\(\int \frac {1}{(27 a^3+27 a^2 b x-4 b^3 x^3)^{5/2}} \, dx\) [14]

Optimal result

Integrand size = 25, antiderivative size = 405 \[ \int \frac {1}{\left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2}} \, dx=-\frac {(3 a-b x) (3 a+2 b x)}{36 a b \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2}}-\frac {11 (3 a-b x) (3 a+2 b x)^2}{1944 a^2 b \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2}}-\frac {11 (3 a-b x) (3 a+2 b x)^3}{7776 a^3 b \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2}}-\frac {77 (3 a-b x) (3 a+2 b x)^4}{139968 a^4 b \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2}}+\frac {385 (3 a-b x) (3 a+2 b x)^5}{3779136 a^5 b \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2}}+\frac {385 (3 a-b x)^2 (3 a+2 b x)^5}{5668704 a^6 b \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2}}-\frac {385 (3 a+2 b x)^5 \left (3-\frac {b x}{a}\right )^{5/2} \text {arctanh}\left (\frac {1}{3} \sqrt {2} \sqrt {3-\frac {b x}{a}}\right )}{8503056 \sqrt {2} a^4 b \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2}} \] Output:

-1/36*(-b*x+3*a)*(2*b*x+3*a)/a/b/(-4*b^3*x^3+27*a^2*b*x+27*a^3)^(5/2)-11/1 
944*(-b*x+3*a)*(2*b*x+3*a)^2/a^2/b/(-4*b^3*x^3+27*a^2*b*x+27*a^3)^(5/2)-11 
/7776*(-b*x+3*a)*(2*b*x+3*a)^3/a^3/b/(-4*b^3*x^3+27*a^2*b*x+27*a^3)^(5/2)- 
77/139968*(-b*x+3*a)*(2*b*x+3*a)^4/a^4/b/(-4*b^3*x^3+27*a^2*b*x+27*a^3)^(5 
/2)+385/3779136*(-b*x+3*a)*(2*b*x+3*a)^5/a^5/b/(-4*b^3*x^3+27*a^2*b*x+27*a 
^3)^(5/2)+385/5668704*(-b*x+3*a)^2*(2*b*x+3*a)^5/a^6/b/(-4*b^3*x^3+27*a^2* 
b*x+27*a^3)^(5/2)-385/17006112*(2*b*x+3*a)^5*(3-b*x/a)^(5/2)*arctanh(1/3*2 
^(1/2)*(3-b*x/a)^(1/2))*2^(1/2)/a^4/b/(-4*b^3*x^3+27*a^2*b*x+27*a^3)^(5/2)
 

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.39 \[ \int \frac {1}{\left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2}} \, dx=\frac {-6 \sqrt {a} \left (134865 a^5-7128 a^4 b x-73359 a^3 b^2 x^2-29106 a^2 b^3 x^3+4620 a b^4 x^4+3080 b^5 x^5\right )-385 \sqrt {2} (3 a-b x)^{3/2} (3 a+2 b x)^4 \text {arctanh}\left (\frac {\sqrt {6 a-2 b x}}{3 \sqrt {a}}\right )}{17006112 a^{13/2} b (3 a+2 b x) \left ((3 a-b x) (3 a+2 b x)^2\right )^{3/2}} \] Input:

Integrate[(27*a^3 + 27*a^2*b*x - 4*b^3*x^3)^(-5/2),x]
 

Output:

(-6*Sqrt[a]*(134865*a^5 - 7128*a^4*b*x - 73359*a^3*b^2*x^2 - 29106*a^2*b^3 
*x^3 + 4620*a*b^4*x^4 + 3080*b^5*x^5) - 385*Sqrt[2]*(3*a - b*x)^(3/2)*(3*a 
 + 2*b*x)^4*ArcTanh[Sqrt[6*a - 2*b*x]/(3*Sqrt[a])])/(17006112*a^(13/2)*b*( 
3*a + 2*b*x)*((3*a - b*x)*(3*a + 2*b*x)^2)^(3/2))
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 345, normalized size of antiderivative = 0.85, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2473, 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.

