\(\int \frac {a+b x}{(3-x^2) \sqrt [3]{1+x^2}} \, dx\) [54]

Optimal result

Integrand size = 24, antiderivative size = 198 \[ \int \frac {a+b x}{\left (3-x^2\right ) \sqrt [3]{1+x^2}} \, dx=-\frac {a \arctan (x)}{6\ 2^{2/3}}+\frac {a \arctan \left (\frac {x}{1+\sqrt [3]{2} \sqrt [3]{1+x^2}}\right )}{2\ 2^{2/3}}-\frac {\sqrt {3} b \arctan \left (\frac {1+\sqrt [3]{2} \sqrt [3]{1+x^2}}{\sqrt {3}}\right )}{2\ 2^{2/3}}-\frac {a \text {arctanh}\left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3} \sqrt {3}}-\frac {a \text {arctanh}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1+x^2}\right )}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {b \log \left (3-x^2\right )}{4\ 2^{2/3}}-\frac {3 b \log \left (2^{2/3}-\sqrt [3]{1+x^2}\right )}{4\ 2^{2/3}} \] Output:

-1/12*a*arctan(x)*2^(1/3)+1/4*a*arctan(x/(1+2^(1/3)*(x^2+1)^(1/3)))*2^(1/3 
)+1/8*b*ln(-x^2+3)*2^(1/3)-3/8*b*ln(2^(2/3)-(x^2+1)^(1/3))*2^(1/3)-1/12*a* 
arctanh(3^(1/2)/x)*2^(1/3)*3^(1/2)-1/12*a*arctanh((1-2^(1/3)*(x^2+1)^(1/3) 
)*3^(1/2)/x)*2^(1/3)*3^(1/2)-1/4*b*arctan(1/3*(1+2^(1/3)*(x^2+1)^(1/3))*3^ 
(1/2))*3^(1/2)*2^(1/3)
                                                                                    
                                                                                    
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1072\) vs. \(2(198)=396\).

Time = 9.75 (sec) , antiderivative size = 1072, normalized size of antiderivative = 5.41 \[ \int \frac {a+b x}{\left (3-x^2\right ) \sqrt [3]{1+x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

((2*(Sqrt[3]*a^4 - 12*a^3*b + 18*Sqrt[3]*a^2*b^2 - 36*a*b^3 + 9*Sqrt[3]*b^ 
4)*ArcTan[(3*b + a*x + 6*2^(1/3)*b*(1 + x^2)^(1/3) - Sqrt[3]*(a + b*x + 2* 
2^(1/3)*a*(1 + x^2)^(1/3)))/(3*b*(Sqrt[3] - x) + a*(-3 + Sqrt[3]*x))])/(Sq 
rt[3]*a^3 - 9*a^2*b + 9*Sqrt[3]*a*b^2 - 9*b^3) - (2*(Sqrt[3]*a^4 + 12*a^3* 
b + 18*Sqrt[3]*a^2*b^2 + 36*a*b^3 + 9*Sqrt[3]*b^4)*ArcTan[(3*b + a*x + 6*2 
^(1/3)*b*(1 + x^2)^(1/3) + Sqrt[3]*(a + b*x + 2*2^(1/3)*a*(1 + x^2)^(1/3)) 
)/(3*b*(Sqrt[3] + x) + a*(3 + Sqrt[3]*x))])/(Sqrt[3]*a^3 + 9*a^2*b + 9*Sqr 
t[3]*a*b^2 + 9*b^3) + (2*(a^4 - 4*Sqrt[3]*a^3*b + 18*a^2*b^2 - 12*Sqrt[3]* 
a*b^3 + 9*b^4)*Log[3*b + a*x - 3*2^(1/3)*b*(1 + x^2)^(1/3) - Sqrt[3]*(a + 
b*x - 2^(1/3)*a*(1 + x^2)^(1/3))])/(Sqrt[3]*a^3 - 9*a^2*b + 9*Sqrt[3]*a*b^ 
2 - 9*b^3) - (2*(a^4 + 4*Sqrt[3]*a^3*b + 18*a^2*b^2 + 12*Sqrt[3]*a*b^3 + 9 
*b^4)*Log[-3*b - a*x + 3*2^(1/3)*b*(1 + x^2)^(1/3) - Sqrt[3]*(a + b*x - 2^ 
(1/3)*a*(1 + x^2)^(1/3))])/(Sqrt[3]*a^3 + 9*a^2*b + 9*Sqrt[3]*a*b^2 + 9*b^ 
3) - ((a^4 - 4*Sqrt[3]*a^3*b + 18*a^2*b^2 - 12*Sqrt[3]*a*b^3 + 9*b^4)*Log[ 
3*a^2 + 9*b^2 + 12*a*b*x + a^2*x^2 + 3*b^2*x^2 + 3*2^(1/3)*(a^2 + 3*b^2 + 
2*a*b*x)*(1 + x^2)^(1/3) + 3*2^(2/3)*(a^2 + 3*b^2)*(1 + x^2)^(2/3) + Sqrt[ 
3]*(-6*a*b - 2*a^2*x - 6*b^2*x - 2*a*b*x^2 - 2^(1/3)*(6*a*b + a^2*x + 3*b^ 
2*x)*(1 + x^2)^(1/3) - 6*2^(2/3)*a*b*(1 + x^2)^(2/3))])/(Sqrt[3]*a^3 - 9*a 
^2*b + 9*Sqrt[3]*a*b^2 - 9*b^3) + ((a^4 + 4*Sqrt[3]*a^3*b + 18*a^2*b^2 + 1 
2*Sqrt[3]*a*b^3 + 9*b^4)*Log[3*a^2 + 9*b^2 + 12*a*b*x + a^2*x^2 + 3*b^2...
 

