Integrand size = 29, antiderivative size = 265 \[ \int \frac {e+f x}{(c+d x) \sqrt {c^3+4 d^3 x^3}} \, dx=\frac {2 (d e-c f) \arctan \left (\frac {\sqrt {3} \sqrt {c} (c+2 d x)}{\sqrt {c^3+4 d^3 x^3}}\right )}{3 \sqrt {3} c^{3/2} d^2}+\frac {\sqrt [3]{2} \sqrt {2+\sqrt {3}} (2 d e+c f) \left (c+2^{2/3} d x\right ) \sqrt {\frac {c^2-2^{2/3} c d x+2 \sqrt [3]{2} d^2 x^2}{\left (\left (1+\sqrt {3}\right ) c+2^{2/3} d x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) c+2^{2/3} d x}{\left (1+\sqrt {3}\right ) c+2^{2/3} d x}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} c d^2 \sqrt {\frac {c \left (c+2^{2/3} d x\right )}{\left (\left (1+\sqrt {3}\right ) c+2^{2/3} d x\right )^2}} \sqrt {c^3+4 d^3 x^3}} \]
2/9*(-c*f+d*e)*arctan((2*d*x+c)*3^(1/2)*c^(1/2)/(4*d^3*x^3+c^3)^(1/2))/c^( 3/2)/d^2*3^(1/2)+1/9*2^(1/3)*(c*f+2*d*e)*(c+2^(2/3)*d*x)*EllipticF((2^(2/3 )*d*x+c*(1-3^(1/2)))/(2^(2/3)*d*x+c*(1+3^(1/2))),I*3^(1/2)+2*I)*(1/2*6^(1/ 2)+1/2*2^(1/2))*((c^2-2^(2/3)*c*d*x+2*2^(1/3)*d^2*x^2)/(2^(2/3)*d*x+c*(1+3 ^(1/2)))^2)^(1/2)*3^(3/4)/c/d^2/(4*d^3*x^3+c^3)^(1/2)/(c*(c+2^(2/3)*d*x)/( 2^(2/3)*d*x+c*(1+3^(1/2)))^2)^(1/2)
Result contains complex when optimal does not.
Time = 11.07 (sec) , antiderivative size = 380, normalized size of antiderivative = 1.43 \[ \int \frac {e+f x}{(c+d x) \sqrt {c^3+4 d^3 x^3}} \, dx=\frac {\sqrt [6]{2} \sqrt {\frac {\sqrt [3]{2} c+2 d x}{\left (1+\sqrt [3]{-1}\right ) c}} \left (-f \sqrt {\frac {\sqrt [3]{-2} c-2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}} \left (\sqrt [3]{-1} \left (2+\sqrt [3]{-2}\right ) c-2 \left (\sqrt [3]{-1}+2^{2/3}\right ) d x\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {\sqrt [3]{2} c+2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}}}{\sqrt [6]{2}}\right ),\sqrt [3]{-1}\right )+\frac {\sqrt [3]{-1} 2^{2/3} \left (1+\sqrt [3]{-1}\right ) (-d e+c f) \sqrt {\frac {\sqrt [3]{2} c+2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}} \sqrt {2^{2/3}-\frac {2 \sqrt [3]{2} d x}{c}+\frac {4 d^2 x^2}{c^2}} \operatorname {EllipticPi}\left (\frac {i \sqrt [3]{2} \sqrt {3}}{2+\sqrt [3]{-2}},\arcsin \left (\frac {\sqrt {\frac {\sqrt [3]{2} c+2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}}}{\sqrt [6]{2}}\right ),\sqrt [3]{-1}\right )}{\sqrt {3}}\right )}{\left (2+\sqrt [3]{-2}\right ) d^2 \sqrt {\frac {\sqrt [3]{2} c+2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}} \sqrt {c^3+4 d^3 x^3}} \]
