\(\int \frac {4 \sqrt {5} a^6 b^2-3 \sqrt {10} a^3 b x^6+2 \sqrt {2} a^3 b^2 x^9+3 \sqrt {5} x^{12}}{a^6 b^2 x^4 (\sqrt {2} a^3 b-x^6)^2} \, dx\) [20]

Optimal result

Integrand size = 82, antiderivative size = 68 \[ \int \frac {4 \sqrt {5} a^6 b^2-3 \sqrt {10} a^3 b x^6+2 \sqrt {2} a^3 b^2 x^9+3 \sqrt {5} x^{12}}{a^6 b^2 x^4 \left (\sqrt {2} a^3 b-x^6\right )^2} \, dx=\frac {-2 \sqrt {10} a^3 b+\sqrt {2} a^3 b^2 x^3+3 \sqrt {5} x^6}{3 a^6 b^2 x^3 \left (\sqrt {2} a^3 b-x^6\right )} \] Output:

1/3*(-2*10^(1/2)*a^3*b+2^(1/2)*a^3*b^2*x^3+3*5^(1/2)*x^6)/a^6/b^2/x^3/(2^( 
1/2)*a^3*b-x^6)
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.96 \[ \int \frac {4 \sqrt {5} a^6 b^2-3 \sqrt {10} a^3 b x^6+2 \sqrt {2} a^3 b^2 x^9+3 \sqrt {5} x^{12}}{a^6 b^2 x^4 \left (\sqrt {2} a^3 b-x^6\right )^2} \, dx=\frac {-\frac {2 \sqrt {5}}{3 x^3}+\frac {-\sqrt {2} a^3 b^2-\sqrt {5} x^3}{3 \left (-\sqrt {2} a^3 b+x^6\right )}}{a^6 b^2} \] Input:

Integrate[(4*Sqrt[5]*a^6*b^2 - 3*Sqrt[10]*a^3*b*x^6 + 2*Sqrt[2]*a^3*b^2*x^ 
9 + 3*Sqrt[5]*x^12)/(a^6*b^2*x^4*(Sqrt[2]*a^3*b - x^6)^2),x]
 

Output:

((-2*Sqrt[5])/(3*x^3) + (-(Sqrt[2]*a^3*b^2) - Sqrt[5]*x^3)/(3*(-(Sqrt[2]*a 
^3*b) + x^6)))/(a^6*b^2)
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.85, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {27, 2368, 27, 281, 15}

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.

\(\displaystyle \int \frac {4 \sqrt {5} a^6 b^2+2 \sqrt {2} a^3 b^2 x^9-3 \sqrt {10} a^3 b x^6+3 \sqrt {5} x^{12}}{a^6 b^2 x^4 \left (\sqrt {2} a^3 b-x^6\right )^2} \, dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {3 \sqrt {5} x^{12}+2 \sqrt {2} a^3 b^2 x^9-3 \sqrt {10} a^3 b x^6+4 \sqrt {5} a^6 b^2}{x^4 \left (\sqrt {2} a^3 b-x^6\right )^2}dx}{a^6 b^2}\)

\(\Big \downarrow \) 2368

\(\displaystyle \frac {\frac {x \left (b x^5+\sqrt {5} x^2\right )}{3 \left (\sqrt {2} a^3 b-x^6\right )}-\frac {\int -\frac {12 a^3 b \left (2 \sqrt {5} a^3 b-\sqrt {10} x^6\right )}{x^4 \left (\sqrt {2} a^3 b-x^6\right )}dx}{6 \sqrt {2} a^3 b}}{a^6 b^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {2} \int \frac {2 \sqrt {5} a^3 b-\sqrt {10} x^6}{x^4 \left (\sqrt {2} a^3 b-x^6\right )}dx+\frac {x \left (b x^5+\sqrt {5} x^2\right )}{3 \left (\sqrt {2} a^3 b-x^6\right )}}{a^6 b^2}\)

\(\Big \downarrow \) 281

\(\displaystyle \frac {2 \sqrt {5} \int \frac {1}{x^4}dx+\frac {x \left (b x^5+\sqrt {5} x^2\right )}{3 \left (\sqrt {2} a^3 b-x^6\right )}}{a^6 b^2}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {\frac {x \left (b x^5+\sqrt {5} x^2\right )}{3 \left (\sqrt {2} a^3 b-x^6\right )}-\frac {2 \sqrt {5}}{3 x^3}}{a^6 b^2}\)

