Integrand size = 42, antiderivative size = 496 \[ \int (e+f x)^2 \sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx=-\frac {\sqrt {1-6 b} \left (36 e^2+\left (5+4 \sqrt {1-6 b}-24 b\right ) f^2+12 e \left (f+2 \sqrt {1-6 b} f\right )\right ) \left (2+\frac {1-6 x}{\sqrt {1-6 b}}\right ) \sqrt {-2 (1-6 b)^{3/2}+3 (1-6 b) (1-6 x)-(1-6 x)^3}}{108 \sqrt {2} \left (1-\frac {1-6 x}{\sqrt {1-6 b}}\right )}+\frac {\sqrt {1-6 b} \left (36 e^2+\left (17+10 \sqrt {1-6 b}-96 b\right ) f^2+12 e \left (f+5 \sqrt {1-6 b} f\right )\right ) \left (2+\frac {1-6 x}{\sqrt {1-6 b}}\right )^2 \sqrt {-2 (1-6 b)^{3/2}+3 (1-6 b) (1-6 x)-(1-6 x)^3}}{540 \sqrt {2} \left (1-\frac {1-6 x}{\sqrt {1-6 b}}\right )}-\frac {\sqrt {1-6 b} f \left (12 \sqrt {1-6 b} e+\left (7+2 \sqrt {1-6 b}-42 b\right ) f\right ) \left (2+\frac {1-6 x}{\sqrt {1-6 b}}\right )^3 \sqrt {-2 (1-6 b)^{3/2}+3 (1-6 b) (1-6 x)-(1-6 x)^3}}{756 \sqrt {2} \left (1-\frac {1-6 x}{\sqrt {1-6 b}}\right )}+\frac {(1-6 b)^{3/2} f^2 \left (2+\frac {1-6 x}{\sqrt {1-6 b}}\right )^4 \sqrt {-2 (1-6 b)^{3/2}+3 (1-6 b) (1-6 x)-(1-6 x)^3}}{972 \sqrt {2} \left (1-\frac {1-6 x}{\sqrt {1-6 b}}\right )} \] Output:
-1/216*(1-6*b)^(1/2)*(36*e^2+(5+4*(1-6*b)^(1/2)-24*b)*f^2+12*e*(f+2*(1-6*b )^(1/2)*f))*(2+(1-6*x)/(1-6*b)^(1/2))*(-2*(1-6*b)^(3/2)+3*(1-6*b)*(1-6*x)- (1-6*x)^3)^(1/2)*2^(1/2)/(1-(1-6*x)/(1-6*b)^(1/2))+1/1080*(1-6*b)^(1/2)*(3 6*e^2+(17+10*(1-6*b)^(1/2)-96*b)*f^2+12*e*(f+5*(1-6*b)^(1/2)*f))*(2+(1-6*x )/(1-6*b)^(1/2))^2*(-2*(1-6*b)^(3/2)+3*(1-6*b)*(1-6*x)-(1-6*x)^3)^(1/2)*2^ (1/2)/(1-(1-6*x)/(1-6*b)^(1/2))-1/1512*(1-6*b)^(1/2)*f*(12*(1-6*b)^(1/2)*e +(7+2*(1-6*b)^(1/2)-42*b)*f)*(2+(1-6*x)/(1-6*b)^(1/2))^3*(-2*(1-6*b)^(3/2) +3*(1-6*b)*(1-6*x)-(1-6*x)^3)^(1/2)*2^(1/2)/(1-(1-6*x)/(1-6*b)^(1/2))+1/19 44*(1-6*b)^(3/2)*f^2*(2+(1-6*x)/(1-6*b)^(1/2))^4*(-2*(1-6*b)^(3/2)+3*(1-6* b)*(1-6*x)-(1-6*x)^3)^(1/2)*2^(1/2)/(1-(1-6*x)/(1-6*b)^(1/2))
Result contains higher order function than in optimal. Order 9 vs. order 2 in optimal.
