\(\int \frac {3 \sqrt {7}+6 \sqrt {11}+(-121 \sqrt {105}-98 \sqrt {165}) x^4}{-12 \sqrt {15}+10200 x^4-118580 \sqrt {15} x^8} \, dx\) [50]

Optimal result

Integrand size = 60, antiderivative size = 117 \[ \int \frac {3 \sqrt {7}+6 \sqrt {11}+\left (-121 \sqrt {105}-98 \sqrt {165}\right ) x^4}{-12 \sqrt {15}+10200 x^4-118580 \sqrt {15} x^8} \, dx=-\frac {\arctan \left (\sqrt [8]{\frac {5}{3}} \sqrt {7} x\right )}{8\ 3^{3/8} 5^{5/8}}-\frac {\arctan \left (\sqrt [8]{\frac {5}{3}} \sqrt {11} x\right )}{4\ 3^{3/8} 5^{5/8}}-\frac {\text {arctanh}\left (\sqrt [8]{\frac {5}{3}} \sqrt {7} x\right )}{8\ 3^{3/8} 5^{5/8}}-\frac {\text {arctanh}\left (\sqrt [8]{\frac {5}{3}} \sqrt {11} x\right )}{4\ 3^{3/8} 5^{5/8}} \] Output:

-1/120*arctan(1/3*5^(1/8)*3^(7/8)*7^(1/2)*x)*3^(5/8)*5^(3/8)-1/60*arctan(1 
/3*5^(1/8)*3^(7/8)*11^(1/2)*x)*3^(5/8)*5^(3/8)-1/120*arctanh(1/3*5^(1/8)*3 
^(7/8)*7^(1/2)*x)*3^(5/8)*5^(3/8)-1/60*arctanh(1/3*5^(1/8)*3^(7/8)*11^(1/2 
)*x)*3^(5/8)*5^(3/8)
 

Mathematica [C] (verified)

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

Time = 0.12 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.01 \[ \int \frac {3 \sqrt {7}+6 \sqrt {11}+\left (-121 \sqrt {105}-98 \sqrt {165}\right ) x^4}{-12 \sqrt {15}+10200 x^4-118580 \sqrt {15} x^8} \, dx=\frac {1}{160} \text {RootSum}\left [3 \sqrt {15}-2550 \text {$\#$1}^4+29645 \sqrt {15} \text {$\#$1}^8\&,\frac {-3 \sqrt {7} \log (x-\text {$\#$1})-6 \sqrt {11} \log (x-\text {$\#$1})+121 \sqrt {105} \log (x-\text {$\#$1}) \text {$\#$1}^4+98 \sqrt {165} \log (x-\text {$\#$1}) \text {$\#$1}^4}{-255 \text {$\#$1}^3+5929 \sqrt {15} \text {$\#$1}^7}\&\right ] \] Input:

Integrate[(3*Sqrt[7] + 6*Sqrt[11] + (-121*Sqrt[105] - 98*Sqrt[165])*x^4)/( 
-12*Sqrt[15] + 10200*x^4 - 118580*Sqrt[15]*x^8),x]
 

Output:

RootSum[3*Sqrt[15] - 2550*#1^4 + 29645*Sqrt[15]*#1^8 & , (-3*Sqrt[7]*Log[x 
 - #1] - 6*Sqrt[11]*Log[x - #1] + 121*Sqrt[105]*Log[x - #1]*#1^4 + 98*Sqrt 
[165]*Log[x - #1]*#1^4)/(-255*#1^3 + 5929*Sqrt[15]*#1^7) & ]/160
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.31, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {1752, 756, 216, 219}

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.

