Integrand size = 40, antiderivative size = 500 \[ \int \frac {1}{(e+f x) \left (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3\right )^2} \, dx=\frac {4}{27 (1-6 b) \left (6 e+f-\sqrt {1-6 b} f\right ) \left (1-\sqrt {1-6 b}-6 x\right )^3}-\frac {2 \left (12 e+\left (2-5 \sqrt {1-6 b}\right ) f\right )}{27 (1-6 b)^{3/2} \left (6 e+f-\sqrt {1-6 b} f\right )^2 \left (1-\sqrt {1-6 b}-6 x\right )^2}+\frac {4 \left (36 e^2+12 e f+7 f^2-36 b f^2-\sqrt {1-6 b} f (24 e+4 f)\right )}{27 (1-6 b)^2 \left (6 e+f-\sqrt {1-6 b} f\right )^3 \left (1-\sqrt {1-6 b}-6 x\right )}+\frac {4}{81 (1-6 b)^2 \left (6 e+f+2 \sqrt {1-6 b} f\right ) \left (1+2 \sqrt {1-6 b}-6 x\right )}+\frac {4 \left (864 e^3+108 \left (4-7 \sqrt {1-6 b}\right ) e^2 f+36 \left (10-7 \sqrt {1-6 b}-48 b\right ) e f^2+\left (52-79 \sqrt {1-6 b}-12 \left (24-29 \sqrt {1-6 b}\right ) b\right ) f^3\right ) \log \left (1-\sqrt {1-6 b}-6 x\right )}{243 (1-6 b)^{5/2} \left (6 e+f-\sqrt {1-6 b} f\right )^4}-\frac {4 \left (24 e+\left (4+11 \sqrt {1-6 b}\right ) f\right ) \log \left (1+2 \sqrt {1-6 b}-6 x\right )}{243 (1-6 b)^{5/2} \left (6 e+f+2 \sqrt {1-6 b} f\right )^2}+\frac {4 f^5 \log (e+f x)}{\left (6 e+f-\sqrt {1-6 b} f\right )^4 \left (6 e+f+2 \sqrt {1-6 b} f\right )^2} \] Output:
4/27/(1-6*b)/(6*e+f-(1-6*b)^(1/2)*f)/(1-(1-6*b)^(1/2)-6*x)^3-2/27*(12*e+(2 -5*(1-6*b)^(1/2))*f)/(1-6*b)^(3/2)/(6*e+f-(1-6*b)^(1/2)*f)^2/(1-(1-6*b)^(1 /2)-6*x)^2+4/27*(36*e^2+12*e*f+7*f^2-36*b*f^2-(1-6*b)^(1/2)*f*(24*e+4*f))/ (1-6*b)^2/(6*e+f-(1-6*b)^(1/2)*f)^3/(1-(1-6*b)^(1/2)-6*x)+4/81/(1-6*b)^2/( 6*e+f+2*(1-6*b)^(1/2)*f)/(1+2*(1-6*b)^(1/2)-6*x)+4/243*(864*e^3+108*(4-7*( 1-6*b)^(1/2))*e^2*f+36*(10-7*(1-6*b)^(1/2)-48*b)*e*f^2+(52-79*(1-6*b)^(1/2 )-12*(24-29*(1-6*b)^(1/2))*b)*f^3)*ln(1-(1-6*b)^(1/2)-6*x)/(1-6*b)^(5/2)/( 6*e+f-(1-6*b)^(1/2)*f)^4-4/243*(24*e+(4+11*(1-6*b)^(1/2))*f)*ln(1+2*(1-6*b )^(1/2)-6*x)/(1-6*b)^(5/2)/(6*e+f+2*(1-6*b)^(1/2)*f)^2+4*f^5*ln(f*x+e)/(6* e+f-(1-6*b)^(1/2)*f)^4/(6*e+f+2*(1-6*b)^(1/2)*f)^2
Leaf count is larger than twice the leaf count of optimal. \(2761\) vs. \(2(500)=1000\).
