\(\int \frac {-14 x+5 x^5}{7-5 x^4+63 x^8} \, dx\) [67]

Optimal result

Integrand size = 24, antiderivative size = 124 \[ \int \frac {-14 x+5 x^5}{7-5 x^4+63 x^8} \, dx=\frac {1}{12} \sqrt {\frac {37}{7}} \arctan \left (\frac {1}{37} \left (-\sqrt {1739}-6 \sqrt {259} x^2\right )\right )+\frac {1}{12} \sqrt {\frac {37}{7}} \arctan \left (\frac {1}{37} \left (\sqrt {1739}-6 \sqrt {259} x^2\right )\right )+\frac {1}{24} \sqrt {\frac {47}{7}} \log \left (7-\sqrt {329} x^2+21 x^4\right )-\frac {1}{24} \sqrt {\frac {47}{7}} \log \left (7+\sqrt {329} x^2+21 x^4\right ) \] Output:

-1/84*259^(1/2)*arctan(1/37*1739^(1/2)+6/37*259^(1/2)*x^2)-1/84*259^(1/2)* 
arctan(-1/37*1739^(1/2)+6/37*259^(1/2)*x^2)+1/168*329^(1/2)*ln(7-329^(1/2) 
*x^2+21*x^4)-1/168*329^(1/2)*ln(7+329^(1/2)*x^2+21*x^4)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.17 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.93 \[ \int \frac {-14 x+5 x^5}{7-5 x^4+63 x^8} \, dx=\frac {\left (1739 i+5 \sqrt {1739}\right ) \arctan \left (\frac {3 x^2}{\sqrt {\frac {1}{14} \left (-5-i \sqrt {1739}\right )}}\right )}{6 \sqrt {24346 \left (-5-i \sqrt {1739}\right )}}+\frac {\left (-1739 i+5 \sqrt {1739}\right ) \arctan \left (\frac {3 x^2}{\sqrt {\frac {1}{14} \left (-5+i \sqrt {1739}\right )}}\right )}{6 \sqrt {24346 \left (-5+i \sqrt {1739}\right )}} \] Input:

Integrate[(-14*x + 5*x^5)/(7 - 5*x^4 + 63*x^8),x]
 

Output:

((1739*I + 5*Sqrt[1739])*ArcTan[(3*x^2)/Sqrt[(-5 - I*Sqrt[1739])/14]])/(6* 
Sqrt[24346*(-5 - I*Sqrt[1739])]) + ((-1739*I + 5*Sqrt[1739])*ArcTan[(3*x^2 
)/Sqrt[(-5 + I*Sqrt[1739])/14]])/(6*Sqrt[24346*(-5 + I*Sqrt[1739])])
 

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.03, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {2027, 1814, 25, 1483, 27, 1142, 25, 1083, 217, 1103}

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.

\(\displaystyle \int \frac {5 x^5-14 x}{63 x^8-5 x^4+7} \, dx\)

\(\Big \downarrow \) 2027

\(\displaystyle \int \frac {x \left (5 x^4-14\right )}{63 x^8-5 x^4+7}dx\)

\(\Big \downarrow \) 1814

\(\displaystyle \frac {1}{2} \int -\frac {14-5 x^4}{63 x^8-5 x^4+7}dx^2\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {1}{2} \int \frac {14-5 x^4}{63 x^8-5 x^4+7}dx^2\)

\(\Big \downarrow \) 1483

\(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {7 \left (2 \sqrt {329}-47 x^2\right )}{21 x^4-\sqrt {329} x^2+7}dx^2}{2 \sqrt {329}}-\frac {\int \frac {7 \left (47 x^2+2 \sqrt {329}\right )}{21 x^4+\sqrt {329} x^2+7}dx^2}{2 \sqrt {329}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (-\frac {1}{2} \sqrt {\frac {7}{47}} \int \frac {2 \sqrt {329}-47 x^2}{21 x^4-\sqrt {329} x^2+7}dx^2-\frac {1}{2} \sqrt {\frac {7}{47}} \int \frac {47 x^2+2 \sqrt {329}}{21 x^4+\sqrt {329} x^2+7}dx^2\right )\)