(2^(1/6)*Sqrt[(2^(1/3)*c + 2*d*x)/((1 + (-1)^(1/3))*c)]*(-(f*Sqrt[((-2)^(1 /3)*c - 2*(-1)^(2/3)*d*x)/((1 + (-1)^(1/3))*c)]*((-1)^(1/3)*(2 + (-2)^(1/3 ))*c - 2*((-1)^(1/3) + 2^(2/3))*d*x)*EllipticF[ArcSin[Sqrt[(2^(1/3)*c + 2* (-1)^(2/3)*d*x)/((1 + (-1)^(1/3))*c)]/2^(1/6)], (-1)^(1/3)]) + ((-1)^(1/3) *2^(2/3)*(1 + (-1)^(1/3))*(-(d*e) + c*f)*Sqrt[(2^(1/3)*c + 2*(-1)^(2/3)*d* x)/((1 + (-1)^(1/3))*c)]*Sqrt[2^(2/3) - (2*2^(1/3)*d*x)/c + (4*d^2*x^2)/c^ 2]*EllipticPi[(I*2^(1/3)*Sqrt[3])/(2 + (-2)^(1/3)), ArcSin[Sqrt[(2^(1/3)*c + 2*(-1)^(2/3)*d*x)/((1 + (-1)^(1/3))*c)]/2^(1/6)], (-1)^(1/3)])/Sqrt[3]) )/((2 + (-2)^(1/3))*d^2*Sqrt[(2^(1/3)*c + 2*(-1)^(2/3)*d*x)/((1 + (-1)^(1/ 3))*c)]*Sqrt[c^3 + 4*d^3*x^3])
Time = 0.58 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2564, 759, 2562, 216}
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.
\(\displaystyle \int \frac {e+f x}{(c+d x) \sqrt {c^3+4 d^3 x^3}} \, dx\) |
\(\Big \downarrow \) 2564 |
\(\displaystyle \frac {(c f+2 d e) \int \frac {1}{\sqrt {c^3+4 d^3 x^3}}dx}{3 c d}+\frac {(d e-c f) \int \frac {c-2 d x}{(c+d x) \sqrt {c^3+4 d^3 x^3}}dx}{3 c d}\) |
\(\Big \downarrow \) 759 |
\(\displaystyle \frac {(d e-c f) \int \frac {c-2 d x}{(c+d x) \sqrt {c^3+4 d^3 x^3}}dx}{3 c d}+\frac {\sqrt [3]{2} \sqrt {2+\sqrt {3}} \left (c+2^{2/3} d x\right ) \sqrt {\frac {c^2-2^{2/3} c d x+2 \sqrt [3]{2} d^2 x^2}{\left (\left (1+\sqrt {3}\right ) c+2^{2/3} d x\right )^2}} (c f+2 d e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) c+2^{2/3} d x}{\left (1+\sqrt {3}\right ) c+2^{2/3} d x}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} c d^2 \sqrt {\frac {c \left (c+2^{2/3} d x\right )}{\left (\left (1+\sqrt {3}\right ) c+2^{2/3} d x\right )^2}} \sqrt {c^3+4 d^3 x^3}}\) |
\(\Big \downarrow \) 2562 |
