\(\int \frac {b-c x+a x^2}{(-b+a x^2) \sqrt {b x+a x^3}} \, dx\) [2049]

Optimal result

Integrand size = 36, antiderivative size = 147 \[ \int \frac {b-c x+a x^2}{\left (-b+a x^2\right ) \sqrt {b x+a x^3}} \, dx=-\frac {\left (2 \sqrt {a} \sqrt {b}+c\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {b x+a x^3}}{b+a x^2}\right )}{2 \sqrt {2} a^{3/4} b^{3/4}}-\frac {\left (2 \sqrt {a} \sqrt {b}-c\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {b x+a x^3}}{b+a x^2}\right )}{2 \sqrt {2} a^{3/4} b^{3/4}} \]

[Out]

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

Rubi [C] (verified)

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

Time = 1.09 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.93, number of steps used = 13, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {2081, 6857, 335, 226, 947, 174, 551} \[ \int \frac {b-c x+a x^2}{\left (-b+a x^2\right ) \sqrt {b x+a x^3}} \, dx=-\frac {\sqrt {x} \left (\frac {c}{\sqrt {a}}+2 \sqrt {b}\right ) \sqrt {\frac {a x^2}{b}+1} \operatorname {EllipticPi}\left (\frac {\sqrt {-a}}{\sqrt {a}},\arcsin \left (\frac {\sqrt [4]{-a} \sqrt {x}}{\sqrt [4]{b}}\right ),-1\right )}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {a x^3+b x}}-\frac {\sqrt {x} \left (2 \sqrt {b}-\frac {c}{\sqrt {a}}\right ) \sqrt {\frac {a x^2}{b}+1} \operatorname {EllipticPi}\left (\frac {\sqrt {a}}{\sqrt {-a}},\arcsin \left (\frac {\sqrt [4]{-a} \sqrt {x}}{\sqrt [4]{b}}\right ),-1\right )}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {a x^3+b x}}+\frac {\sqrt {x} \left (\sqrt {a} x+\sqrt {b}\right ) \sqrt {\frac {a x^2+b}{\left (\sqrt {a} x+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt [4]{b} \sqrt {a x^3+b x}} \]

[In]

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

[Out]

(Sqrt[x]*(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])/(a^(1/4)*b^(1/4)*Sqrt[b*x + a*x^3]) - ((2*Sqrt[b] + c/Sqrt[a])*Sqrt[x]*Sqrt[1 + (a*x^2)/b]*Ell
ipticPi[Sqrt[-a]/Sqrt[a], ArcSin[((-a)^(1/4)*Sqrt[x])/b^(1/4)], -1])/((-a)^(1/4)*b^(1/4)*Sqrt[b*x + a*x^3]) -
((2*Sqrt[b] - c/Sqrt[a])*Sqrt[x]*Sqrt[1 + (a*x^2)/b]*EllipticPi[Sqrt[a]/Sqrt[-a], ArcSin[((-a)^(1/4)*Sqrt[x])/
b^(1/4)], -1])/((-a)^(1/4)*b^(1/4)*Sqrt[b*x + a*x^3])

Rule 174

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Sym
bol] :> Dist[-2, Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + f*(x^2/d), x]]*Sqrt[Simp[(d
*g - c*h)/d + h*(x^2/d), x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && GtQ[(d*e - c
*f)/d, 0]

Rule 226

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))*EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 551

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1/(a*Sqr
t[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b*(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-f/e, -d/c])

Rule 947

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[-c/
a, 2]}, Dist[Sqrt[1 + c*(x^2/a)]/Sqrt[a + c*x^2], Int[1/((d + e*x)*Sqrt[f + g*x]*Sqrt[1 - q*x]*Sqrt[1 + q*x]),
x], x]] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] &&  !GtQ[a, 0]