Time = 17.22 (sec) , antiderivative size = 17362, normalized size of antiderivative = 35.00 \[ \int (e+f x)^2 \sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx=\text {Result too large to show} \] Input:
Integrate[(e + f*x)^2*Sqrt[1 - (1 - 6*b)^(3/2) - 9*b + 54*b*x - 54*x^2 + 1 08*x^3],x]
Output:
Result too large to show
Time = 1.25 (sec) , antiderivative size = 386, normalized size of antiderivative = 0.78, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2489, 27, 86, 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.
| \(\displaystyle \int \sqrt {54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1} (e+f x)^2 \, dx\) |
| \(\Big \downarrow \) 2489 |
| \(\displaystyle -\frac {\sqrt {54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1} \int -157464 \sqrt {2} \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} (e+f x)^2dx}{157464 \sqrt {2} \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1} \int \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} (e+f x)^2dx}{\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \frac {\sqrt {54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1} \int \left (-\frac {f^2 \left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^{7/2}}{36 (6 b-1)^2}+\frac {f \left (-12 e-\left (7 \sqrt {1-6 b}+2\right ) f\right ) \left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^{5/2}}{36 (1-6 b)}+\frac {1}{36} \left (-36 e^2-12 \left (5 \sqrt {1-6 b} f+f\right ) e-\left (-96 b+10 \sqrt {1-6 b}+17\right ) f^2\right ) \left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^{3/2}+\frac {1}{12} (1-6 b)^{3/2} \left (-36 e^2-12 \left (2 \sqrt {1-6 b} f+f\right ) e-\left (-24 b+4 \sqrt {1-6 b}+5\right ) f^2\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}\right )dx}{\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1} \left (-\frac {\left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^{5/2} \left (12 e \left (5 \sqrt {1-6 b} f+f\right )+\left (-96 b+10 \sqrt {1-6 b}+17\right ) f^2+36 e^2\right )}{540 (1-6 b)}-\frac {1}{108} \sqrt {1-6 b} \left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^{3/2} \left (12 e \left (2 \sqrt {1-6 b} f+f\right )+\left (-24 b+4 \sqrt {1-6 b}+5\right ) f^2+36 e^2\right )-\frac {f \left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^{7/2} \left (\left (7 \sqrt {1-6 b}+2\right ) f+12 e\right )}{756 (1-6 b)^2}-\frac {f^2 \left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^{9/2}}{972 (1-6 b)^3}\right )}{\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}\) |
Input:
Int[(e + f*x)^2*Sqrt[1 - (1 - 6*b)^(3/2) - 9*b + 54*b*x - 54*x^2 + 108*x^3 ],x]
Output:
(Sqrt[1 - (1 - 6*b)^(3/2) - 9*b + 54*b*x - 54*x^2 + 108*x^3]*(-1/108*(Sqrt [1 - 6*b]*(36*e^2 + (5 + 4*Sqrt[1 - 6*b] - 24*b)*f^2 + 12*e*(f + 2*Sqrt[1 - 6*b]*f))*(-((1 + 2*Sqrt[1 - 6*b])*(1 - 6*b)) + 6*(1 - 6*b)*x)^(3/2)) - ( (36*e^2 + (17 + 10*Sqrt[1 - 6*b] - 96*b)*f^2 + 12*e*(f + 5*Sqrt[1 - 6*b]*f ))*(-((1 + 2*Sqrt[1 - 6*b])*(1 - 6*b)) + 6*(1 - 6*b)*x)^(5/2))/(540*(1 - 6 *b)) - (f*(12*e + (2 + 7*Sqrt[1 - 6*b])*f)*(-((1 + 2*Sqrt[1 - 6*b])*(1 - 6 *b)) + 6*(1 - 6*b)*x)^(7/2))/(756*(1 - 6*b)^2) - (f^2*(-((1 + 2*Sqrt[1 - 6 *b])*(1 - 6*b)) + 6*(1 - 6*b)*x)^(9/2))/(972*(1 - 6*b)^3)))/(((1 - Sqrt[1 - 6*b])*(1 - 6*b) - 6*(1 - 6*b)*x)*Sqrt[-((1 + 2*Sqrt[1 - 6*b])*(1 - 6*b)) + 6*(1 - 6*b)*x])
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_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Int[((e_.) + (f_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2 + (d_.)*( x_)^3)^(p_), x_Symbol] :> Simp[(a + b*x + c*x^2 + d*x^3)^p/((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)) 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 - 2 7*a^2*d^2, 0] && !IntegerQ[p]
Time = 0.57 (sec) , antiderivative size = 350, normalized size of antiderivative = 0.71
| method | result | size |
| risch | \(\frac {\left (-2835 e^{2}-486 \sqrt {1-6 b}\, e f x +9720 b e f x +864 b x \,f^{2}+6480 \sqrt {1-6 b}\, b e f -6804 e^{2} x -170 f^{2}-7776 e f \,x^{2}+20412 e^{2} x^{2}+4860 \sqrt {1-6 b}\, x^{2} e f +20412 b \,e^{2}+11340 x^{4} f^{2}-351 f^{2} x^{2}-154 \sqrt {1-6 b}\, f^{2}+29160 e f \,x^{3}+1890 x^{3} \sqrt {1-6 b}\, f^{2}+3402 \sqrt {1-6 b}\, x \,e^{2}-567 \sqrt {1-6 b}\, e^{2}+1704 b \,f^{2}-4032 b^{2} f^{2}-810 e f +2268 b \,f^{2} x^{2}-1458 e f x -138 f^{2} x +5184 b e f -135 \sqrt {1-6 b}\, f^{2} x^{2}-1134 \sqrt {1-6 b}\, e f -186 \sqrt {1-6 b}\, x \,f^{2}-2700 f^{2} x^{3}+912 \sqrt {1-6 b}\, b \,f^{2}+1008 \sqrt {1-6 b}\, x b \,f^{2}\right ) \left (-1-2 \sqrt {1-6 b}+6 x \right ) \left (-1+6 x +\sqrt {1-6 b}\right )}{17010 \sqrt {108 x^{3}+54 b x +6 \sqrt {1-6 b}\, b -54 x^{2}-9 b -\sqrt {1-6 b}+1}}\) | \(350\) |
Input:
int((f*x+e)^2*(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3)^(1/2),x,method=_ RETURNVERBOSE)
Output:
1/17010*(-2835*e^2-486*(1-6*b)^(1/2)*e*f*x+9720*b*e*f*x+864*b*x*f^2+6480*( 1-6*b)^(1/2)*b*e*f-6804*e^2*x-170*f^2-7776*e*f*x^2+20412*e^2*x^2+4860*(1-6 *b)^(1/2)*x^2*e*f+20412*b*e^2+11340*x^4*f^2-351*f^2*x^2-154*(1-6*b)^(1/2)* f^2+29160*e*f*x^3+1890*x^3*(1-6*b)^(1/2)*f^2+3402*(1-6*b)^(1/2)*x*e^2-567* (1-6*b)^(1/2)*e^2+1704*b*f^2-4032*b^2*f^2-810*e*f+2268*b*f^2*x^2-1458*e*f* x-138*f^2*x+5184*b*e*f-135*(1-6*b)^(1/2)*f^2*x^2-1134*(1-6*b)^(1/2)*e*f-18 6*(1-6*b)^(1/2)*x*f^2-2700*f^2*x^3+912*(1-6*b)^(1/2)*b*f^2+1008*(1-6*b)^(1 /2)*x*b*f^2)*(-1-2*(1-6*b)^(1/2)+6*x)*(-1+6*x+(1-6*b)^(1/2))/(108*x^3+54*b *x+6*(1-6*b)^(1/2)*b-54*x^2-9*b-(1-6*b)^(1/2)+1)^(1/2)
