3.5.88 \(\int \frac {(-4+x^5) (1+x^5)^{3/4} (2-x^4+2 x^5)}{x^{12}} \, dx\) [488]

3.5.88.1 Optimal result

Integrand size = 30, antiderivative size = 38 \[ \int \frac {\left (-4+x^5\right ) \left (1+x^5\right )^{3/4} \left (2-x^4+2 x^5\right )}{x^{12}} \, dx=\frac {4 \left (1+x^5\right )^{3/4} \left (14-11 x^4+28 x^5-11 x^9+14 x^{10}\right )}{77 x^{11}} \]

output

4/77*(x^5+1)^(3/4)*(14*x^10-11*x^9+28*x^5-11*x^4+14)/x^11
 

3.5.88.2 Mathematica [A] (verified)

Time = 1.89 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.74 \[ \int \frac {\left (-4+x^5\right ) \left (1+x^5\right )^{3/4} \left (2-x^4+2 x^5\right )}{x^{12}} \, dx=\frac {4 \left (1+x^5\right )^{7/4} \left (14-11 x^4+14 x^5\right )}{77 x^{11}} \]

input

Integrate[((-4 + x^5)*(1 + x^5)^(3/4)*(2 - x^4 + 2*x^5))/x^12,x]
 

output

(4*(1 + x^5)^(7/4)*(14 - 11*x^4 + 14*x^5))/(77*x^11)
 

3.5.88.3 Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(89\) vs. \(2(38)=76\).

Time = 0.47 (sec) , antiderivative size = 89, normalized size of antiderivative = 2.34, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {2364, 27, 2374, 9, 27, 2374, 27, 951}

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.

\(\displaystyle \int \frac {\left (x^5-4\right ) \left (x^5+1\right )^{3/4} \left (2 x^5-x^4+2\right )}{x^{12}} \, dx\)

\(\Big \downarrow \) 2364

\(\displaystyle \frac {1}{154} \left (\frac {112}{x^{11}}-\frac {88}{x^7}+\frac {154}{x^6}+\frac {77}{x^2}-\frac {308}{x}\right ) \left (x^5+1\right )^{3/4}-\frac {15}{4} \int \frac {-308 x^{10}+77 x^9+154 x^5-88 x^4+112}{154 x^7 \sqrt [4]{x^5+1}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{154} \left (\frac {112}{x^{11}}-\frac {88}{x^7}+\frac {154}{x^6}+\frac {77}{x^2}-\frac {308}{x}\right ) \left (x^5+1\right )^{3/4}-\frac {15}{616} \int \frac {-308 x^{10}+77 x^9+154 x^5-88 x^4+112}{x^7 \sqrt [4]{x^5+1}}dx\)

\(\Big \downarrow \) 2374

\(\displaystyle \frac {1}{154} \left (\frac {112}{x^{11}}-\frac {88}{x^7}+\frac {154}{x^6}+\frac {77}{x^2}-\frac {308}{x}\right ) \left (x^5+1\right )^{3/4}-\frac {15}{616} \left (-\frac {1}{12} \int \frac {12 \left (308 x^9-77 x^8-112 x^4+88 x^3\right )}{x^6 \sqrt [4]{x^5+1}}dx-\frac {56 \left (x^5+1\right )^{3/4}}{3 x^6}\right )\)

\(\Big \downarrow \) 9

\(\displaystyle \frac {1}{154} \left (\frac {112}{x^{11}}-\frac {88}{x^7}+\frac {154}{x^6}+\frac {77}{x^2}-\frac {308}{x}\right ) \left (x^5+1\right )^{3/4}-\frac {15}{616} \left (-\frac {1}{12} \int \frac {12 \left (308 x^6-77 x^5-112 x+88\right )}{x^3 \sqrt [4]{x^5+1}}dx-\frac {56 \left (x^5+1\right )^{3/4}}{3 x^6}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{154} \left (\frac {112}{x^{11}}-\frac {88}{x^7}+\frac {154}{x^6}+\frac {77}{x^2}-\frac {308}{x}\right ) \left (x^5+1\right )^{3/4}-\frac {15}{616} \left (-\int \frac {308 x^6-77 x^5-112 x+88}{x^3 \sqrt [4]{x^5+1}}dx-\frac {56 \left (x^5+1\right )^{3/4}}{3 x^6}\right )\)

