Integrand size = 36, antiderivative size = 108 \[ \int (e+f x) \left (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3\right ) \, dx=\left (1-\sqrt {1-6 b}-\left (9-6 \sqrt {1-6 b}\right ) b\right ) e x+\frac {1}{2} \left (f-\sqrt {1-6 b} f+b \left (54 e-9 f+6 \sqrt {1-6 b} f\right )\right ) x^2-18 (e-b f) x^3+\frac {27}{2} (2 e-f) x^4+\frac {108 f x^5}{5} \] Output:
(1-(1-6*b)^(1/2)-(9-6*(1-6*b)^(1/2))*b)*e*x+1/2*(f-(1-6*b)^(1/2)*f+b*(54*e -9*f+6*(1-6*b)^(1/2)*f))*x^2-18*(-b*f+e)*x^3+27/2*(2*e-f)*x^4+108/5*f*x^5
Time = 0.04 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.89 \[ \int (e+f x) \left (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3\right ) \, dx=e x \left (1-\sqrt {1-6 b}-18 x^2+27 x^3+3 b \left (-3+2 \sqrt {1-6 b}+9 x\right )\right )+\frac {1}{10} f x^2 \left (5-5 \sqrt {1-6 b}-135 x^2+216 x^3+15 b \left (-3+2 \sqrt {1-6 b}+12 x\right )\right ) \] Input:
Integrate[(e + f*x)*(1 - (1 - 6*b)^(3/2) - 9*b + 54*b*x - 54*x^2 + 108*x^3 ),x]
Output:
e*x*(1 - Sqrt[1 - 6*b] - 18*x^2 + 27*x^3 + 3*b*(-3 + 2*Sqrt[1 - 6*b] + 9*x )) + (f*x^2*(5 - 5*Sqrt[1 - 6*b] - 135*x^2 + 216*x^3 + 15*b*(-3 + 2*Sqrt[1 - 6*b] + 12*x)))/10
Time = 0.57 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.76, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2389, 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.
\(\displaystyle \int \left (54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1\right ) (e+f x) \, dx\) |
\(\Big \downarrow \) 2389 |
\(\displaystyle \int \left (-54 x^2 (e-b f)+x \left (54 b e+(1-6 b)^{3/2} (-f)-9 b f+f\right )-\left ((1-6 b)^{3/2}+9 b-1\right ) e+54 x^3 (2 e-f)+108 f x^4\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -18 x^3 (e-b f)+\frac {1}{2} x^2 \left (54 b e+(1-6 b)^{3/2} (-f)-9 b f+f\right )+\left (-(1-6 b)^{3/2}-9 b+1\right ) e x+\frac {27}{2} x^4 (2 e-f)+\frac {108 f x^5}{5}\) |
Input:
Int[(e + f*x)*(1 - (1 - 6*b)^(3/2) - 9*b + 54*b*x - 54*x^2 + 108*x^3),x]
Output:
(1 - (1 - 6*b)^(3/2) - 9*b)*e*x + ((54*b*e + f - (1 - 6*b)^(3/2)*f - 9*b*f )*x^2)/2 - 18*(e - b*f)*x^3 + (27*(2*e - f)*x^4)/2 + (108*f*x^5)/5
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand [Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p , 0] || EqQ[n, 1])
Time = 0.30 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.70
method | result | size |
default | \(\frac {108 f \,x^{5}}{5}+\frac {\left (108 e -54 f \right ) x^{4}}{4}+\frac {\left (54 b f -54 e \right ) x^{3}}{3}+\frac {\left (54 e b +f \left (1-\left (1-6 b \right )^{\frac {3}{2}}-9 b \right )\right ) x^{2}}{2}+x e \left (1-\left (1-6 b \right )^{\frac {3}{2}}-9 b \right )\) | \(76\) |
norman | \(\left (27 e -\frac {27 f}{2}\right ) x^{4}+\left (3 \sqrt {1-6 b}\, f b +27 e b -\frac {\sqrt {1-6 b}\, f}{2}-\frac {9 b f}{2}+\frac {f}{2}\right ) x^{2}+\left (18 b f -18 e \right ) x^{3}+e \left (6 \sqrt {1-6 b}\, b -\sqrt {1-6 b}-9 b +1\right ) x +\frac {108 f \,x^{5}}{5}\) | \(95\) |
