Integrand size = 33, antiderivative size = 107 \[ \int \sqrt {2+3 x} (f+g x)^2 \sqrt {1+\frac {5 x}{6}-x^2} \, dx=-\frac {13 (2 f+3 g)^2 (3-2 x)^{3/2}}{24 \sqrt {6}}+\frac {(2 f+3 g) (6 f+35 g) (3-2 x)^{5/2}}{40 \sqrt {6}}-\frac {g (12 f+31 g) (3-2 x)^{7/2}}{56 \sqrt {6}}+\frac {g^2 (3-2 x)^{9/2}}{24 \sqrt {6}} \] Output:
-13/144*(2*f+3*g)^2*(3-2*x)^(3/2)*6^(1/2)+1/240*(2*f+3*g)*(6*f+35*g)*(3-2* x)^(5/2)*6^(1/2)-1/336*g*(12*f+31*g)*(3-2*x)^(7/2)*6^(1/2)+1/144*g^2*(3-2* x)^(9/2)*6^(1/2)
Time = 1.02 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.77 \[ \int \sqrt {2+3 x} (f+g x)^2 \sqrt {1+\frac {5 x}{6}-x^2} \, dx=\frac {(-3+2 x) \sqrt {6+5 x-6 x^2} \left (7 f^2 (19+9 x)+6 f g \left (32+32 x+15 x^2\right )+5 g^2 \left (18+18 x+15 x^2+7 x^3\right )\right )}{105 \sqrt {6} \sqrt {2+3 x}} \] Input:
Integrate[Sqrt[2 + 3*x]*(f + g*x)^2*Sqrt[1 + (5*x)/6 - x^2],x]
Output:
((-3 + 2*x)*Sqrt[6 + 5*x - 6*x^2]*(7*f^2*(19 + 9*x) + 6*f*g*(32 + 32*x + 1 5*x^2) + 5*g^2*(18 + 18*x + 15*x^2 + 7*x^3)))/(105*Sqrt[6]*Sqrt[2 + 3*x])
Time = 0.27 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1245, 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.
\(\displaystyle \int \sqrt {3 x+2} \sqrt {-x^2+\frac {5 x}{6}+1} (f+g x)^2 \, dx\) |
\(\Big \downarrow \) 1245 |
\(\displaystyle \int \sqrt {\frac {1}{2}-\frac {x}{3}} (3 x+2) (f+g x)^2dx\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \int \left (-\frac {3}{4} \left (\frac {1}{2}-\frac {x}{3}\right )^{3/2} \left (12 f^2+88 f g+105 g^2\right )+\frac {9}{2} g \left (\frac {1}{2}-\frac {x}{3}\right )^{5/2} (12 f+31 g)+\frac {13}{8} \sqrt {\frac {1}{2}-\frac {x}{3}} (2 f+3 g)^2-81 g^2 \left (\frac {1}{2}-\frac {x}{3}\right )^{7/2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {g (3-2 x)^{7/2} (12 f+31 g)}{56 \sqrt {6}}+\frac {(3-2 x)^{5/2} (2 f+3 g) (6 f+35 g)}{40 \sqrt {6}}-\frac {13 (3-2 x)^{3/2} (2 f+3 g)^2}{24 \sqrt {6}}+\frac {g^2 (3-2 x)^{9/2}}{24 \sqrt {6}}\) |
Input:
Int[Sqrt[2 + 3*x]*(f + g*x)^2*Sqrt[1 + (5*x)/6 - x^2],x]
Output:
(-13*(2*f + 3*g)^2*(3 - 2*x)^(3/2))/(24*Sqrt[6]) + ((2*f + 3*g)*(6*f + 35* g)*(3 - 2*x)^(5/2))/(40*Sqrt[6]) - (g*(12*f + 31*g)*(3 - 2*x)^(7/2))/(56*S qrt[6]) + (g^2*(3 - 2*x)^(9/2))/(24*Sqrt[6])
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[((d_) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Int[(d + e*x)^(m + p)*(f + g*x)^n*(a/d + (c/e)*x)^p, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[a, 0] && GtQ[d, 0] && LtQ[c, 0]
Time = 1.19 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.77
method | result | size |
