Integrand size = 45, antiderivative size = 167 \[ \int \frac {b^2+c x^2+a^2 x^4}{\sqrt {b x+a x^3} \left (-b^2+a^2 x^4\right )} \, dx=\frac {(-2 a b+c) \sqrt {b x+a x^3}}{2 a b \left (b+a x^2\right )}-\frac {(2 a b+c) \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {b x+a x^3}}{b+a x^2}\right )}{4 \sqrt {2} a^{5/4} b^{5/4}}-\frac {(2 a b+c) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {b x+a x^3}}{b+a x^2}\right )}{4 \sqrt {2} a^{5/4} b^{5/4}} \] Output:
1/2*(-2*a*b+c)*(a*x^3+b*x)^(1/2)/a/b/(a*x^2+b)-1/8*(2*a*b+c)*arctan(2^(1/2 )*a^(1/4)*b^(1/4)*(a*x^3+b*x)^(1/2)/(a*x^2+b))*2^(1/2)/a^(5/4)/b^(5/4)-1/8 *(2*a*b+c)*arctanh(2^(1/2)*a^(1/4)*b^(1/4)*(a*x^3+b*x)^(1/2)/(a*x^2+b))*2^ (1/2)/a^(5/4)/b^(5/4)
Time = 1.56 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.02 \[ \int \frac {b^2+c x^2+a^2 x^4}{\sqrt {b x+a x^3} \left (-b^2+a^2 x^4\right )} \, dx=-\frac {\sqrt {x} \left (4 \sqrt [4]{a} \sqrt [4]{b} (2 a b-c) \sqrt {x}+\sqrt {2} (2 a b+c) \sqrt {b+a x^2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {b+a x^2}}\right )+\sqrt {2} (2 a b+c) \sqrt {b+a x^2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {b+a x^2}}\right )\right )}{8 a^{5/4} b^{5/4} \sqrt {x \left (b+a x^2\right )}} \] Input:
Integrate[(b^2 + c*x^2 + a^2*x^4)/(Sqrt[b*x + a*x^3]*(-b^2 + a^2*x^4)),x]
Output:
-1/8*(Sqrt[x]*(4*a^(1/4)*b^(1/4)*(2*a*b - c)*Sqrt[x] + Sqrt[2]*(2*a*b + c) *Sqrt[b + a*x^2]*ArcTan[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/Sqrt[b + a*x^2]] + Sqrt[2]*(2*a*b + c)*Sqrt[b + a*x^2]*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sq rt[x])/Sqrt[b + a*x^2]]))/(a^(5/4)*b^(5/4)*Sqrt[x*(b + a*x^2)])
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.55 (sec) , antiderivative size = 506, normalized size of antiderivative = 3.03, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2467, 25, 1388, 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.
