Integrand size = 27, antiderivative size = 404 \[ \int \frac {1}{\left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{5/2}} \, dx=-\frac {(c-3 d x) (2 c+3 d x)}{36 c d \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{5/2}}-\frac {11 (c-3 d x) (2 c+3 d x)^2}{648 c^2 d \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{5/2}}-\frac {11 (c-3 d x) (2 c+3 d x)^3}{864 c^3 d \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{5/2}}-\frac {77 (c-3 d x) (2 c+3 d x)^4}{5184 c^4 d \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{5/2}}+\frac {385 (c-3 d x) (2 c+3 d x)^5}{46656 c^5 d \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{5/2}}+\frac {385 (c-3 d x)^2 (2 c+3 d x)^5}{46656 c^6 d \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{5/2}}-\frac {385 (2 c+3 d x)^5 \left (1-\frac {3 d x}{c}\right )^{5/2} \text {arctanh}\left (\frac {\sqrt {1-\frac {3 d x}{c}}}{\sqrt {3}}\right )}{46656 \sqrt {3} c^4 d \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{5/2}} \] Output:
-1/36*(-3*d*x+c)*(3*d*x+2*c)/c/d/(-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^(5/2)-11 /648*(-3*d*x+c)*(3*d*x+2*c)^2/c^2/d/(-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^(5/2) -11/864*(-3*d*x+c)*(3*d*x+2*c)^3/c^3/d/(-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^(5 /2)-77/5184*(-3*d*x+c)*(3*d*x+2*c)^4/c^4/d/(-27*d^3*x^3-27*c*d^2*x^2+4*c^3 )^(5/2)+385/46656*(-3*d*x+c)*(3*d*x+2*c)^5/c^5/d/(-27*d^3*x^3-27*c*d^2*x^2 +4*c^3)^(5/2)+385/46656*(-3*d*x+c)^2*(3*d*x+2*c)^5/c^6/d/(-27*d^3*x^3-27*c *d^2*x^2+4*c^3)^(5/2)-385/139968*(3*d*x+2*c)^5*(1-3*d*x/c)^(5/2)*arctanh(1 /3*(1-3*d*x/c)^(1/2)*3^(1/2))*3^(1/2)/c^4/d/(-27*d^3*x^3-27*c*d^2*x^2+4*c^ 3)^(5/2)
Time = 0.37 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.39 \[ \int \frac {1}{\left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{5/2}} \, dx=\frac {(2 c+3 d x) \left (3 \sqrt {c} (c-3 d x) \left (1520 c^5+20988 c^4 d x+12672 c^3 d^2 x^2-101871 c^2 d^3 x^3-187110 c d^4 x^4-93555 d^5 x^5\right )-385 \sqrt {3} (c-3 d x)^{5/2} (2 c+3 d x)^4 \text {arctanh}\left (\frac {\sqrt {c-3 d x}}{\sqrt {3} \sqrt {c}}\right )\right )}{139968 c^{13/2} d \left ((c-3 d x) (2 c+3 d x)^2\right )^{5/2}} \] Input:
Integrate[(4*c^3 - 27*c*d^2*x^2 - 27*d^3*x^3)^(-5/2),x]
Output:
((2*c + 3*d*x)*(3*Sqrt[c]*(c - 3*d*x)*(1520*c^5 + 20988*c^4*d*x + 12672*c^ 3*d^2*x^2 - 101871*c^2*d^3*x^3 - 187110*c*d^4*x^4 - 93555*d^5*x^5) - 385*S qrt[3]*(c - 3*d*x)^(5/2)*(2*c + 3*d*x)^4*ArcTanh[Sqrt[c - 3*d*x]/(Sqrt[3]* Sqrt[c])]))/(139968*c^(13/2)*d*((c - 3*d*x)*(2*c + 3*d*x)^2)^(5/2))
Time = 0.43 (sec) , antiderivative size = 382, normalized size of antiderivative = 0.95, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {2480, 27, 52, 52, 52, 52, 61, 61, 73, 221}
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.