Rule 2081

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 6857

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

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt {b+a x^2}\right ) \int \frac {b-c x+a x^2}{\sqrt {x} \left (-b+a x^2\right ) \sqrt {b+a x^2}} \, dx}{\sqrt {b x+a x^3}} \\ & = \frac {\left (\sqrt {x} \sqrt {b+a x^2}\right ) \int \left (\frac {1}{\sqrt {x} \sqrt {b+a x^2}}+\frac {2 b-c x}{\sqrt {x} \left (-b+a x^2\right ) \sqrt {b+a x^2}}\right ) \, dx}{\sqrt {b x+a x^3}} \\ & = \frac {\left (\sqrt {x} \sqrt {b+a x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+a x^2}} \, dx}{\sqrt {b x+a x^3}}+\frac {\left (\sqrt {x} \sqrt {b+a x^2}\right ) \int \frac {2 b-c x}{\sqrt {x} \left (-b+a x^2\right ) \sqrt {b+a x^2}} \, dx}{\sqrt {b x+a x^3}} \\ & = \frac {\left (\sqrt {x} \sqrt {b+a x^2}\right ) \int \left (-\frac {2 b^{3/2}-\frac {b c}{\sqrt {a}}}{2 b \sqrt {x} \left (\sqrt {b}-\sqrt {a} x\right ) \sqrt {b+a x^2}}-\frac {2 b^{3/2}+\frac {b c}{\sqrt {a}}}{2 b \sqrt {x} \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt {b+a x^2}}\right ) \, dx}{\sqrt {b x+a x^3}}+\frac {\left (2 \sqrt {x} \sqrt {b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {b x+a x^3}} \\ & = \frac {\sqrt {x} \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt {\frac {b+a x^2}{\left (\sqrt {b}+\sqrt {a} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt [4]{b} \sqrt {b x+a x^3}}-\frac {\left (\left (2 \sqrt {b}-\frac {c}{\sqrt {a}}\right ) \sqrt {x} \sqrt {b+a x^2}\right ) \int \frac {1}{\sqrt {x} \left (\sqrt {b}-\sqrt {a} x\right ) \sqrt {b+a x^2}} \, dx}{2 \sqrt {b x+a x^3}}-\frac {\left (\left (2 \sqrt {b}+\frac {c}{\sqrt {a}}\right ) \sqrt {x} \sqrt {b+a x^2}\right ) \int \frac {1}{\sqrt {x} \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt {b+a x^2}} \, dx}{2 \sqrt {b x+a x^3}} \\ & = \frac {\sqrt {x} \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt {\frac {b+a x^2}{\left (\sqrt {b}+\sqrt {a} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt [4]{b} \sqrt {b x+a x^3}}-\frac {\left (\left (2 \sqrt {b}-\frac {c}{\sqrt {a}}\right ) \sqrt {x} \sqrt {1+\frac {a x^2}{b}}\right ) \int \frac {1}{\sqrt {x} \left (\sqrt {b}-\sqrt {a} x\right ) \sqrt {1-\frac {\sqrt {-a} x}{\sqrt {b}}} \sqrt {1+\frac {\sqrt {-a} x}{\sqrt {b}}}} \, dx}{2 \sqrt {b x+a x^3}}-\frac {\left (\left (2 \sqrt {b}+\frac {c}{\sqrt {a}}\right ) \sqrt {x} \sqrt {1+\frac {a x^2}{b}}\right ) \int \frac {1}{\sqrt {x} \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt {1-\frac {\sqrt {-a} x}{\sqrt {b}}} \sqrt {1+\frac {\sqrt {-a} x}{\sqrt {b}}}} \, dx}{2 \sqrt {b x+a x^3}} \\ & = \frac {\sqrt {x} \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt {\frac {b+a x^2}{\left (\sqrt {b}+\sqrt {a} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt [4]{b} \sqrt {b x+a x^3}}+\frac {\left (\left (2 \sqrt {b}-\frac {c}{\sqrt {a}}\right ) \sqrt {x} \sqrt {1+\frac {a x^2}{b}}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt {b}+\sqrt {a} x^2\right ) \sqrt {1-\frac {\sqrt {-a} x^2}{\sqrt {b}}} \sqrt {1+\frac {\sqrt {-a} x^2}{\sqrt {b}}}} \, dx,x,\sqrt {x}\right )}{\sqrt {b x+a x^3}}+\frac {\left (\left (2 \sqrt {b}+\frac {c}{\sqrt {a}}\right ) \sqrt {x} \sqrt {1+\frac {a x^2}{b}}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt {b}-\sqrt {a} x^2\right ) \sqrt {1-\frac {\sqrt {-a} x^2}{\sqrt {b}}} \sqrt {1+\frac {\sqrt {-a} x^2}{\sqrt {b}}}} \, dx,x,\sqrt {x}\right )}{\sqrt {b x+a x^3}} \\ & = \frac {\sqrt {x} \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt {\frac {b+a x^2}{\left (\sqrt {b}+\sqrt {a} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt [4]{b} \sqrt {b x+a x^3}}-\frac {\left (2 \sqrt {b}+\frac {c}{\sqrt {a}}\right ) \sqrt {x} \sqrt {1+\frac {a x^2}{b}} \operatorname {EllipticPi}\left (\frac {\sqrt {-a}}{\sqrt {a}},\arcsin \left (\frac {\sqrt [4]{-a} \sqrt {x}}{\sqrt [4]{b}}\right ),-1\right )}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {b x+a x^3}}-\frac {\left (2 \sqrt {b}-\frac {c}{\sqrt {a}}\right ) \sqrt {x} \sqrt {1+\frac {a x^2}{b}} \operatorname {EllipticPi}\left (\frac {\sqrt {a}}{\sqrt {-a}},\arcsin \left (\frac {\sqrt [4]{-a} \sqrt {x}}{\sqrt [4]{b}}\right ),-1\right )}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {b x+a x^3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.10 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00 \[ \int \frac {b-c x+a x^2}{\left (-b+a x^2\right ) \sqrt {b x+a x^3}} \, dx=-\frac {\sqrt {x} \sqrt {b+a x^2} \left (\left (2 \sqrt {a} \sqrt {b}+c\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {b+a x^2}}\right )+\left (2 \sqrt {a} \sqrt {b}-c\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {b+a x^2}}\right )\right )}{2 \sqrt {2} a^{3/4} b^{3/4} \sqrt {x \left (b+a x^2\right )}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.51 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.34