\(\Big \downarrow \) 2374

\(\displaystyle \frac {1}{154} \left (\frac {112}{x^{11}}-\frac {88}{x^7}+\frac {154}{x^6}+\frac {77}{x^2}-\frac {308}{x}\right ) \left (x^5+1\right )^{3/4}-\frac {15}{616} \left (\frac {1}{4} \int \frac {112 \left (4-11 x^5\right )}{x^2 \sqrt [4]{x^5+1}}dx-\frac {56 \left (x^5+1\right )^{3/4}}{3 x^6}+\frac {44 \left (x^5+1\right )^{3/4}}{x^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{154} \left (\frac {112}{x^{11}}-\frac {88}{x^7}+\frac {154}{x^6}+\frac {77}{x^2}-\frac {308}{x}\right ) \left (x^5+1\right )^{3/4}-\frac {15}{616} \left (28 \int \frac {4-11 x^5}{x^2 \sqrt [4]{x^5+1}}dx-\frac {56 \left (x^5+1\right )^{3/4}}{3 x^6}+\frac {44 \left (x^5+1\right )^{3/4}}{x^2}\right )\)

\(\Big \downarrow \) 951

\(\displaystyle \frac {1}{154} \left (\frac {112}{x^{11}}-\frac {88}{x^7}+\frac {154}{x^6}+\frac {77}{x^2}-\frac {308}{x}\right ) \left (x^5+1\right )^{3/4}-\frac {15}{616} \left (-\frac {112 \left (x^5+1\right )^{3/4}}{x}-\frac {56 \left (x^5+1\right )^{3/4}}{3 x^6}+\frac {44 \left (x^5+1\right )^{3/4}}{x^2}\right )\)

input

Int[((-4 + x^5)*(1 + x^5)^(3/4)*(2 - x^4 + 2*x^5))/x^12,x]
 

output

((112/x^11 - 88/x^7 + 154/x^6 + 77/x^2 - 308/x)*(1 + x^5)^(3/4))/154 - (15 
*((-56*(1 + x^5)^(3/4))/(3*x^6) + (44*(1 + x^5)^(3/4))/x^2 - (112*(1 + x^5 
)^(3/4))/x))/616
 

3.5.88.3.1 Defintions of rubi rules used

rule 9

Int[(u_.)*(Px_)^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> With[{r = Expon[Px, 
x, Min]}, Simp[1/e^(p*r)   Int[u*(e*x)^(m + p*r)*ExpandToSum[Px/x^r, x]^p, 
x], x] /; IGtQ[r, 0]] /; FreeQ[{e, m}, x] && PolyQ[Px, x] && IntegerQ[p] && 
  !MonomialQ[Px, x]
 

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 951

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n 
_)), x_Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), 
 x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[a*d*( 
m + 1) - b*c*(m + n*(p + 1) + 1), 0] && NeQ[m, -1]
 

rule 2364

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Module[{u 
 = IntHide[x^m*Pq, x]}, Simp[u*(a + b*x^n)^p, x] - Simp[b*n*p   Int[x^(m + 
n)*(a + b*x^n)^(p - 1)*ExpandToSum[u/x^(m + 1), x], x], x]] /; FreeQ[{a, b} 
, x] && PolyQ[Pq, x] && IGtQ[n, 0] && GtQ[p, 0] && LtQ[m + Expon[Pq, x] + 1 
, 0]
 

rule 2374

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Wit 
h[{Pq0 = Coeff[Pq, x, 0]}, Simp[Pq0*(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c 
*(m + 1))), x] + Simp[1/(2*a*c*(m + 1))   Int[(c*x)^(m + 1)*ExpandToSum[2*a 
*(m + 1)*((Pq - Pq0)/x) - 2*b*Pq0*(m + n*(p + 1) + 1)*x^(n - 1), x]*(a + b* 
x^n)^p, x], x] /; NeQ[Pq0, 0]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && 
 IGtQ[n, 0] && LtQ[m, -1] && LeQ[n - 1, Expon[Pq, x]]
 