Time = 0.12 (sec) , antiderivative size = 281, normalized size of antiderivative = 0.57 \[ \int (e+f x)^2 \sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx=\frac {{\left (11340 \, f^{2} x^{5} + 270 \, {\left (108 \, e f - 17 \, f^{2}\right )} x^{4} + 54 \, {\left ({\left (77 \, b - 4\right )} f^{2} + 378 \, e^{2} - 234 \, e f\right )} x^{3} - 567 \, {\left (7 \, b - 1\right )} e^{2} + 162 \, {\left (40 \, b^{2} - 19 \, b + 2\right )} e f + 2 \, {\left (792 \, b^{2} - 295 \, b + 27\right )} f^{2} + 3 \, {\left (324 \, {\left (15 \, b - 1\right )} e f + {\left (117 \, b - 19\right )} f^{2} - 3402 \, e^{2}\right )} x^{2} + 2 \, {\left (567 \, {\left (21 \, b - 2\right )} e^{2} + 81 \, {\left (19 \, b - 3\right )} e f - {\left (1512 \, b^{2} - 603 \, b + 58\right )} f^{2}\right )} x + {\left (105 \, {\left (6 \, b - 1\right )} f^{2} x^{2} - 567 \, {\left (6 \, b - 1\right )} e^{2} - 324 \, {\left (6 \, b - 1\right )} e f + 2 \, {\left (336 \, b^{2} - 218 \, b + 27\right )} f^{2} + 10 \, {\left (81 \, {\left (6 \, b - 1\right )} e f + 10 \, {\left (6 \, b - 1\right )} f^{2}\right )} x\right )} \sqrt {-6 \, b + 1}\right )} \sqrt {108 \, x^{3} + 54 \, b x - 54 \, x^{2} + {\left (6 \, b - 1\right )} \sqrt {-6 \, b + 1} - 9 \, b + 1}}{8505 \, {\left (6 \, x^{2} + b - 2 \, x\right )}} \] Input:
integrate((f*x+e)^2*(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3)^(1/2),x, a lgorithm="fricas")
Output:
1/8505*(11340*f^2*x^5 + 270*(108*e*f - 17*f^2)*x^4 + 54*((77*b - 4)*f^2 + 378*e^2 - 234*e*f)*x^3 - 567*(7*b - 1)*e^2 + 162*(40*b^2 - 19*b + 2)*e*f + 2*(792*b^2 - 295*b + 27)*f^2 + 3*(324*(15*b - 1)*e*f + (117*b - 19)*f^2 - 3402*e^2)*x^2 + 2*(567*(21*b - 2)*e^2 + 81*(19*b - 3)*e*f - (1512*b^2 - 6 03*b + 58)*f^2)*x + (105*(6*b - 1)*f^2*x^2 - 567*(6*b - 1)*e^2 - 324*(6*b - 1)*e*f + 2*(336*b^2 - 218*b + 27)*f^2 + 10*(81*(6*b - 1)*e*f + 10*(6*b - 1)*f^2)*x)*sqrt(-6*b + 1))*sqrt(108*x^3 + 54*b*x - 54*x^2 + (6*b - 1)*sqr t(-6*b + 1) - 9*b + 1)/(6*x^2 + b - 2*x)
\[ \int (e+f x)^2 \sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx=\int \left (e + f x\right )^{2} \sqrt {54 b x + 6 b \sqrt {1 - 6 b} - 9 b + 108 x^{3} - 54 x^{2} - \sqrt {1 - 6 b} + 1}\, dx \] Input:
integrate((f*x+e)**2*(1-(1-6*b)**(3/2)-9*b+54*b*x-54*x**2+108*x**3)**(1/2) ,x)
Output:
Integral((e + f*x)**2*sqrt(54*b*x + 6*b*sqrt(1 - 6*b) - 9*b + 108*x**3 - 5 4*x**2 - sqrt(1 - 6*b) + 1), x)
\[ \int (e+f x)^2 \sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx=\int { \sqrt {108 \, x^{3} + 54 \, b x - 54 \, x^{2} - {\left (-6 \, b + 1\right )}^{\frac {3}{2}} - 9 \, b + 1} {\left (f x + e\right )}^{2} \,d x } \] Input:
integrate((f*x+e)^2*(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3)^(1/2),x, a lgorithm="maxima")
Output:
integrate(sqrt(108*x^3 + 54*b*x - 54*x^2 - (-6*b + 1)^(3/2) - 9*b + 1)*(f* x + e)^2, x)
Leaf count of result is larger than twice the leaf count of optimal. 1711 vs. \(2 (416) = 832\).