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1343, 304, 353, 67, 16, 1082, 217}

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.

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

\(\Big \downarrow \) 1343

\(\displaystyle a \int \frac {1}{\left (3-x^2\right ) \sqrt [3]{x^2+1}}dx+b \int \frac {x}{\left (3-x^2\right ) \sqrt [3]{x^2+1}}dx\)

\(\Big \downarrow \) 304

\(\displaystyle b \int \frac {x}{\left (3-x^2\right ) \sqrt [3]{x^2+1}}dx+a \left (\frac {\arctan \left (\frac {x}{\sqrt [3]{2} \sqrt [3]{x^2+1}+1}\right )}{2\ 2^{2/3}}-\frac {\arctan (x)}{6\ 2^{2/3}}-\frac {\text {arctanh}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{x^2+1}\right )}{x}\right )}{2\ 2^{2/3} \sqrt {3}}-\frac {\text {arctanh}\left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3} \sqrt {3}}\right )\)

\(\Big \downarrow \) 353

\(\displaystyle \frac {1}{2} b \int \frac {1}{\left (3-x^2\right ) \sqrt [3]{x^2+1}}dx^2+a \left (\frac {\arctan \left (\frac {x}{\sqrt [3]{2} \sqrt [3]{x^2+1}+1}\right )}{2\ 2^{2/3}}-\frac {\arctan (x)}{6\ 2^{2/3}}-\frac {\text {arctanh}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{x^2+1}\right )}{x}\right )}{2\ 2^{2/3} \sqrt {3}}-\frac {\text {arctanh}\left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3} \sqrt {3}}\right )\)

\(\Big \downarrow \) 67

\(\displaystyle \frac {1}{2} b \left (\frac {3 \int \frac {1}{2^{2/3}-\sqrt [3]{x^2+1}}d\sqrt [3]{x^2+1}}{2\ 2^{2/3}}-\frac {3}{2} \int \frac {1}{x^4+2^{2/3} \sqrt [3]{x^2+1}+2 \sqrt [3]{2}}d\sqrt [3]{x^2+1}+\frac {\log \left (3-x^2\right )}{2\ 2^{2/3}}\right )+a \left (\frac {\arctan \left (\frac {x}{\sqrt [3]{2} \sqrt [3]{x^2+1}+1}\right )}{2\ 2^{2/3}}-\frac {\arctan (x)}{6\ 2^{2/3}}-\frac {\text {arctanh}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{x^2+1}\right )}{x}\right )}{2\ 2^{2/3} \sqrt {3}}-\frac {\text {arctanh}\left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3} \sqrt {3}}\right )\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {1}{2} b \left (-\frac {3}{2} \int \frac {1}{x^4+2^{2/3} \sqrt [3]{x^2+1}+2 \sqrt [3]{2}}d\sqrt [3]{x^2+1}+\frac {\log \left (3-x^2\right )}{2\ 2^{2/3}}-\frac {3 \log \left (2^{2/3}-\sqrt [3]{x^2+1}\right )}{2\ 2^{2/3}}\right )+a \left (\frac {\arctan \left (\frac {x}{\sqrt [3]{2} \sqrt [3]{x^2+1}+1}\right )}{2\ 2^{2/3}}-\frac {\arctan (x)}{6\ 2^{2/3}}-\frac {\text {arctanh}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{x^2+1}\right )}{x}\right )}{2\ 2^{2/3} \sqrt {3}}-\frac {\text {arctanh}\left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3} \sqrt {3}}\right )\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {1}{2} b \left (\frac {3 \int \frac {1}{-x^4-3}d\left (\sqrt [3]{2} \sqrt [3]{x^2+1}+1\right )}{2^{2/3}}+\frac {\log \left (3-x^2\right )}{2\ 2^{2/3}}-\frac {3 \log \left (2^{2/3}-\sqrt [3]{x^2+1}\right )}{2\ 2^{2/3}}\right )+a \left (\frac {\arctan \left (\frac {x}{\sqrt [3]{2} \sqrt [3]{x^2+1}+1}\right )}{2\ 2^{2/3}}-\frac {\arctan (x)}{6\ 2^{2/3}}-\frac {\text {arctanh}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{x^2+1}\right )}{x}\right )}{2\ 2^{2/3} \sqrt {3}}-\frac {\text {arctanh}\left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3} \sqrt {3}}\right )\)