\(\displaystyle \int \frac {\left (-121 \sqrt {105}-98 \sqrt {165}\right ) x^4+6 \sqrt {11}+3 \sqrt {7}}{-118580 \sqrt {15} x^8+10200 x^4-12 \sqrt {15}} \, dx\)

\(\Big \downarrow \) 1752

\(\displaystyle -98 \sqrt {165} \int \frac {1}{2940-118580 \sqrt {15} x^4}dx-121 \sqrt {105} \int \frac {1}{7260-118580 \sqrt {15} x^4}dx\)

\(\Big \downarrow \) 756

\(\displaystyle -121 \sqrt {105} \left (\frac {\int \frac {1}{\sqrt [4]{3}-7 \sqrt [4]{5} x^2}dx}{4840\ 3^{3/4}}+\frac {\int \frac {1}{7 \sqrt [4]{5} x^2+\sqrt [4]{3}}dx}{4840\ 3^{3/4}}\right )-98 \sqrt {165} \left (\frac {\int \frac {1}{\sqrt [4]{3}-11 \sqrt [4]{5} x^2}dx}{1960\ 3^{3/4}}+\frac {\int \frac {1}{11 \sqrt [4]{5} x^2+\sqrt [4]{3}}dx}{1960\ 3^{3/4}}\right )\)

\(\Big \downarrow \) 216

\(\displaystyle -98 \sqrt {165} \left (\frac {\int \frac {1}{\sqrt [4]{3}-11 \sqrt [4]{5} x^2}dx}{1960\ 3^{3/4}}+\frac {\arctan \left (\sqrt [8]{\frac {5}{3}} \sqrt {11} x\right )}{1960\ 3^{7/8} \sqrt [8]{5} \sqrt {11}}\right )-121 \sqrt {105} \left (\frac {\int \frac {1}{\sqrt [4]{3}-7 \sqrt [4]{5} x^2}dx}{4840\ 3^{3/4}}+\frac {\arctan \left (\sqrt [8]{\frac {5}{3}} \sqrt {7} x\right )}{4840\ 3^{7/8} \sqrt [8]{5} \sqrt {7}}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle -121 \sqrt {105} \left (\frac {\arctan \left (\sqrt [8]{\frac {5}{3}} \sqrt {7} x\right )}{4840\ 3^{7/8} \sqrt [8]{5} \sqrt {7}}+\frac {\text {arctanh}\left (\sqrt [8]{\frac {5}{3}} \sqrt {7} x\right )}{4840\ 3^{7/8} \sqrt [8]{5} \sqrt {7}}\right )-98 \sqrt {165} \left (\frac {\arctan \left (\sqrt [8]{\frac {5}{3}} \sqrt {11} x\right )}{1960\ 3^{7/8} \sqrt [8]{5} \sqrt {11}}+\frac {\text {arctanh}\left (\sqrt [8]{\frac {5}{3}} \sqrt {11} x\right )}{1960\ 3^{7/8} \sqrt [8]{5} \sqrt {11}}\right )\)

Input:

Int[(3*Sqrt[7] + 6*Sqrt[11] + (-121*Sqrt[105] - 98*Sqrt[165])*x^4)/(-12*Sq 
rt[15] + 10200*x^4 - 118580*Sqrt[15]*x^8),x]
 

Output:

-121*Sqrt[105]*(ArcTan[(5/3)^(1/8)*Sqrt[7]*x]/(4840*3^(7/8)*5^(1/8)*Sqrt[7 
]) + ArcTanh[(5/3)^(1/8)*Sqrt[7]*x]/(4840*3^(7/8)*5^(1/8)*Sqrt[7])) - 98*S 
qrt[165]*(ArcTan[(5/3)^(1/8)*Sqrt[11]*x]/(1960*3^(7/8)*5^(1/8)*Sqrt[11]) + 
 ArcTanh[(5/3)^(1/8)*Sqrt[11]*x]/(1960*3^(7/8)*5^(1/8)*Sqrt[11]))
 

Defintions of rubi rules used

rule 216

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

rule 219

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

rule 756

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 
]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a)   Int[1/(r - s*x^2), x], x] 
 + Simp[r/(2*a)   Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !GtQ[a 
/b, 0]
 

rule 1752

Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x 
_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) 
   Int[1/(b/2 - q/2 + c*x^n), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q))   I 
nt[1/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2 
, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && (PosQ[b^2 
 - 4*a*c] ||  !IGtQ[n/2, 0])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(191\) vs. \(2(85)=170\).

