Integrand size = 40, antiderivative size = 312 \[ \int (e+f x) \sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx=-\frac {\sqrt {1-6 b} \left (6 e+f+2 \sqrt {1-6 b} f\right ) \left (2+\frac {1-6 x}{\sqrt {1-6 b}}\right ) \sqrt {-2 (1-6 b)^{3/2}+3 (1-6 b) (1-6 x)-(1-6 x)^3}}{18 \sqrt {2} \left (1-\frac {1-6 x}{\sqrt {1-6 b}}\right )}+\frac {\sqrt {1-6 b} \left (6 e+f+5 \sqrt {1-6 b} f\right ) \left (2+\frac {1-6 x}{\sqrt {1-6 b}}\right )^2 \sqrt {-2 (1-6 b)^{3/2}+3 (1-6 b) (1-6 x)-(1-6 x)^3}}{90 \sqrt {2} \left (1-\frac {1-6 x}{\sqrt {1-6 b}}\right )}-\frac {(1-6 b) f \left (2+\frac {1-6 x}{\sqrt {1-6 b}}\right )^3 \sqrt {-2 (1-6 b)^{3/2}+3 (1-6 b) (1-6 x)-(1-6 x)^3}}{126 \sqrt {2} \left (1-\frac {1-6 x}{\sqrt {1-6 b}}\right )} \] Output:
-1/36*(1-6*b)^(1/2)*(6*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))+1/180*(1-6*b)^(1/2)*(6*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))-1/252*(1-6*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))
Result contains higher order function than in optimal. Order 9 vs. order 2 in optimal.
Time = 16.74 (sec) , antiderivative size = 9146, normalized size of antiderivative = 29.31 \[ \int (e+f x) \sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx=\text {Result too large to show} \] Input:
Integrate[(e + f*x)*Sqrt[1 - (1 - 6*b)^(3/2) - 9*b + 54*b*x - 54*x^2 + 108 *x^3],x]
Output:
Result too large to show
Time = 0.86 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.86, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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.
| \(\displaystyle \int \sqrt {54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1} (e+f x) \, dx\) |
| \(\Big \downarrow \) 2489 |
| \(\displaystyle -\frac {\sqrt {54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1} \int -157464 \sqrt {2} \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} (e+f x)dx}{157464 \sqrt {2} \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1} \int \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} (e+f x)dx}{\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \frac {\sqrt {54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1} \int \left (\frac {f \left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^{5/2}}{6 (6 b-1)}+\frac {1}{6} \left (-6 e-5 \sqrt {1-6 b} f-f\right ) \left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^{3/2}+\frac {1}{2} \sqrt {1-6 b} (6 b-1) \left (6 e+2 \sqrt {1-6 b} f+f\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}\right )dx}{\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1} \left (-\frac {\left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^{5/2} \left (5 \sqrt {1-6 b} f+6 e+f\right )}{90 (1-6 b)}-\frac {1}{18} \sqrt {1-6 b} \left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^{3/2} \left (2 \sqrt {1-6 b} f+6 e+f\right )-\frac {f \left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^{7/2}}{126 (1-6 b)^2}\right )}{\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}\) |
Input:
Int[(e + f*x)*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]*(-1/18*(Sqrt[ 1 - 6*b]*(6*e + f + 2*Sqrt[1 - 6*b]*f)*(-((1 + 2*Sqrt[1 - 6*b])*(1 - 6*b)) + 6*(1 - 6*b)*x)^(3/2)) - ((6*e + f + 5*Sqrt[1 - 6*b]*f)*(-((1 + 2*Sqrt[1 - 6*b])*(1 - 6*b)) + 6*(1 - 6*b)*x)^(5/2))/(90*(1 - 6*b)) - (f*(-((1 + 2* Sqrt[1 - 6*b])*(1 - 6*b)) + 6*(1 - 6*b)*x)^(7/2))/(126*(1 - 6*b)^2)))/(((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 = 0.41 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.37
| method | result | size |
| gosper | \(-\frac {\sqrt {1-\left (1-6 b \right )^{\frac {3}{2}}-9 b +54 b x -54 x^{2}+108 x^{3}}\, \left (15 \sqrt {1-6 b}\, f x +30 f \,x^{2}+21 \sqrt {1-6 b}\, e +\sqrt {1-6 b}\, f -20 b f +42 e x -3 f x -7 e +3 f \right ) \left (1+2 \sqrt {1-6 b}-6 x \right )}{105 \left (-1+6 x +\sqrt {1-6 b}\right )}\) | \(116\) |
| risch | \(\frac {\left (30 \sqrt {1-6 b}\, f \,x^{2}+180 f \,x^{3}+40 \sqrt {1-6 b}\, f b +42 \sqrt {1-6 b}\, e x -3 \sqrt {1-6 b}\, f x +60 b f x +252 e \,x^{2}-48 f \,x^{2}-7 \sqrt {1-6 b}\, e -7 \sqrt {1-6 b}\, f +252 e b +32 b f -84 e x -9 f x -35 e -5 f \right ) \left (-1-2 \sqrt {1-6 b}+6 x \right ) \left (-1+6 x +\sqrt {1-6 b}\right )}{210 \sqrt {108 x^{3}+54 b x +6 \sqrt {1-6 b}\, b -54 x^{2}-9 b -\sqrt {1-6 b}+1}}\) | \(181\) |
Input:
int((f*x+e)*(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3)^(1/2),x,method=_RE TURNVERBOSE)
Output:
-1/105*(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3)^(1/2)*(15*(1-6*b)^(1/2) *f*x+30*f*x^2+21*(1-6*b)^(1/2)*e+(1-6*b)^(1/2)*f-20*b*f+42*e*x-3*f*x-7*e+3 *f)*(1+2*(1-6*b)^(1/2)-6*x)/(-1+6*x+(1-6*b)^(1/2))