\(\displaystyle \frac {2 (d e-c f) \int \frac {1}{\frac {3 c (c+2 d x)^2}{c^3+4 d^3 x^3}+1}d\frac {c+2 d x}{c \sqrt {c^3+4 d^3 x^3}}}{3 d^2}+\frac {\sqrt [3]{2} \sqrt {2+\sqrt {3}} \left (c+2^{2/3} d x\right ) \sqrt {\frac {c^2-2^{2/3} c d x+2 \sqrt [3]{2} d^2 x^2}{\left (\left (1+\sqrt {3}\right ) c+2^{2/3} d x\right )^2}} (c f+2 d e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) c+2^{2/3} d x}{\left (1+\sqrt {3}\right ) c+2^{2/3} d x}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} c d^2 \sqrt {\frac {c \left (c+2^{2/3} d x\right )}{\left (\left (1+\sqrt {3}\right ) c+2^{2/3} d x\right )^2}} \sqrt {c^3+4 d^3 x^3}}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {\sqrt [3]{2} \sqrt {2+\sqrt {3}} \left (c+2^{2/3} d x\right ) \sqrt {\frac {c^2-2^{2/3} c d x+2 \sqrt [3]{2} d^2 x^2}{\left (\left (1+\sqrt {3}\right ) c+2^{2/3} d x\right )^2}} (c f+2 d e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) c+2^{2/3} d x}{\left (1+\sqrt {3}\right ) c+2^{2/3} d x}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} c d^2 \sqrt {\frac {c \left (c+2^{2/3} d x\right )}{\left (\left (1+\sqrt {3}\right ) c+2^{2/3} d x\right )^2}} \sqrt {c^3+4 d^3 x^3}}+\frac {2 \arctan \left (\frac {\sqrt {3} \sqrt {c} (c+2 d x)}{\sqrt {c^3+4 d^3 x^3}}\right ) (d e-c f)}{3 \sqrt {3} c^{3/2} d^2}\) |
(2*(d*e - c*f)*ArcTan[(Sqrt[3]*Sqrt[c]*(c + 2*d*x))/Sqrt[c^3 + 4*d^3*x^3]] )/(3*Sqrt[3]*c^(3/2)*d^2) + (2^(1/3)*Sqrt[2 + Sqrt[3]]*(2*d*e + c*f)*(c + 2^(2/3)*d*x)*Sqrt[(c^2 - 2^(2/3)*c*d*x + 2*2^(1/3)*d^2*x^2)/((1 + Sqrt[3]) *c + 2^(2/3)*d*x)^2]*EllipticF[ArcSin[((1 - Sqrt[3])*c + 2^(2/3)*d*x)/((1 + Sqrt[3])*c + 2^(2/3)*d*x)], -7 - 4*Sqrt[3]])/(3*3^(1/4)*c*d^2*Sqrt[(c*(c + 2^(2/3)*d*x))/((1 + Sqrt[3])*c + 2^(2/3)*d*x)^2]*Sqrt[c^3 + 4*d^3*x^3])
3.1.64.3.1 Defintions of rubi rules used
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])
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[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_ Symbol] :> Simp[2*(e/d) Subst[Int[1/(1 + 3*a*x^2), x], x, (1 + 2*d*(x/c)) /Sqrt[a + b*x^3]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && EqQ[b*c^3 - 4*a*d^3, 0] && EqQ[2*d*e + c*f, 0]
Int[((e_.) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x _Symbol] :> Simp[(2*d*e + c*f)/(3*c*d) Int[1/Sqrt[a + b*x^3], x], x] + Si mp[(d*e - c*f)/(3*c*d) Int[(c - 2*d*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] && (EqQ[b*c^3 - 4*a* d^3, 0] || EqQ[b*c^3 + 8*a*d^3, 0]) && NeQ[2*d*e + c*f, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 899 vs. \(2 (216 ) = 432\).