3.5.88.4 Maple [A] (verified)

Time = 1.00 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.61

method	result	size
pseudoelliptic	\(\frac {8 \left (x^{5}-\frac {11}{14} x^{4}+1\right ) \left (x^{5}+1\right )^{\frac {7}{4}}}{11 x^{11}}\)	\(23\)
trager	\(\frac {4 \left (x^{5}+1\right )^{\frac {3}{4}} \left (14 x^{10}-11 x^{9}+28 x^{5}-11 x^{4}+14\right )}{77 x^{11}}\)	\(35\)
gosper	\(\frac {4 \left (x^{4}-x^{3}+x^{2}-x +1\right ) \left (1+x \right ) \left (14 x^{5}-11 x^{4}+14\right ) \left (x^{5}+1\right )^{\frac {3}{4}}}{77 x^{11}}\)	\(44\)
risch	\(\frac {\frac {8}{11} x^{15}+\frac {24}{11} x^{10}+\frac {24}{11} x^{5}+\frac {8}{11}-\frac {4}{7} x^{14}-\frac {8}{7} x^{9}-\frac {4}{7} x^{4}}{x^{11} \left (x^{5}+1\right )^{\frac {1}{4}}}\)	\(45\)
meijerg	\(\frac {\operatorname {hypergeom}\left (\left [-\frac {6}{5}, -\frac {3}{4}\right ], \left [-\frac {1}{5}\right ], -x^{5}\right )}{x^{6}}-\frac {4 \operatorname {hypergeom}\left (\left [-\frac {7}{5}, -\frac {3}{4}\right ], \left [-\frac {2}{5}\right ], -x^{5}\right )}{7 x^{7}}+\frac {8 \operatorname {hypergeom}\left (\left [-\frac {11}{5}, -\frac {3}{4}\right ], \left [-\frac {6}{5}\right ], -x^{5}\right )}{11 x^{11}}-\frac {2 \operatorname {hypergeom}\left (\left [-\frac {3}{4}, -\frac {1}{5}\right ], \left [\frac {4}{5}\right ], -x^{5}\right )}{x}+\frac {\operatorname {hypergeom}\left (\left [-\frac {3}{4}, -\frac {2}{5}\right ], \left [\frac {3}{5}\right ], -x^{5}\right )}{2 x^{2}}\)	\(81\)

input

int((x^5-4)*(x^5+1)^(3/4)*(2*x^5-x^4+2)/x^12,x,method=_RETURNVERBOSE)
 

output

8/11*(x^5-11/14*x^4+1)*(x^5+1)^(7/4)/x^11
 

3.5.88.5 Fricas [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.89 \[ \int \frac {\left (-4+x^5\right ) \left (1+x^5\right )^{3/4} \left (2-x^4+2 x^5\right )}{x^{12}} \, dx=\frac {4 \, {\left (14 \, x^{10} - 11 \, x^{9} + 28 \, x^{5} - 11 \, x^{4} + 14\right )} {\left (x^{5} + 1\right )}^{\frac {3}{4}}}{77 \, x^{11}} \]

input

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

output

4/77*(14*x^10 - 11*x^9 + 28*x^5 - 11*x^4 + 14)*(x^5 + 1)^(3/4)/x^11
 

3.5.88.6 Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 3.04 (sec) , antiderivative size = 192, normalized size of antiderivative = 5.05 \[ \int \frac {\left (-4+x^5\right ) \left (1+x^5\right )^{3/4} \left (2-x^4+2 x^5\right )}{x^{12}} \, dx=\frac {2 \Gamma \left (- \frac {1}{5}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{5} \\ \frac {4}{5} \end {matrix}\middle | {x^{5} e^{i \pi }} \right )}}{5 x \Gamma \left (\frac {4}{5}\right )} - \frac {\Gamma \left (- \frac {2}{5}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {2}{5} \\ \frac {3}{5} \end {matrix}\middle | {x^{5} e^{i \pi }} \right )}}{5 x^{2} \Gamma \left (\frac {3}{5}\right )} - \frac {6 \Gamma \left (- \frac {6}{5}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {6}{5}, - \frac {3}{4} \\ - \frac {1}{5} \end {matrix}\middle | {x^{5} e^{i \pi }} \right )}}{5 x^{6} \Gamma \left (- \frac {1}{5}\right )} + \frac {4 \Gamma \left (- \frac {7}{5}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{5}, - \frac {3}{4} \\ - \frac {2}{5} \end {matrix}\middle | {x^{5} e^{i \pi }} \right )}}{5 x^{7} \Gamma \left (- \frac {2}{5}\right )} - \frac {8 \Gamma \left (- \frac {11}{5}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {11}{5}, - \frac {3}{4} \\ - \frac {6}{5} \end {matrix}\middle | {x^{5} e^{i \pi }} \right )}}{5 x^{11} \Gamma \left (- \frac {6}{5}\right )} \]