\(\displaystyle \int \frac {1}{\left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 2473

\(\displaystyle \frac {31381059609 \sqrt {3} a^{10} (3 a+2 b x)^5 \left (3 a^3-a^2 b x\right )^{5/2} \int \frac {1}{31381059609 \sqrt {3} a^{10} (3 a+2 b x)^5 \left (3 a^3-a^2 b x\right )^{5/2}}dx}{\left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(3 a+2 b x)^5 \left (3 a^3-a^2 b x\right )^{5/2} \int \frac {1}{(3 a+2 b x)^5 \left (3 a^3-a^2 b x\right )^{5/2}}dx}{\left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2}}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {(3 a+2 b x)^5 \left (3 a^3-a^2 b x\right )^{5/2} \left (\frac {11 \int \frac {1}{(3 a+2 b x)^4 \left (3 a^3-a^2 b x\right )^{5/2}}dx}{72 a}-\frac {1}{36 a^3 b (3 a+2 b x)^4 \left (3 a^3-a^2 b x\right )^{3/2}}\right )}{\left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2}}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {(3 a+2 b x)^5 \left (3 a^3-a^2 b x\right )^{5/2} \left (\frac {11 \left (\frac {\int \frac {1}{(3 a+2 b x)^3 \left (3 a^3-a^2 b x\right )^{5/2}}dx}{6 a}-\frac {1}{27 a^3 b (3 a+2 b x)^3 \left (3 a^3-a^2 b x\right )^{3/2}}\right )}{72 a}-\frac {1}{36 a^3 b (3 a+2 b x)^4 \left (3 a^3-a^2 b x\right )^{3/2}}\right )}{\left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2}}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {(3 a+2 b x)^5 \left (3 a^3-a^2 b x\right )^{5/2} \left (\frac {11 \left (\frac {\frac {7 \int \frac {1}{(3 a+2 b x)^2 \left (3 a^3-a^2 b x\right )^{5/2}}dx}{36 a}-\frac {1}{18 a^3 b (3 a+2 b x)^2 \left (3 a^3-a^2 b x\right )^{3/2}}}{6 a}-\frac {1}{27 a^3 b (3 a+2 b x)^3 \left (3 a^3-a^2 b x\right )^{3/2}}\right )}{72 a}-\frac {1}{36 a^3 b (3 a+2 b x)^4 \left (3 a^3-a^2 b x\right )^{3/2}}\right )}{\left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2}}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {(3 a+2 b x)^5 \left (3 a^3-a^2 b x\right )^{5/2} \left (\frac {11 \left (\frac {\frac {7 \left (\frac {5 \int \frac {1}{(3 a+2 b x) \left (3 a^3-a^2 b x\right )^{5/2}}dx}{18 a}-\frac {1}{9 a^3 b (3 a+2 b x) \left (3 a^3-a^2 b x\right )^{3/2}}\right )}{36 a}-\frac {1}{18 a^3 b (3 a+2 b x)^2 \left (3 a^3-a^2 b x\right )^{3/2}}}{6 a}-\frac {1}{27 a^3 b (3 a+2 b x)^3 \left (3 a^3-a^2 b x\right )^{3/2}}\right )}{72 a}-\frac {1}{36 a^3 b (3 a+2 b x)^4 \left (3 a^3-a^2 b x\right )^{3/2}}\right )}{\left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2}}\)

\(\Big \downarrow \) 61

\(\displaystyle \frac {(3 a+2 b x)^5 \left (3 a^3-a^2 b x\right )^{5/2} \left (\frac {11 \left (\frac {\frac {7 \left (\frac {5 \left (\frac {2 \int \frac {1}{(3 a+2 b x) \left (3 a^3-a^2 b x\right )^{3/2}}dx}{9 a^3}+\frac {2}{27 a^3 b \left (3 a^3-a^2 b x\right )^{3/2}}\right )}{18 a}-\frac {1}{9 a^3 b (3 a+2 b x) \left (3 a^3-a^2 b x\right )^{3/2}}\right )}{36 a}-\frac {1}{18 a^3 b (3 a+2 b x)^2 \left (3 a^3-a^2 b x\right )^{3/2}}}{6 a}-\frac {1}{27 a^3 b (3 a+2 b x)^3 \left (3 a^3-a^2 b x\right )^{3/2}}\right )}{72 a}-\frac {1}{36 a^3 b (3 a+2 b x)^4 \left (3 a^3-a^2 b x\right )^{3/2}}\right )}{\left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2}}\)