method	result	size
default	\(\frac {\sqrt {2}\, \left (\ln \left (\frac {-\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} x -\sqrt {\left (a \,x^{2}+b \right ) x}}{\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} x -\sqrt {\left (a \,x^{2}+b \right ) x}}\right ) c \sqrt {a b}+2 \arctan \left (\frac {\sqrt {\left (a \,x^{2}+b \right ) x}\, \sqrt {2}}{2 x \left (a b \right )^{\frac {1}{4}}}\right ) c \sqrt {a b}-2 \ln \left (\frac {-\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} x -\sqrt {\left (a \,x^{2}+b \right ) x}}{\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} x -\sqrt {\left (a \,x^{2}+b \right ) x}}\right ) a b +4 \arctan \left (\frac {\sqrt {\left (a \,x^{2}+b \right ) x}\, \sqrt {2}}{2 x \left (a b \right )^{\frac {1}{4}}}\right ) a b \right )}{8 a b \left (a b \right )^{\frac {1}{4}}}\)	\(197\)
pseudoelliptic	\(\frac {\sqrt {2}\, \left (\ln \left (\frac {-\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} x -\sqrt {\left (a \,x^{2}+b \right ) x}}{\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} x -\sqrt {\left (a \,x^{2}+b \right ) x}}\right ) c \sqrt {a b}+2 \arctan \left (\frac {\sqrt {\left (a \,x^{2}+b \right ) x}\, \sqrt {2}}{2 x \left (a b \right )^{\frac {1}{4}}}\right ) c \sqrt {a b}-2 \ln \left (\frac {-\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} x -\sqrt {\left (a \,x^{2}+b \right ) x}}{\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} x -\sqrt {\left (a \,x^{2}+b \right ) x}}\right ) a b +4 \arctan \left (\frac {\sqrt {\left (a \,x^{2}+b \right ) x}\, \sqrt {2}}{2 x \left (a b \right )^{\frac {1}{4}}}\right ) a b \right )}{8 a b \left (a b \right )^{\frac {1}{4}}}\)	\(197\)
elliptic	\(\frac {\sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \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 )}{a \sqrt {a \,x^{3}+b x}}-\frac {\sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \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}{2 a^{2} \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}+\frac {b \sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \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 )}{\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 {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \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}{2 a^{2} \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}+\frac {\sqrt {a b}}{a}\right )}-\frac {b \sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \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 )}{\sqrt {a b}\, a \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}+\frac {\sqrt {a b}}{a}\right )}\)	\(691\)

[In]

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

[Out]

1/8*2^(1/2)*(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)))*c*(a
*b)^(1/2)+2*arctan(1/2*((a*x^2+b)*x)^(1/2)/x*2^(1/2)/(a*b)^(1/4))*c*(a*b)^(1/2)-2*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)))*a*b+4*arctan(1/2*((a*x^2+b)*x)^(1/2)/x*2^(1/2
)/(a*b)^(1/4))*a*b)/a/b/(a*b)^(1/4)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1553 vs. \(2 (107) = 214\).

