Integrand size = 28, antiderivative size = 87 \[ \int \sqrt {2+3 x} (f+g x)^2 \sqrt {4-9 x^2} \, dx=-\frac {8}{81} (3 f+2 g)^2 (2-3 x)^{3/2}+\frac {2}{135} (3 f+2 g) (3 f+10 g) (2-3 x)^{5/2}-\frac {4}{189} g (3 f+4 g) (2-3 x)^{7/2}+\frac {2}{243} g^2 (2-3 x)^{9/2} \] Output:
-8/81*(3*f+2*g)^2*(2-3*x)^(3/2)+2/135*(3*f+2*g)*(3*f+10*g)*(2-3*x)^(5/2)-4 /189*g*(3*f+4*g)*(2-3*x)^(7/2)+2/243*g^2*(2-3*x)^(9/2)
Time = 0.20 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.85 \[ \int \sqrt {2+3 x} (f+g x)^2 \sqrt {4-9 x^2} \, dx=\frac {2 (-2+3 x) \sqrt {4-9 x^2} \left (189 f^2 (14+9 x)+18 f g \left (88+198 x+135 x^2\right )+5 g^2 \left (64+144 x+270 x^2+189 x^3\right )\right )}{8505 \sqrt {2+3 x}} \] Input:
Integrate[Sqrt[2 + 3*x]*(f + g*x)^2*Sqrt[4 - 9*x^2],x]
Output:
(2*(-2 + 3*x)*Sqrt[4 - 9*x^2]*(189*f^2*(14 + 9*x) + 18*f*g*(88 + 198*x + 1 35*x^2) + 5*g^2*(64 + 144*x + 270*x^2 + 189*x^3)))/(8505*Sqrt[2 + 3*x])
Time = 0.23 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {639, 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.
\(\displaystyle \int \sqrt {3 x+2} \sqrt {4-9 x^2} (f+g x)^2 \, dx\) |
\(\Big \downarrow \) 639 |
\(\displaystyle \int \sqrt {2-3 x} (3 x+2) (f+g x)^2dx\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \int \left (\frac {1}{9} (2-3 x)^{3/2} \left (-9 f^2-36 f g-20 g^2\right )+\frac {2}{9} g (2-3 x)^{5/2} (3 f+4 g)+\frac {4}{9} \sqrt {2-3 x} (3 f+2 g)^2-\frac {1}{9} g^2 (2-3 x)^{7/2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {4}{189} g (2-3 x)^{7/2} (3 f+4 g)+\frac {2}{135} (2-3 x)^{5/2} (3 f+2 g) (3 f+10 g)-\frac {8}{81} (2-3 x)^{3/2} (3 f+2 g)^2+\frac {2}{243} g^2 (2-3 x)^{9/2}\) |
Input:
Int[Sqrt[2 + 3*x]*(f + g*x)^2*Sqrt[4 - 9*x^2],x]
Output:
(-8*(3*f + 2*g)^2*(2 - 3*x)^(3/2))/81 + (2*(3*f + 2*g)*(3*f + 10*g)*(2 - 3 *x)^(5/2))/135 - (4*g*(3*f + 4*g)*(2 - 3*x)^(7/2))/189 + (2*g^2*(2 - 3*x)^ (9/2))/243
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[((c_) + (d_.)*(x_))^(m_.)*((e_) + (f_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^ 2)^(p_.), x_Symbol] :> Int[(c + d*x)^(m + p)*(e + f*x)^n*(a/c + (b/d)*x)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && (I ntegerQ[p] || (GtQ[a, 0] && GtQ[c, 0] && !IntegerQ[m]))
Time = 0.68 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.91
method | result | size |
gosper | \(\frac {2 \left (-2+3 x \right ) \left (945 g^{2} x^{3}+2430 f g \,x^{2}+1350 g^{2} x^{2}+1701 f^{2} x +3564 f g x +720 g^{2} x +2646 f^{2}+1584 f g +320 g^{2}\right ) \sqrt {-9 x^{2}+4}}{8505 \sqrt {3 x +2}}\) | \(79\) |
default | \(\frac {2 \left (-2+3 x \right ) \left (945 g^{2} x^{3}+2430 f g \,x^{2}+1350 g^{2} x^{2}+1701 f^{2} x +3564 f g x +720 g^{2} x +2646 f^{2}+1584 f g +320 g^{2}\right ) \sqrt {-9 x^{2}+4}}{8505 \sqrt {3 x +2}}\) | \(79\) |
