\(\int \frac {a+b x}{(1+x)^{3/2} (1-x+x^2)^{3/2}} \, dx\) [691]

Optimal result

Integrand size = 25, antiderivative size = 304 \[ \int \frac {a+b x}{(1+x)^{3/2} \left (1-x+x^2\right )^{3/2}} \, dx=\frac {2 x (a+b x)}{3 \sqrt {1+x} \sqrt {1-x+x^2}}-\frac {2 b \left (1+x^3\right )}{3 \sqrt {1+x} \left (1+\sqrt {3}+x\right ) \sqrt {1-x+x^2}}+\frac {\sqrt {2-\sqrt {3}} b \sqrt {1+x} \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} E\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{3^{3/4} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1-x+x^2}}+\frac {2 \sqrt {2+\sqrt {3}} \left (a+b-\sqrt {3} b\right ) \sqrt {1+x} \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1-x+x^2}} \] Output:

2/3*x*(b*x+a)/(1+x)^(1/2)/(x^2-x+1)^(1/2)-2/3*b*(x^3+1)/(1+x)^(1/2)/(1+x+3 
^(1/2))/(x^2-x+1)^(1/2)+1/3*3^(1/4)*(1/2*6^(1/2)-1/2*2^(1/2))*b*(1+x)^(1/2 
)*((x^2-x+1)/(1+x+3^(1/2))^2)^(1/2)*EllipticE((1+x-3^(1/2))/(1+x+3^(1/2)), 
I*3^(1/2)+2*I)/((1+x)/(1+x+3^(1/2))^2)^(1/2)/(x^2-x+1)^(1/2)+2/9*(1/2*6^(1 
/2)+1/2*2^(1/2))*(a+b-3^(1/2)*b)*(1+x)^(1/2)*((x^2-x+1)/(1+x+3^(1/2))^2)^( 
1/2)*EllipticF((1+x-3^(1/2))/(1+x+3^(1/2)),I*3^(1/2)+2*I)*3^(3/4)/((1+x)/( 
1+x+3^(1/2))^2)^(1/2)/(x^2-x+1)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 32.18 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.37 \[ \int \frac {a+b x}{(1+x)^{3/2} \left (1-x+x^2\right )^{3/2}} \, dx=\frac {12 \sqrt {-\frac {i}{3 i+\sqrt {3}}} x (a+b x)-12 \sqrt {-\frac {i}{3 i+\sqrt {3}}} b \left (1-x+x^2\right )+3 i \sqrt {2} \left (i+\sqrt {3}\right ) b (1+x)^{3/2} \sqrt {\frac {3 i+\sqrt {3}-\frac {6 i}{1+x}}{3 i+\sqrt {3}}} \sqrt {-\frac {-3 i+\sqrt {3}+\frac {6 i}{1+x}}{3 i-\sqrt {3}}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {6 i}{3 i+\sqrt {3}}}}{\sqrt {1+x}}\right )|\frac {3 i+\sqrt {3}}{3 i-\sqrt {3}}\right )+\sqrt {2} \left (2 i \sqrt {3} a+\left (3-i \sqrt {3}\right ) b\right ) (1+x)^{3/2} \sqrt {\frac {3 i+\sqrt {3}-\frac {6 i}{1+x}}{3 i+\sqrt {3}}} \sqrt {-\frac {-3 i+\sqrt {3}+\frac {6 i}{1+x}}{3 i-\sqrt {3}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {6 i}{3 i+\sqrt {3}}}}{\sqrt {1+x}}\right ),\frac {3 i+\sqrt {3}}{3 i-\sqrt {3}}\right )}{18 \sqrt {-\frac {i}{3 i+\sqrt {3}}} \sqrt {1+x} \sqrt {1-x+x^2}} \] Input:

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

Output:

(12*Sqrt[(-I)/(3*I + Sqrt[3])]*x*(a + b*x) - 12*Sqrt[(-I)/(3*I + Sqrt[3])] 
*b*(1 - x + x^2) + (3*I)*Sqrt[2]*(I + Sqrt[3])*b*(1 + x)^(3/2)*Sqrt[(3*I + 
 Sqrt[3] - (6*I)/(1 + x))/(3*I + Sqrt[3])]*Sqrt[-((-3*I + Sqrt[3] + (6*I)/ 
(1 + x))/(3*I - Sqrt[3]))]*EllipticE[I*ArcSinh[Sqrt[(-6*I)/(3*I + Sqrt[3]) 
]/Sqrt[1 + x]], (3*I + Sqrt[3])/(3*I - Sqrt[3])] + Sqrt[2]*((2*I)*Sqrt[3]* 
a + (3 - I*Sqrt[3])*b)*(1 + x)^(3/2)*Sqrt[(3*I + Sqrt[3] - (6*I)/(1 + x))/ 
(3*I + Sqrt[3])]*Sqrt[-((-3*I + Sqrt[3] + (6*I)/(1 + x))/(3*I - Sqrt[3]))] 
*EllipticF[I*ArcSinh[Sqrt[(-6*I)/(3*I + Sqrt[3])]/Sqrt[1 + x]], (3*I + Sqr 
t[3])/(3*I - Sqrt[3])])/(18*Sqrt[(-I)/(3*I + Sqrt[3])]*Sqrt[1 + x]*Sqrt[1 
- x + x^2])
 

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 298, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1210, 2394, 27, 2417, 759, 2416}

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.

