\(\int \frac {2 b+a x}{(b+a x+x^2) \sqrt [4]{b x^2+a x^3}} \, dx\) [1610]

Optimal result

Integrand size = 33, antiderivative size = 110 \[ \int \frac {2 b+a x}{\left (b+a x+x^2\right ) \sqrt [4]{b x^2+a x^3}} \, dx=-\sqrt {2} \arctan \left (\frac {-\frac {x^2}{\sqrt {2}}+\frac {\sqrt {b x^2+a x^3}}{\sqrt {2}}}{x \sqrt [4]{b x^2+a x^3}}\right )+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{b x^2+a x^3}}{x^2+\sqrt {b x^2+a x^3}}\right ) \]

[Out]

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

Rubi [C] (verified)

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

Time = 0.83 (sec) , antiderivative size = 573, normalized size of antiderivative = 5.21, number of steps used = 13, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2081, 6860, 108, 107, 504, 1232} \[ \int \frac {2 b+a x}{\left (b+a x+x^2\right ) \sqrt [4]{b x^2+a x^3}} \, dx=\frac {\sqrt {2} \sqrt [4]{b} \left (\sqrt {a^2-4 b}+a\right ) \sqrt {-\frac {a x}{b}} \sqrt [4]{a x+b} \operatorname {EllipticPi}\left (-\frac {\sqrt {2} \sqrt {b}}{\sqrt {-a^2-\sqrt {a^2-4 b} a+2 b}},\arcsin \left (\frac {\sqrt [4]{b+a x}}{\sqrt [4]{b}}\right ),-1\right )}{\sqrt {-a \sqrt {a^2-4 b}-a^2+2 b} \sqrt [4]{a x^3+b x^2}}-\frac {\sqrt {2} \sqrt [4]{b} \left (\sqrt {a^2-4 b}+a\right ) \sqrt {-\frac {a x}{b}} \sqrt [4]{a x+b} \operatorname {EllipticPi}\left (\frac {\sqrt {2} \sqrt {b}}{\sqrt {-a^2-\sqrt {a^2-4 b} a+2 b}},\arcsin \left (\frac {\sqrt [4]{b+a x}}{\sqrt [4]{b}}\right ),-1\right )}{\sqrt {-a \sqrt {a^2-4 b}-a^2+2 b} \sqrt [4]{a x^3+b x^2}}+\frac {\sqrt {2} \sqrt [4]{b} \left (a-\sqrt {a^2-4 b}\right ) \sqrt {-\frac {a x}{b}} \sqrt [4]{a x+b} \operatorname {EllipticPi}\left (-\frac {\sqrt {2} \sqrt {b}}{\sqrt {-a^2+\sqrt {a^2-4 b} a+2 b}},\arcsin \left (\frac {\sqrt [4]{b+a x}}{\sqrt [4]{b}}\right ),-1\right )}{\sqrt {a \sqrt {a^2-4 b}-a^2+2 b} \sqrt [4]{a x^3+b x^2}}-\frac {\sqrt {2} \sqrt [4]{b} \left (a-\sqrt {a^2-4 b}\right ) \sqrt {-\frac {a x}{b}} \sqrt [4]{a x+b} \operatorname {EllipticPi}\left (\frac {\sqrt {2} \sqrt {b}}{\sqrt {-a^2+\sqrt {a^2-4 b} a+2 b}},\arcsin \left (\frac {\sqrt [4]{b+a x}}{\sqrt [4]{b}}\right ),-1\right )}{\sqrt {a \sqrt {a^2-4 b}-a^2+2 b} \sqrt [4]{a x^3+b x^2}} \]

[In]

Int[(2*b + a*x)/((b + a*x + x^2)*(b*x^2 + a*x^3)^(1/4)),x]

[Out]

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

Rule 107

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(1/4)), x_Symbol] :> Dist[-4, Subst[
Int[x^2/((b*e - a*f - b*x^4)*Sqrt[c - d*(e/f) + d*(x^4/f)]), x], x, (e + f*x)^(1/4)], x] /; FreeQ[{a, b, c, d,
e, f}, x] && GtQ[-f/(d*e - c*f), 0]