Time = 10.47 (sec) , antiderivative size = 2761, normalized size of antiderivative = 5.52 \[ \int \frac {1}{(e+f x) \left (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3\right )^2} \, dx=\text {Result too large to show} \] Input:
Integrate[1/((e + f*x)*(1 - (1 - 6*b)^(3/2) - 9*b + 54*b*x - 54*x^2 + 108* x^3)^2),x]
Output:
(-2*(-3*e - 3*Sqrt[1 - 6*b]*e - f - Sqrt[1 - 6*b]*f + 3*b*f + 18*e*x + 3*f *x + 3*Sqrt[1 - 6*b]*f*x))/(2187*(6*e^2 + 2*e*f + b*f^2)*(b - 2*x + 6*x^2) ^3) + (36*e^3 - 36*Sqrt[1 - 6*b]*e^3 - 6*e^2*f - 30*Sqrt[1 - 6*b]*e^2*f + 144*b*e^2*f - 12*e*f^2 - 12*Sqrt[1 - 6*b]*e*f^2 + 90*b*e*f^2 + 30*Sqrt[1 - 6*b]*b*e*f^2 - 2*f^3 - 2*Sqrt[1 - 6*b]*f^3 + 15*b*f^3 + 9*Sqrt[1 - 6*b]*b *f^3 - 12*b^2*f^3 - 216*e^3*x - 108*e^2*f*x + 72*Sqrt[1 - 6*b]*e^2*f*x + 2 4*e*f^2*x + 24*Sqrt[1 - 6*b]*e*f^2*x - 252*b*e*f^2*x + 6*f^3*x + 6*Sqrt[1 - 6*b]*f^3*x - 42*b*f^3*x - 24*Sqrt[1 - 6*b]*b*f^3*x)/(1458*(-1 + 6*b)*(6* e^2 + 2*e*f + b*f^2)^2*(b - 2*x + 6*x^2)^2) + (1296*e^5 + 972*e^4*f - 216* Sqrt[1 - 6*b]*e^4*f + 648*b*e^4*f + 180*e^3*f^2 - 72*Sqrt[1 - 6*b]*e^3*f^2 + 1080*b*e^3*f^2 - 432*Sqrt[1 - 6*b]*b*e^3*f^2 + 42*e^2*f^3 + 42*Sqrt[1 - 6*b]*e^2*f^3 - 216*b*e^2*f^3 - 468*Sqrt[1 - 6*b]*b*e^2*f^3 + 1944*b^2*e^2 *f^3 + 26*e*f^4 + 26*Sqrt[1 - 6*b]*e*f^4 - 318*b*e*f^4 - 240*Sqrt[1 - 6*b] *b*e*f^4 + 1152*b^2*e*f^4 + 360*Sqrt[1 - 6*b]*b^2*e*f^4 + 4*f^5 + 4*Sqrt[1 - 6*b]*f^5 - 53*b*f^5 - 41*Sqrt[1 - 6*b]*b*f^5 + 201*b^2*f^5 + 96*Sqrt[1 - 6*b]*b^2*f^5 - 126*b^3*f^5 - 7776*e^5*x - 6480*e^4*f*x + 1296*Sqrt[1 - 6 *b]*e^4*f*x - 1512*e^3*f^2*x + 864*Sqrt[1 - 6*b]*e^3*f^2*x - 3888*b*e^3*f^ 2*x - 36*e^2*f^3*x - 36*Sqrt[1 - 6*b]*e^2*f^3*x - 1944*b*e^2*f^3*x + 1512* Sqrt[1 - 6*b]*b*e^2*f^3*x - 60*e*f^4*x - 60*Sqrt[1 - 6*b]*e*f^4*x + 684*b* e*f^4*x + 504*Sqrt[1 - 6*b]*b*e*f^4*x - 3024*b^2*e*f^4*x - 12*f^5*x - 1...
