Integrand size = 33, antiderivative size = 346 \[ \int \frac {x^2 \left (d+e x+f x^2+g x^3\right )}{\sqrt {a+b x+c x^2}} \, dx=\frac {\left (80 c^2 e-70 b c f+63 b^2 g-64 a c g\right ) x^2 \sqrt {a+b x+c x^2}}{240 c^3}+\frac {(10 c f-9 b g) x^3 \sqrt {a+b x+c x^2}}{40 c^2}+\frac {g x^4 \sqrt {a+b x+c x^2}}{5 c}-\frac {\left (1050 b^3 c f+40 b c^2 (36 c d-55 a f)-945 b^4 g-60 b^2 c (20 c e-49 a g)+256 a c^2 (5 c e-4 a g)-2 c \left (480 c^3 d-40 c^2 (10 b e+9 a f)-315 b^3 g+14 b c (25 b f+46 a g)\right ) x\right ) \sqrt {a+b x+c x^2}}{1920 c^5}+\frac {\left (70 b^4 c f+48 b^2 c^2 (2 c d-5 a f)-32 a c^3 (4 c d-3 a f)-63 b^5 g-40 b^3 c (2 c e-7 a g)+48 a b c^2 (4 c e-5 a g)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{11/2}} \]
1/256*(70*b^4*c*f+48*b^2*c^2*(-5*a*f+2*c*d)-32*a*c^3*(-3*a*f+4*c*d)-63*b^5 *g-40*b^3*c*(-7*a*g+2*c*e)+48*a*b*c^2*(-5*a*g+4*c*e))*arctanh(1/2*(2*c*x+b )/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(11/2)+1/240*(-64*a*c*g+63*b^2*g-70*b*c*f +80*c^2*e)*x^2*(c*x^2+b*x+a)^(1/2)/c^3+1/40*(-9*b*g+10*c*f)*x^3*(c*x^2+b*x +a)^(1/2)/c^2+1/5*g*x^4*(c*x^2+b*x+a)^(1/2)/c-1/1920*(1050*b^3*c*f+40*b*c^ 2*(-55*a*f+36*c*d)-945*b^4*g-60*b^2*c*(-49*a*g+20*c*e)+256*a*c^2*(-4*a*g+5 *c*e)-2*c*(480*c^3*d-40*c^2*(9*a*f+10*b*e)-315*b^3*g+14*b*c*(46*a*g+25*b*f ))*x)*(c*x^2+b*x+a)^(1/2)/c^5
Time = 1.10 (sec) , antiderivative size = 282, normalized size of antiderivative = 0.82 \[ \int \frac {x^2 \left (d+e x+f x^2+g x^3\right )}{\sqrt {a+b x+c x^2}} \, dx=\frac {2 \sqrt {c} \sqrt {a+x (b+c x)} \left (945 b^4 g-210 b^3 c (5 f+3 g x)+4 b^2 c (300 c e-735 a g+7 c x (25 f+18 g x))-8 b c^2 \left (-a (275 f+161 g x)+2 c \left (90 d+x \left (50 e+35 f x+27 g x^2\right )\right )\right )+16 c^2 \left (64 a^2 g-a c (80 e+x (45 f+32 g x))+2 c^2 x (30 d+x (20 e+3 x (5 f+4 g x)))\right )\right )+15 \left (-70 b^4 c f-48 b^2 c^2 (2 c d-5 a f)+32 a c^3 (4 c d-3 a f)+63 b^5 g+40 b^3 c (2 c e-7 a g)+48 a b c^2 (-4 c e+5 a g)\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{3840 c^{11/2}} \]
(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(945*b^4*g - 210*b^3*c*(5*f + 3*g*x) + 4* b^2*c*(300*c*e - 735*a*g + 7*c*x*(25*f + 18*g*x)) - 8*b*c^2*(-(a*(275*f + 161*g*x)) + 2*c*(90*d + x*(50*e + 35*f*x + 27*g*x^2))) + 16*c^2*(64*a^2*g - a*c*(80*e + x*(45*f + 32*g*x)) + 2*c^2*x*(30*d + x*(20*e + 3*x*(5*f + 4* g*x))))) + 15*(-70*b^4*c*f - 48*b^2*c^2*(2*c*d - 5*a*f) + 32*a*c^3*(4*c*d - 3*a*f) + 63*b^5*g + 40*b^3*c*(2*c*e - 7*a*g) + 48*a*b*c^2*(-4*c*e + 5*a* g))*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(3840*c^(11/2))
Time = 0.85 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.07, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2184, 27, 2184, 27, 1236, 27, 1225, 1092, 219}
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.