Time = 0.53 (sec) , antiderivative size = 1553, normalized size of antiderivative = 10.56 \[ \int \frac {b-c x+a x^2}{\left (-b+a x^2\right ) \sqrt {b x+a x^3}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/8*sqrt(1/2)*sqrt(-(a*b*sqrt((16*a^2*b^2 + 8*a*b*c^2 + c^4)/(a^3*b^3)) + 4*c)/(a*b))*log(-(16*a^2*b^4 - b^2*
c^4 + (16*a^4*b^2 - a^2*c^4)*x^4 + 6*(16*a^3*b^3 - a*b*c^4)*x^2 + 4*sqrt(1/2)*(4*a^2*b^3*c + a*b^2*c^3 + (4*a^
3*b^2*c + a^2*b*c^3)*x^2 + 4*(4*a^3*b^3 + a^2*b^2*c^2)*x - 2*(a^4*b^3*x^2 + a^3*b^3*c*x + a^3*b^4)*sqrt((16*a^
2*b^2 + 8*a*b*c^2 + c^4)/(a^3*b^3)))*sqrt(a*x^3 + b*x)*sqrt(-(a*b*sqrt((16*a^2*b^2 + 8*a*b*c^2 + c^4)/(a^3*b^3
)) + 4*c)/(a*b)) - 4*((4*a^4*b^3 - a^3*b^2*c^2)*x^3 + (4*a^3*b^4 - a^2*b^3*c^2)*x)*sqrt((16*a^2*b^2 + 8*a*b*c^
2 + c^4)/(a^3*b^3)))/(a^2*x^4 - 2*a*b*x^2 + b^2)) + 1/8*sqrt(1/2)*sqrt(-(a*b*sqrt((16*a^2*b^2 + 8*a*b*c^2 + c^
4)/(a^3*b^3)) + 4*c)/(a*b))*log(-(16*a^2*b^4 - b^2*c^4 + (16*a^4*b^2 - a^2*c^4)*x^4 + 6*(16*a^3*b^3 - a*b*c^4)
*x^2 - 4*sqrt(1/2)*(4*a^2*b^3*c + a*b^2*c^3 + (4*a^3*b^2*c + a^2*b*c^3)*x^2 + 4*(4*a^3*b^3 + a^2*b^2*c^2)*x -
2*(a^4*b^3*x^2 + a^3*b^3*c*x + a^3*b^4)*sqrt((16*a^2*b^2 + 8*a*b*c^2 + c^4)/(a^3*b^3)))*sqrt(a*x^3 + b*x)*sqrt
(-(a*b*sqrt((16*a^2*b^2 + 8*a*b*c^2 + c^4)/(a^3*b^3)) + 4*c)/(a*b)) - 4*((4*a^4*b^3 - a^3*b^2*c^2)*x^3 + (4*a^
3*b^4 - a^2*b^3*c^2)*x)*sqrt((16*a^2*b^2 + 8*a*b*c^2 + c^4)/(a^3*b^3)))/(a^2*x^4 - 2*a*b*x^2 + b^2)) - 1/8*sqr
t(1/2)*sqrt((a*b*sqrt((16*a^2*b^2 + 8*a*b*c^2 + c^4)/(a^3*b^3)) - 4*c)/(a*b))*log(-(16*a^2*b^4 - b^2*c^4 + (16
*a^4*b^2 - a^2*c^4)*x^4 + 6*(16*a^3*b^3 - a*b*c^4)*x^2 + 4*sqrt(1/2)*(4*a^2*b^3*c + a*b^2*c^3 + (4*a^3*b^2*c +
a^2*b*c^3)*x^2 + 4*(4*a^3*b^3 + a^2*b^2*c^2)*x + 2*(a^4*b^3*x^2 + a^3*b^3*c*x + a^3*b^4)*sqrt((16*a^2*b^2 + 8
*a*b*c^2 + c^4)/(a^3*b^3)))*sqrt(a*x^3 + b*x)*sqrt((a*b*sqrt((16*a^2*b^2 + 8*a*b*c^2 + c^4)/(a^3*b^3)) - 4*c)/
(a*b)) + 4*((4*a^4*b^3 - a^3*b^2*c^2)*x^3 + (4*a^3*b^4 - a^2*b^3*c^2)*x)*sqrt((16*a^2*b^2 + 8*a*b*c^2 + c^4)/(
a^3*b^3)))/(a^2*x^4 - 2*a*b*x^2 + b^2)) + 1/8*sqrt(1/2)*sqrt((a*b*sqrt((16*a^2*b^2 + 8*a*b*c^2 + c^4)/(a^3*b^3
)) - 4*c)/(a*b))*log(-(16*a^2*b^4 - b^2*c^4 + (16*a^4*b^2 - a^2*c^4)*x^4 + 6*(16*a^3*b^3 - a*b*c^4)*x^2 - 4*sq
rt(1/2)*(4*a^2*b^3*c + a*b^2*c^3 + (4*a^3*b^2*c + a^2*b*c^3)*x^2 + 4*(4*a^3*b^3 + a^2*b^2*c^2)*x + 2*(a^4*b^3*
x^2 + a^3*b^3*c*x + a^3*b^4)*sqrt((16*a^2*b^2 + 8*a*b*c^2 + c^4)/(a^3*b^3)))*sqrt(a*x^3 + b*x)*sqrt((a*b*sqrt(
(16*a^2*b^2 + 8*a*b*c^2 + c^4)/(a^3*b^3)) - 4*c)/(a*b)) + 4*((4*a^4*b^3 - a^3*b^2*c^2)*x^3 + (4*a^3*b^4 - a^2*
b^3*c^2)*x)*sqrt((16*a^2*b^2 + 8*a*b*c^2 + c^4)/(a^3*b^3)))/(a^2*x^4 - 2*a*b*x^2 + b^2))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

\text{Hanged}