\(\Big \downarrow \) 61

\(\displaystyle \frac {(3 a+2 b x)^5 \left (3 a^3-a^2 b x\right )^{5/2} \left (\frac {11 \left (\frac {\frac {7 \left (\frac {5 \left (\frac {2 \left (\frac {2 \int \frac {1}{(3 a+2 b x) \sqrt {3 a^3-a^2 b x}}dx}{9 a^3}+\frac {2}{9 a^3 b \sqrt {3 a^3-a^2 b x}}\right )}{9 a^3}+\frac {2}{27 a^3 b \left (3 a^3-a^2 b x\right )^{3/2}}\right )}{18 a}-\frac {1}{9 a^3 b (3 a+2 b x) \left (3 a^3-a^2 b x\right )^{3/2}}\right )}{36 a}-\frac {1}{18 a^3 b (3 a+2 b x)^2 \left (3 a^3-a^2 b x\right )^{3/2}}}{6 a}-\frac {1}{27 a^3 b (3 a+2 b x)^3 \left (3 a^3-a^2 b x\right )^{3/2}}\right )}{72 a}-\frac {1}{36 a^3 b (3 a+2 b x)^4 \left (3 a^3-a^2 b x\right )^{3/2}}\right )}{\left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2}}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {(3 a+2 b x)^5 \left (3 a^3-a^2 b x\right )^{5/2} \left (\frac {11 \left (\frac {\frac {7 \left (\frac {5 \left (\frac {2 \left (\frac {2}{9 a^3 b \sqrt {3 a^3-a^2 b x}}-\frac {4 \int \frac {1}{9 a-\frac {2 \left (3 a^3-a^2 b x\right )}{a^2}}d\sqrt {3 a^3-a^2 b x}}{9 a^5 b}\right )}{9 a^3}+\frac {2}{27 a^3 b \left (3 a^3-a^2 b x\right )^{3/2}}\right )}{18 a}-\frac {1}{9 a^3 b (3 a+2 b x) \left (3 a^3-a^2 b x\right )^{3/2}}\right )}{36 a}-\frac {1}{18 a^3 b (3 a+2 b x)^2 \left (3 a^3-a^2 b x\right )^{3/2}}}{6 a}-\frac {1}{27 a^3 b (3 a+2 b x)^3 \left (3 a^3-a^2 b x\right )^{3/2}}\right )}{72 a}-\frac {1}{36 a^3 b (3 a+2 b x)^4 \left (3 a^3-a^2 b x\right )^{3/2}}\right )}{\left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {(3 a+2 b x)^5 \left (3 a^3-a^2 b x\right )^{5/2} \left (\frac {11 \left (\frac {\frac {7 \left (\frac {5 \left (\frac {2}{27 a^3 b \left (3 a^3-a^2 b x\right )^{3/2}}+\frac {2 \left (\frac {2}{9 a^3 b \sqrt {3 a^3-a^2 b x}}-\frac {2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {3 a^3-a^2 b x}}{3 a^{3/2}}\right )}{27 a^{9/2} b}\right )}{9 a^3}\right )}{18 a}-\frac {1}{9 a^3 b (3 a+2 b x) \left (3 a^3-a^2 b x\right )^{3/2}}\right )}{36 a}-\frac {1}{18 a^3 b (3 a+2 b x)^2 \left (3 a^3-a^2 b x\right )^{3/2}}}{6 a}-\frac {1}{27 a^3 b (3 a+2 b x)^3 \left (3 a^3-a^2 b x\right )^{3/2}}\right )}{72 a}-\frac {1}{36 a^3 b (3 a+2 b x)^4 \left (3 a^3-a^2 b x\right )^{3/2}}\right )}{\left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2}}\)

Input:

Int[(27*a^3 + 27*a^2*b*x - 4*b^3*x^3)^(-5/2),x]
 

Output:

((3*a + 2*b*x)^5*(3*a^3 - a^2*b*x)^(5/2)*(-1/36*1/(a^3*b*(3*a + 2*b*x)^4*( 
3*a^3 - a^2*b*x)^(3/2)) + (11*(-1/27*1/(a^3*b*(3*a + 2*b*x)^3*(3*a^3 - a^2 
*b*x)^(3/2)) + (-1/18*1/(a^3*b*(3*a + 2*b*x)^2*(3*a^3 - a^2*b*x)^(3/2)) + 
(7*(-1/9*1/(a^3*b*(3*a + 2*b*x)*(3*a^3 - a^2*b*x)^(3/2)) + (5*(2/(27*a^3*b 
*(3*a^3 - a^2*b*x)^(3/2)) + (2*(2/(9*a^3*b*Sqrt[3*a^3 - a^2*b*x]) - (2*Sqr 
t[2]*ArcTanh[(Sqrt[2]*Sqrt[3*a^3 - a^2*b*x])/(3*a^(3/2))])/(27*a^(9/2)*b)) 
)/(9*a^3)))/(18*a)))/(36*a))/(6*a)))/(72*a)))/(27*a^3 + 27*a^2*b*x - 4*b^3 
*x^3)^(5/2)
 

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 52

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]
 

rule 61

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]
 

rule 73

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]
 

rule 221

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]
 

rule 2473

Int[((a_.) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_Symbol] :> Simp[(a + b*x + 
d*x^3)^p/((3*a - b*x)^p*(3*a + 2*b*x)^(2*p))   Int[(3*a - b*x)^p*(3*a + 2*b 
*x)^(2*p), x], x] /; FreeQ[{a, b, d, p}, x] && EqQ[4*b^3 + 27*a^2*d, 0] && 
 !IntegerQ[p]
 

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 305, normalized size of antiderivative = 0.75

method	result	size
default	\(-\frac {\left (83160 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-b x +3 a}\, \sqrt {2}}{3 \sqrt {a}}\right ) \left (-b x +3 a \right )^{\frac {3}{2}} a^{2} b^{2} x^{2}+83160 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-b x +3 a}\, \sqrt {2}}{3 \sqrt {a}}\right ) \left (-b x +3 a \right )^{\frac {3}{2}} a^{3} b x +36960 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-b x +3 a}\, \sqrt {2}}{3 \sqrt {a}}\right ) \left (-b x +3 a \right )^{\frac {3}{2}} a \,b^{3} x^{3}+27720 a^{\frac {3}{2}} b^{4} x^{4}+31185 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-b x +3 a}\, \sqrt {2}}{3 \sqrt {a}}\right ) \left (-b x +3 a \right )^{\frac {3}{2}} a^{4}-174636 a^{\frac {5}{2}} b^{3} x^{3}-440154 a^{\frac {7}{2}} b^{2} x^{2}-42768 a^{\frac {9}{2}} b x +809190 a^{\frac {11}{2}}+6160 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-b x +3 a}\, \sqrt {2}}{3 \sqrt {a}}\right ) \left (-b x +3 a \right )^{\frac {3}{2}} b^{4} x^{4}+18480 \sqrt {a}\, b^{5} x^{5}\right ) \left (-b x +3 a \right ) \left (2 b x +3 a \right )}{17006112 a^{\frac {13}{2}} b \left (-4 b^{3} x^{3}+27 b \,a^{2} x +27 a^{3}\right )^{\frac {5}{2}}}\)	\(305\)

Input:

int(1/(-4*b^3*x^3+27*a^2*b*x+27*a^3)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-1/17006112*(83160*2^(1/2)*arctanh(1/3*(-b*x+3*a)^(1/2)*2^(1/2)/a^(1/2))*( 
-b*x+3*a)^(3/2)*a^2*b^2*x^2+83160*2^(1/2)*arctanh(1/3*(-b*x+3*a)^(1/2)*2^( 
1/2)/a^(1/2))*(-b*x+3*a)^(3/2)*a^3*b*x+36960*2^(1/2)*arctanh(1/3*(-b*x+3*a 
)^(1/2)*2^(1/2)/a^(1/2))*(-b*x+3*a)^(3/2)*a*b^3*x^3+27720*a^(3/2)*b^4*x^4+ 
31185*2^(1/2)*arctanh(1/3*(-b*x+3*a)^(1/2)*2^(1/2)/a^(1/2))*(-b*x+3*a)^(3/ 
2)*a^4-174636*a^(5/2)*b^3*x^3-440154*a^(7/2)*b^2*x^2-42768*a^(9/2)*b*x+809 
190*a^(11/2)+6160*2^(1/2)*arctanh(1/3*(-b*x+3*a)^(1/2)*2^(1/2)/a^(1/2))*(- 
b*x+3*a)^(3/2)*b^4*x^4+18480*a^(1/2)*b^5*x^5)*(-b*x+3*a)*(2*b*x+3*a)/a^(13 
/2)/b/(-4*b^3*x^3+27*a^2*b*x+27*a^3)^(5/2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 629, normalized size of antiderivative = 1.55 \[ \int \frac {1}{\left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2}} \, dx=\left [\frac {385 \, \sqrt {2} {\left (32 \, b^{7} x^{7} + 48 \, a b^{6} x^{6} - 432 \, a^{2} b^{5} x^{5} - 1080 \, a^{3} b^{4} x^{4} + 810 \, a^{4} b^{3} x^{3} + 5103 \, a^{5} b^{2} x^{2} + 5832 \, a^{6} b x + 2187 \, a^{7}\right )} \sqrt {a} \log \left (\frac {4 \, b^{2} x^{2} - 24 \, a b x - 45 \, a^{2} + 6 \, \sqrt {2} \sqrt {-4 \, b^{3} x^{3} + 27 \, a^{2} b x + 27 \, a^{3}} \sqrt {a}}{4 \, b^{2} x^{2} + 12 \, a b x + 9 \, a^{2}}\right ) - 12 \, {\left (3080 \, a b^{5} x^{5} + 4620 \, a^{2} b^{4} x^{4} - 29106 \, a^{3} b^{3} x^{3} - 73359 \, a^{4} b^{2} x^{2} - 7128 \, a^{5} b x + 134865 \, a^{6}\right )} \sqrt {-4 \, b^{3} x^{3} + 27 \, a^{2} b x + 27 \, a^{3}}}{34012224 \, {\left (32 \, a^{7} b^{8} x^{7} + 48 \, a^{8} b^{7} x^{6} - 432 \, a^{9} b^{6} x^{5} - 1080 \, a^{10} b^{5} x^{4} + 810 \, a^{11} b^{4} x^{3} + 5103 \, a^{12} b^{3} x^{2} + 5832 \, a^{13} b^{2} x + 2187 \, a^{14} b\right )}}, -\frac {385 \, \sqrt {2} {\left (32 \, b^{7} x^{7} + 48 \, a b^{6} x^{6} - 432 \, a^{2} b^{5} x^{5} - 1080 \, a^{3} b^{4} x^{4} + 810 \, a^{4} b^{3} x^{3} + 5103 \, a^{5} b^{2} x^{2} + 5832 \, a^{6} b x + 2187 \, a^{7}\right )} \sqrt {-a} \arctan \left (\frac {3 \, \sqrt {2} \sqrt {-4 \, b^{3} x^{3} + 27 \, a^{2} b x + 27 \, a^{3}} \sqrt {-a}}{2 \, {\left (2 \, b^{2} x^{2} - 3 \, a b x - 9 \, a^{2}\right )}}\right ) + 6 \, {\left (3080 \, a b^{5} x^{5} + 4620 \, a^{2} b^{4} x^{4} - 29106 \, a^{3} b^{3} x^{3} - 73359 \, a^{4} b^{2} x^{2} - 7128 \, a^{5} b x + 134865 \, a^{6}\right )} \sqrt {-4 \, b^{3} x^{3} + 27 \, a^{2} b x + 27 \, a^{3}}}{17006112 \, {\left (32 \, a^{7} b^{8} x^{7} + 48 \, a^{8} b^{7} x^{6} - 432 \, a^{9} b^{6} x^{5} - 1080 \, a^{10} b^{5} x^{4} + 810 \, a^{11} b^{4} x^{3} + 5103 \, a^{12} b^{3} x^{2} + 5832 \, a^{13} b^{2} x + 2187 \, a^{14} b\right )}}\right ] \] Input:

integrate(1/(-4*b^3*x^3+27*a^2*b*x+27*a^3)^(5/2),x, algorithm="fricas")
                                                                                    
                                                                                    
 

Output:

[1/34012224*(385*sqrt(2)*(32*b^7*x^7 + 48*a*b^6*x^6 - 432*a^2*b^5*x^5 - 10 
80*a^3*b^4*x^4 + 810*a^4*b^3*x^3 + 5103*a^5*b^2*x^2 + 5832*a^6*b*x + 2187* 
a^7)*sqrt(a)*log((4*b^2*x^2 - 24*a*b*x - 45*a^2 + 6*sqrt(2)*sqrt(-4*b^3*x^ 
3 + 27*a^2*b*x + 27*a^3)*sqrt(a))/(4*b^2*x^2 + 12*a*b*x + 9*a^2)) - 12*(30 
80*a*b^5*x^5 + 4620*a^2*b^4*x^4 - 29106*a^3*b^3*x^3 - 73359*a^4*b^2*x^2 - 
7128*a^5*b*x + 134865*a^6)*sqrt(-4*b^3*x^3 + 27*a^2*b*x + 27*a^3))/(32*a^7 
*b^8*x^7 + 48*a^8*b^7*x^6 - 432*a^9*b^6*x^5 - 1080*a^10*b^5*x^4 + 810*a^11 
*b^4*x^3 + 5103*a^12*b^3*x^2 + 5832*a^13*b^2*x + 2187*a^14*b), -1/17006112 
*(385*sqrt(2)*(32*b^7*x^7 + 48*a*b^6*x^6 - 432*a^2*b^5*x^5 - 1080*a^3*b^4* 
x^4 + 810*a^4*b^3*x^3 + 5103*a^5*b^2*x^2 + 5832*a^6*b*x + 2187*a^7)*sqrt(- 
a)*arctan(3/2*sqrt(2)*sqrt(-4*b^3*x^3 + 27*a^2*b*x + 27*a^3)*sqrt(-a)/(2*b 
^2*x^2 - 3*a*b*x - 9*a^2)) + 6*(3080*a*b^5*x^5 + 4620*a^2*b^4*x^4 - 29106* 
a^3*b^3*x^3 - 73359*a^4*b^2*x^2 - 7128*a^5*b*x + 134865*a^6)*sqrt(-4*b^3*x 
^3 + 27*a^2*b*x + 27*a^3))/(32*a^7*b^8*x^7 + 48*a^8*b^7*x^6 - 432*a^9*b^6* 
x^5 - 1080*a^10*b^5*x^4 + 810*a^11*b^4*x^3 + 5103*a^12*b^3*x^2 + 5832*a^13 
*b^2*x + 2187*a^14*b)]
 

Sympy [F]

\[ \int \frac {1}{\left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2}} \, dx=\int \frac {1}{\left (27 a^{3} + 27 a^{2} b x - 4 b^{3} x^{3}\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(1/(-4*b**3*x**3+27*a**2*b*x+27*a**3)**(5/2),x)
 

Output:

Integral((27*a**3 + 27*a**2*b*x - 4*b**3*x**3)**(-5/2), x)
 

Maxima [F]

\[ \int \frac {1}{\left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2}} \, dx=\int { \frac {1}{{\left (-4 \, b^{3} x^{3} + 27 \, a^{2} b x + 27 \, a^{3}\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(-4*b^3*x^3+27*a^2*b*x+27*a^3)^(5/2),x, algorithm="maxima")
 

Output:

integrate((-4*b^3*x^3 + 27*a^2*b*x + 27*a^3)^(-5/2), x)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.49 \[ \int \frac {1}{\left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2}} \, dx=-\frac {385 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-b x + 3 \, a}}{3 \, \sqrt {-a}}\right )}{17006112 \, \sqrt {-a} a^{6} b \mathrm {sgn}\left (-2 \, b x - 3 \, a\right )} - \frac {2 \, {\left (10 \, b x - 33 \, a\right )}}{531441 \, {\left (b x - 3 \, a\right )} \sqrt {-b x + 3 \, a} a^{6} b \mathrm {sgn}\left (-2 \, b x - 3 \, a\right )} + \frac {4120 \, {\left (b x - 3 \, a\right )}^{3} \sqrt {-b x + 3 \, a} + 61836 \, {\left (b x - 3 \, a\right )}^{2} \sqrt {-b x + 3 \, a} a - 316170 \, {\left (-b x + 3 \, a\right )}^{\frac {3}{2}} a^{2} + 557685 \, \sqrt {-b x + 3 \, a} a^{3}}{8503056 \, {\left (2 \, b x + 3 \, a\right )}^{4} a^{6} b \mathrm {sgn}\left (-2 \, b x - 3 \, a\right )} \] Input:

integrate(1/(-4*b^3*x^3+27*a^2*b*x+27*a^3)^(5/2),x, algorithm="giac")
 