| \(\displaystyle \int \frac {x^2 \left (d+e x+f x^2+g x^3\right )}{\sqrt {a+b x+c x^2}} \, dx\) |
| \(\Big \downarrow \) 2184 |
| \(\displaystyle \frac {\int \frac {x^2 \left ((10 c f-9 b g) x^2+2 (5 c e-4 a g) x+10 c d\right )}{2 \sqrt {c x^2+b x+a}}dx}{5 c}+\frac {g x^4 \sqrt {a+b x+c x^2}}{5 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {x^2 \left ((10 c f-9 b g) x^2+2 (5 c e-4 a g) x+10 c d\right )}{\sqrt {c x^2+b x+a}}dx}{10 c}+\frac {g x^4 \sqrt {a+b x+c x^2}}{5 c}\) |
| \(\Big \downarrow \) 2184 |
| \(\displaystyle \frac {\frac {\int \frac {x^2 \left (80 d c^2-60 a f c+54 a b g+\left (63 g b^2-70 c f b+80 c^2 e-64 a c g\right ) x\right )}{2 \sqrt {c x^2+b x+a}}dx}{4 c}+\frac {x^3 \sqrt {a+b x+c x^2} (10 c f-9 b g)}{4 c}}{10 c}+\frac {g x^4 \sqrt {a+b x+c x^2}}{5 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {x^2 \left (2 \left (40 d c^2-30 a f c+27 a b g\right )+\left (63 g b^2-70 c f b+80 c^2 e-64 a c g\right ) x\right )}{\sqrt {c x^2+b x+a}}dx}{8 c}+\frac {x^3 \sqrt {a+b x+c x^2} (10 c f-9 b g)}{4 c}}{10 c}+\frac {g x^4 \sqrt {a+b x+c x^2}}{5 c}\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {\frac {\frac {\int -\frac {x \left (4 a \left (63 g b^2-70 c f b+80 c^2 e-64 a c g\right )-\left (-315 g b^3+14 c (25 b f+46 a g) b+480 c^3 d-40 c^2 (10 b e+9 a f)\right ) x\right )}{2 \sqrt {c x^2+b x+a}}dx}{3 c}+\frac {x^2 \sqrt {a+b x+c x^2} \left (-64 a c g+63 b^2 g-70 b c f+80 c^2 e\right )}{3 c}}{8 c}+\frac {x^3 \sqrt {a+b x+c x^2} (10 c f-9 b g)}{4 c}}{10 c}+\frac {g x^4 \sqrt {a+b x+c x^2}}{5 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {x^2 \sqrt {a+b x+c x^2} \left (-64 a c g+63 b^2 g-70 b c f+80 c^2 e\right )}{3 c}-\frac {\int \frac {x \left (4 a \left (63 g b^2-70 c f b+80 c^2 e-64 a c g\right )-\left (-315 g b^3+14 c (25 b f+46 a g) b+480 c^3 d-40 c^2 (10 b e+9 a f)\right ) x\right )}{\sqrt {c x^2+b x+a}}dx}{6 c}}{8 c}+\frac {x^3 \sqrt {a+b x+c x^2} (10 c f-9 b g)}{4 c}}{10 c}+\frac {g x^4 \sqrt {a+b x+c x^2}}{5 c}\) |
| \(\Big \downarrow \) 1225 |
| \(\displaystyle \frac {\frac {\frac {x^2 \sqrt {a+b x+c x^2} \left (-64 a c g+63 b^2 g-70 b c f+80 c^2 e\right )}{3 c}-\frac {\frac {\sqrt {a+b x+c x^2} \left (-2 c x \left (-40 c^2 (9 a f+10 b e)+14 b c (46 a g+25 b f)-315 b^3 g+480 c^3 d\right )-60 b^2 c (20 c e-49 a g)+40 b c^2 (36 c d-55 a f)+256 a c^2 (5 c e-4 a g)-945 b^4 g+1050 b^3 c f\right )}{4 c^2}-\frac {15 \left (-40 b^3 c (2 c e-7 a g)+48 b^2 c^2 (2 c d-5 a f)+48 a b c^2 (4 c e-5 a g)-32 a c^3 (4 c d-3 a f)-63 b^5 g+70 b^4 c f\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{8 c^2}}{6 c}}{8 c}+\frac {x^3 \sqrt {a+b x+c x^2} (10 c f-9 b g)}{4 c}}{10 c}+\frac {g x^4 \sqrt {a+b x+c x^2}}{5 c}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {\frac {\frac {x^2 \sqrt {a+b x+c x^2} \left (-64 a c g+63 b^2 g-70 b c f+80 c^2 e\right )}{3 c}-\frac {\frac {\sqrt {a+b x+c x^2} \left (-2 c x \left (-40 c^2 (9 a f+10 b e)+14 b c (46 a g+25 b f)-315 b^3 g+480 c^3 d\right )-60 b^2 c (20 c e-49 a g)+40 b c^2 (36 c d-55 a f)+256 a c^2 (5 c e-4 a g)-945 b^4 g+1050 b^3 c f\right )}{4 c^2}-\frac {15 \left (-40 b^3 c (2 c e-7 a g)+48 b^2 c^2 (2 c d-5 a f)+48 a b c^2 (4 c e-5 a g)-32 a c^3 (4 c d-3 a f)-63 b^5 g+70 b^4 c f\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{4 c^2}}{6 c}}{8 c}+\frac {x^3 \sqrt {a+b x+c x^2} (10 c f-9 b g)}{4 c}}{10 c}+\frac {g x^4 \sqrt {a+b x+c x^2}}{5 c}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {x^2 \sqrt {a+b x+c x^2} \left (-64 a c g+63 b^2 g-70 b c f+80 c^2 e\right )}{3 c}-\frac {\frac {\sqrt {a+b x+c x^2} \left (-2 c x \left (-40 c^2 (9 a f+10 b e)+14 b c (46 a g+25 b f)-315 b^3 g+480 c^3 d\right )-60 b^2 c (20 c e-49 a g)+40 b c^2 (36 c d-55 a f)+256 a c^2 (5 c e-4 a g)-945 b^4 g+1050 b^3 c f\right )}{4 c^2}-\frac {15 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-40 b^3 c (2 c e-7 a g)+48 b^2 c^2 (2 c d-5 a f)+48 a b c^2 (4 c e-5 a g)-32 a c^3 (4 c d-3 a f)-63 b^5 g+70 b^4 c f\right )}{8 c^{5/2}}}{6 c}}{8 c}+\frac {x^3 \sqrt {a+b x+c x^2} (10 c f-9 b g)}{4 c}}{10 c}+\frac {g x^4 \sqrt {a+b x+c x^2}}{5 c}\) |
(g*x^4*Sqrt[a + b*x + c*x^2])/(5*c) + (((10*c*f - 9*b*g)*x^3*Sqrt[a + b*x + c*x^2])/(4*c) + (((80*c^2*e - 70*b*c*f + 63*b^2*g - 64*a*c*g)*x^2*Sqrt[a + b*x + c*x^2])/(3*c) - (((1050*b^3*c*f + 40*b*c^2*(36*c*d - 55*a*f) - 94 5*b^4*g - 60*b^2*c*(20*c*e - 49*a*g) + 256*a*c^2*(5*c*e - 4*a*g) - 2*c*(48 0*c^3*d - 40*c^2*(10*b*e + 9*a*f) - 315*b^3*g + 14*b*c*(25*b*f + 46*a*g))* x)*Sqrt[a + b*x + c*x^2])/(4*c^2) - (15*(70*b^4*c*f + 48*b^2*c^2*(2*c*d - 5*a*f) - 32*a*c^3*(4*c*d - 3*a*f) - 63*b^5*g - 40*b^3*c*(2*c*e - 7*a*g) + 48*a*b*c^2*(4*c*e - 5*a*g))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*c^(5/2)))/(6*c))/(8*c))/(10*c)
3.3.80.3.1 Defintions of rubi rules used
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_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( x_)^2)^(p_), x_Symbol] :> Simp[(-(b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p + 3))), x] + Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c , d, e, f, g, p}, x] && !LeQ[p, -1]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Simp[1/(c*(m + 2*p + 2)) Int[(d + e*x)^(m - 1 )*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m *(c*e*f + c*d*g - b*e*g) + e*(p + 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[ {a, b, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && EqQ[f, 0])