Time = 0.62 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.64

method	result	size
default	\(-\frac {\left (-49 \sqrt {7}\, \sqrt {15}-98 \sqrt {11}\, \sqrt {15}+121 \sqrt {105}+98 \sqrt {165}\right ) \sqrt {35}\, \sqrt {\sqrt {5}\, 15^{\frac {1}{4}}}\, \left (\ln \left (\frac {x +\frac {\sqrt {35}\, \sqrt {\sqrt {5}\, 15^{\frac {1}{4}}}}{35}}{x -\frac {\sqrt {35}\, \sqrt {\sqrt {5}\, 15^{\frac {1}{4}}}}{35}}\right )+2 \arctan \left (\frac {x \sqrt {35}}{\sqrt {\sqrt {5}\, 15^{\frac {1}{4}}}}\right )\right )}{604800}+\frac {\left (-121 \sqrt {7}\, \sqrt {15}-242 \sqrt {11}\, \sqrt {15}+121 \sqrt {105}+98 \sqrt {165}\right ) \sqrt {55}\, \sqrt {\sqrt {5}\, 15^{\frac {1}{4}}}\, \left (\ln \left (\frac {x +\frac {\sqrt {55}\, \sqrt {\sqrt {5}\, 15^{\frac {1}{4}}}}{55}}{x -\frac {\sqrt {55}\, \sqrt {\sqrt {5}\, 15^{\frac {1}{4}}}}{55}}\right )+2 \arctan \left (\frac {x \sqrt {55}}{\sqrt {\sqrt {5}\, 15^{\frac {1}{4}}}}\right )\right )}{950400}\)	\(192\)

Input:

int((3*7^(1/2)+6*11^(1/2)+(-121*105^(1/2)-98*165^(1/2))*x^4)/(-12*15^(1/2) 
+10200*x^4-118580*15^(1/2)*x^8),x,method=_RETURNVERBOSE)
 

Output:

-1/604800*(-49*7^(1/2)*15^(1/2)-98*11^(1/2)*15^(1/2)+121*105^(1/2)+98*165^ 
(1/2))*35^(1/2)*(5^(1/2)*15^(1/4))^(1/2)*(ln((x+1/35*35^(1/2)*(5^(1/2)*15^ 
(1/4))^(1/2))/(x-1/35*35^(1/2)*(5^(1/2)*15^(1/4))^(1/2)))+2*arctan(x*35^(1 
/2)/(5^(1/2)*15^(1/4))^(1/2)))+1/950400*(-121*7^(1/2)*15^(1/2)-242*11^(1/2 
)*15^(1/2)+121*105^(1/2)+98*165^(1/2))*55^(1/2)*(5^(1/2)*15^(1/4))^(1/2)*( 
ln((x+1/55*55^(1/2)*(5^(1/2)*15^(1/4))^(1/2))/(x-1/55*55^(1/2)*(5^(1/2)*15 
^(1/4))^(1/2)))+2*arctan(x*55^(1/2)/(5^(1/2)*15^(1/4))^(1/2)))
 

Fricas [F]

\[ \int \frac {3 \sqrt {7}+6 \sqrt {11}+\left (-121 \sqrt {105}-98 \sqrt {165}\right ) x^4}{-12 \sqrt {15}+10200 x^4-118580 \sqrt {15} x^8} \, dx=\int { \frac {x^{4} {\left (98 \, \sqrt {165} + 121 \, \sqrt {105}\right )} - 6 \, \sqrt {11} - 3 \, \sqrt {7}}{4 \, {\left (29645 \, \sqrt {15} x^{8} - 2550 \, x^{4} + 3 \, \sqrt {15}\right )}} \,d x } \] Input:

integrate((3*7^(1/2)+6*11^(1/2)+(-121*105^(1/2)-98*165^(1/2))*x^4)/(-12*15 
^(1/2)+10200*x^4-118580*15^(1/2)*x^8),x, algorithm="fricas")
 