Output:

-385/17006112*sqrt(2)*arctan(1/3*sqrt(2)*sqrt(-b*x + 3*a)/sqrt(-a))/(sqrt( 
-a)*a^6*b*sgn(-2*b*x - 3*a)) - 2/531441*(10*b*x - 33*a)/((b*x - 3*a)*sqrt( 
-b*x + 3*a)*a^6*b*sgn(-2*b*x - 3*a)) + 1/8503056*(4120*(b*x - 3*a)^3*sqrt( 
-b*x + 3*a) + 61836*(b*x - 3*a)^2*sqrt(-b*x + 3*a)*a - 316170*(-b*x + 3*a) 
^(3/2)*a^2 + 557685*sqrt(-b*x + 3*a)*a^3)/((2*b*x + 3*a)^4*a^6*b*sgn(-2*b* 
x - 3*a))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2}} \, dx=\int \frac {1}{{\left (27\,a^3+27\,a^2\,b\,x-4\,b^3\,x^3\right )}^{5/2}} \,d x \] Input:

int(1/(27*a^3 - 4*b^3*x^3 + 27*a^2*b*x)^(5/2),x)
 

Output:

int(1/(27*a^3 - 4*b^3*x^3 + 27*a^2*b*x)^(5/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 614, normalized size of antiderivative = 1.52 \[ \int \frac {1}{\left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2}} \, dx=\frac {93555 \sqrt {a}\, \sqrt {-b x +3 a}\, \sqrt {2}\, \mathrm {log}\left (2 \sqrt {-b x +3 a}-3 \sqrt {a}\, \sqrt {2}\right ) a^{5}+218295 \sqrt {a}\, \sqrt {-b x +3 a}\, \sqrt {2}\, \mathrm {log}\left (2 \sqrt {-b x +3 a}-3 \sqrt {a}\, \sqrt {2}\right ) a^{4} b x +166320 \sqrt {a}\, \sqrt {-b x +3 a}\, \sqrt {2}\, \mathrm {log}\left (2 \sqrt {-b x +3 a}-3 \sqrt {a}\, \sqrt {2}\right ) a^{3} b^{2} x^{2}+27720 \sqrt {a}\, \sqrt {-b x +3 a}\, \sqrt {2}\, \mathrm {log}\left (2 \sqrt {-b x +3 a}-3 \sqrt {a}\, \sqrt {2}\right ) a^{2} b^{3} x^{3}-18480 \sqrt {a}\, \sqrt {-b x +3 a}\, \sqrt {2}\, \mathrm {log}\left (2 \sqrt {-b x +3 a}-3 \sqrt {a}\, \sqrt {2}\right ) a \,b^{4} x^{4}-6160 \sqrt {a}\, \sqrt {-b x +3 a}\, \sqrt {2}\, \mathrm {log}\left (2 \sqrt {-b x +3 a}-3 \sqrt {a}\, \sqrt {2}\right ) b^{5} x^{5}-93555 \sqrt {a}\, \sqrt {-b x +3 a}\, \sqrt {2}\, \mathrm {log}\left (2 \sqrt {-b x +3 a}+3 \sqrt {a}\, \sqrt {2}\right ) a^{5}-218295 \sqrt {a}\, \sqrt {-b x +3 a}\, \sqrt {2}\, \mathrm {log}\left (2 \sqrt {-b x +3 a}+3 \sqrt {a}\, \sqrt {2}\right ) a^{4} b x -166320 \sqrt {a}\, \sqrt {-b x +3 a}\, \sqrt {2}\, \mathrm {log}\left (2 \sqrt {-b x +3 a}+3 \sqrt {a}\, \sqrt {2}\right ) a^{3} b^{2} x^{2}-27720 \sqrt {a}\, \sqrt {-b x +3 a}\, \sqrt {2}\, \mathrm {log}\left (2 \sqrt {-b x +3 a}+3 \sqrt {a}\, \sqrt {2}\right ) a^{2} b^{3} x^{3}+18480 \sqrt {a}\, \sqrt {-b x +3 a}\, \sqrt {2}\, \mathrm {log}\left (2 \sqrt {-b x +3 a}+3 \sqrt {a}\, \sqrt {2}\right ) a \,b^{4} x^{4}+6160 \sqrt {a}\, \sqrt {-b x +3 a}\, \sqrt {2}\, \mathrm {log}\left (2 \sqrt {-b x +3 a}+3 \sqrt {a}\, \sqrt {2}\right ) b^{5} x^{5}-1618380 a^{6}+85536 a^{5} b x +880308 a^{4} b^{2} x^{2}+349272 a^{3} b^{3} x^{3}-55440 a^{2} b^{4} x^{4}-36960 a \,b^{5} x^{5}}{34012224 \sqrt {-b x +3 a}\, a^{7} b \left (-16 b^{5} x^{5}-48 a \,b^{4} x^{4}+72 a^{2} b^{3} x^{3}+432 a^{3} b^{2} x^{2}+567 a^{4} b x +243 a^{5}\right )} \] Input:

int(1/(-4*b^3*x^3+27*a^2*b*x+27*a^3)^(5/2),x)
 

Output:

(93555*sqrt(a)*sqrt(3*a - b*x)*sqrt(2)*log(2*sqrt(3*a - b*x) - 3*sqrt(a)*s 
qrt(2))*a**5 + 218295*sqrt(a)*sqrt(3*a - b*x)*sqrt(2)*log(2*sqrt(3*a - b*x 
) - 3*sqrt(a)*sqrt(2))*a**4*b*x + 166320*sqrt(a)*sqrt(3*a - b*x)*sqrt(2)*l 
og(2*sqrt(3*a - b*x) - 3*sqrt(a)*sqrt(2))*a**3*b**2*x**2 + 27720*sqrt(a)*s 
qrt(3*a - b*x)*sqrt(2)*log(2*sqrt(3*a - b*x) - 3*sqrt(a)*sqrt(2))*a**2*b** 
3*x**3 - 18480*sqrt(a)*sqrt(3*a - b*x)*sqrt(2)*log(2*sqrt(3*a - b*x) - 3*s 
qrt(a)*sqrt(2))*a*b**4*x**4 - 6160*sqrt(a)*sqrt(3*a - b*x)*sqrt(2)*log(2*s 
qrt(3*a - b*x) - 3*sqrt(a)*sqrt(2))*b**5*x**5 - 93555*sqrt(a)*sqrt(3*a - b 
*x)*sqrt(2)*log(2*sqrt(3*a - b*x) + 3*sqrt(a)*sqrt(2))*a**5 - 218295*sqrt( 
a)*sqrt(3*a - b*x)*sqrt(2)*log(2*sqrt(3*a - b*x) + 3*sqrt(a)*sqrt(2))*a**4 
*b*x - 166320*sqrt(a)*sqrt(3*a - b*x)*sqrt(2)*log(2*sqrt(3*a - b*x) + 3*sq 
rt(a)*sqrt(2))*a**3*b**2*x**2 - 27720*sqrt(a)*sqrt(3*a - b*x)*sqrt(2)*log( 
2*sqrt(3*a - b*x) + 3*sqrt(a)*sqrt(2))*a**2*b**3*x**3 + 18480*sqrt(a)*sqrt 
(3*a - b*x)*sqrt(2)*log(2*sqrt(3*a - b*x) + 3*sqrt(a)*sqrt(2))*a*b**4*x**4 
 + 6160*sqrt(a)*sqrt(3*a - b*x)*sqrt(2)*log(2*sqrt(3*a - b*x) + 3*sqrt(a)* 
sqrt(2))*b**5*x**5 - 1618380*a**6 + 85536*a**5*b*x + 880308*a**4*b**2*x**2 
 + 349272*a**3*b**3*x**3 - 55440*a**2*b**4*x**4 - 36960*a*b**5*x**5)/(3401 
2224*sqrt(3*a - b*x)*a**7*b*(243*a**5 + 567*a**4*b*x + 432*a**3*b**2*x**2 
+ 72*a**2*b**3*x**3 - 48*a*b**4*x**4 - 16*b**5*x**5))