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p _), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, S imp[f*(d + e*x)^(m + q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q + 2*p + 1))), x] + Simp[1/(c*e^q*(m + q + 2*p + 1)) Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q - 1) - c *d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[ q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && Pol yQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && !(IGt Q[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Time = 0.84 (sec) , antiderivative size = 314, normalized size of antiderivative = 0.91
| method | result | size |
| risch | \(\frac {\left (384 g \,c^{4} x^{4}-432 b \,c^{3} g \,x^{3}+480 c^{4} f \,x^{3}-512 a \,c^{3} g \,x^{2}+504 b^{2} c^{2} g \,x^{2}-560 b \,c^{3} f \,x^{2}+640 c^{4} e \,x^{2}+1288 a b \,c^{2} g x -720 a \,c^{3} f x -630 g c \,b^{3} x +700 f \,c^{2} b^{2} x -800 c^{3} e b x +960 c^{4} d x +1024 a^{2} c^{2} g -2940 a \,b^{2} c g +2200 a b \,c^{2} f -1280 a \,c^{3} e +945 b^{4} g -1050 b^{3} f c +1200 b^{2} c^{2} e -1440 b d \,c^{3}\right ) \sqrt {c \,x^{2}+b x +a}}{1920 c^{5}}-\frac {\left (240 a^{2} b \,c^{2} g -96 a^{2} c^{3} f -280 a \,b^{3} c g +240 a \,b^{2} c^{2} f -192 a b \,c^{3} e +128 a \,c^{4} d +63 b^{5} g -70 b^{4} c f +80 b^{3} c^{2} e -96 b^{2} c^{3} d \right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{256 c^{\frac {11}{2}}}\) | \(314\) |
| default | \(\text {Expression too large to display}\) | \(1161\) |
1/1920*(384*c^4*g*x^4-432*b*c^3*g*x^3+480*c^4*f*x^3-512*a*c^3*g*x^2+504*b^ 2*c^2*g*x^2-560*b*c^3*f*x^2+640*c^4*e*x^2+1288*a*b*c^2*g*x-720*a*c^3*f*x-6 30*b^3*c*g*x+700*b^2*c^2*f*x-800*b*c^3*e*x+960*c^4*d*x+1024*a^2*c^2*g-2940 *a*b^2*c*g+2200*a*b*c^2*f-1280*a*c^3*e+945*b^4*g-1050*b^3*c*f+1200*b^2*c^2 *e-1440*b*c^3*d)*(c*x^2+b*x+a)^(1/2)/c^5-1/256*(240*a^2*b*c^2*g-96*a^2*c^3 *f-280*a*b^3*c*g+240*a*b^2*c^2*f-192*a*b*c^3*e+128*a*c^4*d+63*b^5*g-70*b^4 *c*f+80*b^3*c^2*e-96*b^2*c^3*d)/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x +a)^(1/2))
Time = 0.37 (sec) , antiderivative size = 701, normalized size of antiderivative = 2.03 \[ \int \frac {x^2 \left (d+e x+f x^2+g x^3\right )}{\sqrt {a+b x+c x^2}} \, dx=\left [-\frac {15 \, {\left (32 \, {\left (3 \, b^{2} c^{3} - 4 \, a c^{4}\right )} d - 16 \, {\left (5 \, b^{3} c^{2} - 12 \, a b c^{3}\right )} e + 2 \, {\left (35 \, b^{4} c - 120 \, a b^{2} c^{2} + 48 \, a^{2} c^{3}\right )} f - {\left (63 \, b^{5} - 280 \, a b^{3} c + 240 \, a^{2} b c^{2}\right )} g\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} + 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, {\left (384 \, c^{5} g x^{4} - 1440 \, b c^{4} d + 48 \, {\left (10 \, c^{5} f - 9 \, b c^{4} g\right )} x^{3} + 8 \, {\left (80 \, c^{5} e - 70 \, b c^{4} f + {\left (63 \, b^{2} c^{3} - 64 \, a c^{4}\right )} g\right )} x^{2} + 80 \, {\left (15 \, b^{2} c^{3} - 16 \, a c^{4}\right )} e - 50 \, {\left (21 \, b^{3} c^{2} - 44 \, a b c^{3}\right )} f + {\left (945 \, b^{4} c - 2940 \, a b^{2} c^{2} + 1024 \, a^{2} c^{3}\right )} g + 2 \, {\left (480 \, c^{5} d - 400 \, b c^{4} e + 10 \, {\left (35 \, b^{2} c^{3} - 36 \, a c^{4}\right )} f - 7 \, {\left (45 \, b^{3} c^{2} - 92 \, a b c^{3}\right )} g\right )} x\right )} \sqrt {c x^{2} + b x + a}}{7680 \, c^{6}}, -\frac {15 \, {\left (32 \, {\left (3 \, b^{2} c^{3} - 4 \, a c^{4}\right )} d - 16 \, {\left (5 \, b^{3} c^{2} - 12 \, a b c^{3}\right )} e + 2 \, {\left (35 \, b^{4} c - 120 \, a b^{2} c^{2} + 48 \, a^{2} c^{3}\right )} f - {\left (63 \, b^{5} - 280 \, a b^{3} c + 240 \, a^{2} b c^{2}\right )} g\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) - 2 \, {\left (384 \, c^{5} g x^{4} - 1440 \, b c^{4} d + 48 \, {\left (10 \, c^{5} f - 9 \, b c^{4} g\right )} x^{3} + 8 \, {\left (80 \, c^{5} e - 70 \, b c^{4} f + {\left (63 \, b^{2} c^{3} - 64 \, a c^{4}\right )} g\right )} x^{2} + 80 \, {\left (15 \, b^{2} c^{3} - 16 \, a c^{4}\right )} e - 50 \, {\left (21 \, b^{3} c^{2} - 44 \, a b c^{3}\right )} f + {\left (945 \, b^{4} c - 2940 \, a b^{2} c^{2} + 1024 \, a^{2} c^{3}\right )} g + 2 \, {\left (480 \, c^{5} d - 400 \, b c^{4} e + 10 \, {\left (35 \, b^{2} c^{3} - 36 \, a c^{4}\right )} f - 7 \, {\left (45 \, b^{3} c^{2} - 92 \, a b c^{3}\right )} g\right )} x\right )} \sqrt {c x^{2} + b x + a}}{3840 \, c^{6}}\right ] \]
[-1/7680*(15*(32*(3*b^2*c^3 - 4*a*c^4)*d - 16*(5*b^3*c^2 - 12*a*b*c^3)*e + 2*(35*b^4*c - 120*a*b^2*c^2 + 48*a^2*c^3)*f - (63*b^5 - 280*a*b^3*c + 240 *a^2*b*c^2)*g)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 + 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) - 4*(384*c^5*g*x^4 - 1440*b*c^4*d + 48* (10*c^5*f - 9*b*c^4*g)*x^3 + 8*(80*c^5*e - 70*b*c^4*f + (63*b^2*c^3 - 64*a *c^4)*g)*x^2 + 80*(15*b^2*c^3 - 16*a*c^4)*e - 50*(21*b^3*c^2 - 44*a*b*c^3) *f + (945*b^4*c - 2940*a*b^2*c^2 + 1024*a^2*c^3)*g + 2*(480*c^5*d - 400*b* c^4*e + 10*(35*b^2*c^3 - 36*a*c^4)*f - 7*(45*b^3*c^2 - 92*a*b*c^3)*g)*x)*s qrt(c*x^2 + b*x + a))/c^6, -1/3840*(15*(32*(3*b^2*c^3 - 4*a*c^4)*d - 16*(5 *b^3*c^2 - 