| \(\displaystyle \int \frac {a^2 x^4+b^2+c x^2}{\sqrt {a x^3+b x} \left (a^2 x^4-b^2\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {a x^2+b} \int -\frac {a^2 x^4+c x^2+b^2}{\sqrt {x} \sqrt {a x^2+b} \left (b^2-a^2 x^4\right )}dx}{\sqrt {a x^3+b x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt {a x^2+b} \int \frac {a^2 x^4+c x^2+b^2}{\sqrt {x} \sqrt {a x^2+b} \left (b^2-a^2 x^4\right )}dx}{\sqrt {a x^3+b x}}\) |
| \(\Big \downarrow \) 1388 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt {a x^2+b} \int \frac {a^2 x^4+c x^2+b^2}{\sqrt {x} \left (b-a x^2\right ) \left (a x^2+b\right )^{3/2}}dx}{\sqrt {a x^3+b x}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a x^2+b} \int \frac {a^2 x^4+c x^2+b^2}{\left (b-a x^2\right ) \left (a x^2+b\right )^{3/2}}d\sqrt {x}}{\sqrt {a x^3+b x}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a x^2+b} \int \left (-\frac {a x^2}{\left (a x^2+b\right )^{3/2}}-\frac {a b+c}{a \left (a x^2+b\right )^{3/2}}+\frac {2 a b^2+c b}{a \left (b-a x^2\right ) \left (a x^2+b\right )^{3/2}}\right )d\sqrt {x}}{\sqrt {a x^3+b x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a x^2+b} \left (\frac {(2 a b+c) \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a x^2+b}}\right )}{8 \sqrt {2} a^{5/4} b^{5/4}}-\frac {(a b+c) \left (\sqrt {a} x+\sqrt {b}\right ) \sqrt {\frac {a x^2+b}{\left (\sqrt {a} x+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 a^{5/4} b^{5/4} \sqrt {a x^2+b}}+\frac {(2 a b+c) \left (\sqrt {a} x+\sqrt {b}\right ) \sqrt {\frac {a x^2+b}{\left (\sqrt {a} x+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 a^{5/4} b^{5/4} \sqrt {a x^2+b}}+\frac {(2 a b+c) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a x^2+b}}\right )}{8 \sqrt {2} a^{5/4} b^{5/4}}-\frac {\left (\sqrt {a} x+\sqrt {b}\right ) \sqrt {\frac {a x^2+b}{\left (\sqrt {a} x+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} \sqrt {a x^2+b}}-\frac {\sqrt {x} (a b+c)}{2 a b \sqrt {a x^2+b}}+\frac {\sqrt {x} (2 a b+c)}{4 a b \sqrt {a x^2+b}}+\frac {\sqrt {x}}{2 \sqrt {a x^2+b}}\right )}{\sqrt {a x^3+b x}}\) |
Input:
Int[(b^2 + c*x^2 + a^2*x^4)/(Sqrt[b*x + a*x^3]*(-b^2 + a^2*x^4)),x]
Output:
(-2*Sqrt[x]*Sqrt[b + a*x^2]*(Sqrt[x]/(2*Sqrt[b + a*x^2]) - ((a*b + c)*Sqrt [x])/(2*a*b*Sqrt[b + a*x^2]) + ((2*a*b + c)*Sqrt[x])/(4*a*b*Sqrt[b + a*x^2 ]) + ((2*a*b + c)*ArcTan[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/Sqrt[b + a*x^2] ])/(8*Sqrt[2]*a^(5/4)*b^(5/4)) + ((2*a*b + c)*ArcTanh[(Sqrt[2]*a^(1/4)*b^( 1/4)*Sqrt[x])/Sqrt[b + a*x^2]])/(8*Sqrt[2]*a^(5/4)*b^(5/4)) - ((Sqrt[b] + Sqrt[a]*x)*Sqrt[(b + a*x^2)/(Sqrt[b] + Sqrt[a]*x)^2]*EllipticF[2*ArcTan[(a ^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(4*a^(1/4)*b^(1/4)*Sqrt[b + a*x^2]) - ((a* b + c)*(Sqrt[b] + Sqrt[a]*x)*Sqrt[(b + a*x^2)/(Sqrt[b] + Sqrt[a]*x)^2]*Ell ipticF[2*ArcTan[(a^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(4*a^(5/4)*b^(5/4)*Sqrt[ b + a*x^2]) + ((2*a*b + c)*(Sqrt[b] + Sqrt[a]*x)*Sqrt[(b + a*x^2)/(Sqrt[b] + Sqrt[a]*x)^2]*EllipticF[2*ArcTan[(a^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(4*a ^(5/4)*b^(5/4)*Sqrt[b + a*x^2])))/Sqrt[b*x + a*x^3]
