Integrand size = 16, antiderivative size = 332 \[ \int \frac {x}{(3+x) \sqrt {1+x^3}} \, dx=-\frac {3 (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \arctan \left (\frac {\sqrt {\frac {13}{2}} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}}}{\sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}}}\right )}{\sqrt {26} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}-\frac {2 \sqrt {2 \left (97+56 \sqrt {3}\right )} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}+\frac {12 \sqrt [4]{3} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticPi}\left (97-56 \sqrt {3},\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{\sqrt {2-\sqrt {3}} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}} \]
-3/26*(1+x)*arctan(1/2*26^(1/2)*((1+x)/(1+x+3^(1/2))^2)^(1/2)/((x^2-x+1)/( 1+x+3^(1/2))^2)^(1/2))*((x^2-x+1)/(1+x+3^(1/2))^2)^(1/2)*26^(1/2)/(x^3+1)^ (1/2)/((1+x)/(1+x+3^(1/2))^2)^(1/2)-12*3^(1/4)*(1+x)*EllipticPi((-1-x+3^(1 /2))/(1+x+3^(1/2)),97-56*3^(1/2),I*3^(1/2)+2*I)*((x^2-x+1)/(1+x+3^(1/2))^2 )^(1/2)/(x^3+1)^(1/2)/(1/2*6^(1/2)-1/2*2^(1/2))/((1+x)/(1+x+3^(1/2))^2)^(1 /2)-2/3*(1+x)*EllipticF((1+x-3^(1/2))/(1+x+3^(1/2)),I*3^(1/2)+2*I)*((x^2-x +1)/(1+x+3^(1/2))^2)^(1/2)*(7*2^(1/2)+4*6^(1/2))*3^(3/4)/(x^3+1)^(1/2)/((1 +x)/(1+x+3^(1/2))^2)^(1/2)
Result contains complex when optimal does not.
Time = 10.23 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.58 \[ \int \frac {x}{(3+x) \sqrt {1+x^3}} \, dx=\frac {2 \sqrt {\frac {1+x}{1+\sqrt [3]{-1}}} \left (-\frac {\left (\sqrt [3]{-1}-x\right ) \sqrt {\frac {\sqrt [3]{-1}-(-1)^{2/3} x}{1+\sqrt [3]{-1}}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {1+(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right ),\sqrt [3]{-1}\right )}{\sqrt {\frac {1+(-1)^{2/3} x}{1+\sqrt [3]{-1}}}}+\frac {3 i \sqrt {1-x+x^2} \operatorname {EllipticPi}\left (\frac {i \sqrt {3}}{3+\sqrt [3]{-1}},\arcsin \left (\sqrt {\frac {1+(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right ),\sqrt [3]{-1}\right )}{3+\sqrt [3]{-1}}\right )}{\sqrt {1+x^3}} \]
(2*Sqrt[(1 + x)/(1 + (-1)^(1/3))]*(-((((-1)^(1/3) - x)*Sqrt[((-1)^(1/3) - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]*EllipticF[ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/( 1 + (-1)^(1/3))]], (-1)^(1/3)])/Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]) + ((3*I)*Sqrt[1 - x + x^2]*EllipticPi[(I*Sqrt[3])/(3 + (-1)^(1/3)), ArcSi n[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(3 + (-1)^(1/3) )))/Sqrt[1 + x^3]
Time = 0.94 (sec) , antiderivative size = 328, normalized size of antiderivative = 0.99, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {2569, 759, 2567, 25, 2538, 412, 435, 104, 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.
