3)^(2/3)]*(a^2 + 
(-1)^(1/3)*(-a^3)^(2/3))*Sqrt[b]) + ((-a^3)^(1/6)*ArcTanh[(Sqrt[a^2 + (-a^ 
3)^(2/3)]*Sqrt[b]*Sqrt[x])/((-a^3)^(1/6)*Sqrt[b^2 + a^2*x^2])])/(3*Sqrt[a^ 
2 + (-a^3)^(2/3)]*Sqrt[b]) - ((b + a*x)*Sqrt[(b^2 + a^2*x^2)/(b + a*x)^2]* 
EllipticF[2*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]], 1/2])/(2*Sqrt[a]*Sqrt[b]*Sq 
rt[b^2 + a^2*x^2]) + (Sqrt[a]*(b + a*x)*Sqrt[(b^2 + a^2*x^2)/(b + a*x)^2]* 
EllipticF[2*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]], 1/2])/(3*(a + (-a^3)^(1/3)) 
*Sqrt[b]*Sqrt[b^2 + a^2*x^2]) + (Sqrt[a]*(b + a*x)*Sqrt[(b^2 + a^2*x^2)/(b 
 + a*x)^2]*EllipticF[2*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]], 1/2])/(3*(a - (- 
1)^(1/3)*(-a^3)^(1/3))*Sqrt[b]*Sqrt[b^2 + a^2*x^2]) + (Sqrt[a]*(a - (-1)^( 
2/3)*(-a^3)^(1/3))*(b + a*x)*Sqrt[(b^2 + a^2*x^2)/(b + a*x)^2]*EllipticF[2 
*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]], 1/2])/(3*(a^2 + (-1)^(1/3)*(-a^3)^(2/3 
))*Sqrt[b]*Sqrt[b^2 + a^2*x^2]) - ((a - (-a^3)^(1/3))*(b + a*x)*Sqrt[(b^2 
+ a^2*x^2)/(b + a*x)^2]*EllipticPi[(a + (-a^3)^(1/3))^2/(4*a*(-a^3)^(1/3)) 
, 2*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]], 1/2])/(6*Sqrt[a]*(a + (-a^3)^(1/...
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

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 7276

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE 
xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ 
[n, 0]
 

Maple [A] (verified)

Time = 1.36 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.56

method	result	size
default	\(\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}\, \sqrt {2}}{2 x \sqrt {a b}}\right )-4 \,\operatorname {arctanh}\left (\frac {\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}{x \sqrt {a b}}\right )}{3 \sqrt {a b}}\)	\(69\)
pseudoelliptic	\(\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}\, \sqrt {2}}{2 x \sqrt {a b}}\right )-4 \,\operatorname {arctanh}\left (\frac {\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}{x \sqrt {a b}}\right )}{3 \sqrt {a b}}\)	\(69\)
elliptic	\(\text {Expression too large to display}\)	\(871\)

Input:

int((a^3*x^3-b^3)/(a^2*x^3+b^2*x)^(1/2)/(a^3*x^3+b^3),x,method=_RETURNVERB 
OSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 222 vs. \(2 (95) = 190\).

