\(\int \frac {-b+a x^2}{(b+c x+a x^2) \sqrt {b x+a x^3}} \, dx\) [451]

Optimal result

Integrand size = 35, antiderivative size = 36 \[ \int \frac {-b+a x^2}{\left (b+c x+a x^2\right ) \sqrt {b x+a x^3}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {c} \sqrt {b x+a x^3}}{b+a x^2}\right )}{\sqrt {c}} \]

[Out]

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

Rubi [C] (verified)

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

Time = 1.32 (sec) , antiderivative size = 289, normalized size of antiderivative = 8.03, number of steps used = 13, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2081, 6860, 335, 226, 947, 174, 551} \[ \int \frac {-b+a x^2}{\left (b+c x+a x^2\right ) \sqrt {b x+a x^3}} \, dx=-\frac {2 \sqrt [4]{b} \sqrt {x} \sqrt {\frac {a x^2}{b}+1} \operatorname {EllipticPi}\left (\frac {2 \sqrt {-a} \sqrt {b}}{c-\sqrt {c^2-4 a b}},\arcsin \left (\frac {\sqrt [4]{-a} \sqrt {x}}{\sqrt [4]{b}}\right ),-1\right )}{\sqrt [4]{-a} \sqrt {a x^3+b x}}-\frac {2 \sqrt [4]{b} \sqrt {x} \sqrt {\frac {a x^2}{b}+1} \operatorname {EllipticPi}\left (\frac {2 \sqrt {-a} \sqrt {b}}{c+\sqrt {c^2-4 a b}},\arcsin \left (\frac {\sqrt [4]{-a} \sqrt {x}}{\sqrt [4]{b}}\right ),-1\right )}{\sqrt [4]{-a} \sqrt {a x^3+b x}}+\frac {\sqrt {x} \left (\sqrt {a} x+\sqrt {b}\right ) \sqrt {\frac {a x^2+b}{\left (\sqrt {a} x+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt [4]{b} \sqrt {a x^3+b x}} \]

[In]

Int[(-b + a*x^2)/((b + c*x + 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*b^(1/4)*Sqrt[x]*Sqrt[1 + (a*x^2)/b]*EllipticPi[(2*Sqr
t[-a]*Sqrt[b])/(c - Sqrt[-4*a*b + c^2]), ArcSin[((-a)^(1/4)*Sqrt[x])/b^(1/4)], -1])/((-a)^(1/4)*Sqrt[b*x + a*x
^3]) - (2*b^(1/4)*Sqrt[x]*Sqrt[1 + (a*x^2)/b]*EllipticPi[(2*Sqrt[-a]*Sqrt[b])/(c + Sqrt[-4*a*b + c^2]), ArcSin
[((-a)^(1/4)*Sqrt[x])/b^(1/4)], -1])/((-a)^(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 6860

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

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

Mathematica [A] (verified)

Time = 1.31 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.64 \[ \int \frac {-b+a x^2}{\left (b+c x+a x^2\right ) \sqrt {b x+a x^3}} \, dx=-\frac {2 \sqrt {x} \sqrt {b+a x^2} \arctan \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b+a x^2}}\right )}{\sqrt {c} \sqrt {x \left (b+a x^2\right )}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 3.76 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.69

method	result	size
default	\(\frac {2 \arctan \left (\frac {\sqrt {x \left (a \,x^{2}+b \right )}}{x \sqrt {c}}\right )}{\sqrt {c}}\)	\(25\)
pseudoelliptic	\(\frac {2 \arctan \left (\frac {\sqrt {x \left (a \,x^{2}+b \right )}}{x \sqrt {c}}\right )}{\sqrt {c}}\)	\(25\)
elliptic	\(\text {Expression too large to display}\)	\(1142\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 159, normalized size of antiderivative = 4.42 \[ \int \frac {-b+a x^2}{\left (b+c x+a x^2\right ) \sqrt {b x+a x^3}} \, dx=\left [-\frac {\sqrt {-c} \log \left (\frac {a^{2} x^{4} - 6 \, a c x^{3} - 6 \, b c x + {\left (2 \, a b + c^{2}\right )} x^{2} + b^{2} - 4 \, \sqrt {a x^{3} + b x} {\left (a x^{2} - c x + b\right )} \sqrt {-c}}{a^{2} x^{4} + 2 \, a c x^{3} + 2 \, b c x + {\left (2 \, a b + c^{2}\right )} x^{2} + b^{2}}\right )}{2 \, c}, \frac {\arctan \left (\frac {\sqrt {a x^{3} + b x} {\left (a x^{2} - c x + b\right )} \sqrt {c}}{2 \, {\left (a c x^{3} + b c x\right )}}\right )}{\sqrt {c}}\right ] \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {-b+a x^2}{\left (b+c x+a x^2\right ) \sqrt {b x+a x^3}} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(c^2-4*a*b>0)', see `assume?` f
or more deta

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 7.38 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.42 \[ \int \frac {-b+a x^2}{\left (b+c x+a x^2\right ) \sqrt {b x+a x^3}} \, dx=\frac {\ln \left (\frac {\frac {b}{2}-\frac {c\,x}{2}+\frac {a\,x^2}{2}+\sqrt {c}\,\sqrt {a\,x^3+b\,x}\,1{}\mathrm {i}}{a\,x^2+c\,x+b}\right )\,1{}\mathrm {i}}{\sqrt {c}} \]

[In]

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

[Out]

(log((b/2 - (c*x)/2 + (a*x^2)/2 + c^(1/2)*(b*x + a*x^3)^(1/2)*1i)/(b + c*x + a*x^2))*1i)/c^(1/2)