orering | \(\frac {2 \left (-2+3 x \right ) \left (945 g^{2} x^{3}+2430 f g \,x^{2}+1350 g^{2} x^{2}+1701 f^{2} x +3564 f g x +720 g^{2} x +2646 f^{2}+1584 f g +320 g^{2}\right ) \sqrt {-9 x^{2}+4}}{8505 \sqrt {3 x +2}}\) | \(79\) |
risch | \(-\frac {2 \sqrt {\frac {-9 x^{2}+4}{3 x +2}}\, \sqrt {3 x +2}\, \left (2835 g^{2} x^{4}+7290 f g \,x^{3}+2160 g^{2} x^{3}+5103 f^{2} x^{2}+5832 f g \,x^{2}-540 g^{2} x^{2}+4536 f^{2} x -2376 f g x -480 g^{2} x -5292 f^{2}-3168 f g -640 g^{2}\right ) \left (-2+3 x \right )}{8505 \sqrt {-9 x^{2}+4}\, \sqrt {2-3 x}}\) | \(126\) |
Input:
int((3*x+2)^(1/2)*(g*x+f)^2*(-9*x^2+4)^(1/2),x,method=_RETURNVERBOSE)
Output:
2/8505*(-2+3*x)*(945*g^2*x^3+2430*f*g*x^2+1350*g^2*x^2+1701*f^2*x+3564*f*g *x+720*g^2*x+2646*f^2+1584*f*g+320*g^2)*(-9*x^2+4)^(1/2)/(3*x+2)^(1/2)
Time = 0.08 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.08 \[ \int \sqrt {2+3 x} (f+g x)^2 \sqrt {4-9 x^2} \, dx=\frac {2 \, {\left (2835 \, g^{2} x^{4} + 270 \, {\left (27 \, f g + 8 \, g^{2}\right )} x^{3} + 27 \, {\left (189 \, f^{2} + 216 \, f g - 20 \, g^{2}\right )} x^{2} - 5292 \, f^{2} - 3168 \, f g - 640 \, g^{2} + 24 \, {\left (189 \, f^{2} - 99 \, f g - 20 \, g^{2}\right )} x\right )} \sqrt {-9 \, x^{2} + 4}}{8505 \, \sqrt {3 \, x + 2}} \] Input:
integrate((2+3*x)^(1/2)*(g*x+f)^2*(-9*x^2+4)^(1/2),x, algorithm="fricas")
Output:
2/8505*(2835*g^2*x^4 + 270*(27*f*g + 8*g^2)*x^3 + 27*(189*f^2 + 216*f*g - 20*g^2)*x^2 - 5292*f^2 - 3168*f*g - 640*g^2 + 24*(189*f^2 - 99*f*g - 20*g^ 2)*x)*sqrt(-9*x^2 + 4)/sqrt(3*x + 2)
\[ \int \sqrt {2+3 x} (f+g x)^2 \sqrt {4-9 x^2} \, dx=\int \sqrt {- \left (3 x - 2\right ) \left (3 x + 2\right )} \left (f + g x\right )^{2} \sqrt {3 x + 2}\, dx \] Input:
integrate((2+3*x)**(1/2)*(g*x+f)**2*(-9*x**2+4)**(1/2),x)
Output:
Integral(sqrt(-(3*x - 2)*(3*x + 2))*(f + g*x)**2*sqrt(3*x + 2), x)
Time = 0.06 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.93 \[ \int \sqrt {2+3 x} (f+g x)^2 \sqrt {4-9 x^2} \, dx=\frac {2}{45} \, {\left (27 \, x^{2} + 24 \, x - 28\right )} f^{2} \sqrt {-3 \, x + 2} + \frac {4}{945} \, {\left (405 \, x^{3} + 324 \, x^{2} - 132 \, x - 176\right )} f g \sqrt {-3 \, x + 2} + \frac {2}{1701} \, {\left (567 \, x^{4} + 432 \, x^{3} - 108 \, x^{2} - 96 \, x - 128\right )} g^{2} \sqrt {-3 \, x + 2} \] Input:
integrate((2+3*x)^(1/2)*(g*x+f)^2*(-9*x^2+4)^(1/2),x, algorithm="maxima")
Output:
2/45*(27*x^2 + 24*x - 28)*f^2*sqrt(-3*x + 2) + 4/945*(405*x^3 + 324*x^2 - 132*x - 176)*f*g*sqrt(-3*x + 2) + 2/1701*(567*x^4 + 432*x^3 - 108*x^2 - 96 *x - 128)*g^2*sqrt(-3*x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 241 vs. \(2 (71) = 142\).
