Integrand size = 119, antiderivative size = 58 \[ \int \frac {-500+192 \sqrt {7}+952 x+360 \sqrt {7} x+672 x^2+252 \sqrt {7} x^2+196 x^3+84 \sqrt {7} x^3+49 x^4}{\left (16-6 \sqrt {7}+14 x+6 \sqrt {7} x+7 x^2\right )^2 \left (2 \sqrt {7}+630 x+238 \sqrt {7} x+147 x^2+56 \sqrt {7} x^2\right )} \, dx=-\frac {\left (8-3 \sqrt {7}\right ) \left (3+\sqrt {7}+\sqrt {7} x\right )}{7 \left (2 \left (8-3 \sqrt {7}\right )+2 \left (7+3 \sqrt {7}\right ) x+7 x^2\right )} \] Output:
-1/7*(8-3*7^(1/2))*(3+7^(1/2)+7^(1/2)*x)/(16-6*7^(1/2)+2*(7+3*7^(1/2))*x+7 *x^2)
Time = 0.17 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.22 \[ \int \frac {-500+192 \sqrt {7}+952 x+360 \sqrt {7} x+672 x^2+252 \sqrt {7} x^2+196 x^3+84 \sqrt {7} x^3+49 x^4}{\left (16-6 \sqrt {7}+14 x+6 \sqrt {7} x+7 x^2\right )^2 \left (2 \sqrt {7}+630 x+238 \sqrt {7} x+147 x^2+56 \sqrt {7} x^2\right )} \, dx=-\frac {3832073521103764228193740275217176667475959453642040802037565598915+1448387648936598037907455646451256190184263094401177986262766639167 \sqrt {7}+\left (1795812989971104599350193324478071660768904403465226048627570115051+678753510377553209614515650246368335569018350058938251136665161288 \sqrt {7}\right ) x}{7 \left (36049049892983023583045990401+13625260145284667680742546880 \sqrt {7}\right )^2 \left (261152656+98706426 \sqrt {7}+\left (62169690254+23497934214 \sqrt {7}\right ) x+7 \left (2081028097+786554688 \sqrt {7}\right ) x^2\right )} \] Input:
Integrate[(-500 + 192*Sqrt[7] + 952*x + 360*Sqrt[7]*x + 672*x^2 + 252*Sqrt [7]*x^2 + 196*x^3 + 84*Sqrt[7]*x^3 + 49*x^4)/((16 - 6*Sqrt[7] + 14*x + 6*S qrt[7]*x + 7*x^2)^2*(2*Sqrt[7] + 630*x + 238*Sqrt[7]*x + 147*x^2 + 56*Sqrt [7]*x^2)),x]
Output:
-1/7*(3832073521103764228193740275217176667475959453642040802037565598915 + 1448387648936598037907455646451256190184263094401177986262766639167*Sqrt [7] + (1795812989971104599350193324478071660768904403465226048627570115051 + 678753510377553209614515650246368335569018350058938251136665161288*Sqrt [7])*x)/((36049049892983023583045990401 + 13625260145284667680742546880*Sq rt[7])^2*(261152656 + 98706426*Sqrt[7] + (62169690254 + 23497934214*Sqrt[7 ])*x + 7*(2081028097 + 786554688*Sqrt[7])*x^2))
Time = 0.87 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.09, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.084, Rules used = {6, 6, 6, 6, 6, 6, 2019, 2126, 2191, 24}
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.