Time = 0.96 (sec) , antiderivative size = 900, normalized size of antiderivative = 3.40
method | result | size |
default | \(\text {Expression too large to display}\) | \(900\) |
elliptic | \(\text {Expression too large to display}\) | \(900\) |
2*f/d*((1/4*2^(1/3)-1/4*I*3^(1/2)*2^(1/3))*c/d-(1/4*2^(1/3)+1/4*I*3^(1/2)* 2^(1/3))*c/d)*((x-(1/4*2^(1/3)+1/4*I*3^(1/2)*2^(1/3))*c/d)/((1/4*2^(1/3)-1 /4*I*3^(1/2)*2^(1/3))*c/d-(1/4*2^(1/3)+1/4*I*3^(1/2)*2^(1/3))*c/d))^(1/2)* ((x+1/2*2^(1/3)*c/d)/((1/4*2^(1/3)+1/4*I*3^(1/2)*2^(1/3))*c/d+1/2*2^(1/3)* c/d))^(1/2)*((x-(1/4*2^(1/3)-1/4*I*3^(1/2)*2^(1/3))*c/d)/((1/4*2^(1/3)+1/4 *I*3^(1/2)*2^(1/3))*c/d-(1/4*2^(1/3)-1/4*I*3^(1/2)*2^(1/3))*c/d))^(1/2)/(4 *d^3*x^3+c^3)^(1/2)*EllipticF(((x-(1/4*2^(1/3)+1/4*I*3^(1/2)*2^(1/3))*c/d) /((1/4*2^(1/3)-1/4*I*3^(1/2)*2^(1/3))*c/d-(1/4*2^(1/3)+1/4*I*3^(1/2)*2^(1/ 3))*c/d))^(1/2),(((1/4*2^(1/3)+1/4*I*3^(1/2)*2^(1/3))*c/d-(1/4*2^(1/3)-1/4 *I*3^(1/2)*2^(1/3))*c/d)/((1/4*2^(1/3)+1/4*I*3^(1/2)*2^(1/3))*c/d+1/2*2^(1 /3)*c/d))^(1/2))+2*(-c*f+d*e)/d^2*((1/4*2^(1/3)-1/4*I*3^(1/2)*2^(1/3))*c/d -(1/4*2^(1/3)+1/4*I*3^(1/2)*2^(1/3))*c/d)*((x-(1/4*2^(1/3)+1/4*I*3^(1/2)*2 ^(1/3))*c/d)/((1/4*2^(1/3)-1/4*I*3^(1/2)*2^(1/3))*c/d-(1/4*2^(1/3)+1/4*I*3 ^(1/2)*2^(1/3))*c/d))^(1/2)*((x+1/2*2^(1/3)*c/d)/((1/4*2^(1/3)+1/4*I*3^(1/ 2)*2^(1/3))*c/d+1/2*2^(1/3)*c/d))^(1/2)*((x-(1/4*2^(1/3)-1/4*I*3^(1/2)*2^( 1/3))*c/d)/((1/4*2^(1/3)+1/4*I*3^(1/2)*2^(1/3))*c/d-(1/4*2^(1/3)-1/4*I*3^( 1/2)*2^(1/3))*c/d))^(1/2)/(4*d^3*x^3+c^3)^(1/2)/((1/4*2^(1/3)+1/4*I*3^(1/2 )*2^(1/3))*c/d+c/d)*EllipticPi(((x-(1/4*2^(1/3)+1/4*I*3^(1/2)*2^(1/3))*c/d )/((1/4*2^(1/3)-1/4*I*3^(1/2)*2^(1/3))*c/d-(1/4*2^(1/3)+1/4*I*3^(1/2)*2^(1 /3))*c/d))^(1/2),((1/4*2^(1/3)+1/4*I*3^(1/2)*2^(1/3))*c/d-(1/4*2^(1/3)-...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.32 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.47 \[ \int \frac {e+f x}{(c+d x) \sqrt {c^3+4 d^3 x^3}} \, dx=\left [\frac {\sqrt {3} {\left (d^{3} e - c d^{2} f\right )} \sqrt {-c} \log \left (\frac {2 \, d^{6} x^{6} - 36 \, c d^{5} x^{5} - 18 \, c^{2} d^{4} x^{4} + 28 \, c^{3} d^{3} x^{3} + 18 \, c^{4} d^{2} x^{2} - c^{6} - \sqrt {3} {\left (4 \, d^{4} x^{4} - 