Time = 0.11 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.50 \[ \int (e+f x) \sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx=\frac {{\left (180 \, f x^{4} + 6 \, {\left (42 \, e - 13 \, f\right )} x^{3} + 6 \, {\left ({\left (15 \, b - 1\right )} f - 21 \, e\right )} x^{2} - 7 \, {\left (7 \, b - 1\right )} e + {\left (40 \, b^{2} - 19 \, b + 2\right )} f + {\left (14 \, {\left (21 \, b - 2\right )} e + {\left (19 \, b - 3\right )} f\right )} x + {\left (5 \, {\left (6 \, b - 1\right )} f x - 7 \, {\left (6 \, b - 1\right )} e - 2 \, {\left (6 \, b - 1\right )} f\right )} \sqrt {-6 \, b + 1}\right )} \sqrt {108 \, x^{3} + 54 \, b x - 54 \, x^{2} + {\left (6 \, b - 1\right )} \sqrt {-6 \, b + 1} - 9 \, b + 1}}{105 \, {\left (6 \, x^{2} + b - 2 \, x\right )}} \] Input:
integrate((f*x+e)*(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3)^(1/2),x, alg orithm="fricas")
Output:
1/105*(180*f*x^4 + 6*(42*e - 13*f)*x^3 + 6*((15*b - 1)*f - 21*e)*x^2 - 7*( 7*b - 1)*e + (40*b^2 - 19*b + 2)*f + (14*(21*b - 2)*e + (19*b - 3)*f)*x + (5*(6*b - 1)*f*x - 7*(6*b - 1)*e - 2*(6*b - 1)*f)*sqrt(-6*b + 1))*sqrt(108 *x^3 + 54*b*x - 54*x^2 + (6*b - 1)*sqrt(-6*b + 1) - 9*b + 1)/(6*x^2 + b - 2*x)
\[ \int (e+f x) \sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx=\int \left (e + f x\right ) \sqrt {54 b x + 6 b \sqrt {1 - 6 b} - 9 b + 108 x^{3} - 54 x^{2} - \sqrt {1 - 6 b} + 1}\, dx \] Input:
integrate((f*x+e)*(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)*sqrt(54*b*x + 6*b*sqrt(1 - 6*b) - 9*b + 108*x**3 - 54*x **2 - sqrt(1 - 6*b) + 1), x)
\[ \int (e+f x) \sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx=\int { \sqrt {108 \, x^{3} + 54 \, b x - 54 \, x^{2} - {\left (-6 \, b + 1\right )}^{\frac {3}{2}} - 9 \, b + 1} {\left (f x + e\right )} \,d x } \] Input:
integrate((f*x+e)*(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3)^(1/2),x, alg orithm="maxima")
Output:
integrate(sqrt(108*x^3 + 54*b*x - 54*x^2 - (-6*b + 1)^(3/2) - 9*b + 1)*(f* x + e), x)
Leaf count of result is larger than twice the leaf count of optimal. 933 vs. \(2 (259) = 518\).
Time = 0.13 (sec) , antiderivative size = 933, normalized size of antiderivative = 2.99 \[ \int (e+f x) \sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)*(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3)^(1/2),x, alg orithm="giac")
Output:
1/1890*sqrt(1/2)*(420*((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*f*sgn(6*x + sqrt(-6*b + 1) - 1) - 210*((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*sqr t(-6*b + 1) - 1))*sqrt(-6*b + 1)*e*sgn(6*x + sqrt(-6*b + 1) - 1) - 7*(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)*f*sgn(6*x + sqrt(-6*b + 1) - 1) + 35*((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)*f*sgn(6*x + sqrt(-6*b + 1) - 1) + 7560*b*e*sqrt(6*x - 2*sqrt(-6*b + 1) - 1)*sgn(6*x + sqrt(-6*b + 1) - 1) + 42*(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))*e*sgn(6*x + sqrt(-6*b + 1) - 1) - 420*((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))*e*sgn(6*x + sqrt(-6*b + 1) - 1) + 3*(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*sqrt(-6 *b + 1) - 5)*(6*x - 2*sqrt(-6*b + 1) - 1)^(3/2) - 35*(2*(24*b - 7)*sqrt(-6 *b + 1) + 72*b - 13)*sqrt(6*x - 2*sqrt(-6*b + 1) - 1))*f*sgn(6*x + sqrt...
Timed out. \[ \int (e+f x) \sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx=\int \left (e+f\,x\right )\,\sqrt {54\,b\,x-9\,b-{\left (1-6\,b\right )}^{3/2}-54\,x^2+108\,x^3+1} \,d x \] Input:
int((e + f*x)*(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)*(54*b*x - 9*b - (1 - 6*b)^(3/2) - 54*x^2 + 108*x^3 + 1)^(1/2 ), x)
\[ \int (e+f x) \sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx=\text {too large to display} \] Input:
int((f*x+e)*(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3)^(1/2),x)
Output:
( - 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 + 5*sqrt(6*sqrt( - 6*b + 1)*b - sq rt( - 6*b + 1) + 54*b*x - 9*b + 108*x**3 - 54*x**2 + 1)*sqrt( - 6*b + 1)*f - 1512*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 + 18*sqrt( - 6*b + 1)*int(sq rt(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 + 126*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*e - 39*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*f + 21*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)*e + 6*sqrt( - 6* b + 1)*int(sqrt(6*sqrt( - 6*b + 1)*b - sqrt( - 6*b + 1) + 54*b*x - 9*b + 1 08*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)*f + 3240*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)*x**3)/(...