| \(\displaystyle \int \frac {49 x^4+84 \sqrt {7} x^3+196 x^3+252 \sqrt {7} x^2+672 x^2+360 \sqrt {7} x+952 x+192 \sqrt {7}-500}{\left (7 x^2+6 \sqrt {7} x+14 x-6 \sqrt {7}+16\right )^2 \left (56 \sqrt {7} x^2+147 x^2+238 \sqrt {7} x+630 x+2 \sqrt {7}\right )} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {49 x^4+84 \sqrt {7} x^3+196 x^3+252 \sqrt {7} x^2+672 x^2+360 \sqrt {7} x+952 x+192 \sqrt {7}-500}{\left (7 x^2+\left (14+6 \sqrt {7}\right ) x-6 \sqrt {7}+16\right )^2 \left (56 \sqrt {7} x^2+147 x^2+238 \sqrt {7} x+630 x+2 \sqrt {7}\right )}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {49 x^4+84 \sqrt {7} x^3+196 x^3+252 \sqrt {7} x^2+672 x^2+360 \sqrt {7} x+952 x+192 \sqrt {7}-500}{\left (7 x^2+\left (14+6 \sqrt {7}\right ) x-6 \sqrt {7}+16\right )^2 \left (56 \sqrt {7} x^2+147 x^2+\left (630+238 \sqrt {7}\right ) x+2 \sqrt {7}\right )}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {49 x^4+84 \sqrt {7} x^3+196 x^3+252 \sqrt {7} x^2+672 x^2+360 \sqrt {7} x+952 x+192 \sqrt {7}-500}{\left (7 x^2+\left (14+6 \sqrt {7}\right ) x-6 \sqrt {7}+16\right )^2 \left (\left (147+56 \sqrt {7}\right ) x^2+\left (630+238 \sqrt {7}\right ) x+2 \sqrt {7}\right )}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {49 x^4+84 \sqrt {7} x^3+196 x^3+252 \sqrt {7} x^2+672 x^2+\left (952+360 \sqrt {7}\right ) x+192 \sqrt {7}-500}{\left (7 x^2+\left (14+6 \sqrt {7}\right ) x-6 \sqrt {7}+16\right )^2 \left (\left (147+56 \sqrt {7}\right ) x^2+\left (630+238 \sqrt {7}\right ) x+2 \sqrt {7}\right )}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {49 x^4+84 \sqrt {7} x^3+196 x^3+\left (672+252 \sqrt {7}\right ) x^2+\left (952+360 \sqrt {7}\right ) x+192 \sqrt {7}-500}{\left (7 x^2+\left (14+6 \sqrt {7}\right ) x-6 \sqrt {7}+16\right )^2 \left (\left (147+56 \sqrt {7}\right ) x^2+\left (630+238 \sqrt {7}\right ) x+2 \sqrt {7}\right )}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {49 x^4+\left (196+84 \sqrt {7}\right ) x^3+\left (672+252 \sqrt {7}\right ) x^2+\left (952+360 \sqrt {7}\right ) x+192 \sqrt {7}-500}{\left (7 x^2+\left (14+6 \sqrt {7}\right ) x-6 \sqrt {7}+16\right )^2 \left (\left (147+56 \sqrt {7}\right ) x^2+\left (630+238 \sqrt {7}\right ) x+2 \sqrt {7}\right )}dx\) |
| \(\Big \downarrow \) 2019 |
| \(\displaystyle \int \frac {7 x^2+\left (14+6 \sqrt {7}\right ) x+18 \sqrt {7}+16}{\left (7 x^2+\left (14+6 \sqrt {7}\right ) x-6 \sqrt {7}+16\right ) \left (\left (147+56 \sqrt {7}\right ) x^2+\left (630+238 \sqrt {7}\right ) x+2 \sqrt {7}\right )}dx\) |
| \(\Big \downarrow \) 2126 |
| \(\displaystyle -\frac {1}{7} \left (21-8 \sqrt {7}\right ) \int \frac {7 x^2+2 \left (7+3 \sqrt {7}\right ) x+2 \left (8+9 \sqrt {7}\right )}{\left (7 x^2+2 \left (7+3 \sqrt {7}\right ) x+2 \left (8-3 \sqrt {7}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 2191 |
| \(\displaystyle -\frac {1}{7} \left (21-8 \sqrt {7}\right ) \left (-\frac {\int 0dx}{336 \sqrt {7}}-\frac {\sqrt {7} x+\sqrt {7}+3}{\sqrt {7} \left (7 x^2+2 \left (7+3 \sqrt {7}\right ) x+2 \left (8-3 \sqrt {7}\right )\right )}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\left (21-8 \sqrt {7}\right ) \left (\sqrt {7} x+\sqrt {7}+3\right )}{7 \sqrt {7} \left (7 x^2+2 \left (7+3 \sqrt {7}\right ) x+2 \left (8-3 \sqrt {7}\right )\right )}\) |
Input:
Int[(-500 + 192*Sqrt[7] + 952*x + 360*Sqrt[7]*x + 672*x^2 + 252*Sqrt[7]*x^ 2 + 196*x^3 + 84*Sqrt[7]*x^3 + 49*x^4)/((16 - 6*Sqrt[7] + 14*x + 6*Sqrt[7] *x + 7*x^2)^2*(2*Sqrt[7] + 630*x + 238*Sqrt[7]*x + 147*x^2 + 56*Sqrt[7]*x^ 2)),x]
Output:
((21 - 8*Sqrt[7])*(3 + Sqrt[7] + Sqrt[7]*x))/(7*Sqrt[7]*(2*(8 - 3*Sqrt[7]) + 2*(7 + 3*Sqrt[7])*x + 7*x^2))
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px , Qx, x]^p*Qx^(p + q), x] /; FreeQ[q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]
Int[(Px_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.)*((d_) + (e_.)*(x_) + (f_ .)*(x_)^2)^(q_.), x_Symbol] :> Simp[(c/f)^p Int[Px*(d + e*x + f*x^2)^(p + q), x], x] /; FreeQ[{a, b, c, d, e, f, p, q}, x] && PolyQ[Px, x] && EqQ[c* d - a*f, 0] && EqQ[b*d - a*e, 0] && (IntegerQ[p] || GtQ[c/f, 0]) && ( !Inte gerQ[q] || LeafCount[d + e*x + f*x^2] <= LeafCount[a + b*x + c*x^2])
Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[P q, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 1]}, Simp[(b*f - 2*a*g + (2*c*f - b*g)*x)*((a + b*x + c*x^2)^ (p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)) Int [(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (2*p + 3)* (2*c*f - b*g), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^ 2 - 4*a*c, 0] && LtQ[p, -1]
Time = 0.36 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.67
| method | result | size |
| risch | \(\frac {\left (\frac {3}{7}-\frac {8 \sqrt {7}}{49}\right ) x +\frac {\sqrt {7}}{49}-\frac {3}{49}}{x^{2}+\frac {6 \sqrt {7}\, x}{7}+2 x +\frac {16}{7}-\frac {6 \sqrt {7}}{7}}\) | \(39\) |
| default | \(\frac {-\frac {x}{7}-\frac {3 \sqrt {7}}{49}-\frac {1}{7}}{\left (8 \sqrt {7}+21\right ) \left (x^{2}+\frac {6 \sqrt {7}\, x}{7}+2 x +\frac {16}{7}-\frac {6 \sqrt {7}}{7}\right )}\) | \(42\) |
| norman | \(\frac {\left (192+\frac {506 \sqrt {7}}{7}\right ) x +\left (123-21 \sqrt {7}\right ) x^{2}+\left (63+6 \sqrt {7}\right ) x^{3}+\left (\frac {21}{2}+\frac {7 \sqrt {7}}{2}\right ) x^{4}}{49 x^{4}+196 x^{3}+168 x^{2}+952 x +4}\) | \(67\) |
| parallelrisch | \(\frac {-90-56 \sqrt {7}\, x^{3}+21 \sqrt {7}\, x^{2}+147 x^{3}-258 \sqrt {7}\, x -63 x^{2}+34 \sqrt {7}+672 x}{7 \left (16-6 \sqrt {7}+14 x +6 \sqrt {7}\, x +7 x^{2}\right )^{2}}\) | \(68\) |
| gosper | \(-\frac {\left (-500+192 \sqrt {7}+952 x +360 \sqrt {7}\, x +672 x^{2}+252 \sqrt {7}\, x^{2}+196 x^{3}+84 \sqrt {7}\, x^{3}+49 x^{4}\right ) \left (7+3 \sqrt {7}+7 x \right )}{7 \left (2 \sqrt {7}+630 x +238 \sqrt {7}\, x +147 x^{2}+56 \sqrt {7}\, x^{2}\right ) \left (16-6 \sqrt {7}+14 x +6 \sqrt {7}\, x +7 x^{2}\right ) \left (6 \sqrt {7}\, x +7 x^{2}+18 \sqrt {7}+14 x +16\right )}\) | \(136\) |
| orering | \(-\frac {\left (-500+192 \sqrt {7}+952 x +360 \sqrt {7}\, x +672 x^{2}+252 \sqrt {7}\, x^{2}+196 x^{3}+84 \sqrt {7}\, x^{3}+49 x^{4}\right ) \left (7+3 \sqrt {7}+7 x \right )}{7 \left (2 \sqrt {7}+630 x +238 \sqrt {7}\, x +147 x^{2}+56 \sqrt {7}\, x^{2}\right ) \left (16-6 \sqrt {7}+14 x +6 \sqrt {7}\, x +7 x^{2}\right ) \left (6 \sqrt {7}\, x +7 x^{2}+18 \sqrt {7}+14 x +16\right )}\) | \(136\) |
Input:
int((-500+192*7^(1/2)+952*x+360*7^(1/2)*x+672*x^2+252*7^(1/2)*x^2+196*x^3+ 84*7^(1/2)*x^3+49*x^4)/(16-6*7^(1/2)+14*x+6*7^(1/2)*x+7*x^2)^2/(2*7^(1/2)+ 630*x+238*7^(1/2)*x+147*x^2+56*7^(1/2)*x^2),x,method=_RETURNVERBOSE)
Output:
((3/7-8/49*7^(1/2))*x+1/49*7^(1/2)-3/49)/(x^2+6/7*7^(1/2)*x+2*x+16/7-6/7*7 ^(1/2))