Time = 4.26 (sec) , antiderivative size = 518, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2488, 27, 27, 99, 2009}
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.
| \(\displaystyle \int \frac {1}{\left (54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1\right )^2 (e+f x)} \, dx\) |
| \(\Big \downarrow \) 2488 |
| \(\displaystyle 9836602018824134393856 (1-6 b)^6 \int \frac {1}{2459150504706033598464 \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^4 \left (\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)-6 (1-6 b) x\right )^2 (e+f x)}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 4 (1-6 b)^6 \int \frac {1}{(1-6 b)^2 \left (-6 x+2 \sqrt {1-6 b}+1\right )^2 \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^4 (e+f x)}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 4 (1-6 b)^4 \int \frac {1}{\left (-6 x+2 \sqrt {1-6 b}+1\right )^2 \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^4 (e+f x)}dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle 4 (1-6 b)^4 \int \left (\frac {f^6}{(1-6 b)^4 \left (6 e+\left (1-\sqrt {1-6 b}\right ) f\right )^4 \left (6 e+2 \sqrt {1-6 b} f+f\right )^2 (e+f x)}+\frac {2 \left (-864 e^3-108 \left (4-7 \sqrt {1-6 b}\right ) f e^2-36 \left (-48 b-7 \sqrt {1-6 b}+10\right ) f^2 e-\left (-12 \left (24-29 \sqrt {1-6 b}\right ) b-79 \sqrt {1-6 b}+52\right ) f^3\right )}{81 (1-6 b)^{13/2} \left (6 e+\left (1-\sqrt {1-6 b}\right ) f\right )^4 \left (-6 x-\sqrt {1-6 b}+1\right )}+\frac {2 \left (24 e+\left (11 \sqrt {1-6 b}+4\right ) f\right )}{81 (1-6 b)^{13/2} \left (6 e+2 \sqrt {1-6 b} f+f\right )^2 \left (-6 x+2 \sqrt {1-6 b}+1\right )}+\frac {2 \left (36 e^2+12 \left (1-2 \sqrt {1-6 b}\right ) f e+\left (-36 b-4 \sqrt {1-6 b}+7\right ) f^2\right )}{9 (1-6 b)^6 \left (6 e-\sqrt {1-6 b} f+f\right )^3 \left (-6 x-\sqrt {1-6 b}+1\right )^2}+\frac {2}{27 (6 b-1)^6 \left (6 e+2 \sqrt {1-6 b} f+f\right ) \left (-6 x+2 \sqrt {1-6 b}+1\right )^2}+\frac {2 \left (-12 e-\left (2-5 \sqrt {1-6 b}\right ) f\right )}{9 (1-6 b)^{11/2} \left (6 e+\left (1-\sqrt {1-6 b}\right ) f\right )^2 \left (-6 x-\sqrt {1-6 b}+1\right )^3}-\frac {2}{3 (6 b-1)^5 \left (6 e-\sqrt {1-6 b} f+f\right ) \left (6 x+\sqrt {1-6 b}-1\right )^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 4 (1-6 b)^4 \left (\frac {12 \left (1-2 \sqrt {1-6 b}\right ) e f+\left (-36 b-4 \sqrt {1-6 b}+7\right ) f^2+36 e^2}{27 (1-6 b)^6 \left (-\sqrt {1-6 b}-6 x+1\right ) \left (-\sqrt {1-6 b} f+6 