\(\Big \downarrow \) 1142

\(\displaystyle \frac {1}{2} \left (-\frac {1}{2} \sqrt {\frac {7}{47}} \left (\frac {37}{6} \sqrt {\frac {47}{7}} \int \frac {1}{21 x^4-\sqrt {329} x^2+7}dx^2-\frac {47}{42} \int -\frac {\sqrt {329}-42 x^2}{21 x^4-\sqrt {329} x^2+7}dx^2\right )-\frac {1}{2} \sqrt {\frac {7}{47}} \left (\frac {37}{6} \sqrt {\frac {47}{7}} \int \frac {1}{21 x^4+\sqrt {329} x^2+7}dx^2+\frac {47}{42} \int \frac {42 x^2+\sqrt {329}}{21 x^4+\sqrt {329} x^2+7}dx^2\right )\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{2} \left (-\frac {1}{2} \sqrt {\frac {7}{47}} \left (\frac {37}{6} \sqrt {\frac {47}{7}} \int \frac {1}{21 x^4-\sqrt {329} x^2+7}dx^2+\frac {47}{42} \int \frac {\sqrt {329}-42 x^2}{21 x^4-\sqrt {329} x^2+7}dx^2\right )-\frac {1}{2} \sqrt {\frac {7}{47}} \left (\frac {37}{6} \sqrt {\frac {47}{7}} \int \frac {1}{21 x^4+\sqrt {329} x^2+7}dx^2+\frac {47}{42} \int \frac {42 x^2+\sqrt {329}}{21 x^4+\sqrt {329} x^2+7}dx^2\right )\right )\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {1}{2} \left (-\frac {1}{2} \sqrt {\frac {7}{47}} \left (\frac {47}{42} \int \frac {\sqrt {329}-42 x^2}{21 x^4-\sqrt {329} x^2+7}dx^2-\frac {37}{3} \sqrt {\frac {47}{7}} \int \frac {1}{-x^4-259}d\left (42 x^2-\sqrt {329}\right )\right )-\frac {1}{2} \sqrt {\frac {7}{47}} \left (\frac {47}{42} \int \frac {42 x^2+\sqrt {329}}{21 x^4+\sqrt {329} x^2+7}dx^2-\frac {37}{3} \sqrt {\frac {47}{7}} \int \frac {1}{-x^4-259}d\left (42 x^2+\sqrt {329}\right )\right )\right )\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{2} \left (-\frac {1}{2} \sqrt {\frac {7}{47}} \left (\frac {47}{42} \int \frac {\sqrt {329}-42 x^2}{21 x^4-\sqrt {329} x^2+7}dx^2+\frac {1}{21} \sqrt {1739} \arctan \left (\frac {42 x^2-\sqrt {329}}{\sqrt {259}}\right )\right )-\frac {1}{2} \sqrt {\frac {7}{47}} \left (\frac {47}{42} \int \frac {42 x^2+\sqrt {329}}{21 x^4+\sqrt {329} x^2+7}dx^2+\frac {1}{21} \sqrt {1739} \arctan \left (\frac {42 x^2+\sqrt {329}}{\sqrt {259}}\right )\right )\right )\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {1}{2} \left (-\frac {1}{2} \sqrt {\frac {7}{47}} \left (\frac {1}{21} \sqrt {1739} \arctan \left (\frac {42 x^2-\sqrt {329}}{\sqrt {259}}\right )-\frac {47}{42} \log \left (21 x^4-\sqrt {329} x^2+7\right )\right )-\frac {1}{2} \sqrt {\frac {7}{47}} \left (\frac {1}{21} \sqrt {1739} \arctan \left (\frac {42 x^2+\sqrt {329}}{\sqrt {259}}\right )+\frac {47}{42} \log \left (21 x^4+\sqrt {329} x^2+7\right )\right )\right )\)

Input:

Int[(-14*x + 5*x^5)/(7 - 5*x^4 + 63*x^8),x]
 

Output:

(-1/2*(Sqrt[7/47]*((Sqrt[1739]*ArcTan[(-Sqrt[329] + 42*x^2)/Sqrt[259]])/21 
 - (47*Log[7 - Sqrt[329]*x^2 + 21*x^4])/42)) - (Sqrt[7/47]*((Sqrt[1739]*Ar 
cTan[(Sqrt[329] + 42*x^2)/Sqrt[259]])/21 + (47*Log[7 + Sqrt[329]*x^2 + 21* 
x^4])/42))/2)/2
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, 
x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1142

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[(2*c*d - b*e)/(2*c)   Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) 
Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
 

rule 1483

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r)   In 
t[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Simp[1/(2*c*q*r)   Int[(d*r 
 + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && N 
eQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]
 

rule 1814

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_)*((d_) + (e 
_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k   Sub 
st[Int[x^((m + 1)/k - 1)*(d + e*x^(n/k))^q*(a + b*x^(n/k) + c*x^(2*(n/k)))^ 
p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, c, d, e, p, q}, x] && EqQ[n2, 
 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && IntegerQ[m]
 

rule 2027

Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x_Symbol] :> Int[x^ 
(p*r)*(a + b*x^(s - r))^p*Fx, x] /; FreeQ[{a, b, r, s}, x] && IntegerQ[p] & 
& PosQ[s - r] &&  !(EqQ[p, 1] && EqQ[u, 1])
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.09 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.23

method	result	size
risch	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (63 \textit {\_Z}^{4}-5 \textit {\_Z}^{2}+7\right )}{\sum }\textit {\_R} \ln \left (x^{2}-\textit {\_R} \right )\right )}{4}\)	\(29\)
default	\(-\frac {\sqrt {47}\, \sqrt {7}\, \ln \left (21 x^{4}+\sqrt {47}\, \sqrt {7}\, x^{2}+7\right )}{168}-\frac {\sqrt {259}\, \arctan \left (\frac {\left (42 x^{2}+\sqrt {47}\, \sqrt {7}\right ) \sqrt {259}}{259}\right )}{84}+\frac {\sqrt {47}\, \sqrt {7}\, \ln \left (21 x^{4}-\sqrt {47}\, \sqrt {7}\, x^{2}+7\right )}{168}-\frac {\sqrt {259}\, \arctan \left (\frac {\left (42 x^{2}-\sqrt {47}\, \sqrt {7}\right ) \sqrt {259}}{259}\right )}{84}\)	\(104\)

Input:

int((5*x^5-14*x)/(63*x^8-5*x^4+7),x,method=_RETURNVERBOSE)
 

Output:

1/4*sum(_R*ln(x^2-_R),_R=RootOf(63*_Z^4-5*_Z^2+7))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.71 \[ \int \frac {-14 x+5 x^5}{7-5 x^4+63 x^8} \, dx=-\frac {1}{12} \, \sqrt {\frac {37}{7}} \arctan \left (\frac {42}{37} \, \sqrt {\frac {37}{7}} x^{2} + \frac {7}{37} \, \sqrt {\frac {47}{7}} \sqrt {\frac {37}{7}}\right ) + \frac {1}{12} \, \sqrt {\frac {37}{7}} \arctan \left (-\frac {42}{37} \, \sqrt {\frac {37}{7}} x^{2} + \frac {7}{37} \, \sqrt {\frac {47}{7}} \sqrt {\frac {37}{7}}\right ) - \frac {1}{24} \, \sqrt {\frac {47}{7}} \log \left (3 \, x^{4} + \sqrt {\frac {47}{7}} x^{2} + 1\right ) + \frac {1}{24} \, \sqrt {\frac {47}{7}} \log \left (3 \, x^{4} - \sqrt {\frac {47}{7}} x^{2} + 1\right ) \] Input:

integrate((5*x^5-14*x)/(63*x^8-5*x^4+7),x, algorithm="fricas")
 

Output:

-1/12*sqrt(37/7)*arctan(42/37*sqrt(37/7)*x^2 + 7/37*sqrt(47/7)*sqrt(37/7)) 
 + 1/12*sqrt(37/7)*arctan(-42/37*sqrt(37/7)*x^2 + 7/37*sqrt(47/7)*sqrt(37/ 
7)) - 1/24*sqrt(47/7)*log(3*x^4 + sqrt(47/7)*x^2 + 1) + 1/24*sqrt(47/7)*lo 
g(3*x^4 - sqrt(47/7)*x^2 + 1)
 

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.84 \[ \int \frac {-14 x+5 x^5}{7-5 x^4+63 x^8} \, dx=\frac {\sqrt {329} \log {\left (x^{4} - \frac {\sqrt {329} x^{2}}{21} + \frac {1}{3} \right )}}{168} - \frac {\sqrt {329} \log {\left (x^{4} + \frac {\sqrt {329} x^{2}}{21} + \frac {1}{3} \right )}}{168} - \frac {\sqrt {259} \operatorname {atan}{\left (\frac {6 \sqrt {259} x^{2}}{37} - \frac {\sqrt {1739}}{37} \right )}}{84} - \frac {\sqrt {259} \operatorname {atan}{\left (\frac {6 \sqrt {259} x^{2}}{37} + \frac {\sqrt {1739}}{37} \right )}}{84} \] Input:

integrate((5*x**5-14*x)/(63*x**8-5*x**4+7),x)
 

Output:

sqrt(329)*log(x**4 - sqrt(329)*x**2/21 + 1/3)/168 - sqrt(329)*log(x**4 + s 
qrt(329)*x**2/21 + 1/3)/168 - sqrt(259)*atan(6*sqrt(259)*x**2/37 - sqrt(17 
39)/37)/84 - sqrt(259)*atan(6*sqrt(259)*x**2/37 + sqrt(1739)/37)/84
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int \frac {-14 x+5 x^5}{7-5 x^4+63 x^8} \, dx=\int { \frac {5 \, x^{5} - 14 \, x}{63 \, x^{8} - 5 \, x^{4} + 7} \,d x } \] Input:

integrate((5*x^5-14*x)/(63*x^8-5*x^4+7),x, algorithm="maxima")
 

Output:

integrate((5*x^5 - 14*x)/(63*x^8 - 5*x^4 + 7), x)
 

Giac [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.16 \[ \int \frac {-14 x+5 x^5}{7-5 x^4+63 x^8} \, dx=\frac {1}{1036} \, {\left (5 \, \sqrt {259} x^{4} - 14 \, \sqrt {259}\right )} \arctan \left (\frac {3}{259} \, \sqrt {777} \left (\frac {1}{9}\right )^{\frac {3}{4}} {\left (42 \, x^{2} + \sqrt {987} \left (\frac {1}{9}\right )^{\frac {1}{4}}\right )}\right ) + \frac {1}{1036} \, {\left (5 \, \sqrt {259} x^{4} - 14 \, \sqrt {259}\right )} \arctan \left (\frac {3}{259} \, \sqrt {777} \left (\frac {1}{9}\right )^{\frac {3}{4}} {\left (42 \, x^{2} - \sqrt {987} \left (\frac {1}{9}\right )^{\frac {1}{4}}\right )}\right ) + \frac {1}{2632} \, {\left (5 \, \sqrt {329} x^{4} - 14 \, \sqrt {329}\right )} \log \left (x^{4} + \frac {1}{21} \, \sqrt {987} \left (\frac {1}{9}\right )^{\frac {1}{4}} x^{2} + \frac {1}{3}\right ) - \frac {1}{2632} \, {\left (5 \, \sqrt {329} x^{4} - 14 \, \sqrt {329}\right )} \log \left (x^{4} - \frac {1}{21} \, \sqrt {987} \left (\frac {1}{9}\right )^{\frac {1}{4}} x^{2} + \frac {1}{3}\right ) \] Input:

integrate((5*x^5-14*x)/(63*x^8-5*x^4+7),x, algorithm="giac")
 