input

integrate((x**5-4)*(x**5+1)**(3/4)*(2*x**5-x**4+2)/x**12,x)
 

output

2*gamma(-1/5)*hyper((-3/4, -1/5), (4/5,), x**5*exp_polar(I*pi))/(5*x*gamma 
(4/5)) - gamma(-2/5)*hyper((-3/4, -2/5), (3/5,), x**5*exp_polar(I*pi))/(5* 
x**2*gamma(3/5)) - 6*gamma(-6/5)*hyper((-6/5, -3/4), (-1/5,), x**5*exp_pol 
ar(I*pi))/(5*x**6*gamma(-1/5)) + 4*gamma(-7/5)*hyper((-7/5, -3/4), (-2/5,) 
, x**5*exp_polar(I*pi))/(5*x**7*gamma(-2/5)) - 8*gamma(-11/5)*hyper((-11/5 
, -3/4), (-6/5,), x**5*exp_polar(I*pi))/(5*x**11*gamma(-6/5))
 

3.5.88.7 Maxima [A] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.32 \[ \int \frac {\left (-4+x^5\right ) \left (1+x^5\right )^{3/4} \left (2-x^4+2 x^5\right )}{x^{12}} \, dx=\frac {4 \, {\left (14 \, x^{10} - 11 \, x^{9} + 28 \, x^{5} - 11 \, x^{4} + 14\right )} {\left (x^{4} - x^{3} + x^{2} - x + 1\right )}^{\frac {3}{4}} {\left (x + 1\right )}^{\frac {3}{4}}}{77 \, x^{11}} \]

input

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

output

4/77*(14*x^10 - 11*x^9 + 28*x^5 - 11*x^4 + 14)*(x^4 - x^3 + x^2 - x + 1)^( 
3/4)*(x + 1)^(3/4)/x^11
 

3.5.88.8 Giac [F]

\[ \int \frac {\left (-4+x^5\right ) \left (1+x^5\right )^{3/4} \left (2-x^4+2 x^5\right )}{x^{12}} \, dx=\int { \frac {{\left (2 \, x^{5} - x^{4} + 2\right )} {\left (x^{5} + 1\right )}^{\frac {3}{4}} {\left (x^{5} - 4\right )}}{x^{12}} \,d x } \]

input

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

output

integrate((2*x^5 - x^4 + 2)*(x^5 + 1)^(3/4)*(x^5 - 4)/x^12, x)
 

3.5.88.9 Mupad [B] (verification not implemented)

Time = 5.75 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.61 \[ \int \frac {\left (-4+x^5\right ) \left (1+x^5\right )^{3/4} \left (2-x^4+2 x^5\right )}{x^{12}} \, dx=\frac {8\,{\left (x^5+1\right )}^{3/4}}{11\,x}-\frac {4\,{\left (x^5+1\right )}^{3/4}}{7\,x^2}+\frac {16\,{\left (x^5+1\right )}^{3/4}}{11\,x^6}-\frac {4\,{\left (x^5+1\right )}^{3/4}}{7\,x^7}+\frac {8\,{\left (x^5+1\right )}^{3/4}}{11\,x^{11}} \]

input

int(((x^5 + 1)^(3/4)*(x^5 - 4)*(2*x^5 - x^4 + 2))/x^12,x)
 

output

(8*(x^5 + 1)^(3/4))/(11*x) - (4*(x^5 + 1)^(3/4))/(7*x^2) + (16*(x^5 + 1)^( 
3/4))/(11*x^6) - (4*(x^5 + 1)^(3/4))/(7*x^7) + (8*(x^5 + 1)^(3/4))/(11*x^1 
1)