Input:

Int[(4*Sqrt[5]*a^6*b^2 - 3*Sqrt[10]*a^3*b*x^6 + 2*Sqrt[2]*a^3*b^2*x^9 + 3* 
Sqrt[5]*x^12)/(a^6*b^2*x^4*(Sqrt[2]*a^3*b - x^6)^2),x]
 

Output:

((-2*Sqrt[5])/(3*x^3) + (x*(Sqrt[5]*x^2 + b*x^5))/(3*(Sqrt[2]*a^3*b - x^6) 
))/(a^6*b^2)
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

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 281

Int[(u_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_ 
Symbol] :> Simp[(b/d)^p   Int[u*(c + d*x^n)^(p + q), x], x] /; FreeQ[{a, b, 
 c, d, n, p, q}, x] && EqQ[b*c - a*d, 0] && IntegerQ[p] &&  !(IntegerQ[q] & 
& SimplerQ[a + b*x^n, c + d*x^n])
 

rule 2368

Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = 
Expon[Pq, x]}, Module[{Q = PolynomialQuotient[a*b^(Floor[(q - 1)/n] + 1)*x^ 
m*Pq, a + b*x^n, x], R = PolynomialRemainder[a*b^(Floor[(q - 1)/n] + 1)*x^m 
*Pq, a + b*x^n, x], i}, Simp[(-x)*R*((a + b*x^n)^(p + 1)/(a^2*n*(p + 1)*b^( 
Floor[(q - 1)/n] + 1))), x] + Simp[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)) 
   Int[x^m*(a + b*x^n)^(p + 1)*ExpandToSum[(n*(p + 1)*Q)/x^m + Sum[((n*(p + 
 1) + i + 1)/a)*Coeff[R, x, i]*x^(i - m), {i, 0, n - 1}], x], x], x]]] /; F 
reeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && ILtQ[m, 0]
 

Maple [A] (verified)

Time = 1.52 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.85

method	result	size
default	\(\frac {-\frac {2 \sqrt {5}}{3 x^{3}}+\frac {\sqrt {2}\, \left (\frac {\sqrt {5}\, x^{3}}{2}+\frac {\sqrt {2}\, a^{3} b^{2}}{2}\right )}{-\frac {3 x^{6} \sqrt {2}}{2}+3 a^{3} b}}{a^{6} b^{2}}\)	\(58\)
risch	\(\frac {\frac {\sqrt {2}\, \sqrt {5}\, x^{6}}{2}+\frac {a^{3} b^{2} x^{3}}{3}-\frac {2 \sqrt {5}\, a^{3} b}{3}}{a^{6} b^{2} x^{3} \left (-\frac {x^{6} \sqrt {2}}{2}+a^{3} b \right )}\)	\(59\)
parallelrisch	\(-\frac {-\sqrt {2}\, a^{3} b^{2} x^{3}-3 \sqrt {5}\, x^{6}+2 \sqrt {2}\, a^{3} b \sqrt {5}}{3 a^{6} b^{2} x^{3} \left (\sqrt {2}\, a^{3} b -x^{6}\right )}\)	\(63\)
orering	\(\frac {\left (\sqrt {5}\, \sqrt {2}\, a^{3} b^{2} x^{3}+15 x^{6}-10 \sqrt {2}\, a^{3} b \right ) \left (4 \sqrt {5}\, a^{6} b^{2}-3 \sqrt {10}\, a^{3} b \,x^{6}+2 \sqrt {2}\, a^{3} b^{2} x^{9}+3 \sqrt {5}\, x^{12}\right )}{3 x^{3} \left (\sqrt {2}\, a^{3} b -x^{6}\right ) \left (2 \sqrt {5}\, \sqrt {2}\, a^{3} b^{2} x^{9}+15 x^{12}-15 \sqrt {2}\, a^{3} b \,x^{6}+20 a^{6} b^{2}\right ) a^{6} b^{2}}\)	\(150\)
gosper	\(-\frac {\left (x^{6} \sqrt {2}-2 a^{3} b \right ) \left (2 \sqrt {5}\, a^{3} b^{2} x^{3}+15 x^{6} \sqrt {2}-20 a^{3} b \right ) \left (4 \sqrt {5}\, a^{6} b^{2}-3 \sqrt {10}\, a^{3} b \,x^{6}+2 \sqrt {2}\, a^{3} b^{2} x^{9}+3 \sqrt {5}\, x^{12}\right )}{6 x^{3} \left (2 \sqrt {5}\, \sqrt {2}\, a^{3} b^{2} x^{9}+15 x^{12}-15 \sqrt {2}\, a^{3} b \,x^{6}+20 a^{6} b^{2}\right ) a^{6} b^{2} \left (\sqrt {2}\, a^{3} b -x^{6}\right )^{2}}\)	\(162\)