Output:

1/1036*(5*sqrt(259)*x^4 - 14*sqrt(259))*arctan(3/259*sqrt(777)*(1/9)^(3/4) 
*(42*x^2 + sqrt(987)*(1/9)^(1/4))) + 1/1036*(5*sqrt(259)*x^4 - 14*sqrt(259 
))*arctan(3/259*sqrt(777)*(1/9)^(3/4)*(42*x^2 - sqrt(987)*(1/9)^(1/4))) + 
1/2632*(5*sqrt(329)*x^4 - 14*sqrt(329))*log(x^4 + 1/21*sqrt(987)*(1/9)^(1/ 
4)*x^2 + 1/3) - 1/2632*(5*sqrt(329)*x^4 - 14*sqrt(329))*log(x^4 - 1/21*sqr 
t(987)*(1/9)^(1/4)*x^2 + 1/3)
 

Mupad [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.95 \[ \int \frac {-14 x+5 x^5}{7-5 x^4+63 x^8} \, dx=\mathrm {atan}\left (\frac {136732606894\,\sqrt {259}\,x^2}{2233402022407689\,\left (-\frac {26294732095}{319057431772527}+\frac {\sqrt {259}\,\sqrt {329}\,5258946419{}\mathrm {i}}{2233402022407689}\right )}-\frac {\sqrt {329}\,x^2\,84143142704{}\mathrm {i}}{2233402022407689\,\left (-\frac {26294732095}{319057431772527}+\frac {\sqrt {259}\,\sqrt {329}\,5258946419{}\mathrm {i}}{2233402022407689}\right )}\right )\,\left (\frac {\sqrt {259}}{84}-\frac {\sqrt {329}\,1{}\mathrm {i}}{84}\right )-\mathrm {atan}\left (\frac {136732606894\,\sqrt {259}\,x^2}{2233402022407689\,\left (\frac {26294732095}{319057431772527}+\frac {\sqrt {259}\,\sqrt {329}\,5258946419{}\mathrm {i}}{2233402022407689}\right )}+\frac {\sqrt {329}\,x^2\,84143142704{}\mathrm {i}}{2233402022407689\,\left (\frac {26294732095}{319057431772527}+\frac {\sqrt {259}\,\sqrt {329}\,5258946419{}\mathrm {i}}{2233402022407689}\right )}\right )\,\left (\frac {\sqrt {259}}{84}+\frac {\sqrt {329}\,1{}\mathrm {i}}{84}\right ) \] Input:

int(-(14*x - 5*x^5)/(63*x^8 - 5*x^4 + 7),x)
 

Output:

atan((136732606894*259^(1/2)*x^2)/(2233402022407689*((259^(1/2)*329^(1/2)* 
5258946419i)/2233402022407689 - 26294732095/319057431772527)) - (329^(1/2) 
*x^2*84143142704i)/(2233402022407689*((259^(1/2)*329^(1/2)*5258946419i)/22 
33402022407689 - 26294732095/319057431772527)))*(259^(1/2)/84 - (329^(1/2) 
*1i)/84) - atan((136732606894*259^(1/2)*x^2)/(2233402022407689*((259^(1/2) 
*329^(1/2)*5258946419i)/2233402022407689 + 26294732095/319057431772527)) + 
 (329^(1/2)*x^2*84143142704i)/(2233402022407689*((259^(1/2)*329^(1/2)*5258 
946419i)/2233402022407689 + 26294732095/319057431772527)))*(259^(1/2)/84 + 
 (329^(1/2)*1i)/84)
 

Reduce [F]

\[ \int \frac {-14 x+5 x^5}{7-5 x^4+63 x^8} \, dx=\int \frac {5 x^{5}-14 x}{63 x^{8}-5 x^{4}+7}d x \] Input:

int((5*x^5-14*x)/(63*x^8-5*x^4+7),x)
 

Output:

int((5*x^5-14*x)/(63*x^8-5*x^4+7),x)