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && (Integer Q[p] || (GtQ[a, 0] && GtQ[d, 0]))
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]
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]
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]
Time = 0.96 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.86
| method | result | size |
| default | \(-\frac {8 x \left (a b -\frac {c}{2}\right ) \left (a b \right )^{\frac {1}{4}}+\left (a b +\frac {c}{2}\right ) \left (\ln \left (\frac {-\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} x -\sqrt {\left (a \,x^{2}+b \right ) x}}{\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} x -\sqrt {\left (a \,x^{2}+b \right ) x}}\right )-2 \arctan \left (\frac {\sqrt {\left (a \,x^{2}+b \right ) x}\, \sqrt {2}}{2 x \left (a b \right )^{\frac {1}{4}}}\right )\right ) \sqrt {2}\, \sqrt {\left (a \,x^{2}+b \right ) x}}{8 \sqrt {\left (a \,x^{2}+b \right ) x}\, \left (a b \right )^{\frac {1}{4}} b a}\) | \(144\) |
| pseudoelliptic | \(-\frac {8 x \left (a b -\frac {c}{2}\right ) \left (a b \right )^{\frac {1}{4}}+\left (a b +\frac {c}{2}\right ) \left (\ln \left (\frac {-\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} x -\sqrt {\left (a \,x^{2}+b \right ) x}}{\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} x -\sqrt {\left (a \,x^{2}+b \right ) x}}\right )-2 \arctan \left (\frac {\sqrt {\left (a \,x^{2}+b \right ) x}\, \sqrt {2}}{2 x \left (a b \right )^{\frac {1}{4}}}\right )\right ) \sqrt {2}\, \sqrt {\left (a \,x^{2}+b \right ) x}}{8 \sqrt {\left (a \,x^{2}+b \right ) x}\, \left (a b \right )^{\frac {1}{4}} b a}\) | \(144\) |
| elliptic | \(-\frac {x \left (2 a b -c \right )}{2 a b \sqrt {\left (x^{2}+\frac {b}{a}\right ) a x}}+\frac {\sqrt {-a b}\, \sqrt {1+\frac {x a}{\sqrt {-a b}}}\, \sqrt {2-\frac {2 x a}{\sqrt {-a b}}}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{2 a \sqrt {a \,x^{3}+b x}}+\frac {\sqrt {-a b}\, \sqrt {1+\frac {x a}{\sqrt {-a b}}}\, \sqrt {2-\frac {2 x a}{\sqrt {-a b}}}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) c}{4 a^{2} \sqrt {a \,x^{3}+b x}\, b}+\frac {b \sqrt {-a b}\, \sqrt {1+\frac {x a}{\sqrt {-a b}}}\, \sqrt {2-\frac {2 x a}{\sqrt {-a b}}}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {a b}\, a \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}+\frac {\sqrt {-a b}\, \sqrt {1+\frac {x a}{\sqrt {-a b}}}\, \sqrt {2-\frac {2 x a}{\sqrt {-a b}}}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right ) c}{4 a^{2} \sqrt {a b}\, \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}-\frac {b \sqrt {-a b}\, \sqrt {1+\frac {x a}{\sqrt {-a b}}}\, \sqrt {2-\frac {2 x a}{\sqrt {-a b}}}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}+\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {a b}\, a \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}+\frac {\sqrt {a b}}{a}\right )}-\frac {\sqrt {-a b}\, \sqrt {1+\frac {x a}{\sqrt {-a b}}}\, \sqrt {2-\frac {2 x a}{\sqrt {-a b}}}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}+\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right ) c}{4 a^{2} \sqrt {a b}\, \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}+\frac {\sqrt {a b}}{a}\right )}\) | \(827\) |
Input:
int((a^2*x^4+c*x^2+b^2)/(a*x^3+b*x)^(1/2)/(a^2*x^4-b^2),x,method=_RETURNVE RBOSE)
Output:
-1/8/((a*x^2+b)*x)^(1/2)/(a*b)^(1/4)*(8*x*(a*b-1/2*c)*(a*b)^(1/4)+(a*b+1/2 *c)*(ln((-2^(1/2)*(a*b)^(1/4)*x-((a*x^2+b)*x)^(1/2))/(2^(1/2)*(a*b)^(1/4)* x-((a*x^2+b)*x)^(1/2)))-2*arctan(1/2*((a*x^2+b)*x)^(1/2)/x*2^(1/2)/(a*b)^( 1/4)))*2^(1/2)*((a*x^2+b)*x)^(1/2))/b/a
Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 1877, normalized size of antiderivative = 11.24 \[ \int \frac {b^2+c x^2+a^2 x^4}{\sqrt {b x+a x^3} \left (-b^2+a^2 x^4\right )} \, dx=\text {Too large to display} \] Input:
integrate((a^2*x^4+c*x^2+b^2)/(a*x^3+b*x)^(1/2)/(a^2*x^4-b^2),x, algorithm ="fricas")
Output:
-1/16*((1/4)^(1/4)*(a^2*b*x^2 + a*b^2)*((16*a^4*b^4 + 32*a^3*b^3*c + 24*a^ 2*b^2*c^2 + 8*a*b*c^3 + c^4)/(a^5*b^5))^(1/4)*log((8*a^3*b^5 + 12*a^2*b^4* c + 6*a*b^3*c^2 + b^2*c^3 + (8*a^5*b^3 + 12*a^4*b^2*c + 6*a^3*b*c^2 + a^2* c^3)*x^4 + 6*(8*a^4*b^4 + 12*a^3*b^3*c + 6*a^2*b^2*c^2 + a*b*c^3)*x^2 + 8* sqrt(a*x^3 + b*x)*((1/4)^(1/4)*(4*a^4*b^4 + 4*a^3*b^3*c + a^2*b^2*c^2)*x*( (16*a^4*b^4 + 32*a^3*b^3*c + 24*a^2*b^2*c^2 + 8*a*b*c^3 + c^4)/(a^5*b^5))^ (1/4) + (1/4)^(3/4)*(a^5*b^4*x^2 + a^4*b^5)*((16*a^4*b^4 + 32*a^3*b^3*c + 24*a^2*b^2*c^2 + 8*a*b*c^3 + c^4)/(a^5*b^5))^(3/4)) + 4*((2*a^5*b^4 + a^4* b^3*c)*x^3 + (2*a^4*b^5 + a^3*b^4*c)*x)*sqrt((16*a^4*b^4 + 32*a^3*b^3*c + 24*a^2*b^2*c^2 + 8*a*b*c^3 + c^4)/(a^5*b^5)))/(a^2*x^4 - 2*a*b*x^2 + b^2)) - (1/4)^(1/4)*(a^2*b*x^2 + a*b^2)*((16*a^4*b^4 + 32*a^3*b^3*c + 24*a^2*b^ 2*c^2 + 8*a*b*c^3 + c^4)/(a^5*b^5))^(1/4)*log((8*a^3*b^5 + 12*a^2*b^4*c + 6*a*b^3*c^2 + b^2*c^3 + (8*a^5*b^3 + 12*a^4*b^2*c + 6*a^3*b*c^2 + a^2*c^3) *x^4 + 6*(8*a^4*b^4 + 12*a^3*b^3*c + 6*a^2*b^2*c^2 + a*b*c^3)*x^2 - 8*sqrt (a*x^3 + b*x)*((1/4)^(1/4)*(4*a^4*b^4 + 4*a^3*b^3*c + a^2*b^2*c^2)*x*((16* a^4*b^4 + 32*a^3*b^3*c + 24*a^2*b^2*c^2 + 8*a*b*c^3 + c^4)/(a^5*b^5))^(1/4 ) + (1/4)^(3/4)*(a^5*b^4*x^2 + a^4*b^5)*((16*a^4*b^4 + 32*a^3*b^3*c + 24*a ^2*b^2*c^2 + 8*a*b*c^3 + c^4)/(a^5*b^5))^(3/4)) + 4*((2*a^5*b^4 + a^4*b^3* c)*x^3 + (2*a^4*b^5 + a^3*b^4*c)*x)*sqrt((16*a^4*b^4 + 32*a^3*b^3*c + 24*a ^2*b^2*c^2 + 8*a*b*c^3 + c^4)/(a^5*b^5)))/(a^2*x^4 - 2*a*b*x^2 + b^2)) ...