10 \, c d^{3} x^{3} - 18 \, c^{2} d^{2} x^{2} - 8 \, c^{3} d x - c^{4}\right )} \sqrt {4 \, d^{3} x^{3} + c^{3}} \sqrt {-c}}{d^{6} x^{6} + 6 \, c d^{5} x^{5} + 15 \, c^{2} d^{4} x^{4} + 20 \, c^{3} d^{3} x^{3} + 15 \, c^{4} d^{2} x^{2} + 6 \, c^{5} d x + c^{6}}\right ) + 6 \, \sqrt {d^{3}} {\left (2 \, c d e + c^{2} f\right )} {\rm weierstrassPInverse}\left (0, -\frac {c^{3}}{d^{3}}, x\right )}{18 \, c^{2} d^{4}}, -\frac {\sqrt {3} {\left (d^{3} e - c d^{2} f\right )} \sqrt {c} \arctan \left (\frac {\sqrt {3} \sqrt {4 \, d^{3} x^{3} + c^{3}} {\left (2 \, d^{3} x^{3} - 6 \, c d^{2} x^{2} - 6 \, c^{2} d x - c^{3}\right )} \sqrt {c}}{3 \, {\left (8 \, c d^{4} x^{4} + 4 \, c^{2} d^{3} x^{3} + 2 \, c^{4} d x + c^{5}\right )}}\right ) - 3 \, \sqrt {d^{3}} {\left (2 \, c d e + c^{2} f\right )} {\rm weierstrassPInverse}\left (0, -\frac {c^{3}}{d^{3}}, x\right )}{9 \, c^{2} d^{4}}\right ] \]
[1/18*(sqrt(3)*(d^3*e - c*d^2*f)*sqrt(-c)*log((2*d^6*x^6 - 36*c*d^5*x^5 - 18*c^2*d^4*x^4 + 28*c^3*d^3*x^3 + 18*c^4*d^2*x^2 - c^6 - sqrt(3)*(4*d^4*x^ 4 - 10*c*d^3*x^3 - 18*c^2*d^2*x^2 - 8*c^3*d*x - c^4)*sqrt(4*d^3*x^3 + c^3) *sqrt(-c))/(d^6*x^6 + 6*c*d^5*x^5 + 15*c^2*d^4*x^4 + 20*c^3*d^3*x^3 + 15*c ^4*d^2*x^2 + 6*c^5*d*x + c^6)) + 6*sqrt(d^3)*(2*c*d*e + c^2*f)*weierstrass PInverse(0, -c^3/d^3, x))/(c^2*d^4), -1/9*(sqrt(3)*(d^3*e - c*d^2*f)*sqrt( c)*arctan(1/3*sqrt(3)*sqrt(4*d^3*x^3 + c^3)*(2*d^3*x^3 - 6*c*d^2*x^2 - 6*c ^2*d*x - c^3)*sqrt(c)/(8*c*d^4*x^4 + 4*c^2*d^3*x^3 + 2*c^4*d*x + c^5)) - 3 *sqrt(d^3)*(2*c*d*e + c^2*f)*weierstrassPInverse(0, -c^3/d^3, x))/(c^2*d^4 )]
\[ \int \frac {e+f x}{(c+d x) \sqrt {c^3+4 d^3 x^3}} \, dx=\int \frac {e + f x}{\left (c + d x\right ) \sqrt {c^{3} + 4 d^{3} x^{3}}}\, dx \]
\[ \int \frac {e+f x}{(c+d x) \sqrt {c^3+4 d^3 x^3}} \, dx=\int { \frac {f x + e}{\sqrt {4 \, d^{3} x^{3} + c^{3}} {\left (d x + c\right )}} \,d x } \]
\[ \int \frac {e+f x}{(c+d x) \sqrt {c^3+4 d^3 x^3}} \, dx=\int { \frac {f x + e}{\sqrt {4 \, d^{3} x^{3} + c^{3}} {\left (d x + c\right )}} \,d x } \]
Timed out. \[ \int \frac {e+f x}{(c+d x) \sqrt {c^3+4 d^3 x^3}} \, dx=\int \frac {e+f\,x}{\sqrt {c^3+4\,d^3\,x^3}\,\left (c+d\,x\right )} \,d x \]