Input:

int((4*5^(1/2)*a^6*b^2-3*10^(1/2)*a^3*b*x^6+2*2^(1/2)*a^3*b^2*x^9+3*5^(1/2 
)*x^12)/a^6/b^2/x^4/(2^(1/2)*a^3*b-x^6)^2,x,method=_RETURNVERBOSE)
 

Output:

1/a^6/b^2*(-2/3*5^(1/2)/x^3+1/3*2^(1/2)*(1/2*5^(1/2)*x^3+1/2*2^(1/2)*a^3*b 
^2)/(-1/2*x^6*2^(1/2)+a^3*b))
 

Fricas [F(-1)]

Timed out. \[ \int \frac {4 \sqrt {5} a^6 b^2-3 \sqrt {10} a^3 b x^6+2 \sqrt {2} a^3 b^2 x^9+3 \sqrt {5} x^{12}}{a^6 b^2 x^4 \left (\sqrt {2} a^3 b-x^6\right )^2} \, dx=\text {Timed out} \] Input:

integrate((4*5^(1/2)*a^6*b^2-3*10^(1/2)*a^3*b*x^6+2*2^(1/2)*a^3*b^2*x^9+3* 
5^(1/2)*x^12)/a^6/b^2/x^4/(2^(1/2)*a^3*b-x^6)^2,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {4 \sqrt {5} a^6 b^2-3 \sqrt {10} a^3 b x^6+2 \sqrt {2} a^3 b^2 x^9+3 \sqrt {5} x^{12}}{a^6 b^2 x^4 \left (\sqrt {2} a^3 b-x^6\right )^2} \, dx=\text {Timed out} \] Input:

integrate((4*5**(1/2)*a**6*b**2-3*10**(1/2)*a**3*b*x**6+2*2**(1/2)*a**3*b* 
*2*x**9+3*5**(1/2)*x**12)/a**6/b**2/x**4/(2**(1/2)*a**3*b-x**6)**2,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 133 vs. \(2 (57) = 114\).

Time = 0.10 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.96 \[ \int \frac {4 \sqrt {5} a^6 b^2-3 \sqrt {10} a^3 b x^6+2 \sqrt {2} a^3 b^2 x^9+3 \sqrt {5} x^{12}}{a^6 b^2 x^4 \left (\sqrt {2} a^3 b-x^6\right )^2} \, dx=\frac {\frac {3 \cdot 2^{\frac {3}{4}} {\left (\sqrt {10} \sqrt {2} - 2 \, \sqrt {5}\right )} \log \left (\frac {x^{3} - \sqrt {\sqrt {2} a b} a}{x^{3} + \sqrt {\sqrt {2} a b} a}\right )}{\sqrt {a b} a} - \frac {4 \, {\left (4 \, \sqrt {2} a^{3} b^{2} x^{3} - 3 \, {\left (\sqrt {10} \sqrt {2} - 6 \, \sqrt {5}\right )} x^{6} - 8 \, \sqrt {5} \sqrt {2} a^{3} b\right )}}{x^{9} - \sqrt {2} a^{3} b x^{3}}}{48 \, a^{6} b^{2}} \] Input:

integrate((4*5^(1/2)*a^6*b^2-3*10^(1/2)*a^3*b*x^6+2*2^(1/2)*a^3*b^2*x^9+3* 
5^(1/2)*x^12)/a^6/b^2/x^4/(2^(1/2)*a^3*b-x^6)^2,x, algorithm="maxima")
 