Time = 0.07 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.02 \[ \int \frac {-500+192 \sqrt {7}+952 x+360 \sqrt {7} x+672 x^2+252 \sqrt {7} x^2+196 x^3+84 \sqrt {7} x^3+49 x^4}{\left (16-6 \sqrt {7}+14 x+6 \sqrt {7} x+7 x^2\right )^2 \left (2 \sqrt {7}+630 x+238 \sqrt {7} x+147 x^2+56 \sqrt {7} x^2\right )} \, dx=\frac {147 \, x^{3} + 609 \, x^{2} - \sqrt {7} {\left (56 \, x^{3} + 231 \, x^{2} - 30 \, x + 2\right )} - 84 \, x - 6}{7 \, {\left (49 \, x^{4} + 196 \, x^{3} + 168 \, x^{2} + 952 \, x + 4\right )}} \] Input:
integrate((-500+192*7^(1/2)+952*x+360*7^(1/2)*x+672*x^2+252*7^(1/2)*x^2+19 6*x^3+84*7^(1/2)*x^3+49*x^4)/(16-6*7^(1/2)+14*x+6*7^(1/2)*x+7*x^2)^2/(2*7^ (1/2)+630*x+238*7^(1/2)*x+147*x^2+56*7^(1/2)*x^2),x, algorithm="fricas")
Output:
1/7*(147*x^3 + 609*x^2 - sqrt(7)*(56*x^3 + 231*x^2 - 30*x + 2) - 84*x - 6) /(49*x^4 + 196*x^3 + 168*x^2 + 952*x + 4)
Time = 0.83 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.84 \[ \int \frac {-500+192 \sqrt {7}+952 x+360 \sqrt {7} x+672 x^2+252 \sqrt {7} x^2+196 x^3+84 \sqrt {7} x^3+49 x^4}{\left (16-6 \sqrt {7}+14 x+6 \sqrt {7} x+7 x^2\right )^2 \left (2 \sqrt {7}+630 x+238 \sqrt {7} x+147 x^2+56 \sqrt {7} x^2\right )} \, dx=\frac {x \left (- 282008183944355983519396809697240962337292909995418119 \sqrt {7} - 746123522401724069574392731052928214064440761469781472\right ) - 1592148074234792020132583160144651101076319491456035829 - 601775407830809156194136551577067339793481807768181607 \sqrt {7}}{x^{2} \cdot \left (220227365899240983767039280866098904703691612654261779032 + 83238120294316877473517323964458401451190740411634225925 \sqrt {7}\right ) + x \left (939883453564383232375182505518948218114527667778328913614 + 355242554216554598175925460099858721219831434526921405306 \sqrt {7}\right ) + 3948114575220983769271555335761373472722100739935853666 + 1492247044803448139148785462105856428128881522939562944 \sqrt {7}} \] Input:
integrate((-500+192*7**(1/2)+952*x+360*7**(1/2)*x+672*x**2+252*7**(1/2)*x* *2+196*x**3+84*7**(1/2)*x**3+49*x**4)/(16-6*7**(1/2)+14*x+6*7**(1/2)*x+7*x **2)**2/(2*7**(1/2)+630*x+238*7**(1/2)*x+147*x**2+56*7**(1/2)*x**2),x)
Output:
(x*(-282008183944355983519396809697240962337292909995418119*sqrt(7) - 7461 23522401724069574392731052928214064440761469781472) - 15921480742347920201 32583160144651101076319491456035829 - 601775407830809156194136551577067339 793481807768181607*sqrt(7))/(x**2*(220227365899240983767039280866098904703 691612654261779032 + 83238120294316877473517323964458401451190740411634225 925*sqrt(7)) + x*(93988345356438323237518250551894821811452766777832891361 4 + 355242554216554598175925460099858721219831434526921405306*sqrt(7)) + 3 948114575220983769271555335761373472722100739935853666 + 14922470448034481 39148785462105856428128881522939562944*sqrt(7))
Exception generated. \[ \int \frac {-500+192 \sqrt {7}+952 x+360 \sqrt {7} x+672 x^2+252 \sqrt {7} x^2+196 x^3+84 \sqrt {7} x^3+49 x^4}{\left (16-6 \sqrt {7}+14 x+6 \sqrt {7} x+7 x^2\right )^2 \left (2 \sqrt {7}+630 x+238 \sqrt {7} x+147 x^2+56 \sqrt {7} x^2\right )} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((-500+192*7^(1/2)+952*x+360*7^(1/2)*x+672*x^2+252*7^(1/2)*x^2+19 6*x^3+84*7^(1/2)*x^3+49*x^4)/(16-6*7^(1/2)+14*x+6*7^(1/2)*x+7*x^2)^2/(2*7^ (1/2)+630*x+238*7^(1/2)*x+147*x^2+56*7^(1/2)*x^2),x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati ve exponent.