\(\displaystyle \int \frac {x}{(x+3) \sqrt {x^3+1}} \, dx\) |
\(\Big \downarrow \) 2569 |
\(\displaystyle \frac {3 \int \frac {x+\sqrt {3}+1}{(x+3) \sqrt {x^3+1}}dx}{2-\sqrt {3}}-\frac {\left (1+\sqrt {3}\right ) \int \frac {1}{\sqrt {x^3+1}}dx}{2-\sqrt {3}}\) |
\(\Big \downarrow \) 759 |
\(\displaystyle \frac {3 \int \frac {x+\sqrt {3}+1}{(x+3) \sqrt {x^3+1}}dx}{2-\sqrt {3}}-\frac {2 \left (1+\sqrt {3}\right ) \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \left (2-\sqrt {3}\right ) \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}\) |
\(\Big \downarrow \) 2567 |
\(\displaystyle \frac {12 \sqrt [4]{3} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \int -\frac {1}{\sqrt {1-\frac {\left (x-\sqrt {3}+1\right )^2}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {\frac {\left (x-\sqrt {3}+1\right )^2}{\left (x+\sqrt {3}+1\right )^2}-4 \sqrt {3}+7} \left (-\frac {\left (2-\sqrt {3}\right ) \left (x-\sqrt {3}+1\right )}{x+\sqrt {3}+1}+\sqrt {3}+2\right )}d\left (-\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )}{\sqrt {2-\sqrt {3}} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\frac {2 \left (1+\sqrt {3}\right ) \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \left (2-\sqrt {3}\right ) \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {12 \sqrt [4]{3} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \int \frac {1}{\sqrt {1-\frac {\left (x-\sqrt {3}+1\right )^2}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {\frac {\left (x-\sqrt {3}+1\right )^2}{\left (x+\sqrt {3}+1\right )^2}-4 \sqrt {3}+7} \left (-\frac {\left (2-\sqrt {3}\right ) \left (x-\sqrt {3}+1\right )}{x+\sqrt {3}+1}+\sqrt {3}+2\right )}d\left (-\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )}{\sqrt {2-\sqrt {3}} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\frac {2 \left (1+\sqrt {3}\right ) \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \left (2-\sqrt {3}\right ) \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}\) |
\(\Big \downarrow \) 2538 |
\(\displaystyle \frac {12 \sqrt [4]{3} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \left (\left (2-\sqrt {3}\right ) \int -\frac {x-\sqrt {3}+1}{\left (x+\sqrt {3}+1\right ) \sqrt {1-\frac {\left (x-\sqrt {3}+1\right )^2}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {\frac {\left (x-\sqrt {3}+1\right )^2}{\left (x+\sqrt {3}+1\right )^2}-4 \sqrt {3}+7} \left (-\frac {\left (7-4 \sqrt {3}\right ) \left (x-\sqrt {3}+1\right )^2}{\left (x+\sqrt {3}+1\right )^2}+4 \sqrt {3}+7\right )}d\left (-\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )-\left (2+\sqrt {3}\right ) \int \frac {1}{\sqrt {1-\frac {\left (x-\sqrt {3}+1\right )^2}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {\frac {\left (x-\sqrt {3}+1\right )^2}{\left (x+\sqrt {3}+1\right )^2}-4 \sqrt {3}+7} \left (-\frac {\left (7-4 \sqrt {3}\right ) \left (x-\sqrt {3}+1\right )^2}{\left (x+\sqrt {3}+1\right )^2}+4 \sqrt {3}+7\right )}d\left (-\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )\right )}{\sqrt {2-\sqrt {3}} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\frac {2 \left (1+\sqrt {3}\right ) \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \left (2-\sqrt {3}\right ) \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}\) |
\(\Big \downarrow \) 412 |
\(\displaystyle \frac {12 \sqrt [4]{3} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \left (\left (2-\sqrt {3}\right ) \int -\frac {x-\sqrt {3}+1}{\left (x+\sqrt {3}+1\right ) \sqrt {1-\frac {\left (x-\sqrt {3}+1\right )^2}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {\frac {\left (x-\sqrt {3}+1\right )^2}{\left (x+\sqrt {3}+1\right )^2}-4 \sqrt {3}+7} \left (-\frac {\left (7-4 \sqrt {3}\right ) \left (x-\sqrt {3}+1\right )^2}{\left (x+\sqrt {3}+1\right )^2}+4 \sqrt {3}+7\right )}d\left (-\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )+\sqrt {7-4 \sqrt {3}} \left (2+\sqrt {3}\right ) \operatorname {EllipticPi}\left (97-56 \sqrt {3},\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )\right )}{\sqrt {2-\sqrt {3}} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\frac {2 \left (1+\sqrt {3}\right ) \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \left (2-\sqrt {3}\right ) \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}\) |