Time = 0.17 (sec) , antiderivative size = 1711, normalized size of antiderivative = 3.45 \[ \int (e+f x)^2 \sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^2*(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3)^(1/2),x, a lgorithm="giac")
Output:
1/34020*sqrt(1/2)*(15120*((6*x - 2*sqrt(-6*b + 1) - 1)^(3/2) + 6*sqrt(-6*b + 1)*sqrt(6*x - 2*sqrt(-6*b + 1) - 1) + 3*sqrt(6*x - 2*sqrt(-6*b + 1) - 1 ))*b*e*f*sgn(6*x + sqrt(-6*b + 1) - 1) + 252*(3*(6*x - 2*sqrt(-6*b + 1) - 1)^(5/2) + 10*(6*x - 2*sqrt(-6*b + 1) - 1)^(3/2)*(2*sqrt(-6*b + 1) + 1) - 15*(24*b - 4*sqrt(-6*b + 1) - 5)*sqrt(6*x - 2*sqrt(-6*b + 1) - 1))*b*f^2*s gn(6*x + sqrt(-6*b + 1) - 1) - 3780*((6*x - 2*sqrt(-6*b + 1) - 1)^(3/2) + 6*sqrt(-6*b + 1)*sqrt(6*x - 2*sqrt(-6*b + 1) - 1) + 3*sqrt(6*x - 2*sqrt(-6 *b + 1) - 1))*sqrt(-6*b + 1)*e^2*sgn(6*x + sqrt(-6*b + 1) - 1) - 252*(3*(6 *x - 2*sqrt(-6*b + 1) - 1)^(5/2) + 10*(6*x - 2*sqrt(-6*b + 1) - 1)^(3/2)*( 2*sqrt(-6*b + 1) + 1) - 15*(24*b - 4*sqrt(-6*b + 1) - 5)*sqrt(6*x - 2*sqrt (-6*b + 1) - 1))*sqrt(-6*b + 1)*e*f*sgn(6*x + sqrt(-6*b + 1) - 1) + 1260*( (6*x - 2*sqrt(-6*b + 1) - 1)^(3/2) + 6*sqrt(-6*b + 1)*sqrt(6*x - 2*sqrt(-6 *b + 1) - 1) + 3*sqrt(6*x - 2*sqrt(-6*b + 1) - 1))*sqrt(-6*b + 1)*e*f*sgn( 6*x + sqrt(-6*b + 1) - 1) - 9*(5*(6*x - 2*sqrt(-6*b + 1) - 1)^(7/2) + 21*( 6*x - 2*sqrt(-6*b + 1) - 1)^(5/2)*(2*sqrt(-6*b + 1) + 1) - 35*(24*b - 4*sq rt(-6*b + 1) - 5)*(6*x - 2*sqrt(-6*b + 1) - 1)^(3/2) - 35*(2*(24*b - 7)*sq rt(-6*b + 1) + 72*b - 13)*sqrt(6*x - 2*sqrt(-6*b + 1) - 1))*sqrt(-6*b + 1) *f^2*sgn(6*x + sqrt(-6*b + 1) - 1) + 21*(3*(6*x - 2*sqrt(-6*b + 1) - 1)^(5 /2) + 10*(6*x - 2*sqrt(-6*b + 1) - 1)^(3/2)*(2*sqrt(-6*b + 1) + 1) - 15*(2 4*b - 4*sqrt(-6*b + 1) - 5)*sqrt(6*x - 2*sqrt(-6*b + 1) - 1))*sqrt(-6*b...