e+f\right )^3}+\frac {\left (108 \left (4-7 \sqrt {1-6 b}\right ) e^2 f+36 \left (-48 b-7 \sqrt {1-6 b}+10\right ) e f^2+\left (-12 \left (24-29 \sqrt {1-6 b}\right ) b-79 \sqrt {1-6 b}+52\right ) f^3+864 e^3\right ) \log \left (-\sqrt {1-6 b}-6 x+1\right )}{243 (1-6 b)^{13/2} \left (-\sqrt {1-6 b} f+6 e+f\right )^4}+\frac {f^5 \log (e+f x)}{(1-6 b)^4 \left (-\sqrt {1-6 b} f+6 e+f\right )^4 \left (2 \sqrt {1-6 b} f+6 e+f\right )^2}+\frac {1}{81 (1-6 b)^6 \left (2 \sqrt {1-6 b}-6 x+1\right ) \left (2 \sqrt {1-6 b} f+6 e+f\right )}-\frac {\left (2-5 \sqrt {1-6 b}\right ) f+12 e}{54 (1-6 b)^{11/2} \left (-\sqrt {1-6 b}-6 x+1\right )^2 \left (-\sqrt {1-6 b} f+6 e+f\right )^2}+\frac {1}{27 (1-6 b)^5 \left (-\sqrt {1-6 b}-6 x+1\right )^3 \left (-\sqrt {1-6 b} f+6 e+f\right )}-\frac {\left (\left (11 \sqrt {1-6 b}+4\right ) f+24 e\right ) \log \left (2 \sqrt {1-6 b}-6 x+1\right )}{243 (1-6 b)^{13/2} \left (2 \sqrt {1-6 b} f+6 e+f\right )^2}\right )\) |
Input:
Int[1/((e + f*x)*(1 - (1 - 6*b)^(3/2) - 9*b + 54*b*x - 54*x^2 + 108*x^3)^2 ),x]
Output:
4*(1 - 6*b)^4*(1/(27*(1 - 6*b)^5*(6*e + f - Sqrt[1 - 6*b]*f)*(1 - Sqrt[1 - 6*b] - 6*x)^3) - (12*e + (2 - 5*Sqrt[1 - 6*b])*f)/(54*(1 - 6*b)^(11/2)*(6 *e + f - Sqrt[1 - 6*b]*f)^2*(1 - Sqrt[1 - 6*b] - 6*x)^2) + (36*e^2 + 12*(1 - 2*Sqrt[1 - 6*b])*e*f + (7 - 4*Sqrt[1 - 6*b] - 36*b)*f^2)/(27*(1 - 6*b)^ 6*(6*e + f - Sqrt[1 - 6*b]*f)^3*(1 - Sqrt[1 - 6*b] - 6*x)) + 1/(81*(1 - 6* b)^6*(6*e + f + 2*Sqrt[1 - 6*b]*f)*(1 + 2*Sqrt[1 - 6*b] - 6*x)) + ((864*e^ 3 + 108*(4 - 7*Sqrt[1 - 6*b])*e^2*f + 36*(10 - 7*Sqrt[1 - 6*b] - 48*b)*e*f ^2 + (52 - 79*Sqrt[1 - 6*b] - 12*(24 - 29*Sqrt[1 - 6*b])*b)*f^3)*Log[1 - S qrt[1 - 6*b] - 6*x])/(243*(1 - 6*b)^(13/2)*(6*e + f - Sqrt[1 - 6*b]*f)^4) - ((24*e + (4 + 11*Sqrt[1 - 6*b])*f)*Log[1 + 2*Sqrt[1 - 6*b] - 6*x])/(243* (1 - 6*b)^(13/2)*(6*e + f + 2*Sqrt[1 - 6*b]*f)^2) + (f^5*Log[e + f*x])/((1 - 6*b)^4*(6*e + f - Sqrt[1 - 6*b]*f)^4*(6*e + f + 2*Sqrt[1 - 6*b]*f)^2))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[((e_.) + (f_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2 + (d_.)*( x_)^3)^(p_), x_Symbol] :> Simp[1/(4^p*(c^2 - 3*b*d)^(3*p)) Int[(e + f*x)^ m*(c^3 - 4*b*c*d + 9*a*d^2 + d*(c^2 - 3*b*d)*x)^p*(b*c - 9*a*d + 2*(c^2 - 3 *b*d)*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && NeQ[c^2 - 3 *b*d, 0] && EqQ[b^2*c^2 - 4*a*c^3 - 4*b^3*d + 18*a*b*c*d - 27*a^2*d^2, 0] & & ILtQ[p, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(2311\) vs. \(2(440)=880\).