Time = 0.13 (sec) , antiderivative size = 421, normalized size of antiderivative = 3.42 \[ \int \frac {-b^3+a^3 x^3}{\sqrt {b^2 x+a^2 x^3} \left (b^3+a^3 x^3\right )} \, dx=\left [-\frac {\sqrt {2} a b \sqrt {\frac {1}{a b}} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {a^{2} x^{3} + b^{2} x} a b \sqrt {\frac {1}{a b}}}{a^{2} x^{2} - 2 \, a b x + b^{2}}\right ) - 2 \, \sqrt {a b} \log \left (\frac {a^{4} x^{4} + 6 \, a^{3} b x^{3} + 3 \, a^{2} b^{2} x^{2} + 6 \, a b^{3} x + b^{4} - 4 \, \sqrt {a^{2} x^{3} + b^{2} x} {\left (a^{2} x^{2} + a b x + b^{2}\right )} \sqrt {a b}}{a^{4} x^{4} - 2 \, a^{3} b x^{3} + 3 \, a^{2} b^{2} x^{2} - 2 \, a b^{3} x + b^{4}}\right )}{6 \, a b}, \frac {\sqrt {2} a b \sqrt {-\frac {1}{a b}} \log \left (\frac {a^{4} x^{4} - 12 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} - 12 \, a b^{3} x + b^{4} + 4 \, \sqrt {2} {\left (a^{3} b x^{2} - 2 \, a^{2} b^{2} x + a b^{3}\right )} \sqrt {a^{2} x^{3} + b^{2} x} \sqrt {-\frac {1}{a b}}}{a^{4} x^{4} + 4 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a b^{3} x + b^{4}}\right ) + 8 \, \sqrt {-a b} \arctan \left (\frac {\sqrt {a^{2} x^{3} + b^{2} x} {\left (a^{2} x^{2} + a b x + b^{2}\right )} \sqrt {-a b}}{2 \, {\left (a^{3} b x^{3} + a b^{3} x\right )}}\right )}{12 \, a b}\right ] \] Input:

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

Output:

[-1/6*(sqrt(2)*a*b*sqrt(1/(a*b))*arctan(2*sqrt(2)*sqrt(a^2*x^3 + b^2*x)*a* 
b*sqrt(1/(a*b))/(a^2*x^2 - 2*a*b*x + b^2)) - 2*sqrt(a*b)*log((a^4*x^4 + 6* 
a^3*b*x^3 + 3*a^2*b^2*x^2 + 6*a*b^3*x + b^4 - 4*sqrt(a^2*x^3 + b^2*x)*(a^2 
*x^2 + a*b*x + b^2)*sqrt(a*b))/(a^4*x^4 - 2*a^3*b*x^3 + 3*a^2*b^2*x^2 - 2* 
a*b^3*x + b^4)))/(a*b), 1/12*(sqrt(2)*a*b*sqrt(-1/(a*b))*log((a^4*x^4 - 12 
*a^3*b*x^3 + 6*a^2*b^2*x^2 - 12*a*b^3*x + b^4 + 4*sqrt(2)*(a^3*b*x^2 - 2*a 
^2*b^2*x + a*b^3)*sqrt(a^2*x^3 + b^2*x)*sqrt(-1/(a*b)))/(a^4*x^4 + 4*a^3*b 
*x^3 + 6*a^2*b^2*x^2 + 4*a*b^3*x + b^4)) + 8*sqrt(-a*b)*arctan(1/2*sqrt(a^ 
2*x^3 + b^2*x)*(a^2*x^2 + a*b*x + b^2)*sqrt(-a*b)/(a^3*b*x^3 + a*b^3*x)))/ 
(a*b)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \frac {-b^3+a^3 x^3}{\sqrt {b^2 x+a^2 x^3} \left (b^3+a^3 x^3\right )} \, dx=\left (\int \frac {\sqrt {x}\, \sqrt {a^{2} x^{2}+b^{2}}\, x^{2}}{a^{5} x^{5}+a^{3} b^{2} x^{3}+a^{2} b^{3} x^{2}+b^{5}}d x \right ) a^{3}-\left (\int \frac {\sqrt {x}\, \sqrt {a^{2} x^{2}+b^{2}}}{a^{5} x^{6}+a^{3} b^{2} x^{4}+a^{2} b^{3} x^{3}+b^{5} x}d x \right ) b^{3} \] Input:

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

Output:

int((sqrt(x)*sqrt(a**2*x**2 + b**2)*x**2)/(a**5*x**5 + a**3*b**2*x**3 + a* 
*2*b**3*x**2 + b**5),x)*a**3 - int((sqrt(x)*sqrt(a**2*x**2 + b**2))/(a**5* 
x**6 + a**3*b**2*x**4 + a**2*b**3*x**3 + b**5*x),x)*b**3