parallelrisch | \(\frac {108 f \,x^{5}}{5}+27 x^{4} e -\frac {27 f \,x^{4}}{2}+18 x^{3} b f -18 e \,x^{3}+3 x^{2} \sqrt {1-6 b}\, f b +27 x^{2} e b -\frac {\sqrt {1-6 b}\, f \,x^{2}}{2}-\frac {9 b f \,x^{2}}{2}+\frac {f \,x^{2}}{2}+6 \sqrt {1-6 b}\, x b e -\sqrt {1-6 b}\, e x -9 x e b +e x\) | \(111\) |
gosper | \(-\frac {\left (\left (1-6 b \right )^{\frac {3}{2}}-108 x^{3}-54 b x +54 x^{2}+9 b -1\right ) x \left (216 f \,x^{4}+30 \sqrt {1-6 b}\, b f x +180 b f \,x^{2}+270 e \,x^{3}-135 f \,x^{3}+60 \sqrt {1-6 b}\, b e -5 \sqrt {1-6 b}\, f x +270 x e b -45 b f x -180 e \,x^{2}-10 \sqrt {1-6 b}\, e -90 e b +5 f x +10 e \right )}{10 \left (108 x^{3}+54 b x +6 \sqrt {1-6 b}\, b -54 x^{2}-9 b -\sqrt {1-6 b}+1\right )}\) | \(167\) |
Input:
int((f*x+e)*(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3),x,method=_RETURNVE RBOSE)
Output:
108/5*f*x^5+1/4*(108*e-54*f)*x^4+1/3*(54*b*f-54*e)*x^3+1/2*(54*e*b+f*(1-(1 -6*b)^(3/2)-9*b))*x^2+x*e*(1-(1-6*b)^(3/2)-9*b)
Time = 0.07 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.81 \[ \int (e+f x) \left (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3\right ) \, dx=\frac {108}{5} \, f x^{5} + \frac {27}{2} \, {\left (2 \, e - f\right )} x^{4} + 18 \, {\left (b f - e\right )} x^{3} - {\left (9 \, b - 1\right )} e x + \frac {1}{2} \, {\left (54 \, b e - {\left (9 \, b - 1\right )} f\right )} x^{2} + \frac {1}{2} \, {\left ({\left (6 \, b - 1\right )} f x^{2} + 2 \, {\left (6 \, b - 1\right )} e x\right )} \sqrt {-6 \, b + 1} \] Input:
integrate((f*x+e)*(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3),x, algorithm ="fricas")
Output:
108/5*f*x^5 + 27/2*(2*e - f)*x^4 + 18*(b*f - e)*x^3 - (9*b - 1)*e*x + 1/2* (54*b*e - (9*b - 1)*f)*x^2 + 1/2*((6*b - 1)*f*x^2 + 2*(6*b - 1)*e*x)*sqrt( -6*b + 1)
Time = 0.05 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.99 \[ \int (e+f x) \left (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3\right ) \, dx=\frac {108 f x^{5}}{5} + x^{4} \cdot \left (27 e - \frac {27 f}{2}\right ) + x^{3} \cdot \left (18 b f - 18 e\right ) + x^{2} \cdot \left (27 b e + 3 b f \sqrt {1 - 6 b} - \frac {9 b f}{2} - \frac {f \sqrt {1 - 6 b}}{2} + \frac {f}{2}\right ) + x \left (6 b e \sqrt {1 - 6 b} - 9 b e - e \sqrt {1 - 6 b} + e\right ) \] Input:
integrate((f*x+e)*(1-(1-6*b)**(3/2)-9*b+54*b*x-54*x**2+108*x**3),x)
Output:
108*f*x**5/5 + x**4*(27*e - 27*f/2) + x**3*(18*b*f - 18*e) + x**2*(27*b*e + 3*b*f*sqrt(1 - 6*b) - 9*b*f/2 - f*sqrt(1 - 6*b)/2 + f/2) + x*(6*b*e*sqrt (1 - 6*b) - 9*b*e - e*sqrt(1 - 6*b) + e)