| \(\displaystyle \int \frac {1}{\left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 2480 |
| \(\displaystyle \frac {94538006609626678356576 c^{10} d^{15} (2 c+3 d x)^5 \left (c^3 d^6-3 c^2 d^7 x\right )^{5/2} \int \frac {1}{94538006609626678356576 c^{10} d^{15} (2 c+3 d x)^5 \left (c^3 d^6-3 c^2 d^7 x\right )^{5/2}}dx}{\left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(2 c+3 d x)^5 \left (c^3 d^6-3 c^2 d^7 x\right )^{5/2} \int \frac {1}{(2 c+3 d x)^5 \left (c^3 d^6-3 c^2 d^7 x\right )^{5/2}}dx}{\left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{5/2}}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {(2 c+3 d x)^5 \left (c^3 d^6-3 c^2 d^7 x\right )^{5/2} \left (\frac {11 \int \frac {1}{(2 c+3 d x)^4 \left (c^3 d^6-3 c^2 d^7 x\right )^{5/2}}dx}{24 c}-\frac {1}{36 c^3 d^7 (2 c+3 d x)^4 \left (c^3 d^6-3 c^2 d^7 x\right )^{3/2}}\right )}{\left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{5/2}}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {(2 c+3 d x)^5 \left (c^3 d^6-3 c^2 d^7 x\right )^{5/2} \left (\frac {11 \left (\frac {\int \frac {1}{(2 c+3 d x)^3 \left (c^3 d^6-3 c^2 d^7 x\right )^{5/2}}dx}{2 c}-\frac {1}{27 c^3 d^7 (2 c+3 d x)^3 \left (c^3 d^6-3 c^2 d^7 x\right )^{3/2}}\right )}{24 c}-\frac {1}{36 c^3 d^7 (2 c+3 d x)^4 \left (c^3 d^6-3 c^2 d^7 x\right )^{3/2}}\right )}{\left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{5/2}}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {(2 c+3 d x)^5 \left (c^3 d^6-3 c^2 d^7 x\right )^{5/2} \left (\frac {11 \left (\frac {\frac {7 \int \frac {1}{(2 c+3 d x)^2 \left (c^3 d^6-3 c^2 d^7 x\right )^{5/2}}dx}{12 c}-\frac {1}{18 c^3 d^7 (2 c+3 d x)^2 \left (c^3 d^6-3 c^2 d^7 x\right )^{3/2}}}{2 c}-\frac {1}{27 c^3 d^7 (2 c+3 d x)^3 \left (c^3 d^6-3 c^2 d^7 x\right )^{3/2}}\right )}{24 c}-\frac {1}{36 c^3 d^7 (2 c+3 d x)^4 \left (c^3 d^6-3 c^2 d^7 x\right )^{3/2}}\right )}{\left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{5/2}}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {(2 c+3 d x)^5 \left (c^3 d^6-3 c^2 d^7 x\right )^{5/2} \left (\frac {11 \left (\frac {\frac {7 \left (\frac {5 \int \frac {1}{(2 c+3 d x) \left (c^3 d^6-3 c^2 d^7 x\right )^{5/2}}dx}{6 c}-\frac {1}{9 c^3 d^7 (2 c+3 d x) \left (c^3 d^6-3 c^2 d^7 x\right )^{3/2}}\right )}{12 c}-\frac {1}{18 c^3 d^7 (2 c+3 d x)^2 \left (c^3 d^6-3 c^2 d^7 x\right )^{3/2}}}{2 c}-\frac {1}{27 c^3 d^7 (2 c+3 d x)^3 \left (c^3 d^6-3 c^2 d^7 x\right )^{3/2}}\right )}{24 c}-\frac {1}{36 c^3 d^7 (2 c+3 d x)^4 \left (c^3 d^6-3 c^2 d^7 x\right )^{3/2}}\right )}{\left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{5/2}}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {(2 c+3 d x)^5 \left (c^3 d^6-3 c^2 d^7 x\right )^{5/2} \left (\frac {11 \left (\frac {\frac {7 \left (\frac {5 \left (\frac {\int \frac {1}{(2 c+3 d x) \left (c^3 d^6-3 c^2 d^7 x\right )^{3/2}}dx}{3 c^3 d^6}+\frac {2}{27 c^3 d^7 \left (c^3 d^6-3 c^2 d^7 x\right )^{3/2}}\right )}{6 c}-\frac {1}{9 c^3 d^7 (2 c+3 d x) \left (c^3 d^6-3 c^2 d^7 x\right )^{3/2}}\right )}{12 c}-\frac {1}{18 c^3 d^7 (2 c+3 d x)^2 \left (c^3 d^6-3 c^2 d^7 x\right )^{3/2}}}{2 c}-\frac {1}{27 c^3 d^7 (2 c+3 d x)^3 \left (c^3 d^6-3 c^2 d^7 x\right )^{3/2}}\right )}{24 c}-\frac {1}{36 c^3 d^7 (2 c+3 d x)^4 \left (c^3 d^6-3 c^2 d^7 x\right )^{3/2}}\right )}{\left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{5/2}}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {(2 c+3 d x)^5 \left (c^3 d^6-3 c^2 d^7 x\right )^{5/2} \left (\frac {11 \left (\frac {\frac {7 \left (\frac {5 \left (\frac {\frac {\int \frac {1}{(2 c+3 d x) \sqrt {c^3 d^6-3 c^2 d^7 x}}dx}{3 c^3 d^6}+\frac {2}{9 c^3 d^7 \sqrt {c^3 d^6-3 c^2 d^7 x}}}{3 c^3 d^6}+\frac {2}{27 c^3 d^7 \left (c^3 d^6-3 c^2 d^7 x\right )^{3/2}}\right )}{6 c}-\frac {1}{9 c^3 d^7 (2 c+3 d x) \left (c^3 d^6-3 c^2 d^7 x\right )^{3/2}}\right )}{12 c}-\frac {1}{18 c^3 d^7 (2 c+3 d x)^2 \left (c^3 d^6-3 c^2 d^7 x\right )^{3/2}}}{2 c}-\frac {1}{27 c^3 d^7 (2 c+3 d x)^3 \left (c^3 d^6-3 c^2 d^7 x\right )^{3/2}}\right )}{24 c}-\frac {1}{36 c^3 d^7 (2 c+3 d x)^4 \left (c^3 d^6-3 c^2 d^7 x\right )^{3/2}}\right )}{\left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{5/2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {(2 c+3 d x)^5 \left (c^3 d^6-3 c^2 d^7 x\right )^{5/2} \left (\frac {11 \left (\frac {\frac {7 \left (\frac {5 \left (\frac {\frac {2}{9 c^3 d^7 \sqrt {c^3 d^6-3 c^2 d^7 x}}-\frac {2 \int \frac {1}{3 c-\frac {c^3 d^6-3 c^2 d^7 x}{c^2 d^6}}d\sqrt {c^3 d^6-3 c^2 d^7 x}}{9 c^5 d^{13}}}{3 c^3 d^6}+\frac {2}{27 c^3 d^7 \left (c^3 d^6-3 c^2 d^7 x\right )^{3/2}}\right )}{6 c}-\frac {1}{9 c^3 d^7 (2 c+3 d x) \left (c^3 d^6-3 c^2 d^7 x\right )^{3/2}}\right )}{12 c}-\frac {1}{18 c^3 d^7 (2 c+3 d x)^2 \left (c^3 d^6-3 c^2 d^7 x\right )^{3/2}}}{2 c}-\frac {1}{27 c^3 d^7 (2 c+3 d x)^3 \left (c^3 d^6-3 c^2 d^7 x\right )^{3/2}}\right )}{24 c}-\frac {1}{36 c^3 d^7 (2 c+3 d x)^4 \left (c^3 d^6-3 c^2 d^7 x\right )^{3/2}}\right )}{\left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{5/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {(2 c+3 d x)^5 \left (c^3 d^6-3 c^2 d^7 x\right )^{5/2} \left (\frac {11 \left (\frac {\frac {7 \left (\frac {5 \left (\frac {\frac {2}{9 c^3 d^7 \sqrt {c^3 d^6-3 c^2 d^7 x}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {c^3 d^6-3 c^2 d^7 x}}{\sqrt {3} c^{3/2} d^3}\right )}{9 \sqrt {3} c^{9/2} d^{10}}}{3 c^3 d^6}+\frac {2}{27 c^3 d^7 \left (c^3 d^6-3 c^2 d^7 x\right )^{3/2}}\right )}{6 c}-\frac {1}{9 c^3 d^7 (2 c+3 d x) \left (c^3 d^6-3 c^2 d^7 x\right )^{3/2}}\right )}{12 c}-\frac {1}{18 c^3 d^7 (2 c+3 d x)^2 \left (c^3 d^6-3 c^2 d^7 x\right )^{3/2}}}{2 c}-\frac {1}{27 c^3 d^7 (2 c+3 d x)^3 \left (c^3 d^6-3 c^2 d^7 x\right )^{3/2}}\right )}{24 c}-\frac {1}{36 c^3 d^7 (2 c+3 d x)^4 \left (c^3 d^6-3 c^2 d^7 x\right )^{3/2}}\right )}{\left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{5/2}}\) |
Input:
Int[(4*c^3 - 27*c*d^2*x^2 - 27*d^3*x^3)^(-5/2),x]
Output:
((2*c + 3*d*x)^5*(c^3*d^6 - 3*c^2*d^7*x)^(5/2)*(-1/36*1/(c^3*d^7*(2*c + 3* d*x)^4*(c^3*d^6 - 3*c^2*d^7*x)^(3/2)) + (11*(-1/27*1/(c^3*d^7*(2*c + 3*d*x )^3*(c^3*d^6 - 3*c^2*d^7*x)^(3/2)) + (-1/18*1/(c^3*d^7*(2*c + 3*d*x)^2*(c^ 3*d^6 - 3*c^2*d^7*x)^(3/2)) + (7*(-1/9*1/(c^3*d^7*(2*c + 3*d*x)*(c^3*d^6 - 3*c^2*d^7*x)^(3/2)) + (5*(2/(27*c^3*d^7*(c^3*d^6 - 3*c^2*d^7*x)^(3/2)) + (2/(9*c^3*d^7*Sqrt[c^3*d^6 - 3*c^2*d^7*x]) - (2*ArcTanh[Sqrt[c^3*d^6 - 3*c ^2*d^7*x]/(Sqrt[3]*c^(3/2)*d^3)])/(9*Sqrt[3]*c^(9/2)*d^10))/(3*c^3*d^6)))/ (6*c)))/(12*c))/(2*c)))/(24*c)))/(4*c^3 - 27*c*d^2*x^2 - 27*d^3*x^3)^(5/2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(Px_)^(p_), x_Symbol] :> With[{a = Coeff[Px, x, 0], b = Coeff[Px, x, 1] , c = Coeff[Px, x, 2], d = Coeff[Px, x, 3]}, Simp[Px^p/((c^3 - 4*b*c*d + 9* a*d^2 + d*(c^2 - 3*b*d)*x)^p*(b*c - 9*a*d + 2*(c^2 - 3*b*d)*x)^(2*p)) Int [(c^3 - 4*b*c*d + 9*a*d^2 + d*(c^2 - 3*b*d)*x)^p*(b*c - 9*a*d + 2*(c^2 - 3* b*d)*x)^(2*p), x], x] /; EqQ[b^2*c^2 - 4*a*c^3 - 4*b^3*d + 18*a*b*c*d - 27* a^2*d^2, 0] && NeQ[c^2 - 3*b*d, 0]] /; FreeQ[p, x] && PolyQ[Px, x, 3] && ! IntegerQ[p]
Time = 0.20 (sec) , antiderivative size = 285, normalized size of antiderivative = 0.71
| method | result | size |
| default | \(\frac {\left (38016 c^{\frac {7}{2}} d^{2} x^{2}+62964 c^{\frac {9}{2}} d x -36960 \,\operatorname {arctanh}\left (\frac {\sqrt {-3 d x +c}\, \sqrt {3}}{3 \sqrt {c}}\right ) \sqrt {3}\, \left (-3 d x +c \right )^{\frac {3}{2}} c^{3} d x -83160 \,\operatorname {arctanh}\left (\frac {\sqrt {-3 d x +c}\, \sqrt {3}}{3 \sqrt {c}}\right ) \sqrt {3}\, \left (-3 d x +c \right )^{\frac {3}{2}} c^{2} d^{2} x^{2}-83160 \,\operatorname {arctanh}\left (\frac {\sqrt {-3 d x +c}\, \sqrt {3}}{3 \sqrt {c}}\right ) \sqrt {3}\, \left (-3 d x +c \right )^{\frac {3}{2}} c \,d^{3} x^{3}-6160 \,\operatorname {arctanh}\left (\frac {\sqrt {-3 d x +c}\, \sqrt {3}}{3 \sqrt {c}}\right ) \sqrt {3}\, \left (-3 d x +c \right )^{\frac {3}{2}} c^{4}-561330 c^{\frac {3}{2}} d^{4} x^{4}-305613 c^{\frac {5}{2}} d^{3} x^{3}-31185 \,\operatorname {arctanh}\left (\frac {\sqrt {-3 d x +c}\, \sqrt {3}}{3 \sqrt {c}}\right ) \sqrt {3}\, \left (-3 d x +c \right )^{\frac {3}{2}} d^{4} x^{4}+4560 c^{\frac {11}{2}}-280665 \sqrt {c}\, d^{5} x^{5}\right ) \left (-3 d x +c \right ) \left (3 d x +2 c \right )}{139968 c^{\frac {13}{2}} d \left (-27 d^{3} x^{3}-27 c \,d^{2} x^{2}+4 c^{3}\right )^{\frac {5}{2}}}\) | \(285\) |
Input:
int(1/(-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/139968*(38016*c^(7/2)*d^2*x^2+62964*c^(9/2)*d*x-36960*arctanh(1/3*(-3*d* x+c)^(1/2)*3^(1/2)/c^(1/2))*3^(1/2)*(-3*d*x+c)^(3/2)*c^3*d*x-83160*arctanh (1/3*(-3*d*x+c)^(1/2)*3^(1/2)/c^(1/2))*3^(1/2)*(-3*d*x+c)^(3/2)*c^2*d^2*x^ 2-83160*arctanh(1/3*(-3*d*x+c)^(1/2)*3^(1/2)/c^(1/2))*3^(1/2)*(-3*d*x+c)^( 3/2)*c*d^3*x^3-6160*arctanh(1/3*(-3*d*x+c)^(1/2)*3^(1/2)/c^(1/2))*3^(1/2)* (-3*d*x+c)^(3/2)*c^4-561330*c^(3/2)*d^4*x^4-305613*c^(5/2)*d^3*x^3-31185*a rctanh(1/3*(-3*d*x+c)^(1/2)*3^(1/2)/c^(1/2))*3^(1/2)*(-3*d*x+c)^(3/2)*d^4* x^4+4560*c^(11/2)-280665*c^(1/2)*d^5*x^5)*(-3*d*x+c)*(3*d*x+2*c)/c^(13/2)/ d/(-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^(5/2)
Time = 0.15 (sec) , antiderivative size = 636, normalized size of antiderivative = 1.57 \[ \int \frac {1}{\left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{5/2}} \, dx=\left [\frac {385 \, \sqrt {3} {\left (2187 \, d^{7} x^{7} + 5832 \, c d^{6} x^{6} + 5103 \, c^{2} d^{5} x^{5} + 810 \, c^{3} d^{4} x^{4} - 1080 \, c^{4} d^{3} x^{3} - 432 \, c^{5} d^{2} x^{2} + 48 \, c^{6} d x + 32 \, c^{7}\right )} \sqrt {c} \log \left (\frac {9 \, d^{2} x^{2} - 6 \, c d x - 8 \, c^{2} + 2 \, \sqrt {3} \sqrt {-27 \, d^{3} x^{3} - 27 \, c d^{2} x^{2} + 4 \, c^{3}} \sqrt {c}}{9 \, d^{2} x^{2} + 12 \, c d x + 4 \, c^{2}}\right ) - 6 \, {\left (93555 \, c d^{5} x^{5} + 187110 \, c^{2} d^{4} x^{4} + 101871 \, c^{3} d^{3} x^{3} - 12672 \, c^{4} d^{2} x^{2} - 20988 \, c^{5} d x - 1520 \, c^{6}\right )} \sqrt {-27 \, d^{3} x^{3} - 27 \, c d^{2} x^{2} + 4 \, c^{3}}}{279936 \, {\left (2187 \, c^{7} d^{8} x^{7} + 5832 \, c^{8} d^{7} x^{6} + 5103 \, c^{9} d^{6} x^{5} + 810 \, c^{10} d^{5} x^{4} - 1080 \, c^{11} d^{4} x^{3} - 432 \, c^{12} d^{3} x^{2} + 48 \, c^{13} d^{2} x + 32 \, c^{14} d\right )}}, -\frac {385 \, \sqrt {3} {\left (2187 \, d^{7} x^{7} + 5832 \, c d^{6} x^{6} + 5103 \, c^{2} d^{5} x^{5} + 810 \, c^{3} d^{4} x^{4} - 1080 \, c^{4} d^{3} x^{3} - 432 \, c^{5} d^{2} x^{2} + 48 \, c^{6} d x + 32 \, c^{7}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {3} \sqrt {-27 \, d^{3} x^{3} - 27 \, c d^{2} x^{2} + 4 \, c^{3}} \sqrt {-c}}{9 \, d^{2} x^{2} + 3 \, c d x - 2 \, c^{2}}\right ) + 3 \, {\left (93555 \, c d^{5} x^{5} + 187110 \, c^{2} d^{4} x^{4} + 101871 \, c^{3} d^{3} x^{3} - 12672 \, c^{4} d^{2} x^{2} - 20988 \, c^{5} d x - 1520 \, c^{6}\right )} \sqrt {-27 \, d^{3} x^{3} - 27 \, c d^{2} x^{2} + 4 \, c^{3}}}{139968 \, {\left (2187 \, c^{7} d^{8} x^{7} + 5832 \, c^{8} d^{7} x^{6} + 5103 \, c^{9} d^{6} x^{5} + 810 \, c^{10} d^{5} x^{4} - 1080 \, c^{11} d^{4} x^{3} - 432 \, c^{12} d^{3} x^{2} + 48 \, c^{13} d^{2} x + 32 \, c^{14} d\right )}}\right ] \] Input:
integrate(1/(-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^(5/2),x, algorithm="fricas")
Output:
[1/279936*(385*sqrt(3)*(2187*d^7*x^7 + 5832*c*d^6*x^6 + 5103*c^2*d^5*x^5 + 810*c^3*d^4*x^4 - 1080*c^4*d^3*x^3 - 432*c^5*d^2*x^2 + 48*c^6*d*x + 32*c^ 7)*sqrt(c)*log((9*d^2*x^2 - 6*c*d*x - 8*c^2 + 2*sqrt(3)*sqrt(-27*d^3*x^3 - 27*c*d^2*x^2 + 4*c^3)*sqrt(c))/(9*d^2*x^2 + 12*c*d*x + 4*c^2)) - 6*(93555 *c*d^5*x^5 + 187110*c^2*d^4*x^4 + 101871*c^3*d^3*x^3 - 12672*c^4*d^2*x^2 - 20988*c^5*d*x - 1520*c^6)*sqrt(-27*d^3*x^3 - 27*c*d^2*x^2 + 4*c^3))/(2187 *c^7*d^8*x^7 + 5832*c^8*d^7*x^6 + 5103*c^9*d^6*x^5 + 810*c^10*d^5*x^4 - 10 80*c^11*d^4*x^3 - 432*c^12*d^3*x^2 + 48*c^13*d^2*x + 32*c^14*d), -1/139968 *(385*sqrt(3)*(2187*d^7*x^7 + 5832*c*d^6*x^6 + 5103*c^2*d^5*x^5 + 810*c^3* d^4*x^4 - 1080*c^4*d^3*x^3 - 432*c^5*d^2*x^2 + 48*c^6*d*x + 32*c^7)*sqrt(- c)*arctan(sqrt(3)*sqrt(-27*d^3*x^3 - 27*c*d^2*x^2 + 4*c^3)*sqrt(-c)/(9*d^2 *x^2 + 3*c*d*x - 2*c^2)) + 3*(93555*c*d^5*x^5 + 187110*c^2*d^4*x^4 + 10187 1*c^3*d^3*x^3 - 12672*c^4*d^2*x^2 - 20988*c^5*d*x - 1520*c^6)*sqrt(-27*d^3 *x^3 - 27*c*d^2*x^2 + 4*c^3))/(2187*c^7*d^8*x^7 + 5832*c^8*d^7*x^6 + 5103* c^9*d^6*x^5 + 810*c^10*d^5*x^4 - 1080*c^11*d^4*x^3 - 432*c^12*d^3*x^2 + 48 *c^13*d^2*x + 32*c^14*d)]
\[ \int \frac {1}{\left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{5/2}} \, dx=\int \frac {1}{\left (4 c^{3} - 27 c d^{2} x^{2} - 27 d^{3} x^{3}\right )^{\frac {5}{2}}}\, dx \] Input:
integrate(1/(-27*d**3*x**3-27*c*d**2*x**2+4*c**3)**(5/2),x)
Output:
Integral((4*c**3 - 27*c*d**2*x**2 - 27*d**3*x**3)**(-5/2), x)
\[ \int \frac {1}{\left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{5/2}} \, dx=\int { \frac {1}{{\left (-27 \, d^{3} x^{3} - 27 \, c d^{2} x^{2} + 4 \, c^{3}\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^(5/2),x, algorithm="maxima")
Output:
integrate((-27*d^3*x^3 - 27*c*d^2*x^2 + 4*c^3)^(-5/2), x)
Time = 0.12 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.47 \[ \int \frac {1}{\left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{5/2}} \, dx=-\frac {385 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt {-3 \, d x + c}}{3 \, \sqrt {-c}}\right )}{139968 \, \sqrt {-c} c^{6} d \mathrm {sgn}\left (-3 \, d x - 2 \, c\right )} - \frac {2 \, {\left (5 \, d x - 2 \, c\right )}}{729 \, {\left (3 \, d x - c\right )} \sqrt {-3 \, d x + c} c^{6} d \mathrm {sgn}\left (-3 \, d x - 2 \, c\right )} + \frac {515 \, {\left (3 \, d x - c\right )}^{3} \sqrt {-3 \, d x + c} + 5153 \, {\left (3 \, d x - c\right )}^{2} \sqrt {-3 \, d x + c} c - 17565 \, {\left (-3 \, d x + c\right )}^{\frac {3}{2}} c^{2} + 20655 \, \sqrt {-3 \, d x + c} c^{3}}{139968 \, {\left (3 \, d x + 2 \, c\right )}^{4} c^{6} d \mathrm {sgn}\left (-3 \, d x - 2 \, c\right )} \] Input:
integrate(1/(-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^(5/2),x, algorithm="giac")
Output:
-385/139968*sqrt(3)*arctan(1/3*sqrt(3)*sqrt(-3*d*x + c)/sqrt(-c))/(sqrt(-c )*c^6*d*sgn(-3*d*x - 2*c)) - 2/729*(5*d*x - 2*c)/((3*d*x - c)*sqrt(-3*d*x + c)*c^6*d*sgn(-3*d*x - 2*c)) + 1/139968*(515*(3*d*x - c)^3*sqrt(-3*d*x + c) + 5153*(3*d*x - c)^2*sqrt(-3*d*x + c)*c - 17565*(-3*d*x + c)^(3/2)*c^2 + 20655*sqrt(-3*d*x + c)*c^3)/((3*d*x + 2*c)^4*c^6*d*sgn(-3*d*x - 2*c))
Timed out. \[ \int \frac {1}{\left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{5/2}} \, dx=\int \frac {1}{{\left (4\,c^3-27\,c\,d^2\,x^2-27\,d^3\,x^3\right )}^{5/2}} \,d x \] Input:
int(1/(4*c^3 - 27*d^3*x^3 - 27*c*d^2*x^2)^(5/2),x)
Output:
int(1/(4*c^3 - 27*d^3*x^3 - 27*c*d^2*x^2)^(5/2), x)
Time = 0.16 (sec) , antiderivative size = 534, normalized size of antiderivative = 1.32 \[ \int \frac {1}{\left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{5/2}} \, dx=\frac {6160 \sqrt {c}\, \sqrt {-3 d x +c}\, \sqrt {3}\, \mathrm {log}\left (\sqrt {-3 d x +c}-\sqrt {c}\, \sqrt {3}\right ) c^{5}+18480 \sqrt {c}\, \sqrt {-3 d x +c}\, \sqrt {3}\, \mathrm {log}\left (\sqrt {-3 d x +c}-\sqrt {c}\, \sqrt {3}\right ) c^{4} d x -27720 \sqrt {c}\, \sqrt {-3 d x +c}\, \sqrt {3}\, \mathrm {log}\left (\sqrt {-3 d x +c}-\sqrt {c}\, \sqrt {3}\right ) c^{3} d^{2} x^{2}-166320 \sqrt {c}\, \sqrt {-3 d x +c}\, \sqrt {3}\, \mathrm {log}\left (\sqrt {-3 d x +c}-\sqrt {c}\, \sqrt {3}\right ) c^{2} d^{3} x^{3}-218295 \sqrt {c}\, \sqrt {-3 d x +c}\, \sqrt {3}\, \mathrm {log}\left (\sqrt {-3 d x +c}-\sqrt {c}\, \sqrt {3}\right ) c \,d^{4} x^{4}-93555 \sqrt {c}\, \sqrt {-3 d x +c}\, \sqrt {3}\, \mathrm {log}\left (\sqrt {-3 d x +c}-\sqrt {c}\, \sqrt {3}\right ) d^{5} x^{5}-6160 \sqrt {c}\, \sqrt {-3 d x +c}\, \sqrt {3}\, \mathrm {log}\left (\sqrt {-3 d x +c}+\sqrt {c}\, \sqrt {3}\right ) c^{5}-18480 \sqrt {c}\, \sqrt {-3 d x +c}\, \sqrt {3}\, \mathrm {log}\left (\sqrt {-3 d x +c}+\sqrt {c}\, \sqrt {3}\right ) c^{4} d x +27720 \sqrt {c}\, \sqrt {-3 d x +c}\, \sqrt {3}\, \mathrm {log}\left (\sqrt {-3 d x +c}+\sqrt {c}\, \sqrt {3}\right ) c^{3} d^{2} x^{2}+166320 \sqrt {c}\, \sqrt {-3 d x +c}\, \sqrt {3}\, \mathrm {log}\left (\sqrt {-3 d x +c}+\sqrt {c}\, \sqrt {3}\right ) c^{2} d^{3} x^{3}+218295 \sqrt {c}\, \sqrt {-3 d x +c}\, \sqrt {3}\, \mathrm {log}\left (\sqrt {-3 d x +c}+\sqrt {c}\, \sqrt {3}\right ) c \,d^{4} x^{4}+93555 \sqrt {c}\, \sqrt {-3 d x +c}\, \sqrt {3}\, \mathrm {log}\left (\sqrt {-3 d x +c}+\sqrt {c}\, \sqrt {3}\right ) d^{5} x^{5}+9120 c^{6}+125928 c^{5} d x +76032 c^{4} d^{2} x^{2}-611226 c^{3} d^{3} x^{3}-1122660 c^{2} d^{4} x^{4}-561330 c \,d^{5} x^{5}}{279936 \sqrt {-3 d x +c}\, c^{7} d \left (-243 d^{5} x^{5}-567 c \,d^{4} x^{4}-432 c^{2} d^{3} x^{3}-72 c^{3} d^{2} x^{2}+48 c^{4} d x +16 c^{5}\right )} \] Input:
int(1/(-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^(5/2),x)
Output:
(6160*sqrt(c)*sqrt(c - 3*d*x)*sqrt(3)*log(sqrt(c - 3*d*x) - sqrt(c)*sqrt(3 ))*c**5 + 18480*sqrt(c)*sqrt(c - 3*d*x)*sqrt(3)*log(sqrt(c - 3*d*x) - sqrt (c)*sqrt(3))*c**4*d*x - 27720*sqrt(c)*sqrt(c - 3*d*x)*sqrt(3)*log(sqrt(c - 3*d*x) - sqrt(c)*sqrt(3))*c**3*d**2*x**2 - 166320*sqrt(c)*sqrt(c - 3*d*x) *sqrt(3)*log(sqrt(c - 3*d*x) - sqrt(c)*sqrt(3))*c**2*d**3*x**3 - 218295*sq rt(c)*sqrt(c - 3*d*x)*sqrt(3)*log(sqrt(c - 3*d*x) - sqrt(c)*sqrt(3))*c*d** 4*x**4 - 93555*sqrt(c)*sqrt(c - 3*d*x)*sqrt(3)*log(sqrt(c - 3*d*x) - sqrt( c)*sqrt(3))*d**5*x**5 - 6160*sqrt(c)*sqrt(c - 3*d*x)*sqrt(3)*log(sqrt(c - 3*d*x) + sqrt(c)*sqrt(3))*c**5 - 18480*sqrt(c)*sqrt(c - 3*d*x)*sqrt(3)*log (sqrt(c - 3*d*x) + sqrt(c)*sqrt(3))*c**4*d*x + 27720*sqrt(c)*sqrt(c - 3*d* x)*sqrt(3)*log(sqrt(c - 3*d*x) + sqrt(c)*sqrt(3))*c**3*d**2*x**2 + 166320* sqrt(c)*sqrt(c - 3*d*x)*sqrt(3)*log(sqrt(c - 3*d*x) + sqrt(c)*sqrt(3))*c** 2*d**3*x**3 + 218295*sqrt(c)*sqrt(c - 3*d*x)*sqrt(3)*log(sqrt(c - 3*d*x) + sqrt(c)*sqrt(3))*c*d**4*x**4 + 93555*sqrt(c)*sqrt(c - 3*d*x)*sqrt(3)*log( sqrt(c - 3*d*x) + sqrt(c)*sqrt(3))*d**5*x**5 + 9120*c**6 + 125928*c**5*d*x + 76032*c**4*d**2*x**2 - 611226*c**3*d**3*x**3 - 1122660*c**2*d**4*x**4 - 561330*c*d**5*x**5)/(279936*sqrt(c - 3*d*x)*c**7*d*(16*c**5 + 48*c**4*d*x - 72*c**3*d**2*x**2 - 432*c**2*d**3*x**3 - 567*c*d**4*x**4 - 243*d**5*x** 5))