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

\(\Big \downarrow \) 1210

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

\(\Big \downarrow \) 2394

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2417

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

\(\Big \downarrow \) 759

\(\displaystyle \frac {\sqrt {x^3+1} \left (\frac {1}{3} \left (\frac {2 \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \left (a-\sqrt {3} b+b\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-b \int \frac {x-\sqrt {3}+1}{\sqrt {x^3+1}}dx\right )+\frac {2 x (a+b x)}{3 \sqrt {x^3+1}}\right )}{\sqrt {x+1} \sqrt {x^2-x+1}}\)

\(\Big \downarrow \) 2416

\(\displaystyle \frac {\sqrt {x^3+1} \left (\frac {1}{3} \left (\frac {2 \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \left (a-\sqrt {3} b+b\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-b \left (\frac {2 \sqrt {x^3+1}}{x+\sqrt {3}+1}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{\sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}\right )\right )+\frac {2 x (a+b x)}{3 \sqrt {x^3+1}}\right )}{\sqrt {x+1} \sqrt {x^2-x+1}}\)

Input:

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

Output:

(Sqrt[1 + x^3]*((2*x*(a + b*x))/(3*Sqrt[1 + x^3]) + (-(b*((2*Sqrt[1 + x^3] 
)/(1 + Sqrt[3] + x) - (3^(1/4)*Sqrt[2 - Sqrt[3]]*(1 + x)*Sqrt[(1 - x + x^2 
)/(1 + Sqrt[3] + x)^2]*EllipticE[ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x 
)], -7 - 4*Sqrt[3]])/(Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 + x^3]))) + 
 (2*Sqrt[2 + Sqrt[3]]*(a + b - Sqrt[3]*b)*(1 + x)*Sqrt[(1 - x + x^2)/(1 + 
Sqrt[3] + x)^2]*EllipticF[ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 
- 4*Sqrt[3]])/(3^(1/4)*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 + x^3]))/3 
))/(Sqrt[1 + x]*Sqrt[1 - x + x^2])
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 759

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], 
s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 + Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s 
*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[s* 
((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 - Sqrt[3])*s 
+ r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x] & 
& PosQ[a]
 

rule 1210

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_) 
 + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^FracPart[p]*((a + b*x + 
c*x^2)^FracPart[p]/(a*d + c*e*x^3)^FracPart[p])   Int[(f + g*x)^n*(a*d + c* 
e*x^3)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[b*d + a*e, 
 0] && EqQ[c*d + b*e, 0] && EqQ[m, p]
 

rule 2394

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(-x)*Pq*((a + b 
*x^n)^(p + 1)/(a*n*(p + 1))), x] + Simp[1/(a*n*(p + 1))   Int[ExpandToSum[n 
*(p + 1)*Pq + D[x*Pq, x], x]*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x 
] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && LtQ[Expon[Pq, x], n - 1]
 

rule 2416

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N 
umer[Simplify[(1 - Sqrt[3])*(d/c)]], s = Denom[Simplify[(1 - Sqrt[3])*(d/c) 
]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 + Sqrt[3])*s + r*x))), x] - S 
imp[3^(1/4)*Sqrt[2 - Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/( 
(1 + Sqrt[3])*s + r*x)^2]/(r^2*Sqrt[a + b*x^3]*Sqrt[s*((s + r*x)/((1 + Sqrt 
[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3]) 
*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && Eq 
Q[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]
 

rule 2417

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N 
umer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(c*r - (1 - Sqrt[3])*d*s)/r 
  Int[1/Sqrt[a + b*x^3], x], x] + Simp[d/r   Int[((1 - Sqrt[3])*s + r*x)/Sq 
rt[a + b*x^3], x], x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && NeQ[b*c^3 - 
2*(5 - 3*Sqrt[3])*a*d^3, 0]
 

Maple [A] (verified)