Exception generated. \[ \int \frac {-500+192 \sqrt {7}+952 x+360 \sqrt {7} x+672 x^2+252 \sqrt {7} x^2+196 x^3+84 \sqrt {7} x^3+49 x^4}{\left (16-6 \sqrt {7}+14 x+6 \sqrt {7} x+7 x^2\right )^2 \left (2 \sqrt {7}+630 x+238 \sqrt {7} x+147 x^2+56 \sqrt {7} x^2\right )} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((-500+192*7^(1/2)+952*x+360*7^(1/2)*x+672*x^2+252*7^(1/2)*x^2+19 6*x^3+84*7^(1/2)*x^3+49*x^4)/(16-6*7^(1/2)+14*x+6*7^(1/2)*x+7*x^2)^2/(2*7^ (1/2)+630*x+238*7^(1/2)*x+147*x^2+56*7^(1/2)*x^2),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%%%{%%{[-168,0]:[1,0,-7]%%},[2]%%%}+%%%{%%{[-336,-1008]:[1,0,- 7]%%},[1]
Time = 9.95 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.72 \[ \int \frac {-500+192 \sqrt {7}+952 x+360 \sqrt {7} x+672 x^2+252 \sqrt {7} x^2+196 x^3+84 \sqrt {7} x^3+49 x^4}{\left (16-6 \sqrt {7}+14 x+6 \sqrt {7} x+7 x^2\right )^2 \left (2 \sqrt {7}+630 x+238 \sqrt {7} x+147 x^2+56 \sqrt {7} x^2\right )} \, dx=-\frac {\frac {x\,\left (8\,\sqrt {7}-21\right )}{7}-\frac {\sqrt {7}}{7}+\frac {3}{7}}{7\,x^2+\left (6\,\sqrt {7}+14\right )\,x-6\,\sqrt {7}+16} \] Input:
int((952*x + 360*7^(1/2)*x + 192*7^(1/2) + 252*7^(1/2)*x^2 + 84*7^(1/2)*x^ 3 + 672*x^2 + 196*x^3 + 49*x^4 - 500)/((14*x + 6*7^(1/2)*x - 6*7^(1/2) + 7 *x^2 + 16)^2*(630*x + 238*7^(1/2)*x + 2*7^(1/2) + 56*7^(1/2)*x^2 + 147*x^2 )),x)
Output:
-((x*(8*7^(1/2) - 21))/7 - 7^(1/2)/7 + 3/7)/(7*x^2 - 6*7^(1/2) + x*(6*7^(1 /2) + 14) + 16)
Time = 0.16 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.05 \[ \int \frac {-500+192 \sqrt {7}+952 x+360 \sqrt {7} x+672 x^2+252 \sqrt {7} x^2+196 x^3+84 \sqrt {7} x^3+49 x^4}{\left (16-6 \sqrt {7}+14 x+6 \sqrt {7} x+7 x^2\right )^2 \left (2 \sqrt {7}+630 x+238 \sqrt {7} x+147 x^2+56 \sqrt {7} x^2\right )} \, dx=\frac {392 \sqrt {7}\, x^{4}-5124 \sqrt {7}\, x^{2}+8456 \sqrt {7}\, x -24 \sqrt {7}-1029 x^{4}+13524 x^{2}-22344 x -252}{9604 x^{4}+38416 x^{3}+32928 x^{2}+186592 x +784} \] Input:
int((-500+192*7^(1/2)+952*x+360*7^(1/2)*x+672*x^2+252*7^(1/2)*x^2+196*x^3+ 84*7^(1/2)*x^3+49*x^4)/(16-6*7^(1/2)+14*x+6*7^(1/2)*x+7*x^2)^2/(2*7^(1/2)+ 630*x+238*7^(1/2)*x+147*x^2+56*7^(1/2)*x^2),x)
Output:
(392*sqrt(7)*x**4 - 5124*sqrt(7)*x**2 + 8456*sqrt(7)*x - 24*sqrt(7) - 1029 *x**4 + 13524*x**2 - 22344*x - 252)/(196*(49*x**4 + 196*x**3 + 168*x**2 + 952*x + 4))