Integrand size = 24, antiderivative size = 249 \[ \int \frac {1}{(c+d x) \sqrt {c^3+4 d^3 x^3}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {3} \sqrt {c} (c+2 d x)}{\sqrt {c^3+4 d^3 x^3}}\right )}{3 \sqrt {3} c^{3/2} d}+\frac {2 \sqrt [3]{2} \sqrt {2+\sqrt {3}} \left (c+2^{2/3} d x\right ) \sqrt {\frac {c^2-2^{2/3} c d x+2 \sqrt [3]{2} d^2 x^2}{\left (\left (1+\sqrt {3}\right ) c+2^{2/3} d x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) c+2^{2/3} d x}{\left (1+\sqrt {3}\right ) c+2^{2/3} d x}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} c d \sqrt {\frac {c \left (c+2^{2/3} d x\right )}{\left (\left (1+\sqrt {3}\right ) c+2^{2/3} d x\right )^2}} \sqrt {c^3+4 d^3 x^3}} \]
2/9*arctan((2*d*x+c)*3^(1/2)*c^(1/2)/(4*d^3*x^3+c^3)^(1/2))/c^(3/2)/d*3^(1 /2)+2/9*2^(1/3)*(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/(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 = 10.17 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.68 \[ \int \frac {1}{(c+d x) \sqrt {c^3+4 d^3 x^3}} \, dx=-\frac {i 2^{5/6} \sqrt {\frac {\sqrt [3]{2} c+2 d x}{\left (1+\sqrt [3]{-1}\right ) c}} \sqrt {2^{2/3}-\frac {2 \sqrt [3]{2} d x}{c}+\frac {4 d^2 x^2}{c^2}} \operatorname {EllipticPi}\left (\frac {i \sqrt [3]{2} \sqrt {3}}{2+\sqrt [3]{-2}},\arcsin \left (\frac {\sqrt {\frac {\sqrt [3]{2} c+2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}}}{\sqrt [6]{2}}\right ),\sqrt [3]{-1}\right )}{\left (2+\sqrt [3]{-2}\right ) d \sqrt {c^3+4 d^3 x^3}} \]
((-I)*2^(5/6)*Sqrt[(2^(1/3)*c + 2*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)])/((2 + (-2)^(1/3))*d*Sqrt[c^3 + 4*d^3*x^3])
Time = 0.52 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2559, 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.
| \(\displaystyle \int \frac {1}{(c+d x) \sqrt {c^3+4 d^3 x^3}} \, dx\) |
| \(\Big \downarrow \) 2559 |
| \(\displaystyle \frac {2 \int \frac {1}{\sqrt {c^3+4 d^3 x^3}}dx}{3 c}+\frac {\int \frac {c-2 d x}{(c+d x) \sqrt {c^3+4 d^3 x^3}}dx}{3 c}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {\int \frac {c-2 d x}{(c+d x) \sqrt {c^3+4 d^3 x^3}}dx}{3 c}+\frac {2 \sqrt [3]{2} \sqrt {2+\sqrt {3}} \left (c+2^{2/3} d x\right ) \sqrt {\frac {c^2-2^{2/3} c d x+2 \sqrt [3]{2} d^2 x^2}{\left (\left (1+\sqrt {3}\right ) c+2^{2/3} d x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) c+2^{2/3} d x}{\left (1+\sqrt {3}\right ) c+2^{2/3} d x}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} c d \sqrt {\frac {c \left (c+2^{2/3} d x\right )}{\left (\left (1+\sqrt {3}\right ) c+2^{2/3} d x\right )^2}} \sqrt {c^3+4 d^3 x^3}}\) |
| \(\Big \downarrow \) 2562 |
| \(\displaystyle \frac {2 \int \frac {1}{\frac {3 c (c+2 d x)^2}{c^3+4 d^3 x^3}+1}d\frac {c+2 d x}{c \sqrt {c^3+4 d^3 x^3}}}{3 d}+\frac {2 \sqrt [3]{2} \sqrt {2+\sqrt {3}} \left (c+2^{2/3} d x\right ) \sqrt {\frac {c^2-2^{2/3} c d x+2 \sqrt [3]{2} d^2 x^2}{\left (\left (1+\sqrt {3}\right ) c+2^{2/3} d x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) c+2^{2/3} d x}{\left (1+\sqrt {3}\right ) c+2^{2/3} d x}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} c d \sqrt {\frac {c \left (c+2^{2/3} d x\right )}{\left (\left (1+\sqrt {3}\right ) c+2^{2/3} d x\right )^2}} \sqrt {c^3+4 d^3 x^3}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {2 \sqrt [3]{2} \sqrt {2+\sqrt {3}} \left (c+2^{2/3} d x\right ) \sqrt {\frac {c^2-2^{2/3} c d x+2 \sqrt [3]{2} d^2 x^2}{\left (\left (1+\sqrt {3}\right ) c+2^{2/3} d x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) c+2^{2/3} d x}{\left (1+\sqrt {3}\right ) c+2^{2/3} d x}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} c d \sqrt {\frac {c \left (c+2^{2/3} d x\right )}{\left (\left (1+\sqrt {3}\right ) c+2^{2/3} d x\right )^2}} \sqrt {c^3+4 d^3 x^3}}+\frac {2 \arctan \left (\frac {\sqrt {3} \sqrt {c} (c+2 d x)}{\sqrt {c^3+4 d^3 x^3}}\right )}{3 \sqrt {3} c^{3/2} d}\) |