Output:

0
 

Sympy [F(-2)]

Exception generated. \[ \int \frac {3 \sqrt {7}+6 \sqrt {11}+\left (-121 \sqrt {105}-98 \sqrt {165}\right ) x^4}{-12 \sqrt {15}+10200 x^4-118580 \sqrt {15} x^8} \, dx=\text {Exception raised: PolynomialError} \] Input:

integrate((3*7**(1/2)+6*11**(1/2)+(-121*105**(1/2)-98*165**(1/2))*x**4)/(- 
12*15**(1/2)+10200*x**4-118580*15**(1/2)*x**8),x)
 

Output:

Exception raised: PolynomialError >> 1/(8876619644311361811449478210059404 
03892090960648942070894847733907374495278002081072424698386397863110524278 
65545878613259174768870299313417544282422774969138741079283023024302083445 
868952121128899
 

Maxima [F]

\[ \int \frac {3 \sqrt {7}+6 \sqrt {11}+\left (-121 \sqrt {105}-98 \sqrt {165}\right ) x^4}{-12 \sqrt {15}+10200 x^4-118580 \sqrt {15} x^8} \, dx=\int { \frac {x^{4} {\left (98 \, \sqrt {165} + 121 \, \sqrt {105}\right )} - 6 \, \sqrt {11} - 3 \, \sqrt {7}}{4 \, {\left (29645 \, \sqrt {15} x^{8} - 2550 \, x^{4} + 3 \, \sqrt {15}\right )}} \,d x } \] Input:

integrate((3*7^(1/2)+6*11^(1/2)+(-121*105^(1/2)-98*165^(1/2))*x^4)/(-12*15 
^(1/2)+10200*x^4-118580*15^(1/2)*x^8),x, algorithm="maxima")
 

Output:

1/4*integrate((x^4*(98*sqrt(165) + 121*sqrt(105)) - 6*sqrt(11) - 3*sqrt(7) 
)/(29645*sqrt(15)*x^8 - 2550*x^4 + 3*sqrt(15)), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {3 \sqrt {7}+6 \sqrt {11}+\left (-121 \sqrt {105}-98 \sqrt {165}\right ) x^4}{-12 \sqrt {15}+10200 x^4-118580 \sqrt {15} x^8} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((3*7^(1/2)+6*11^(1/2)+(-121*105^(1/2)-98*165^(1/2))*x^4)/(-12*15 
^(1/2)+10200*x^4-118580*15^(1/2)*x^8),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{%%{[1274,0,-1480290,0,427747950,0,-19021101750,0]:[1,0,-11 
40,0,3163
 

Mupad [B] (verification not implemented)

Time = 18.53 (sec) , antiderivative size = 1889, normalized size of antiderivative = 16.15 \[ \int \frac {3 \sqrt {7}+6 \sqrt {11}+\left (-121 \sqrt {105}-98 \sqrt {165}\right ) x^4}{-12 \sqrt {15}+10200 x^4-118580 \sqrt {15} x^8} \, dx=\text {Too large to display} \] Input:

int(-(3*7^(1/2) - x^4*(121*105^(1/2) + 98*165^(1/2)) + 6*11^(1/2))/(12*15^ 
(1/2) + 118580*15^(1/2)*x^8 - 10200*x^4),x)
 