\[ \int \frac {b^2+c x^2+a^2 x^4}{\sqrt {b x+a x^3} \left (-b^2+a^2 x^4\right )} \, dx=\int \frac {a^{2} x^{4} + b^{2} + c x^{2}}{\sqrt {x \left (a x^{2} + b\right )} \left (a x^{2} - b\right ) \left (a x^{2} + b\right )}\, dx \] Input:
integrate((a**2*x**4+c*x**2+b**2)/(a*x**3+b*x)**(1/2)/(a**2*x**4-b**2),x)
Output:
Integral((a**2*x**4 + b**2 + c*x**2)/(sqrt(x*(a*x**2 + b))*(a*x**2 - b)*(a *x**2 + b)), x)
\[ \int \frac {b^2+c x^2+a^2 x^4}{\sqrt {b x+a x^3} \left (-b^2+a^2 x^4\right )} \, dx=\int { \frac {a^{2} x^{4} + c x^{2} + b^{2}}{{\left (a^{2} x^{4} - b^{2}\right )} \sqrt {a x^{3} + b x}} \,d x } \] Input:
integrate((a^2*x^4+c*x^2+b^2)/(a*x^3+b*x)^(1/2)/(a^2*x^4-b^2),x, algorithm ="maxima")
Output:
integrate((a^2*x^4 + c*x^2 + b^2)/((a^2*x^4 - b^2)*sqrt(a*x^3 + b*x)), x)
\[ \int \frac {b^2+c x^2+a^2 x^4}{\sqrt {b x+a x^3} \left (-b^2+a^2 x^4\right )} \, dx=\int { \frac {a^{2} x^{4} + c x^{2} + b^{2}}{{\left (a^{2} x^{4} - b^{2}\right )} \sqrt {a x^{3} + b x}} \,d x } \] Input:
integrate((a^2*x^4+c*x^2+b^2)/(a*x^3+b*x)^(1/2)/(a^2*x^4-b^2),x, algorithm ="giac")
Output:
integrate((a^2*x^4 + c*x^2 + b^2)/((a^2*x^4 - b^2)*sqrt(a*x^3 + b*x)), x)
Timed out. \[ \int \frac {b^2+c x^2+a^2 x^4}{\sqrt {b x+a x^3} \left (-b^2+a^2 x^4\right )} \, dx=\text {Hanged} \] Input:
int(-(c*x^2 + b^2 + a^2*x^4)/((b^2 - a^2*x^4)*(b*x + a*x^3)^(1/2)),x)
Output:
\text{Hanged}
\[ \int \frac {b^2+c x^2+a^2 x^4}{\sqrt {b x+a x^3} \left (-b^2+a^2 x^4\right )} \, dx=\frac {2 \sqrt {x}\, \sqrt {a \,x^{2}+b}\, c +2 \left (\int \frac {\sqrt {a \,x^{2}+b}}{\sqrt {x}\, a^{3} x^{6}+\sqrt {x}\, a^{2} b \,x^{4}-\sqrt {x}\, a \,b^{2} x^{2}-\sqrt {x}\, b^{3}}d x \right ) a^{2} b^{3} x^{2}+2 \left (\int \frac {\sqrt {a \,x^{2}+b}}{\sqrt {x}\, a^{3} x^{6}+\sqrt {x}\, a^{2} b \,x^{4}-\sqrt {x}\, a \,b^{2} x^{2}-\sqrt {x}\, b^{3}}d x \right ) a \,b^{4}+\left (\int \frac {\sqrt {a \,x^{2}+b}}{\sqrt {x}\, a^{3} x^{6}+\sqrt {x}\, a^{2} b \,x^{4}-\sqrt {x}\, a \,b^{2} x^{2}-\sqrt {x}\, b^{3}}d x \right ) a \,b^{2} c \,x^{2}+\left (\int \frac {\sqrt {a \,x^{2}+b}}{\sqrt {x}\, a^{3} x^{6}+\sqrt {x}\, a^{2} b \,x^{4}-\sqrt {x}\, a \,b^{2} x^{2}-\sqrt {x}\, b^{3}}d x \right ) b^{3} c +2 \left (\int \frac {\sqrt {x}\, \sqrt {a \,x^{2}+b}\, x^{3}}{a^{3} x^{6}+a^{2} b \,x^{4}-a \,b^{2} x^{2}-b^{3}}d x \right ) a^{4} b \,x^{2}+2 \left (\int \frac {\sqrt {x}\, \sqrt {a \,x^{2}+b}\, x^{3}}{a^{3} x^{6}+a^{2} b \,x^{4}-a \,b^{2} x^{2}-b^{3}}d x \right ) a^{3} b^{2}+\left (\int \frac {\sqrt {x}\, \sqrt {a \,x^{2}+b}\, x^{3}}{a^{3} x^{6}+a^{2} b \,x^{4}-a \,b^{2} x^{2}-b^{3}}d x \right ) a^{3} c \,x^{2}+\left (\int \frac {\sqrt {x}\, \sqrt {a \,x^{2}+b}\, x^{3}}{a^{3} x^{6}+a^{2} b \,x^{4}-a \,b^{2} x^{2}-b^{3}}d x \right ) a^{2} b c}{2 a b \left (a \,x^{2}+b \right )} \] Input:
int((a^2*x^4+c*x^2+b^2)/(a*x^3+b*x)^(1/2)/(a^2*x^4-b^2),x)
Output:
(2*sqrt(x)*sqrt(a*x**2 + b)*c + 2*int(sqrt(a*x**2 + b)/(sqrt(x)*a**3*x**6 + sqrt(x)*a**2*b*x**4 - sqrt(x)*a*b**2*x**2 - sqrt(x)*b**3),x)*a**2*b**3*x **2 + 2*int(sqrt(a*x**2 + b)/(sqrt(x)*a**3*x**6 + sqrt(x)*a**2*b*x**4 - sq rt(x)*a*b**2*x**2 - sqrt(x)*b**3),x)*a*b**4 + int(sqrt(a*x**2 + b)/(sqrt(x )*a**3*x**6 + sqrt(x)*a**2*b*x**4 - sqrt(x)*a*b**2*x**2 - sqrt(x)*b**3),x) *a*b**2*c*x**2 + int(sqrt(a*x**2 + b)/(sqrt(x)*a**3*x**6 + sqrt(x)*a**2*b* x**4 - sqrt(x)*a*b**2*x**2 - sqrt(x)*b**3),x)*b**3*c + 2*int((sqrt(x)*sqrt (a*x**2 + b)*x**3)/(a**3*x**6 + a**2*b*x**4 - a*b**2*x**2 - b**3),x)*a**4* b*x**2 + 2*int((sqrt(x)*sqrt(a*x**2 + b)*x**3)/(a**3*x**6 + a**2*b*x**4 - a*b**2*x**2 - b**3),x)*a**3*b**2 + int((sqrt(x)*sqrt(a*x**2 + b)*x**3)/(a* *3*x**6 + a**2*b*x**4 - a*b**2*x**2 - b**3),x)*a**3*c*x**2 + int((sqrt(x)* sqrt(a*x**2 + b)*x**3)/(a**3*x**6 + a**2*b*x**4 - a*b**2*x**2 - b**3),x)*a **2*b*c)/(2*a*b*(a*x**2 + b))