Time = 0.03 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.67 \[ \int (e+f x) \left (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3\right ) \, dx=\frac {108}{5} \, f x^{5} + \frac {27}{2} \, {\left (2 \, e - f\right )} x^{4} + 18 \, {\left (b f - e\right )} x^{3} - {\left ({\left (-6 \, b + 1\right )}^{\frac {3}{2}} + 9 \, b - 1\right )} e x + \frac {1}{2} \, {\left (54 \, b e - {\left ({\left (-6 \, b + 1\right )}^{\frac {3}{2}} + 9 \, b - 1\right )} f\right )} x^{2} \] Input:
integrate((f*x+e)*(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3),x, algorithm ="maxima")
Output:
108/5*f*x^5 + 27/2*(2*e - f)*x^4 + 18*(b*f - e)*x^3 - ((-6*b + 1)^(3/2) + 9*b - 1)*e*x + 1/2*(54*b*e - ((-6*b + 1)^(3/2) + 9*b - 1)*f)*x^2
Time = 0.11 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.83 \[ \int (e+f x) \left (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3\right ) \, dx=\frac {108}{5} \, f x^{5} + 18 \, b f x^{3} + 27 \, e x^{4} - \frac {27}{2} \, f x^{4} + 27 \, b e x^{2} + \frac {1}{2} \, {\left (6 \, b \sqrt {-6 \, b + 1} - 9 \, b - \sqrt {-6 \, b + 1} + 1\right )} f x^{2} - 18 \, e x^{3} + {\left ({\left (6 \, b - 1\right )} \sqrt {-6 \, b + 1} - 9 \, b + 1\right )} e x \] Input:
integrate((f*x+e)*(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3),x, algorithm ="giac")
Output:
108/5*f*x^5 + 18*b*f*x^3 + 27*e*x^4 - 27/2*f*x^4 + 27*b*e*x^2 + 1/2*(6*b*s qrt(-6*b + 1) - 9*b - sqrt(-6*b + 1) + 1)*f*x^2 - 18*e*x^3 + ((6*b - 1)*sq rt(-6*b + 1) - 9*b + 1)*e*x
Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.68 \[ \int (e+f x) \left (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3\right ) \, dx=\frac {108\,f\,x^5}{5}+\left (27\,e-\frac {27\,f}{2}\right )\,x^4+\left (18\,b\,f-18\,e\right )\,x^3+\left (\frac {f}{2}+27\,b\,e-\frac {9\,b\,f}{2}-\frac {f\,{\left (1-6\,b\right )}^{3/2}}{2}\right )\,x^2-e\,\left (9\,b+{\left (1-6\,b\right )}^{3/2}-1\right )\,x \] Input:
int(-(e + f*x)*(9*b - 54*b*x + (1 - 6*b)^(3/2) + 54*x^2 - 108*x^3 - 1),x)
Output:
x^4*(27*e - (27*f)/2) - x^3*(18*e - 18*b*f) + (108*f*x^5)/5 + x^2*(f/2 + 2 7*b*e - (9*b*f)/2 - (f*(1 - 6*b)^(3/2))/2) - e*x*(9*b + (1 - 6*b)^(3/2) - 1)
Time = 0.17 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.89 \[ \int (e+f x) \left (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3\right ) \, dx=\frac {x \left (60 \sqrt {-6 b +1}\, b e +30 \sqrt {-6 b +1}\, b f x -10 \sqrt {-6 b +1}\, e -5 \sqrt {-6 b +1}\, f x +270 b e x -90 b e +180 b f \,x^{2}-45 b f x +270 e \,x^{3}-180 e \,x^{2}+10 e +216 f \,x^{4}-135 f \,x^{3}+5 f x \right )}{10} \] Input:
int((f*x+e)*(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3),x)
Output:
(x*(60*sqrt( - 6*b + 1)*b*e + 30*sqrt( - 6*b + 1)*b*f*x - 10*sqrt( - 6*b + 1)*e - 5*sqrt( - 6*b + 1)*f*x + 270*b*e*x - 90*b*e + 180*b*f*x**2 - 45*b* f*x + 270*e*x**3 - 180*e*x**2 + 10*e + 216*f*x**4 - 135*f*x**3 + 5*f*x))/1 0