Rule 108

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(1/4)), x_Symbol] :> Dist[Sqrt[(-f)*
((c + d*x)/(d*e - c*f))]/Sqrt[c + d*x], Int[1/((a + b*x)*Sqrt[(-c)*(f/(d*e - c*f)) - d*f*(x/(d*e - c*f))]*(e +
f*x)^(1/4)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] &&  !GtQ[-f/(d*e - c*f), 0]

Rule 504

Int[(x_)^2/(((a_) + (b_.)*(x_)^4)*Sqrt[(c_) + (d_.)*(x_)^4]), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s
= Denominator[Rt[-a/b, 2]]}, Dist[s/(2*b), Int[1/((r + s*x^2)*Sqrt[c + d*x^4]), x], x] - Dist[s/(2*b), Int[1/
((r - s*x^2)*Sqrt[c + d*x^4]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 1232

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[-c/a, 4]}, Simp[(1/(d*Sqrt[
a]*q))*EllipticPi[-e/(d*q^2), ArcSin[q*x], -1], x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && 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 [4]{b+a x}\right ) \int \frac {2 b+a x}{\sqrt {x} \sqrt [4]{b+a x} \left (b+a x+x^2\right )} \, dx}{\sqrt [4]{b x^2+a x^3}} \\ & = \frac {\left (\sqrt {x} \sqrt [4]{b+a x}\right ) \int \left (\frac {a-\sqrt {a^2-4 b}}{\sqrt {x} \left (a-\sqrt {a^2-4 b}+2 x\right ) \sqrt [4]{b+a x}}+\frac {a+\sqrt {a^2-4 b}}{\sqrt {x} \left (a+\sqrt {a^2-4 b}+2 x\right ) \sqrt [4]{b+a x}}\right ) \, dx}{\sqrt [4]{b x^2+a x^3}} \\ & = \frac {\left (\left (a-\sqrt {a^2-4 b}\right ) \sqrt {x} \sqrt [4]{b+a x}\right ) \int \frac {1}{\sqrt {x} \left (a-\sqrt {a^2-4 b}+2 x\right ) \sqrt [4]{b+a x}} \, dx}{\sqrt [4]{b x^2+a x^3}}+\frac {\left (\left (a+\sqrt {a^2-4 b}\right ) \sqrt {x} \sqrt [4]{b+a x}\right ) \int \frac {1}{\sqrt {x} \left (a+\sqrt {a^2-4 b}+2 x\right ) \sqrt [4]{b+a x}} \, dx}{\sqrt [4]{b x^2+a x^3}} \\ & = \frac {\left (\left (a-\sqrt {a^2-4 b}\right ) \sqrt {-\frac {a x}{b}} \sqrt [4]{b+a x}\right ) \int \frac {1}{\sqrt {-\frac {a x}{b}} \left (a-\sqrt {a^2-4 b}+2 x\right ) \sqrt [4]{b+a x}} \, dx}{\sqrt [4]{b x^2+a x^3}}+\frac {\left (\left (a+\sqrt {a^2-4 b}\right ) \sqrt {-\frac {a x}{b}} \sqrt [4]{b+a x}\right ) \int \frac {1}{\sqrt {-\frac {a x}{b}} \left (a+\sqrt {a^2-4 b}+2 x\right ) \sqrt [4]{b+a x}} \, dx}{\sqrt [4]{b x^2+a x^3}} \\ & = -\frac {\left (4 \left (a-\sqrt {a^2-4 b}\right ) \sqrt {-\frac {a x}{b}} \sqrt [4]{b+a x}\right ) \text {Subst}\left (\int \frac {x^2}{\left (-a \left (a-\sqrt {a^2-4 b}\right )+2 b-2 x^4\right ) \sqrt {1-\frac {x^4}{b}}} \, dx,x,\sqrt [4]{b+a x}\right )}{\sqrt [4]{b x^2+a x^3}}-\frac {\left (4 \left (a+\sqrt {a^2-4 b}\right ) \sqrt {-\frac {a x}{b}} \sqrt [4]{b+a x}\right ) \text {Subst}\left (\int \frac {x^2}{\left (-a \left (a+\sqrt {a^2-4 b}\right )+2 b-2 x^4\right ) \sqrt {1-\frac {x^4}{b}}} \, dx,x,\sqrt [4]{b+a x}\right )}{\sqrt [4]{b x^2+a x^3}} \\ & = -\frac {\left (\sqrt {2} \left (a-\sqrt {a^2-4 b}\right ) \sqrt {-\frac {a x}{b}} \sqrt [4]{b+a x}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-a^2+a \sqrt {a^2-4 b}+2 b}-\sqrt {2} x^2\right ) \sqrt {1-\frac {x^4}{b}}} \, dx,x,\sqrt [4]{b+a x}\right )}{\sqrt [4]{b x^2+a x^3}}+\frac {\left (\sqrt {2} \left (a-\sqrt {a^2-4 b}\right ) \sqrt {-\frac {a x}{b}} \sqrt [4]{b+a x}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-a^2+a \sqrt {a^2-4 b}+2 b}+\sqrt {2} x^2\right ) \sqrt {1-\frac {x^4}{b}}} \, dx,x,\sqrt [4]{b+a x}\right )}{\sqrt [4]{b x^2+a x^3}}-\frac {\left (\sqrt {2} \left (a+\sqrt {a^2-4 b}\right ) \sqrt {-\frac {a x}{b}} \sqrt [4]{b+a x}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-a^2-a \sqrt {a^2-4 b}+2 b}-\sqrt {2} x^2\right ) \sqrt {1-\frac {x^4}{b}}} \, dx,x,\sqrt [4]{b+a x}\right )}{\sqrt [4]{b x^2+a x^3}}+\frac {\left (\sqrt {2} \left (a+\sqrt {a^2-4 b}\right ) \sqrt {-\frac {a x}{b}} \sqrt [4]{b+a x}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-a^2-a \sqrt {a^2-4 b}+2 b}+\sqrt {2} x^2\right ) \sqrt {1-\frac {x^4}{b}}} \, dx,x,\sqrt [4]{b+a x}\right )}{\sqrt [4]{b x^2+a x^3}} \\ & = \frac {\sqrt {2} \left (a+\sqrt {a^2-4 b}\right ) \sqrt [4]{b} \sqrt {-\frac {a x}{b}} \sqrt [4]{b+a x} \operatorname {EllipticPi}\left (-\frac {\sqrt {2} \sqrt {b}}{\sqrt {-a^2-a \sqrt {a^2-4 b}+2 b}},\arcsin \left (\frac {\sqrt [4]{b+a x}}{\sqrt [4]{b}}\right ),-1\right )}{\sqrt {-a^2-a \sqrt {a^2-4 b}+2 b} \sqrt [4]{b x^2+a x^3}}-\frac {\sqrt {2} \left (a+\sqrt {a^2-4 b}\right ) \sqrt [4]{b} \sqrt {-\frac {a x}{b}} \sqrt [4]{b+a x} \operatorname {EllipticPi}\left (\frac {\sqrt {2} \sqrt {b}}{\sqrt {-a^2-a \sqrt {a^2-4 b}+2 b}},\arcsin \left (\frac {\sqrt [4]{b+a x}}{\sqrt [4]{b}}\right ),-1\right )}{\sqrt {-a^2-a \sqrt {a^2-4 b}+2 b} \sqrt [4]{b x^2+a x^3}}+\frac {\sqrt {2} \left (a-\sqrt {a^2-4 b}\right ) \sqrt [4]{b} \sqrt {-\frac {a x}{b}} \sqrt [4]{b+a x} \operatorname {EllipticPi}\left (-\frac {\sqrt {2} \sqrt {b}}{\sqrt {-a^2+a \sqrt {a^2-4 b}+2 b}},\arcsin \left (\frac {\sqrt [4]{b+a x}}{\sqrt [4]{b}}\right ),-1\right )}{\sqrt {-a^2+a \sqrt {a^2-4 b}+2 b} \sqrt [4]{b x^2+a x^3}}-\frac {\sqrt {2} \left (a-\sqrt {a^2-4 b}\right ) \sqrt [4]{b} \sqrt {-\frac {a x}{b}} \sqrt [4]{b+a x} \operatorname {EllipticPi}\left (\frac {\sqrt {2} \sqrt {b}}{\sqrt {-a^2+a \sqrt {a^2-4 b}+2 b}},\arcsin \left (\frac {\sqrt [4]{b+a x}}{\sqrt [4]{b}}\right ),-1\right )}{\sqrt {-a^2+a \sqrt {a^2-4 b}+2 b} \sqrt [4]{b x^2+a x^3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.95 \[ \int \frac {2 b+a x}{\left (b+a x+x^2\right ) \sqrt [4]{b x^2+a x^3}} \, dx=\frac {\sqrt {2} \sqrt {x} \sqrt [4]{b+a x} \left (-\arctan \left (\frac {-x+\sqrt {b+a x}}{\sqrt {2} \sqrt {x} \sqrt [4]{b+a x}}\right )+\text {arctanh}\left (\frac {\sqrt {2} \sqrt {x} \sqrt [4]{b+a x}}{x+\sqrt {b+a x}}\right )\right )}{\sqrt [4]{x^2 (b+a x)}} \]