Time = 18.92 (sec) , antiderivative size = 2312, normalized size of antiderivative = 4.62
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2312\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(383514\) |
Input:
int(1/(f*x+e)/(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3)^2,x,method=_RETU RNVERBOSE)
Output:
f^5/((1-6*b)^(3/2)*f^3+54*b*e*f^2+9*b*f^3+108*e^3+54*e^2*f-f^3)^2*ln(f*x+e )+54/((1-6*b)^(3/2)*f^3+54*b*e*f^2+9*b*f^3+108*e^3+54*e^2*f-f^3)^2*((-1/26 244*(-2520*(1-6*b)^(1/2)*b^2*f^5-10800*(1-6*b)^(1/2)*b*e^2*f^3-3600*(1-6*b )^(1/2)*b*e*f^4+540*(1-6*b)^(1/2)*b*f^5+8856*f^4*e*b^2+1476*b^2*f^5-6480*( 1-6*b)^(1/2)*e^4*f-4320*(1-6*b)^(1/2)*e^3*f^2+720*(1-6*b)^(1/2)*e^2*f^3+48 0*(1-6*b)^(1/2)*e*f^4-25*(1-6*b)^(1/2)*f^5+25920*b*e^3*f^2+12960*b*e^2*f^3 -792*b*e*f^4-372*b*f^5+31104*e^5+25920*e^4*f+4320*e^3*f^2-720*e^2*f^3+6*e* f^4+25*f^5)/(36*b^2-12*b+1)*x^3-1/52488*((1-6*b)^(1/2)-2)*(-1206*(1-6*b)^( 1/2)*b^2*f^5+245*(1-6*b)^(1/2)*b*f^5+15552*e^5+3024*b^3*f^4*e+25920*b^2*e^ 3*f^2-4968*b^2*(1-6*b)^(1/2)*e^2*f^3-6480*b*(1-6*b)^(1/2)*e^4*f+288*(1-6*b )^(1/2)*f^4*e*b^2-4320*(1-6*b)^(1/2)*b*e^3*f^2+24624*b^2*e^2*f^3+25920*e^4 *f*b-8*(1-6*b)^(1/2)*f^5-576*b^3*(1-6*b)^(1/2)*f^5+12960*e^4*f+4392*b^3*f^ 5-3240*(1-6*b)^(1/2)*e^4*f-2160*(1-6*b)^(1/2)*e^3*f^2+522*(1-6*b)^(1/2)*e^ 2*f^3+348*(1-6*b)^(1/2)*e*f^4+2160*e^3*f^2-36*e^2*f^3+138*e*f^4-768*b^2*f^ 5-26*b*f^5-2172*b*e*f^4+17280*b*e^3*f^2+10440*f^4*e*b^2-6624*(1-6*b)^(1/2) *b*e^2*f^3-2616*(1-6*b)^(1/2)*b*e*f^4+1872*b*e^2*f^3+8*f^5+31104*b*e^5)/(2 *b+1)/(36*b^2-12*b+1)*x^2-1/314928*(6*b+1-2*(1-6*b)^(1/2))*(-1752*(1-6*b)^ (1/2)*b^2*f^5+219*(1-6*b)^(1/2)*b*f^5-15552*e^5+48384*b^3*f^4*e+158112*b^2 *e^3*f^2+287712*b^3*e^2*f^3+155520*e^4*f*b^2-99792*(1-6*b)^(1/2)*b^3*e^2*f ^3-38880*(1-6*b)^(1/2)*b^2*e^4*f-7992*b^2*(1-6*b)^(1/2)*e^2*f^3-38880*b...