Timed out. \[ \int (e+f x)^2 \sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx=\int {\left (e+f\,x\right )}^2\,\sqrt {54\,b\,x-9\,b-{\left (1-6\,b\right )}^{3/2}-54\,x^2+108\,x^3+1} \,d x \] Input:
int((e + f*x)^2*(54*b*x - 9*b - (1 - 6*b)^(3/2) - 54*x^2 + 108*x^3 + 1)^(1 /2),x)
Output:
int((e + f*x)^2*(54*b*x - 9*b - (1 - 6*b)^(3/2) - 54*x^2 + 108*x^3 + 1)^(1 /2), x)
\[ \int (e+f x)^2 \sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx=\text {too large to display} \] Input:
int((f*x+e)^2*(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3)^(1/2),x)
Output:
( - 1080*sqrt(6*sqrt( - 6*b + 1)*b - sqrt( - 6*b + 1) + 54*b*x - 9*b + 108 *x**3 - 54*x**2 + 1)*sqrt( - 6*b + 1)*b*e*f + 30*sqrt(6*sqrt( - 6*b + 1)*b - sqrt( - 6*b + 1) + 54*b*x - 9*b + 108*x**3 - 54*x**2 + 1)*sqrt( - 6*b + 1)*b*f**2 + 180*sqrt(6*sqrt( - 6*b + 1)*b - sqrt( - 6*b + 1) + 54*b*x - 9 *b + 108*x**3 - 54*x**2 + 1)*sqrt( - 6*b + 1)*e*f - 5*sqrt(6*sqrt( - 6*b + 1)*b - sqrt( - 6*b + 1) + 54*b*x - 9*b + 108*x**3 - 54*x**2 + 1)*sqrt( - 6*b + 1)*f**2 + 4536*sqrt( - 6*b + 1)*int(sqrt(6*sqrt( - 6*b + 1)*b - sqrt ( - 6*b + 1) + 54*b*x - 9*b + 108*x**3 - 54*x**2 + 1)/(8*b**2 + 60*b*x**2 - 20*b*x - b + 72*x**4 - 48*x**3 + 2*x**2 + 2*x),x)*b**3*f**2 - 27216*sqrt ( - 6*b + 1)*int(sqrt(6*sqrt( - 6*b + 1)*b - sqrt( - 6*b + 1) + 54*b*x - 9 *b + 108*x**3 - 54*x**2 + 1)/(8*b**2 + 60*b*x**2 - 20*b*x - b + 72*x**4 - 48*x**3 + 2*x**2 + 2*x),x)*b**2*e**2 + 648*sqrt( - 6*b + 1)*int(sqrt(6*sqr t( - 6*b + 1)*b - sqrt( - 6*b + 1) + 54*b*x - 9*b + 108*x**3 - 54*x**2 + 1 )/(8*b**2 + 60*b*x**2 - 20*b*x - b + 72*x**4 - 48*x**3 + 2*x**2 + 2*x),x)* b**2*e*f - 270*sqrt( - 6*b + 1)*int(sqrt(6*sqrt( - 6*b + 1)*b - sqrt( - 6* b + 1) + 54*b*x - 9*b + 108*x**3 - 54*x**2 + 1)/(8*b**2 + 60*b*x**2 - 20*b *x - b + 72*x**4 - 48*x**3 + 2*x**2 + 2*x),x)*b**2*f**2 + 2268*sqrt( - 6*b + 1)*int(sqrt(6*sqrt( - 6*b + 1)*b - sqrt( - 6*b + 1) + 54*b*x - 9*b + 10 8*x**3 - 54*x**2 + 1)/(8*b**2 + 60*b*x**2 - 20*b*x - b + 72*x**4 - 48*x**3 + 2*x**2 + 2*x),x)*b*e**2 - 1404*sqrt( - 6*b + 1)*int(sqrt(6*sqrt( - 6...