\(\Big \downarrow \) 217

\(\displaystyle a \left (\frac {\arctan \left (\frac {x}{\sqrt [3]{2} \sqrt [3]{x^2+1}+1}\right )}{2\ 2^{2/3}}-\frac {\arctan (x)}{6\ 2^{2/3}}-\frac {\text {arctanh}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{x^2+1}\right )}{x}\right )}{2\ 2^{2/3} \sqrt {3}}-\frac {\text {arctanh}\left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3} \sqrt {3}}\right )+\frac {1}{2} b \left (-\frac {\sqrt {3} \arctan \left (\frac {\sqrt [3]{2} \sqrt [3]{x^2+1}+1}{\sqrt {3}}\right )}{2^{2/3}}+\frac {\log \left (3-x^2\right )}{2\ 2^{2/3}}-\frac {3 \log \left (2^{2/3}-\sqrt [3]{x^2+1}\right )}{2\ 2^{2/3}}\right )\)

Input:

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

Output:

a*(-1/6*ArcTan[x]/2^(2/3) + ArcTan[x/(1 + 2^(1/3)*(1 + x^2)^(1/3))]/(2*2^( 
2/3)) - ArcTanh[Sqrt[3]/x]/(2*2^(2/3)*Sqrt[3]) - ArcTanh[(Sqrt[3]*(1 - 2^( 
1/3)*(1 + x^2)^(1/3)))/x]/(2*2^(2/3)*Sqrt[3])) + (b*(-((Sqrt[3]*ArcTan[(1 
+ 2^(1/3)*(1 + x^2)^(1/3))/Sqrt[3]])/2^(2/3)) + Log[3 - x^2]/(2*2^(2/3)) - 
 (3*Log[2^(2/3) - (1 + x^2)^(1/3)])/(2*2^(2/3))))/2
 

Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 67

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[ 
{q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q), x 
] + (Simp[3/(2*b)   Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/3)], 
 x] - Simp[3/(2*b*q)   Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] / 
; FreeQ[{a, b, c, d}, x] && PosQ[(b*c - a*d)/b]
 

rule 217

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

rule 304

Int[1/(((a_) + (b_.)*(x_)^2)^(1/3)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Wit 
h[{q = Rt[b/a, 2]}, Simp[q*(ArcTanh[Sqrt[3]/(q*x)]/(2*2^(2/3)*Sqrt[3]*a^(1/ 
3)*d)), x] + (-Simp[q*(ArcTan[(a^(1/3)*q*x)/(a^(1/3) + 2^(1/3)*(a + b*x^2)^ 
(1/3))]/(2*2^(2/3)*a^(1/3)*d)), x] + Simp[q*(ArcTan[q*x]/(6*2^(2/3)*a^(1/3) 
*d)), x] + Simp[q*(ArcTanh[Sqrt[3]*((a^(1/3) - 2^(1/3)*(a + b*x^2)^(1/3))/( 
a^(1/3)*q*x))]/(2*2^(2/3)*Sqrt[3]*a^(1/3)*d)), x])] /; FreeQ[{a, b, c, d}, 
x] && NeQ[b*c - a*d, 0] && EqQ[b*c + 3*a*d, 0] && PosQ[b/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 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {a+b x}{\left (3-x^2\right ) \sqrt [3]{1+x^2}} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (trace 0)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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