Output:

atan((((39715937915904*x)/17683290930758754796256905891884705078125 + ((17 
*15^(1/2))/147456000 - (5^(1/2)*888446500935303168^(1/2))/5349660268953600 
)^(1/4)*((940369969152*7^(1/2)*15^(1/2))/360883488382831730535855222283361 
328125 + (117546246144*11^(1/2)*15^(1/2))/14614290025420458509303228009822 
0703125 - ((4092730840461606912*15^(1/2)*x)/707331637230350191850276235675 
388203125 + ((34665798542819328*7^(1/2))/288706790706265384428684177826689 
0625 + (69331597085638656*11^(1/2))/1169143202033636680744258240785765625) 
*((17*15^(1/2))/147456000 - (5^(1/2)*888446500935303168^(1/2))/53496602689 
53600)^(1/4))*((17*15^(1/2))/147456000 - (5^(1/2)*888446500935303168^(1/2) 
)/5349660268953600)^(3/4)))*((17*15^(1/2))/147456000 - (5^(1/2)*8884465009 
35303168^(1/2))/5349660268953600)^(1/4)*1i + ((39715937915904*x)/176832909 
30758754796256905891884705078125 - ((17*15^(1/2))/147456000 - (5^(1/2)*888 
446500935303168^(1/2))/5349660268953600)^(1/4)*((940369969152*7^(1/2)*15^( 
1/2))/360883488382831730535855222283361328125 + (117546246144*11^(1/2)*15^ 
(1/2))/146142900254204585093032280098220703125 + ((4092730840461606912*15^ 
(1/2)*x)/707331637230350191850276235675388203125 - ((34665798542819328*7^( 
1/2))/2887067907062653844286841778266890625 + (69331597085638656*11^(1/2)) 
/1169143202033636680744258240785765625)*((17*15^(1/2))/147456000 - (5^(1/2 
)*888446500935303168^(1/2))/5349660268953600)^(1/4))*((17*15^(1/2))/147456 
000 - (5^(1/2)*888446500935303168^(1/2))/5349660268953600)^(3/4)))*((17...
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.91 \[ \int \frac {3 \sqrt {7}+6 \sqrt {11}+\left (-121 \sqrt {105}-98 \sqrt {165}\right ) x^4}{-12 \sqrt {15}+10200 x^4-118580 \sqrt {15} x^8} \, dx=\frac {\sqrt {3}\, 15^{\frac {1}{8}} 5^{\frac {1}{4}} \left (-4 \mathit {atan} \left (\frac {11 \,5^{\frac {1}{4}} x 15^{\frac {7}{8}}}{15 \sqrt {11}}\right )-2 \mathit {atan} \left (\frac {7 \,5^{\frac {1}{4}} x 15^{\frac {7}{8}}}{15 \sqrt {7}}\right )+2 \,\mathrm {log}\left (-15^{\frac {1}{8}} \sqrt {11}+11 \,5^{\frac {1}{4}} x \right )+\mathrm {log}\left (-15^{\frac {1}{8}} \sqrt {7}+7 \,5^{\frac {1}{4}} x \right )-2 \,\mathrm {log}\left (15^{\frac {1}{8}} \sqrt {11}+11 \,5^{\frac {1}{4}} x \right )-\mathrm {log}\left (15^{\frac {1}{8}} \sqrt {7}+7 \,5^{\frac {1}{4}} x \right )\right )}{240} \] Input:

int((3*7^(1/2)+6*11^(1/2)+(-121*105^(1/2)-98*165^(1/2))*x^4)/(-12*15^(1/2) 
+10200*x^4-118580*15^(1/2)*x^8),x)
 

Output:

(sqrt(3)*15**(1/8)*5**(1/4)*( - 4*atan((11*5**(1/4)*x)/(15**(1/8)*sqrt(11) 
)) - 2*atan((7*5**(1/4)*x)/(15**(1/8)*sqrt(7))) + 2*log( - 15**(1/8)*sqrt( 
11) + 11*5**(1/4)*x) + log( - 15**(1/8)*sqrt(7) + 7*5**(1/4)*x) - 2*log(15 
**(1/8)*sqrt(11) + 11*5**(1/4)*x) - log(15**(1/8)*sqrt(7) + 7*5**(1/4)*x)) 
)/240