\(\int \frac {\sqrt {b x+a x^3}}{-b^2+a^2 x^4} \, dx\) [1731]

Optimal result

Integrand size = 29, antiderivative size = 117 \[ \int \frac {\sqrt {b x+a x^3}}{-b^2+a^2 x^4} \, dx=\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {b x+a x^3}}{b+a x^2}\right )}{2 \sqrt {2} a^{3/4} b^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {b x+a x^3}}{b+a x^2}\right )}{2 \sqrt {2} a^{3/4} b^{3/4}} \]

[Out]

1/4*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*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 [A] (verified)

Time = 0.23 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.39, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {2081, 1268, 477, 504, 1225, 226, 1713, 211, 214} \[ \int \frac {\sqrt {b x+a x^3}}{-b^2+a^2 x^4} \, dx=\frac {\sqrt {a x^3+b x} \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a x^2+b}}\right )}{2 \sqrt {2} a^{3/4} b^{3/4} \sqrt {x} \sqrt {a x^2+b}}-\frac {\sqrt {a x^3+b x} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a x^2+b}}\right )}{2 \sqrt {2} a^{3/4} b^{3/4} \sqrt {x} \sqrt {a x^2+b}} \]

[In]

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

[Out]

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

Rule 211

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

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

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 477

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno
minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/e^n))^p*(c + d*(x^(k*n)/e^n))^q, x], x, (e*
x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege
rQ[p]

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 1225

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

Rule 1268

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[(f*x)^m*(d +
e*x^2)^(q + p)*(a/d + (c/e)*x^2)^p, x] /; FreeQ[{a, c, d, e, f, q, m, q}, x] && EqQ[c*d^2 + a*e^2, 0] && Integ
erQ[p]

Rule 1713

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Dist[A, Subst[Int[1/
(d + 2*a*e*x^2), x], x, x/Sqrt[a + c*x^4]], x] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ
[c*d^2 - a*e^2, 0] && EqQ[B*d + A*e, 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]

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

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {b x+a x^3}}{-b^2+a^2 x^4} \, dx=\frac {\sqrt {x} \sqrt {b+a x^2} \left (\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {b+a x^2}}\right )-\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {b+a x^2}}\right )\right )}{2 \sqrt {2} a^{3/4} b^{3/4} \sqrt {x \left (b+a x^2\right )}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.84

method	result	size
default	\(-\frac {\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} \left (\ln \left (\frac {-\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} x -\sqrt {\left (a \,x^{2}+b \right ) x}}{\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} x -\sqrt {\left (a \,x^{2}+b \right ) x}}\right )+2 \arctan \left (\frac {\sqrt {\left (a \,x^{2}+b \right ) x}\, \sqrt {2}}{2 x \left (a b \right )^{\frac {1}{4}}}\right )\right )}{8 a b}\)	\(98\)
pseudoelliptic	\(-\frac {\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} \left (\ln \left (\frac {-\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} x -\sqrt {\left (a \,x^{2}+b \right ) x}}{\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} x -\sqrt {\left (a \,x^{2}+b \right ) x}}\right )+2 \arctan \left (\frac {\sqrt {\left (a \,x^{2}+b \right ) x}\, \sqrt {2}}{2 x \left (a b \right )^{\frac {1}{4}}}\right )\right )}{8 a b}\)	\(98\)
elliptic	\(\frac {\sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{2 a^{2} \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}+\frac {\sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}+\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{2 a^{2} \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}+\frac {\sqrt {a b}}{a}\right )}\)	\(296\)

[In]

int((a*x^3+b*x)^(1/2)/(a^2*x^4-b^2),x,method=_RETURNVERBOSE)

[Out]

-1/8*2^(1/2)*(a*b)^(1/4)*(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)))/a/b