[In]

Integrate[(2*b + a*x)/((b + a*x + x^2)*(b*x^2 + a*x^3)^(1/4)),x]

[Out]

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

Maple [F]

\[\int \frac {a x +2 b}{\left (a x +x^{2}+b \right ) \left (a \,x^{3}+b \,x^{2}\right )^{\frac {1}{4}}}d x\]

[In]

int((a*x+2*b)/(a*x+x^2+b)/(a*x^3+b*x^2)^(1/4),x)

[Out]

int((a*x+2*b)/(a*x+x^2+b)/(a*x^3+b*x^2)^(1/4),x)

Fricas [F(-1)]

Timed out. \[ \int \frac {2 b+a x}{\left (b+a x+x^2\right ) \sqrt [4]{b x^2+a x^3}} \, dx=\text {Timed out} \]

[In]

integrate((a*x+2*b)/(a*x+x^2+b)/(a*x^3+b*x^2)^(1/4),x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

\[ \int \frac {2 b+a x}{\left (b+a x+x^2\right ) \sqrt [4]{b x^2+a x^3}} \, dx=\int \frac {a x + 2 b}{\sqrt [4]{x^{2} \left (a x + b\right )} \left (a x + b + x^{2}\right )}\, dx \]

[In]

integrate((a*x+2*b)/(a*x+x**2+b)/(a*x**3+b*x**2)**(1/4),x)

[Out]

Integral((a*x + 2*b)/((x**2*(a*x + b))**(1/4)*(a*x + b + x**2)), x)

Maxima [F]

\[ \int \frac {2 b+a x}{\left (b+a x+x^2\right ) \sqrt [4]{b x^2+a x^3}} \, dx=\int { \frac {a x + 2 \, b}{{\left (a x^{3} + b x^{2}\right )}^{\frac {1}{4}} {\left (a x + x^{2} + b\right )}} \,d x } \]

[In]

integrate((a*x+2*b)/(a*x+x^2+b)/(a*x^3+b*x^2)^(1/4),x, algorithm="maxima")

[Out]

integrate((a*x + 2*b)/((a*x^3 + b*x^2)^(1/4)*(a*x + x^2 + b)), x)

Giac [F]

\[ \int \frac {2 b+a x}{\left (b+a x+x^2\right ) \sqrt [4]{b x^2+a x^3}} \, dx=\int { \frac {a x + 2 \, b}{{\left (a x^{3} + b x^{2}\right )}^{\frac {1}{4}} {\left (a x + x^{2} + b\right )}} \,d x } \]

[In]

integrate((a*x+2*b)/(a*x+x^2+b)/(a*x^3+b*x^2)^(1/4),x, algorithm="giac")

[Out]

integrate((a*x + 2*b)/((a*x^3 + b*x^2)^(1/4)*(a*x + x^2 + b)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {2 b+a x}{\left (b+a x+x^2\right ) \sqrt [4]{b x^2+a x^3}} \, dx=\int \frac {2\,b+a\,x}{{\left (a\,x^3+b\,x^2\right )}^{1/4}\,\left (x^2+a\,x+b\right )} \,d x \]

[In]

int((2*b + a*x)/((a*x^3 + b*x^2)^(1/4)*(b + a*x + x^2)),x)

[Out]

int((2*b + a*x)/((a*x^3 + b*x^2)^(1/4)*(b + a*x + x^2)), x)