Output:

1/48*(3*2^(3/4)*(sqrt(10)*sqrt(2) - 2*sqrt(5))*log((x^3 - sqrt(sqrt(2)*a*b 
)*a)/(x^3 + sqrt(sqrt(2)*a*b)*a))/(sqrt(a*b)*a) - 4*(4*sqrt(2)*a^3*b^2*x^3 
 - 3*(sqrt(10)*sqrt(2) - 6*sqrt(5))*x^6 - 8*sqrt(5)*sqrt(2)*a^3*b)/(x^9 - 
sqrt(2)*a^3*b*x^3))/(a^6*b^2)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.82 \[ \int \frac {4 \sqrt {5} a^6 b^2-3 \sqrt {10} a^3 b x^6+2 \sqrt {2} a^3 b^2 x^9+3 \sqrt {5} x^{12}}{a^6 b^2 x^4 \left (\sqrt {2} a^3 b-x^6\right )^2} \, dx=-\frac {2 \, a^{3} b^{2} x^{3} + 3 \, \sqrt {10} x^{6} - 4 \, \sqrt {5} a^{3} b}{3 \, {\left (\sqrt {2} x^{9} - 2 \, a^{3} b x^{3}\right )} a^{6} b^{2}} \] Input:

integrate((4*5^(1/2)*a^6*b^2-3*10^(1/2)*a^3*b*x^6+2*2^(1/2)*a^3*b^2*x^9+3* 
5^(1/2)*x^12)/a^6/b^2/x^4/(2^(1/2)*a^3*b-x^6)^2,x, algorithm="giac")
 

Output:

-1/3*(2*a^3*b^2*x^3 + 3*sqrt(10)*x^6 - 4*sqrt(5)*a^3*b)/((sqrt(2)*x^9 - 2* 
a^3*b*x^3)*a^6*b^2)
 

Mupad [B] (verification not implemented)

Time = 11.27 (sec) , antiderivative size = 358, normalized size of antiderivative = 5.26 \[ \int \frac {4 \sqrt {5} a^6 b^2-3 \sqrt {10} a^3 b x^6+2 \sqrt {2} a^3 b^2 x^9+3 \sqrt {5} x^{12}}{a^6 b^2 x^4 \left (\sqrt {2} a^3 b-x^6\right )^2} \, dx=\frac {2\,\sqrt {2}\,\sqrt {5}}{3\,\left (a^3\,b\,x^9-\sqrt {2}\,a^6\,b^2\,x^3\right )}-\frac {3\,\sqrt {5}\,x^6}{2\,\left (a^6\,b^2\,x^9-\sqrt {2}\,a^9\,b^3\,x^3\right )}-\frac {\sqrt {2}\,x^3}{3\,\left (a^3\,x^9-\sqrt {2}\,a^6\,b\,x^3\right )}+\frac {\sqrt {2}\,\sqrt {10}\,x^6}{4\,\left (a^6\,b^2\,x^9-\sqrt {2}\,a^9\,b^3\,x^3\right )}-\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {2^{3/4}\,\sqrt {5}\,x^3\,135{}\mathrm {i}}{2\,a^{39/2}\,b^{13/2}\,\left (\frac {135\,\sqrt {5}}{a^{18}\,b^6}-\frac {135\,\sqrt {2}\,\sqrt {10}}{2\,a^{18}\,b^6}\right )}-\frac {2^{1/4}\,\sqrt {10}\,x^3\,135{}\mathrm {i}}{2\,a^{39/2}\,b^{13/2}\,\left (\frac {135\,\sqrt {5}}{a^{18}\,b^6}-\frac {135\,\sqrt {2}\,\sqrt {10}}{2\,a^{18}\,b^6}\right )}\right )\,1{}\mathrm {i}}{4\,a^{15/2}\,b^{5/2}}+\frac {2^{1/4}\,\sqrt {10}\,\mathrm {atan}\left (\frac {2^{3/4}\,\sqrt {5}\,x^3\,135{}\mathrm {i}}{2\,a^{39/2}\,b^{13/2}\,\left (\frac {135\,\sqrt {5}}{a^{18}\,b^6}-\frac {135\,\sqrt {2}\,\sqrt {10}}{2\,a^{18}\,b^6}\right )}-\frac {2^{1/4}\,\sqrt {10}\,x^3\,135{}\mathrm {i}}{2\,a^{39/2}\,b^{13/2}\,\left (\frac {135\,\sqrt {5}}{a^{18}\,b^6}-\frac {135\,\sqrt {2}\,\sqrt {10}}{2\,a^{18}\,b^6}\right )}\right )\,1{}\mathrm {i}}{4\,a^{15/2}\,b^{5/2}} \] Input:

int((3*5^(1/2)*x^12 + 4*5^(1/2)*a^6*b^2 + 2*2^(1/2)*a^3*b^2*x^9 - 3*10^(1/ 
2)*a^3*b*x^6)/(a^6*b^2*x^4*(x^6 - 2^(1/2)*a^3*b)^2),x)
 

Output:

(2*2^(1/2)*5^(1/2))/(3*(a^3*b*x^9 - 2^(1/2)*a^6*b^2*x^3)) - (3*5^(1/2)*x^6 
)/(2*(a^6*b^2*x^9 - 2^(1/2)*a^9*b^3*x^3)) - (2^(1/2)*x^3)/(3*(a^3*x^9 - 2^ 
(1/2)*a^6*b*x^3)) + (2^(1/2)*10^(1/2)*x^6)/(4*(a^6*b^2*x^9 - 2^(1/2)*a^9*b 
^3*x^3)) - (2^(3/4)*5^(1/2)*atan((2^(3/4)*5^(1/2)*x^3*135i)/(2*a^(39/2)*b^ 
(13/2)*((135*5^(1/2))/(a^18*b^6) - (135*2^(1/2)*10^(1/2))/(2*a^18*b^6))) - 
 (2^(1/4)*10^(1/2)*x^3*135i)/(2*a^(39/2)*b^(13/2)*((135*5^(1/2))/(a^18*b^6 
) - (135*2^(1/2)*10^(1/2))/(2*a^18*b^6))))*1i)/(4*a^(15/2)*b^(5/2)) + (2^( 
1/4)*10^(1/2)*atan((2^(3/4)*5^(1/2)*x^3*135i)/(2*a^(39/2)*b^(13/2)*((135*5 
^(1/2))/(a^18*b^6) - (135*2^(1/2)*10^(1/2))/(2*a^18*b^6))) - (2^(1/4)*10^( 
1/2)*x^3*135i)/(2*a^(39/2)*b^(13/2)*((135*5^(1/2))/(a^18*b^6) - (135*2^(1/ 
2)*10^(1/2))/(2*a^18*b^6))))*1i)/(4*a^(15/2)*b^(5/2))
 

Reduce [B] (verification not implemented)

Time = 0.45 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.06 \[ \int \frac {4 \sqrt {5} a^6 b^2-3 \sqrt {10} a^3 b x^6+2 \sqrt {2} a^3 b^2 x^9+3 \sqrt {5} x^{12}}{a^6 b^2 x^4 \left (\sqrt {2} a^3 b-x^6\right )^2} \, dx=\frac {\sqrt {10}\, a^{3} b \,x^{6}-4 \sqrt {5}\, a^{6} b^{2}+3 \sqrt {5}\, x^{12}+\sqrt {2}\, a^{3} b^{2} x^{9}+b \,x^{15}}{3 a^{6} b^{2} x^{3} \left (-x^{12}+2 a^{6} b^{2}\right )} \] Input:

int((4*5^(1/2)*a^6*b^2-3*10^(1/2)*a^3*b*x^6+2*2^(1/2)*a^3*b^2*x^9+3*5^(1/2 
)*x^12)/a^6/b^2/x^4/(2^(1/2)*a^3*b-x^6)^2,x)
 

Output:

(sqrt(10)*a**3*b*x**6 - 4*sqrt(5)*a**6*b**2 + 3*sqrt(5)*x**12 + sqrt(2)*a* 
*3*b**2*x**9 + b*x**15)/(3*a**6*b**2*x**3*(2*a**6*b**2 - x**12))