(2*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]]*(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]*Ellip ticF[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*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])
3.1.9.3.1 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[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> Simp[2/ (3*c) Int[1/Sqrt[a + b*x^3], x], x] + Simp[1/(3*c) Int[(c - 2*d*x)/((c + d*x)*Sqrt[a + b*x^3]), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^3 - 4* a*d^3, 0]
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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 494 vs. \(2 (200 ) = 400\).
Time = 1.38 (sec) , antiderivative size = 495, normalized size of antiderivative = 1.99
| method | result | size |
| default | \(\frac {2 \left (\frac {\left (\frac {2^{\frac {1}{3}}}{4}-\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}-\frac {\left (\frac {2^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}\right ) \sqrt {\frac {x -\frac {\left (\frac {2^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}}{\frac {\left (\frac {2^{\frac {1}{3}}}{4}-\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}-\frac {\left (\frac {2^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}}}\, \sqrt {\frac {x +\frac {2^{\frac {1}{3}} c}{2 d}}{\frac {\left (\frac {2^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}+\frac {2^{\frac {1}{3}} c}{2 d}}}\, \sqrt {\frac {x -\frac {\left (\frac {2^{\frac {1}{3}}}{4}-\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}}{\frac {\left (\frac {2^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}-\frac {\left (\frac {2^{\frac {1}{3}}}{4}-\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}}}\, \Pi \left (\sqrt {\frac {x -\frac {\left (\frac {2^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}}{\frac {\left (\frac {2^{\frac {1}{3}}}{4}-\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}-\frac {\left (\frac {2^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}}}, \frac {\frac {\left (\frac {2^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}-\frac {\left (\frac {2^{\frac {1}{3}}}{4}-\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}}{\frac {\left (\frac {2^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}+\frac {c}{d}}, \sqrt {\frac {\frac {\left (\frac {2^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}-\frac {\left (\frac {2^{\frac {1}{3}}}{4}-\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}}{\frac {\left (\frac {2^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}+\frac {2^{\frac {1}{3}} c}{2 d}}}\right )}{d \sqrt {4 d^{3} x^{3}+c^{3}}\, \left (\frac {\left (\frac {2^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}+\frac {c}{d}\right )}\) | \(495\) |
| elliptic | \(\frac {2 \left (\frac {\left (\frac {2^{\frac {1}{3}}}{4}-\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}-\frac {\left (\frac {2^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}\right ) \sqrt {\frac {x -\frac {\left (\frac {2^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}}{\frac {\left (\frac {2^{\frac {1}{3}}}{4}-\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}-\frac {\left (\frac {2^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}}}\, \sqrt {\frac {x +\frac {2^{\frac {1}{3}} c}{2 d}}{\frac {\left (\frac {2^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}+\frac {2^{\frac {1}{3}} c}{2 d}}}\, \sqrt {\frac {x -\frac {\left (\frac {2^{\frac {1}{3}}}{4}-\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}}{\frac {\left (\frac {2^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}-\frac {\left (\frac {2^{\frac {1}{3}}}{4}-\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}}}\, \Pi \left (\sqrt {\frac {x -\frac {\left (\frac {2^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}}{\frac {\left (\frac {2^{\frac {1}{3}}}{4}-\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}-\frac {\left (\frac {2^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}}}, \frac {\frac {\left (\frac {2^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}-\frac {\left (\frac {2^{\frac {1}{3}}}{4}-\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}}{\frac {\left (\frac {2^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}+\frac {c}{d}}, \sqrt {\frac {\frac {\left (\frac {2^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}-\frac {\left (\frac {2^{\frac {1}{3}}}{4}-\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}}{\frac {\left (\frac {2^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}+\frac {2^{\frac {1}{3}} c}{2 d}}}\right )}{d \sqrt {4 d^{3} x^{3}+c^{3}}\, \left (\frac {\left (\frac {2^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}+\frac {c}{d}\right )}\) | \(495\) |
2/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)/((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)-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+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)/((1/4*2^(1/3)+1/4*I*3^(1 /2)*2^(1/3))*c/d+1/2*2^(1/3)*c/d))^(1/2))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.16 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.40 \[ \int \frac {1}{(c+d x) \sqrt {c^3+4 d^3 x^3}} \, dx=\left [-\frac {\sqrt {3} \sqrt {-c} d^{2} \log \left (\frac {2 \, d^{6} x^{6} - 36 \, c d^{5} x^{5} - 18 \, c^{2} d^{4} x^{4} + 28 \, c^{3} d^{3} x^{3} + 18 \, c^{4} d^{2} x^{2} - c^{6} + \sqrt {3} {\left (4 \, d^{4} x^{4} - 10 \, c d^{3} x^{3} - 18 \, c^{2} d^{2} x^{2} - 8 \, c^{3} d x - c^{4}\right )} \sqrt {4 \, d^{3} x^{3} + c^{3}} \sqrt {-c}}{d^{6} x^{6} + 6 \, c d^{5} x^{5} + 15 \, c^{2} d^{4} x^{4} + 20 \, c^{3} d^{3} x^{3} + 15 \, c^{4} d^{2} x^{2} + 6 \, c^{5} d x + c^{6}}\right ) - 12 \, c \sqrt {d^{3}} {\rm weierstrassPInverse}\left (0, -\frac {c^{3}}{d^{3}}, x\right )}{18 \, c^{2} d^{3}}, -\frac {\sqrt {3} \sqrt {c} d^{2} \arctan \left (\frac {\sqrt {3} \sqrt {4 \, d^{3} x^{3} + c^{3}} {\left (2 \, d^{3} x^{3} - 6 \, c d^{2} x^{2} - 6 \, c^{2} d x - c^{3}\right )} \sqrt {c}}{3 \, {\left (8 \, c d^{4} x^{4} + 4 \, c^{2} d^{3} x^{3} + 2 \, c^{4} d x + c^{5}\right )}}\right ) - 6 \, c \sqrt {d^{3}} {\rm weierstrassPInverse}\left (0, -\frac {c^{3}}{d^{3}}, x\right )}{9 \, c^{2} d^{3}}\right ] \]
[-1/18*(sqrt(3)*sqrt(-c)*d^2*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)) - 12*c*sqrt(d^3)*weierstrassPInverse(0, -c^3/d^3, x))/(c ^2*d^3), -1/9*(sqrt(3)*sqrt(c)*d^2*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)) - 6*c*sqrt(d^3)*weierstrassPInverse(0, -c^3/d ^3, x))/(c^2*d^3)]
\[ \int \frac {1}{(c+d x) \sqrt {c^3+4 d^3 x^3}} \, dx=\int \frac {1}{\left (c + d x\right ) \sqrt {c^{3} + 4 d^{3} x^{3}}}\, dx \]
\[ \int \frac {1}{(c+d x) \sqrt {c^3+4 d^3 x^3}} \, dx=\int { \frac {1}{\sqrt {4 \, d^{3} x^{3} + c^{3}} {\left (d x + c\right )}} \,d x } \]
\[ \int \frac {1}{(c+d x) \sqrt {c^3+4 d^3 x^3}} \, dx=\int { \frac {1}{\sqrt {4 \, d^{3} x^{3} + c^{3}} {\left (d x + c\right )}} \,d x } \]
Timed out. \[ \int \frac {1}{(c+d x) \sqrt {c^3+4 d^3 x^3}} \, dx=\int \frac {1}{\sqrt {c^3+4\,d^3\,x^3}\,\left (c+d\,x\right )} \,d x \]