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.36 (sec) , antiderivative size = 589, normalized size of antiderivative = 5.03 \[ \int \frac {\sqrt {b x+a x^3}}{-b^2+a^2 x^4} \, dx=-\frac {1}{8} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a^{3} b^{3}}\right )^{\frac {1}{4}} \log \left (\frac {a^{2} x^{4} + 6 \, a b x^{2} + b^{2} + 4 \, {\left (4 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a^{3} b^{3} x \left (\frac {1}{a^{3} b^{3}}\right )^{\frac {3}{4}} + \left (\frac {1}{4}\right )^{\frac {1}{4}} {\left (a^{2} b x^{2} + a b^{2}\right )} \left (\frac {1}{a^{3} b^{3}}\right )^{\frac {1}{4}}\right )} \sqrt {a x^{3} + b x} + 4 \, {\left (a^{3} b^{2} x^{3} + a^{2} b^{3} x\right )} \sqrt {\frac {1}{a^{3} b^{3}}}}{a^{2} x^{4} - 2 \, a b x^{2} + b^{2}}\right ) + \frac {1}{8} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a^{3} b^{3}}\right )^{\frac {1}{4}} \log \left (\frac {a^{2} x^{4} + 6 \, a b x^{2} + b^{2} - 4 \, {\left (4 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a^{3} b^{3} x \left (\frac {1}{a^{3} b^{3}}\right )^{\frac {3}{4}} + \left (\frac {1}{4}\right )^{\frac {1}{4}} {\left (a^{2} b x^{2} + a b^{2}\right )} \left (\frac {1}{a^{3} b^{3}}\right )^{\frac {1}{4}}\right )} \sqrt {a x^{3} + b x} + 4 \, {\left (a^{3} b^{2} x^{3} + a^{2} b^{3} x\right )} \sqrt {\frac {1}{a^{3} b^{3}}}}{a^{2} x^{4} - 2 \, a b x^{2} + b^{2}}\right ) - \frac {1}{8} i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a^{3} b^{3}}\right )^{\frac {1}{4}} \log \left (\frac {a^{2} x^{4} + 6 \, a b x^{2} + b^{2} - 4 \, {\left (4 i \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a^{3} b^{3} x \left (\frac {1}{a^{3} b^{3}}\right )^{\frac {3}{4}} + \left (\frac {1}{4}\right )^{\frac {1}{4}} {\left (-i \, a^{2} b x^{2} - i \, a b^{2}\right )} \left (\frac {1}{a^{3} b^{3}}\right )^{\frac {1}{4}}\right )} \sqrt {a x^{3} + b x} - 4 \, {\left (a^{3} b^{2} x^{3} + a^{2} b^{3} x\right )} \sqrt {\frac {1}{a^{3} b^{3}}}}{a^{2} x^{4} - 2 \, a b x^{2} + b^{2}}\right ) + \frac {1}{8} i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a^{3} b^{3}}\right )^{\frac {1}{4}} \log \left (\frac {a^{2} x^{4} + 6 \, a b x^{2} + b^{2} - 4 \, {\left (-4 i \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a^{3} b^{3} x \left (\frac {1}{a^{3} b^{3}}\right )^{\frac {3}{4}} + \left (\frac {1}{4}\right )^{\frac {1}{4}} {\left (i \, a^{2} b x^{2} + i \, a b^{2}\right )} \left (\frac {1}{a^{3} b^{3}}\right )^{\frac {1}{4}}\right )} \sqrt {a x^{3} + b x} - 4 \, {\left (a^{3} b^{2} x^{3} + a^{2} b^{3} x\right )} \sqrt {\frac {1}{a^{3} b^{3}}}}{a^{2} x^{4} - 2 \, a b x^{2} + b^{2}}\right ) \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\sqrt {b x+a x^3}}{-b^2+a^2 x^4} \, dx=\int { \frac {\sqrt {a x^{3} + b x}}{a^{2} x^{4} - b^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\sqrt {b x+a x^3}}{-b^2+a^2 x^4} \, dx=\int { \frac {\sqrt {a x^{3} + b x}}{a^{2} x^{4} - b^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {b x+a x^3}}{-b^2+a^2 x^4} \, dx=\text {Hanged} \]

[In]

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

[Out]

\text{Hanged}