gosper | \(\frac {\left (2 x -3\right ) \left (35 g^{2} x^{3}+90 f g \,x^{2}+75 g^{2} x^{2}+63 f^{2} x +192 f g x +90 g^{2} x +133 f^{2}+192 f g +90 g^{2}\right ) \sqrt {-36 x^{2}+30 x +36}}{630 \sqrt {3 x +2}}\) | \(82\) |
default | \(\frac {\left (2 x -3\right ) \left (35 g^{2} x^{3}+90 f g \,x^{2}+75 g^{2} x^{2}+63 f^{2} x +192 f g x +90 g^{2} x +133 f^{2}+192 f g +90 g^{2}\right ) \sqrt {-36 x^{2}+30 x +36}}{630 \sqrt {3 x +2}}\) | \(82\) |
orering | \(\frac {\left (2 x -3\right ) \left (35 g^{2} x^{3}+90 f g \,x^{2}+75 g^{2} x^{2}+63 f^{2} x +192 f g x +90 g^{2} x +133 f^{2}+192 f g +90 g^{2}\right ) \sqrt {-36 x^{2}+30 x +36}}{630 \sqrt {3 x +2}}\) | \(82\) |
risch | \(-\frac {\sqrt {\frac {-36 x^{2}+30 x +36}{3 x +2}}\, \sqrt {3 x +2}\, \left (70 g^{2} x^{4}+180 f g \,x^{3}+45 g^{2} x^{3}+126 f^{2} x^{2}+114 f g \,x^{2}-45 g^{2} x^{2}+77 f^{2} x -192 f g x -90 g^{2} x -399 f^{2}-576 f g -270 g^{2}\right ) \left (2 x -3\right )}{105 \sqrt {-36 x^{2}+30 x +36}\, \sqrt {-12 x +18}}\) | \(132\) |
Input:
int(1/6*(3*x+2)^(1/2)*(g*x+f)^2*(-36*x^2+30*x+36)^(1/2),x,method=_RETURNVE RBOSE)
Output:
1/630*(2*x-3)*(35*g^2*x^3+90*f*g*x^2+75*g^2*x^2+63*f^2*x+192*f*g*x+90*g^2* x+133*f^2+192*f*g+90*g^2)*(-36*x^2+30*x+36)^(1/2)/(3*x+2)^(1/2)
Time = 0.07 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.88 \[ \int \sqrt {2+3 x} (f+g x)^2 \sqrt {1+\frac {5 x}{6}-x^2} \, dx=\frac {{\left (70 \, g^{2} x^{4} + 45 \, {\left (4 \, f g + g^{2}\right )} x^{3} + 3 \, {\left (42 \, f^{2} + 38 \, f g - 15 \, g^{2}\right )} x^{2} - 399 \, f^{2} - 576 \, f g - 270 \, g^{2} + {\left (77 \, f^{2} - 192 \, f g - 90 \, g^{2}\right )} x\right )} \sqrt {-36 \, x^{2} + 30 \, x + 36}}{630 \, \sqrt {3 \, x + 2}} \] Input:
integrate(1/6*(2+3*x)^(1/2)*(g*x+f)^2*(-36*x^2+30*x+36)^(1/2),x, algorithm ="fricas")
Output:
1/630*(70*g^2*x^4 + 45*(4*f*g + g^2)*x^3 + 3*(42*f^2 + 38*f*g - 15*g^2)*x^ 2 - 399*f^2 - 576*f*g - 270*g^2 + (77*f^2 - 192*f*g - 90*g^2)*x)*sqrt(-36* x^2 + 30*x + 36)/sqrt(3*x + 2)
\[ \int \sqrt {2+3 x} (f+g x)^2 \sqrt {1+\frac {5 x}{6}-x^2} \, dx=\frac {\sqrt {6} \left (\int f^{2} \sqrt {3 x + 2} \sqrt {- 6 x^{2} + 5 x + 6}\, dx + \int g^{2} x^{2} \sqrt {3 x + 2} \sqrt {- 6 x^{2} + 5 x + 6}\, dx + \int 2 f g x \sqrt {3 x + 2} \sqrt {- 6 x^{2} + 5 x + 6}\, dx\right )}{6} \] Input:
integrate(1/6*(2+3*x)**(1/2)*(g*x+f)**2*(-36*x**2+30*x+36)**(1/2),x)
Output:
sqrt(6)*(Integral(f**2*sqrt(3*x + 2)*sqrt(-6*x**2 + 5*x + 6), x) + Integra l(g**2*x**2*sqrt(3*x + 2)*sqrt(-6*x**2 + 5*x + 6), x) + Integral(2*f*g*x*s qrt(3*x + 2)*sqrt(-6*x**2 + 5*x + 6), x))/6
Result contains complex when optimal does not.
