\(\int \frac {a+b x}{\sqrt [4]{-1-x^2} (2+x^2)} \, dx\) [70]

Optimal result

Integrand size = 24, antiderivative size = 88 \[ \int \frac {a+b x}{\sqrt [4]{-1-x^2} \left (2+x^2\right )} \, dx=\frac {a \arctan \left (\frac {x}{\sqrt {2} \sqrt [4]{-1-x^2}}\right )}{2 \sqrt {2}}+b \arctan \left (\sqrt [4]{-1-x^2}\right )+\frac {a \text {arctanh}\left (\frac {x}{\sqrt {2} \sqrt [4]{-1-x^2}}\right )}{2 \sqrt {2}}-b \text {arctanh}\left (\sqrt [4]{-1-x^2}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1024, 407, 455, 65, 304, 209, 212} \[ \int \frac {a+b x}{\sqrt [4]{-1-x^2} \left (2+x^2\right )} \, dx=\frac {a \arctan \left (\frac {x}{\sqrt {2} \sqrt [4]{-x^2-1}}\right )}{2 \sqrt {2}}+\frac {a \text {arctanh}\left (\frac {x}{\sqrt {2} \sqrt [4]{-x^2-1}}\right )}{2 \sqrt {2}}+b \arctan \left (\sqrt [4]{-x^2-1}\right )-b \text {arctanh}\left (\sqrt [4]{-x^2-1}\right ) \]

[In]

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

[Out]

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

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 304

Int[(x_)^2/((a_) + (b_.)*(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), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !
GtQ[a/b, 0]

Rule 407

Int[1/(((a_) + (b_.)*(x_)^2)^(1/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> With[{q = Rt[-b^2/a, 4]}, Simp[(b/(2*S
qrt[2]*a*d*q))*ArcTan[q*(x/(Sqrt[2]*(a + b*x^2)^(1/4)))], x] + Simp[(b/(2*Sqrt[2]*a*d*q))*ArcTanh[q*(x/(Sqrt[2
]*(a + b*x^2)^(1/4)))], x]] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c - 2*a*d, 0] && NegQ[b^2/a]

Rule 455

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 1024

Int[((g_) + (h_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_)*((d_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Dist[g, Int[(a + c
*x^2)^p*(d + f*x^2)^q, x], x] + Dist[h, Int[x*(a + c*x^2)^p*(d + f*x^2)^q, x], x] /; FreeQ[{a, c, d, f, g, h,
p, q}, x]

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

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 10.18 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.84 \[ \int \frac {a+b x}{\sqrt [4]{-1-x^2} \left (2+x^2\right )} \, dx=\frac {x \left (b x \sqrt [4]{1+x^2} \operatorname {AppellF1}\left (1,\frac {1}{4},1,2,-x^2,-\frac {x^2}{2}\right )-\frac {24 a \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{4},1,\frac {3}{2},-x^2,-\frac {x^2}{2}\right )}{\left (2+x^2\right ) \left (-6 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{4},1,\frac {3}{2},-x^2,-\frac {x^2}{2}\right )+x^2 \left (2 \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},2,\frac {5}{2},-x^2,-\frac {x^2}{2}\right )+\operatorname {AppellF1}\left (\frac {3}{2},\frac {5}{4},1,\frac {5}{2},-x^2,-\frac {x^2}{2}\right )\right )\right )}\right )}{4 \sqrt [4]{-1-x^2}} \]

[In]

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

[Out]

(x*(b*x*(1 + x^2)^(1/4)*AppellF1[1, 1/4, 1, 2, -x^2, -1/2*x^2] - (24*a*AppellF1[1/2, 1/4, 1, 3/2, -x^2, -1/2*x
^2])/((2 + x^2)*(-6*AppellF1[1/2, 1/4, 1, 3/2, -x^2, -1/2*x^2] + x^2*(2*AppellF1[3/2, 1/4, 2, 5/2, -x^2, -1/2*
x^2] + AppellF1[3/2, 5/4, 1, 5/2, -x^2, -1/2*x^2])))))/(4*(-1 - x^2)^(1/4))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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