Time = 0.14 (sec) , antiderivative size = 241, normalized size of antiderivative = 2.77 \[ \int \sqrt {2+3 x} (f+g x)^2 \sqrt {4-9 x^2} \, dx=\frac {2}{45} \, {\left (3 \, {\left (3 \, x - 2\right )}^{2} \sqrt {-3 \, x + 2} - 20 \, {\left (-3 \, x + 2\right )}^{\frac {3}{2}} + 60 \, \sqrt {-3 \, x + 2}\right )} f^{2} + \frac {4}{315} \, {\left (5 \, {\left (3 \, x - 2\right )}^{3} \sqrt {-3 \, x + 2} + 42 \, {\left (3 \, x - 2\right )}^{2} \sqrt {-3 \, x + 2} - 140 \, {\left (-3 \, x + 2\right )}^{\frac {3}{2}} + 280 \, \sqrt {-3 \, x + 2}\right )} f g + \frac {16}{27} \, {\left ({\left (-3 \, x + 2\right )}^{\frac {3}{2}} - 6 \, \sqrt {-3 \, x + 2}\right )} f g + \frac {2}{8505} \, {\left (35 \, {\left (3 \, x - 2\right )}^{4} \sqrt {-3 \, x + 2} + 360 \, {\left (3 \, x - 2\right )}^{3} \sqrt {-3 \, x + 2} + 1512 \, {\left (3 \, x - 2\right )}^{2} \sqrt {-3 \, x + 2} - 3360 \, {\left (-3 \, x + 2\right )}^{\frac {3}{2}} + 5040 \, \sqrt {-3 \, x + 2}\right )} g^{2} - \frac {8}{405} \, {\left (3 \, {\left (3 \, x - 2\right )}^{2} \sqrt {-3 \, x + 2} - 20 \, {\left (-3 \, x + 2\right )}^{\frac {3}{2}} + 60 \, \sqrt {-3 \, x + 2}\right )} g^{2} - \frac {8}{3} \, f^{2} \sqrt {-3 \, x + 2} \] Input:
integrate((2+3*x)^(1/2)*(g*x+f)^2*(-9*x^2+4)^(1/2),x, algorithm="giac")
Output:
2/45*(3*(3*x - 2)^2*sqrt(-3*x + 2) - 20*(-3*x + 2)^(3/2) + 60*sqrt(-3*x + 2))*f^2 + 4/315*(5*(3*x - 2)^3*sqrt(-3*x + 2) + 42*(3*x - 2)^2*sqrt(-3*x + 2) - 140*(-3*x + 2)^(3/2) + 280*sqrt(-3*x + 2))*f*g + 16/27*((-3*x + 2)^( 3/2) - 6*sqrt(-3*x + 2))*f*g + 2/8505*(35*(3*x - 2)^4*sqrt(-3*x + 2) + 360 *(3*x - 2)^3*sqrt(-3*x + 2) + 1512*(3*x - 2)^2*sqrt(-3*x + 2) - 3360*(-3*x + 2)^(3/2) + 5040*sqrt(-3*x + 2))*g^2 - 8/405*(3*(3*x - 2)^2*sqrt(-3*x + 2) - 20*(-3*x + 2)^(3/2) + 60*sqrt(-3*x + 2))*g^2 - 8/3*f^2*sqrt(-3*x + 2)
Time = 6.50 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.90 \[ \int \sqrt {2+3 x} (f+g x)^2 \sqrt {4-9 x^2} \, dx=\frac {2\,\left (3\,x-2\right )\,\sqrt {4-9\,x^2}\,\left (1701\,f^2\,x+2646\,f^2+2430\,f\,g\,x^2+3564\,f\,g\,x+1584\,f\,g+945\,g^2\,x^3+1350\,g^2\,x^2+720\,g^2\,x+320\,g^2\right )}{8505\,\sqrt {3\,x+2}} \] Input:
int((f + g*x)^2*(3*x + 2)^(1/2)*(4 - 9*x^2)^(1/2),x)
Output:
(2*(3*x - 2)*(4 - 9*x^2)^(1/2)*(1584*f*g + 1701*f^2*x + 720*g^2*x + 2646*f ^2 + 320*g^2 + 1350*g^2*x^2 + 945*g^2*x^3 + 2430*f*g*x^2 + 3564*f*g*x))/(8 505*(3*x + 2)^(1/2))
Time = 0.18 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.99 \[ \int \sqrt {2+3 x} (f+g x)^2 \sqrt {4-9 x^2} \, dx=\frac {2 \sqrt {-3 x +2}\, \left (2835 g^{2} x^{4}+7290 f g \,x^{3}+2160 g^{2} x^{3}+5103 f^{2} x^{2}+5832 f g \,x^{2}-540 g^{2} x^{2}+4536 f^{2} x -2376 f g x -480 g^{2} x -5292 f^{2}-3168 f g -640 g^{2}\right )}{8505} \] Input:
int((2+3*x)^(1/2)*(g*x+f)^2*(-9*x^2+4)^(1/2),x)
Output:
(2*sqrt( - 3*x + 2)*(5103*f**2*x**2 + 4536*f**2*x - 5292*f**2 + 7290*f*g*x **3 + 5832*f*g*x**2 - 2376*f*g*x - 3168*f*g + 2835*g**2*x**4 + 2160*g**2*x **3 - 540*g**2*x**2 - 480*g**2*x - 640*g**2))/8505