\(\Big \downarrow \) 435 |
\(\displaystyle \frac {12 \sqrt [4]{3} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \left (\frac {1}{2} \left (2-\sqrt {3}\right ) \int \frac {1}{\sqrt {\frac {\left (x-\sqrt {3}+1\right )^2}{\left (x+\sqrt {3}+1\right )^2}-4 \sqrt {3}+7} \sqrt {\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}+1} \left (\frac {\left (7-4 \sqrt {3}\right ) \left (x-\sqrt {3}+1\right )}{x+\sqrt {3}+1}+4 \sqrt {3}+7\right )}d\frac {\left (x-\sqrt {3}+1\right )^2}{\left (x+\sqrt {3}+1\right )^2}+\sqrt {7-4 \sqrt {3}} \left (2+\sqrt {3}\right ) \operatorname {EllipticPi}\left (97-56 \sqrt {3},\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )\right )}{\sqrt {2-\sqrt {3}} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\frac {2 \left (1+\sqrt {3}\right ) \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \left (2-\sqrt {3}\right ) \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}\) |
\(\Big \downarrow \) 104 |
\(\displaystyle \frac {12 \sqrt [4]{3} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \left (\left (2-\sqrt {3}\right ) \int \frac {1}{-\frac {52 \left (2-\sqrt {3}\right ) \sqrt {\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}+1}}{\sqrt {\frac {\left (x-\sqrt {3}+1\right )^2}{\left (x+\sqrt {3}+1\right )^2}-4 \sqrt {3}+7}}-8 \sqrt {3}}d\frac {\sqrt {\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}+1}}{\sqrt {\frac {\left (x-\sqrt {3}+1\right )^2}{\left (x+\sqrt {3}+1\right )^2}-4 \sqrt {3}+7}}+\sqrt {7-4 \sqrt {3}} \left (2+\sqrt {3}\right ) \operatorname {EllipticPi}\left (97-56 \sqrt {3},\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )\right )}{\sqrt {2-\sqrt {3}} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\frac {2 \left (1+\sqrt {3}\right ) \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \left (2-\sqrt {3}\right ) \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {12 \sqrt [4]{3} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \left (\sqrt {7-4 \sqrt {3}} \left (2+\sqrt {3}\right ) \operatorname {EllipticPi}\left (97-56 \sqrt {3},\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )+\frac {\sqrt {\frac {1}{26} \left (2-\sqrt {3}\right )} \arctan \left (\frac {\sqrt {\frac {13}{2} \left (2-\sqrt {3}\right )} \left (x-\sqrt {3}+1\right )}{\sqrt [4]{3} \left (x+\sqrt {3}+1\right )}\right )}{4 \sqrt [4]{3}}\right )}{\sqrt {2-\sqrt {3}} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\frac {2 \left (1+\sqrt {3}\right ) \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \left (2-\sqrt {3}\right ) \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}\) |
(-2*(1 + Sqrt[3])*Sqrt[2 + Sqrt[3]]*(1 + x)*Sqrt[(1 - x + x^2)/(1 + Sqrt[3 ] + x)^2]*EllipticF[ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*Sq rt[3]])/(3^(1/4)*(2 - Sqrt[3])*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 + x^3]) + (12*3^(1/4)*(1 + x)*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*((Sqrt [(2 - Sqrt[3])/26]*ArcTan[(Sqrt[(13*(2 - Sqrt[3]))/2]*(1 - Sqrt[3] + x))/( 3^(1/4)*(1 + Sqrt[3] + x))])/(4*3^(1/4)) + Sqrt[7 - 4*Sqrt[3]]*(2 + Sqrt[3 ])*EllipticPi[97 - 56*Sqrt[3], ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)] , -7 - 4*Sqrt[3]]))/(Sqrt[2 - Sqrt[3]]*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*S qrt[1 + x^3])
3.2.60.3.1 Defintions of rubi rules used
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
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])
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* (c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] && !( !GtQ[f/e, 0] && S implerSqrtQ[-f/e, -d/c])
Int[(x_)^(m_.)*((a_.) + (b_.)*(x_)^2)^(p_.)*((c_.) + (d_.)*(x_)^2)^(q_.)*(( e_.) + (f_.)*(x_)^2)^(r_.), x_Symbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2) *(a + b*x)^p*(c + d*x)^q*(e + f*x)^r, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, f, p, q, r}, x] && IntegerQ[(m - 1)/2]
