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

3.1.70.1 Optimal result

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

output

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*arctanh(1/2*x/(-x^2-1)^(1/4)*2^(1/2))*2^ 
(1/2)
 

3.1.70.2 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 \[ \int \frac {a+b x}{\sqrt [4]{-1-x^2} \left (2+x^2\right )} \, dx=\frac {x \left (b x \sqrt [4]{1+x^2} \operatorname {AppellF1}\left (1,\frac {1}{4},1,2,-x^2,-\frac {x^2}{2}\right )-\frac {24 a \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{4},1,\frac {3}{2},-x^2,-\frac {x^2}{2}\right )}{\left (2+x^2\right ) \left (-6 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{4},1,\frac {3}{2},-x^2,-\frac {x^2}{2}\right )+x^2 \left (2 \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},2,\frac {5}{2},-x^2,-\frac {x^2}{2}\right )+\operatorname {AppellF1}\left (\frac {3}{2},\frac {5}{4},1,\frac {5}{2},-x^2,-\frac {x^2}{2}\right )\right )\right )}\right )}{4 \sqrt [4]{-1-x^2}} \]

input

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

output

(x*(b*x*(1 + x^2)^(1/4)*AppellF1[1, 1/4, 1, 2, -x^2, -1/2*x^2] - (24*a*App 
ellF1[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))
 

3.1.70.3 Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.09, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1343, 309, 353, 73, 827, 216, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 1343

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

\(\Big \downarrow \) 309

\(\displaystyle b \int \frac {x}{\sqrt [4]{-x^2-1} \left (x^2+2\right )}dx+a \left (\frac {\arctan \left (\frac {x}{\sqrt {2} \sqrt [4]{-x^2-1}}\right )}{2 \sqrt {2}}+\frac {\text {arctanh}\left (\frac {x}{\sqrt {2} \sqrt [4]{-x^2-1}}\right )}{2 \sqrt {2}}\right )\)

\(\Big \downarrow \) 353

\(\displaystyle \frac {1}{2} b \int \frac {1}{\sqrt [4]{-x^2-1} \left (x^2+2\right )}dx^2+a \left (\frac {\arctan \left (\frac {x}{\sqrt {2} \sqrt [4]{-x^2-1}}\right )}{2 \sqrt {2}}+\frac {\text {arctanh}\left (\frac {x}{\sqrt {2} \sqrt [4]{-x^2-1}}\right )}{2 \sqrt {2}}\right )\)

\(\Big \downarrow \) 73

\(\displaystyle a \left (\frac {\arctan \left (\frac {x}{\sqrt {2} \sqrt [4]{-x^2-1}}\right )}{2 \sqrt {2}}+\frac {\text {arctanh}\left (\frac {x}{\sqrt {2} \sqrt [4]{-x^2-1}}\right )}{2 \sqrt {2}}\right )-2 b \int \frac {x^4}{1-x^8}d\sqrt [4]{-x^2-1}\)

\(\Big \downarrow \) 827

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

\(\Big \downarrow \) 216

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

\(\Big \downarrow \) 219

\(\displaystyle a \left (\frac {\arctan \left (\frac {x}{\sqrt {2} \sqrt [4]{-x^2-1}}\right )}{2 \sqrt {2}}+\frac {\text {arctanh}\left (\frac {x}{\sqrt {2} \sqrt [4]{-x^2-1}}\right )}{2 \sqrt {2}}\right )-2 b \left (\frac {1}{2} \text {arctanh}\left (\sqrt [4]{-x^2-1}\right )-\frac {1}{2} \arctan \left (\sqrt [4]{-x^2-1}\right )\right )\)

input

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

output

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

3.1.70.3.1 Defintions of rubi rules used

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[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] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 216

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

rule 219

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] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 309

Int[1/(((a_) + (b_.)*(x_)^2)^(1/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Wit 
h[{q = Rt[-b^2/a, 4]}, Simp[(b/(2*Sqrt[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 353

Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] 
 :> Simp[1/2   Subst[Int[(a + b*x)^p*(c + d*x)^q, x], x, x^2], x] /; FreeQ[ 
{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0]
 

rule 827

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 
 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b)   Int[1/(r + s*x^2), x], 
x] - Simp[s/(2*b)   Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !GtQ 
[a/b, 0]
 

rule 1343

Int[((g_) + (h_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_)*((d_) + (f_.)*(x_)^2)^(q 
_), x_Symbol] :> Simp[g   Int[(a + c*x^2)^p*(d + f*x^2)^q, x], x] + Simp[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]
 

3.1.70.4 Maple [F]

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

input

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

output

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

3.1.70.5 Fricas [F(-1)]

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

input

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

output

Timed out
 

3.1.70.6 Sympy [F]

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

input

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

output

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

3.1.70.7 Maxima [F]

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

input

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

output

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

3.1.70.8 Giac [F]

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

input

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

output

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

3.1.70.9 Mupad [F(-1)]

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

input

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

output

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