Time = 0.16 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.46 \[ \int \sqrt {2+3 x} (f+g x)^2 \sqrt {1+\frac {5 x}{6}-x^2} \, dx=\frac {1}{90} \, {\left (18 i \, \sqrt {3} \sqrt {2} x^{2} + 11 i \, \sqrt {3} \sqrt {2} x - 57 i \, \sqrt {3} \sqrt {2}\right )} f^{2} \sqrt {2 \, x - 3} + \frac {1}{105} \, {\left (30 i \, \sqrt {3} \sqrt {2} x^{3} + 19 i \, \sqrt {3} \sqrt {2} x^{2} - 32 i \, \sqrt {3} \sqrt {2} x - 96 i \, \sqrt {3} \sqrt {2}\right )} f g \sqrt {2 \, x - 3} + \frac {1}{126} \, {\left (14 i \, \sqrt {3} \sqrt {2} x^{4} + 9 i \, \sqrt {3} \sqrt {2} x^{3} - 9 i \, \sqrt {3} \sqrt {2} x^{2} - 18 i \, \sqrt {3} \sqrt {2} x - 54 i \, \sqrt {3} \sqrt {2}\right )} g^{2} \sqrt {2 \, x - 3} \] Input:
integrate(1/6*(2+3*x)^(1/2)*(g*x+f)^2*(-36*x^2+30*x+36)^(1/2),x, algorithm ="maxima")
Output:
1/90*(18*I*sqrt(3)*sqrt(2)*x^2 + 11*I*sqrt(3)*sqrt(2)*x - 57*I*sqrt(3)*sqr t(2))*f^2*sqrt(2*x - 3) + 1/105*(30*I*sqrt(3)*sqrt(2)*x^3 + 19*I*sqrt(3)*s qrt(2)*x^2 - 32*I*sqrt(3)*sqrt(2)*x - 96*I*sqrt(3)*sqrt(2))*f*g*sqrt(2*x - 3) + 1/126*(14*I*sqrt(3)*sqrt(2)*x^4 + 9*I*sqrt(3)*sqrt(2)*x^3 - 9*I*sqrt (3)*sqrt(2)*x^2 - 18*I*sqrt(3)*sqrt(2)*x - 54*I*sqrt(3)*sqrt(2))*g^2*sqrt( 2*x - 3)
Leaf count of result is larger than twice the leaf count of optimal. 359 vs. \(2 (83) = 166\).
Time = 0.27 (sec) , antiderivative size = 359, normalized size of antiderivative = 3.36 \[ \int \sqrt {2+3 x} (f+g x)^2 \sqrt {1+\frac {5 x}{6}-x^2} \, dx=\frac {1}{5040} \, \sqrt {6} {\left (252 \, {\left ({\left (2 \, x - 3\right )}^{2} \sqrt {-2 \, x + 3} - 10 \, {\left (-2 \, x + 3\right )}^{\frac {3}{2}} + 45 \, \sqrt {-2 \, x + 3}\right )} f^{2} + 700 \, {\left ({\left (-2 \, x + 3\right )}^{\frac {3}{2}} - 9 \, \sqrt {-2 \, x + 3}\right )} f^{2} + 36 \, {\left (5 \, {\left (2 \, x - 3\right )}^{3} \sqrt {-2 \, x + 3} + 63 \, {\left (2 \, x - 3\right )}^{2} \sqrt {-2 \, x + 3} - 315 \, {\left (-2 \, x + 3\right )}^{\frac {3}{2}} + 945 \, \sqrt {-2 \, x + 3}\right )} f g - 420 \, {\left ({\left (2 \, x - 3\right )}^{2} \sqrt {-2 \, x + 3} - 10 \, {\left (-2 \, x + 3\right )}^{\frac {3}{2}} + 45 \, \sqrt {-2 \, x + 3}\right )} f g + 1680 \, {\left ({\left (-2 \, x + 3\right )}^{\frac {3}{2}} - 9 \, \sqrt {-2 \, x + 3}\right )} f g + {\left (35 \, {\left (2 \, x - 3\right )}^{4} \sqrt {-2 \, x + 3} + 540 \, {\left (2 \, x - 3\right )}^{3} \sqrt {-2 \, x + 3} + 3402 \, {\left (2 \, x - 3\right )}^{2} \sqrt {-2 \, x + 3} - 11340 \, {\left (-2 \, x + 3\right )}^{\frac {3}{2}} + 25515 \, \sqrt {-2 \, x + 3}\right )} g^{2} - 15 \, {\left (5 \, {\left (2 \, x - 3\right )}^{3} \sqrt {-2 \, x + 3} + 63 \, {\left (2 \, x - 3\right )}^{2} \sqrt {-2 \, x + 3} - 315 \, {\left (-2 \, x + 3\right )}^{\frac {3}{2}} + 945 \, \sqrt {-2 \, x + 3}\right )} g^{2} - 252 \, {\left ({\left (2 \, x - 3\right )}^{2} \sqrt {-2 \, x + 3} - 10 \, {\left (-2 \, x + 3\right )}^{\frac {3}{2}} + 45 \, \sqrt {-2 \, x + 3}\right )} g^{2} - 5040 \, f^{2} \sqrt {-2 \, x + 3}\right )} \] Input:
integrate(1/6*(2+3*x)^(1/2)*(g*x+f)^2*(-36*x^2+30*x+36)^(1/2),x, algorithm ="giac")
Output:
1/5040*sqrt(6)*(252*((2*x - 3)^2*sqrt(-2*x + 3) - 10*(-2*x + 3)^(3/2) + 45 *sqrt(-2*x + 3))*f^2 + 700*((-2*x + 3)^(3/2) - 9*sqrt(-2*x + 3))*f^2 + 36* (5*(2*x - 3)^3*sqrt(-2*x + 3) + 63*(2*x - 3)^2*sqrt(-2*x + 3) - 315*(-2*x + 3)^(3/2) + 945*sqrt(-2*x + 3))*f*g - 420*((2*x - 3)^2*sqrt(-2*x + 3) - 1 0*(-2*x + 3)^(3/2) + 45*sqrt(-2*x + 3))*f*g + 1680*((-2*x + 3)^(3/2) - 9*s qrt(-2*x + 3))*f*g + (35*(2*x - 3)^4*sqrt(-2*x + 3) + 540*(2*x - 3)^3*sqrt (-2*x + 3) + 3402*(2*x - 3)^2*sqrt(-2*x + 3) - 11340*(-2*x + 3)^(3/2) + 25 515*sqrt(-2*x + 3))*g^2 - 15*(5*(2*x - 3)^3*sqrt(-2*x + 3) + 63*(2*x - 3)^ 2*sqrt(-2*x + 3) - 315*(-2*x + 3)^(3/2) + 945*sqrt(-2*x + 3))*g^2 - 252*(( 2*x - 3)^2*sqrt(-2*x + 3) - 10*(-2*x + 3)^(3/2) + 45*sqrt(-2*x + 3))*g^2 - 5040*f^2*sqrt(-2*x + 3))
Time = 0.28 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.76 \[ \int \sqrt {2+3 x} (f+g x)^2 \sqrt {1+\frac {5 x}{6}-x^2} \, dx=\frac {\left (2\,x-3\right )\,\sqrt {-36\,x^2+30\,x+36}\,\left (63\,f^2\,x+133\,f^2+90\,f\,g\,x^2+192\,f\,g\,x+192\,f\,g+35\,g^2\,x^3+75\,g^2\,x^2+90\,g^2\,x+90\,g^2\right )}{630\,\sqrt {3\,x+2}} \] Input:
int(((f + g*x)^2*(3*x + 2)^(1/2)*(30*x - 36*x^2 + 36)^(1/2))/6,x)
Output:
((2*x - 3)*(30*x - 36*x^2 + 36)^(1/2)*(192*f*g + 63*f^2*x + 90*g^2*x + 133 *f^2 + 90*g^2 + 75*g^2*x^2 + 35*g^2*x^3 + 90*f*g*x^2 + 192*f*g*x))/(630*(3 *x + 2)^(1/2))
Time = 0.22 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.82 \[ \int \sqrt {2+3 x} (f+g x)^2 \sqrt {1+\frac {5 x}{6}-x^2} \, dx=\frac {\sqrt {-2 x +3}\, \sqrt {6}\, \left (70 g^{2} x^{4}+180 f g \,x^{3}+45 g^{2} x^{3}+126 f^{2} x^{2}+114 f g \,x^{2}-45 g^{2} x^{2}+77 f^{2} x -192 f g x -90 g^{2} x -399 f^{2}-576 f g -270 g^{2}\right )}{630} \] Input:
int(1/6*(2+3*x)^(1/2)*(g*x+f)^2*(-36*x^2+30*x+36)^(1/2),x)
Output:
(sqrt( - 2*x + 3)*sqrt(6)*(126*f**2*x**2 + 77*f**2*x - 399*f**2 + 180*f*g* x**3 + 114*f*g*x**2 - 192*f*g*x - 576*f*g + 70*g**2*x**4 + 45*g**2*x**3 - 45*g**2*x**2 - 90*g**2*x - 270*g**2))/630