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]
Int[1/(((a_) + (b_.)*(x_))*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_) ^2]), x_Symbol] :> Simp[a Int[1/((a^2 - b^2*x^2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x], x] - Simp[b Int[x/((a^2 - b^2*x^2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x]
Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_ Symbol] :> With[{q = Simplify[(1 + Sqrt[3])*(f/e)]}, Simp[4*3^(1/4)*Sqrt[2 - Sqrt[3]]*f*(1 + q*x)*(Sqrt[(1 - q*x + q^2*x^2)/(1 + Sqrt[3] + q*x)^2]/(q* Sqrt[a + b*x^3]*Sqrt[(1 + q*x)/(1 + Sqrt[3] + q*x)^2])) Subst[Int[1/(((1 - Sqrt[3])*d - c*q + ((1 + Sqrt[3])*d - c*q)*x)*Sqrt[1 - x^2]*Sqrt[7 - 4*Sq rt[3] + x^2]), x], x, (-1 + Sqrt[3] - q*x)/(1 + Sqrt[3] + q*x)], x]] /; Fre eQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && EqQ[b*e^3 - 2*(5 + 3*Sqrt [3])*a*f^3, 0] && NeQ[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]
Int[((e_.) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x _Symbol] :> With[{q = Rt[b/a, 3]}, Simp[((1 + Sqrt[3])*f - e*q)/((1 + Sqrt[ 3])*d - c*q) Int[1/Sqrt[a + b*x^3], x], x] + Simp[(d*e - c*f)/((1 + Sqrt[ 3])*d - c*q) Int[(1 + Sqrt[3] + q*x)/((c + d*x)*Sqrt[a + b*x^3]), x], x]] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && NeQ[b^2*c^6 - 20*a *b*c^3*d^3 - 8*a^2*d^6, 0] && NeQ[b^2*e^6 - 20*a*b*e^3*f^3 - 8*a^2*f^6, 0]
Time = 0.96 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.72
method | result | size |
default | \(\frac {2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, F\left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}-\frac {3 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \Pi \left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {3}{4}+\frac {i \sqrt {3}}{4}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}\) | \(240\) |
elliptic | \(\frac {2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, F\left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}-\frac {3 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \Pi \left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {3}{4}+\frac {i \sqrt {3}}{4}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}\) | \(240\) |
2*(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))-3*(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)*Ellip ticPi(((x+1)/(3/2-1/2*I*3^(1/2)))^(1/2),-3/4+1/4*I*3^(1/2),((-3/2+1/2*I*3^ (1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2))
\[ \int \frac {x}{(3+x) \sqrt {1+x^3}} \, dx=\int { \frac {x}{\sqrt {x^{3} + 1} {\left (x + 3\right )}} \,d x } \]
\[ \int \frac {x}{(3+x) \sqrt {1+x^3}} \, dx=\int \frac {x}{\sqrt {\left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (x + 3\right )}\, dx \]
\[ \int \frac {x}{(3+x) \sqrt {1+x^3}} \, dx=\int { \frac {x}{\sqrt {x^{3} + 1} {\left (x + 3\right )}} \,d x } \]
\[ \int \frac {x}{(3+x) \sqrt {1+x^3}} \, dx=\int { \frac {x}{\sqrt {x^{3} + 1} {\left (x + 3\right )}} \,d x } \]
Time = 0.27 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.62 \[ \int \frac {x}{(3+x) \sqrt {1+x^3}} \, dx=\frac {\left (3+\sqrt {3}\,1{}\mathrm {i}\right )\,\sqrt {\frac {x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\left (2\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )-3\,\Pi \left (-\frac {3}{4}-\frac {\sqrt {3}\,1{}\mathrm {i}}{4};\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )\right )}{2\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]
((3^(1/2)*1i + 3)*((x + (3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 - 3/2))^(1/2 )*((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(((3^(1/2)*1i)/2 - x + 1/2)/((3^( 1/2)*1i)/2 + 3/2))^(1/2)*(2*ellipticF(asin(((x + 1)/((3^(1/2)*1i)/2 + 3/2) )^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2)) - 3*ellipticPi(- (3^(1/2)*1i)/4 - 3/4, asin(((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^ (1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2))))/(2*(x^3 - x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) - ((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/ 2 + 1/2))^(1/2))