Timed out. \[ \int \frac {1}{(e+f x) \left (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3\right )^2} \, dx=\text {Timed out} \] Input:
integrate(1/(f*x+e)/(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3)^2,x, algor ithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {1}{(e+f x) \left (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3\right )^2} \, dx=\text {Timed out} \] Input:
integrate(1/(f*x+e)/(1-(1-6*b)**(3/2)-9*b+54*b*x-54*x**2+108*x**3)**2,x)
Output:
Timed out
\[ \int \frac {1}{(e+f x) \left (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3\right )^2} \, dx=\int { \frac {1}{{\left (108 \, x^{3} + 54 \, b x - 54 \, x^{2} - {\left (-6 \, b + 1\right )}^{\frac {3}{2}} - 9 \, b + 1\right )}^{2} {\left (f x + e\right )}} \,d x } \] Input:
integrate(1/(f*x+e)/(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3)^2,x, algor ithm="maxima")
Output:
f^5*log(f*x + e)/(2916*(4*b + 1)*e^4*f^2 + 216*((-6*b + 1)^(3/2) + 36*b - 1)*e^3*f^3 + 108*(27*b^2 + (-6*b + 1)^(3/2) + 9*b - 1)*e^2*f^4 + 108*(((-6 *b + 1)^(3/2) - 1)*b + 9*b^2)*e*f^5 - ((6*b - 1)^3 - 18*((-6*b + 1)^(3/2) - 1)*b - 81*b^2 + 2*(-6*b + 1)^(3/2) - 1)*f^6 + 11664*e^6 + 11664*e^5*f) + 1/3*(18*(72*b^2 - (-6*b + 1)^(3/2) - 18*b + 1)*e^2 - 3*(12*((-6*b + 1)^(3 /2) + 2)*b - 108*b^2 + (-6*b + 1)^(3/2) - 1)*e*f - ((6*b - 1)^3 - 648*b^3 + 3*(4*(-6*b + 1)^(3/2) - 13)*b + 270*b^2 - (-6*b + 1)^(3/2) + 2)*f^2 + 18 *(36*(6*b - 1)*e^2 - 6*((-6*b + 1)^(3/2) - 12*b + 2)*e*f + (72*b^2 - (-6*b + 1)^(3/2) - 18*b + 1)*f^2)*x^2 + 3*(36*((-6*b + 1)^(3/2) - 12*b + 2)*e^2 + 6*(36*b^2 + 4*(-6*b + 1)^(3/2) - 36*b + 5)*e*f + (12*((-6*b + 1)^(3/2) + 2)*b - 108*b^2 + (-6*b + 1)^(3/2) - 1)*f^2)*x)/(108*((-6*b + 1)^(9/2) + 108*(2*(-6*b + 1)^(3/2) - 11)*b^3 + 1944*b^4 + (6*b - 1)^3 - 54*(2*(-6*b + 1)^(3/2) - 5)*b^2 - 9*((6*b - 1)^3 - 2*(-6*b + 1)^(3/2) + 3)*b - (-6*b + 1)^(3/2) + 1)*e^3 + 54*((-6*b + 1)^(9/2) + 108*(2*(-6*b + 1)^(3/2) - 11)*b ^3 + 1944*b^4 + (6*b - 1)^3 - 54*(2*(-6*b + 1)^(3/2) - 5)*b^2 - 9*((6*b - 1)^3 - 2*(-6*b + 1)^(3/2) + 3)*b - (-6*b + 1)^(3/2) + 1)*e^2*f + 54*(108*( 2*(-6*b + 1)^(3/2) - 11)*b^4 + 1944*b^5 - 54*(2*(-6*b + 1)^(3/2) - 5)*b^3 - 9*((6*b - 1)^3 - 2*(-6*b + 1)^(3/2) + 3)*b^2 + ((-6*b + 1)^(9/2) + (6*b - 1)^3 - (-6*b + 1)^(3/2) + 1)*b)*e*f^2 + ((6*b - 1)^6 + 972*(4*(-6*b + 1) ^(3/2) - 13)*b^4 + 17496*b^5 - 2*(-6*b + 1)^(9/2) - 54*(4*(6*b - 1)^3 +...
Exception generated. \[ \int \frac {1}{(e+f x) \left (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3\right )^2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/(f*x+e)/(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3)^2,x, algor ithm="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%%%{%%%{17006112,[3]%%%}+%%%{-8503056,[2]%%%}+%%%{1417176,[1]% %%}+%%%{-
Timed out. \[ \int \frac {1}{(e+f x) \left (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3\right )^2} \, dx=\text {Hanged} \] Input:
int(1/((e + f*x)*(9*b - 54*b*x + (1 - 6*b)^(3/2) + 54*x^2 - 108*x^3 - 1)^2 ),x)
Output:
\text{Hanged}
\[ \int \frac {1}{(e+f x) \left (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3\right )^2} \, dx=\int \frac {1}{\left (f x +e \right ) \left (1-\left (-6 b +1\right )^{\frac {3}{2}}-9 b +54 b x -54 x^{2}+108 x^{3}\right )^{2}}d x \] Input:
int(1/(f*x+e)/(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3)^2,x)
Output:
int(1/(f*x+e)/(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3)^2,x)