Time = 5.61 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.12

method	result	size
elliptic	\(\frac {\sqrt {\left (x +1\right ) \left (x^{2}-x +1\right )}\, \left (-\frac {2 \left (-\frac {1}{3} b \,x^{2}-\frac {1}{3} a x \right )}{\sqrt {x^{3}+1}}-\frac {2 b \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (\left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )+\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )\right )}{3 \sqrt {x^{3}+1}}+\frac {2 a \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {x^{3}+1}}\right )}{\sqrt {x +1}\, \sqrt {x^{2}-x +1}}\)	\(341\)
risch	\(\frac {2 x \left (b x +a \right )}{3 \sqrt {x +1}\, \sqrt {x^{2}-x +1}}+\frac {\left (-\frac {2 b \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (\left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )+\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )\right )}{3 \sqrt {x^{3}+1}}+\frac {2 a \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {x^{3}+1}}\right ) \sqrt {\left (x +1\right ) \left (x^{2}-x +1\right )}}{\sqrt {x +1}\, \sqrt {x^{2}-x +1}}\)	\(345\)
default	\(-\frac {\sqrt {x +1}\, \sqrt {x^{2}-x +1}\, \left (i \sqrt {3}\, \sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{i \sqrt {3}-3}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}, \sqrt {-\frac {i \sqrt {3}-3}{i \sqrt {3}+3}}\right ) a +i \sqrt {3}\, \sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{i \sqrt {3}-3}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}, \sqrt {-\frac {i \sqrt {3}-3}{i \sqrt {3}+3}}\right ) b -3 \sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{i \sqrt {3}-3}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}, \sqrt {-\frac {i \sqrt {3}-3}{i \sqrt {3}+3}}\right ) a +3 \sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{i \sqrt {3}-3}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}, \sqrt {-\frac {i \sqrt {3}-3}{i \sqrt {3}+3}}\right ) b -6 \sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{i \sqrt {3}-3}}\, \operatorname {EllipticE}\left (\sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}, \sqrt {-\frac {i \sqrt {3}-3}{i \sqrt {3}+3}}\right ) b -2 b \,x^{2}-2 a x \right )}{3 \left (x^{3}+1\right )}\)	\(584\)

Input:

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

Output:

((x+1)*(x^2-x+1))^(1/2)/(x+1)^(1/2)/(x^2-x+1)^(1/2)*(-2*(-1/3*b*x^2-1/3*a* 
x)/(x^3+1)^(1/2)-2/3*b*(3/2-1/2*I*3^(1/2))*((x+1)/(3/2-1/2*I*3^(1/2)))^(1/ 
2)*((x-1/2-1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2)*((x-1/2+1/2*I*3^(1/2 
))/(-3/2+1/2*I*3^(1/2)))^(1/2)/(x^3+1)^(1/2)*((-3/2-1/2*I*3^(1/2))*Ellipti 
cE(((x+1)/(3/2-1/2*I*3^(1/2)))^(1/2),((-3/2+1/2*I*3^(1/2))/(-3/2-1/2*I*3^( 
1/2)))^(1/2))+(1/2+1/2*I*3^(1/2))*EllipticF(((x+1)/(3/2-1/2*I*3^(1/2)))^(1 
/2),((-3/2+1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2)))+2/3*a*(3/2-1/2*I*3 
^(1/2))*((x+1)/(3/2-1/2*I*3^(1/2)))^(1/2)*((x-1/2-1/2*I*3^(1/2))/(-3/2-1/2 
*I*3^(1/2)))^(1/2)*((x-1/2+1/2*I*3^(1/2))/(-3/2+1/2*I*3^(1/2)))^(1/2)/(x^3 
+1)^(1/2)*EllipticF(((x+1)/(3/2-1/2*I*3^(1/2)))^(1/2),((-3/2+1/2*I*3^(1/2) 
)/(-3/2-1/2*I*3^(1/2)))^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.20 \[ \int \frac {a+b x}{(1+x)^{3/2} \left (1-x+x^2\right )^{3/2}} \, dx=\frac {2 \, {\left ({\left (b x^{2} + a x\right )} \sqrt {x^{2} - x + 1} \sqrt {x + 1} + {\left (a x^{3} + a\right )} {\rm weierstrassPInverse}\left (0, -4, x\right ) + {\left (b x^{3} + b\right )} {\rm weierstrassZeta}\left (0, -4, {\rm weierstrassPInverse}\left (0, -4, x\right )\right )\right )}}{3 \, {\left (x^{3} + 1\right )}} \] Input:

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

Output:

2/3*((b*x^2 + a*x)*sqrt(x^2 - x + 1)*sqrt(x + 1) + (a*x^3 + a)*weierstrass 
PInverse(0, -4, x) + (b*x^3 + b)*weierstrassZeta(0, -4, weierstrassPInvers 
e(0, -4, x)))/(x^3 + 1)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {a+b x}{(1+x)^{3/2} \left (1-x+x^2\right )^{3/2}} \, dx=\left (\int \frac {\sqrt {x +1}\, \sqrt {x^{2}-x +1}\, x}{x^{6}+2 x^{3}+1}d x \right ) b +\left (\int \frac {\sqrt {x +1}\, \sqrt {x^{2}-x +1}}{x^{6}+2 x^{3}+1}d x \right ) a \] Input:

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

Output:

int((sqrt(x + 1)*sqrt(x**2 - x + 1)*x)/(x**6 + 2*x**3 + 1),x)*b + int((sqr 
t(x + 1)*sqrt(x**2 - x + 1))/(x**6 + 2*x**3 + 1),x)*a