12*a*b*c^3)*e + 2*(35*b^4*c - 120*a*b^2*c^2 + 48*a^2*c^3)*f - ( 63*b^5 - 280*a*b^3*c + 240*a^2*b*c^2)*g)*sqrt(-c)*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(-c)/(c^2*x^2 + b*c*x + a*c)) - 2*(384*c^5*g*x^4 - 1440*b*c^4*d + 48*(10*c^5*f - 9*b*c^4*g)*x^3 + 8*(80*c^5*e - 70*b*c^4*f + (63*b^2*c^3 - 64*a*c^4)*g)*x^2 + 80*(15*b^2*c^3 - 16*a*c^4)*e - 50*(21*b ^3*c^2 - 44*a*b*c^3)*f + (945*b^4*c - 2940*a*b^2*c^2 + 1024*a^2*c^3)*g + 2 *(480*c^5*d - 400*b*c^4*e + 10*(35*b^2*c^3 - 36*a*c^4)*f - 7*(45*b^3*c^2 - 92*a*b*c^3)*g)*x)*sqrt(c*x^2 + b*x + a))/c^6]
Time = 0.87 (sec) , antiderivative size = 700, normalized size of antiderivative = 2.02 \[ \int \frac {x^2 \left (d+e x+f x^2+g x^3\right )}{\sqrt {a+b x+c x^2}} \, dx=\begin {cases} \left (- \frac {a \left (- \frac {3 a \left (- \frac {9 b g}{10 c} + f\right )}{4 c} - \frac {5 b \left (- \frac {4 a g}{5 c} - \frac {7 b \left (- \frac {9 b g}{10 c} + f\right )}{8 c} + e\right )}{6 c} + d\right )}{2 c} - \frac {b \left (- \frac {2 a \left (- \frac {4 a g}{5 c} - \frac {7 b \left (- \frac {9 b g}{10 c} + f\right )}{8 c} + e\right )}{3 c} - \frac {3 b \left (- \frac {3 a \left (- \frac {9 b g}{10 c} + f\right )}{4 c} - \frac {5 b \left (- \frac {4 a g}{5 c} - \frac {7 b \left (- \frac {9 b g}{10 c} + f\right )}{8 c} + e\right )}{6 c} + d\right )}{4 c}\right )}{2 c}\right ) \left (\begin {cases} \frac {\log {\left (b + 2 \sqrt {c} \sqrt {a + b x + c x^{2}} + 2 c x \right )}}{\sqrt {c}} & \text {for}\: a - \frac {b^{2}}{4 c} \neq 0 \\\frac {\left (\frac {b}{2 c} + x\right ) \log {\left (\frac {b}{2 c} + x \right )}}{\sqrt {c \left (\frac {b}{2 c} + x\right )^{2}}} & \text {otherwise} \end {cases}\right ) + \sqrt {a + b x + c x^{2}} \left (\frac {g x^{4}}{5 c} + \frac {x^{3} \left (- \frac {9 b g}{10 c} + f\right )}{4 c} + \frac {x^{2} \left (- \frac {4 a g}{5 c} - \frac {7 b \left (- \frac {9 b g}{10 c} + f\right )}{8 c} + e\right )}{3 c} + \frac {x \left (- \frac {3 a \left (- \frac {9 b g}{10 c} + f\right )}{4 c} - \frac {5 b \left (- \frac {4 a g}{5 c} - \frac {7 b \left (- \frac {9 b g}{10 c} + f\right )}{8 c} + e\right )}{6 c} + d\right )}{2 c} + \frac {- \frac {2 a \left (- \frac {4 a g}{5 c} - \frac {7 b \left (- \frac {9 b g}{10 c} + f\right )}{8 c} + e\right )}{3 c} - \frac {3 b \left (- \frac {3 a \left (- \frac {9 b g}{10 c} + f\right )}{4 c} - \frac {5 b \left (- \frac {4 a g}{5 c} - \frac {7 b \left (- \frac {9 b g}{10 c} + f\right )}{8 c} + e\right )}{6 c} + d\right )}{4 c}}{c}\right ) & \text {for}\: c \neq 0 \\\frac {2 \left (\frac {g \left (a + b x\right )^{\frac {11}{2}}}{11 b^{3}} + \frac {\left (a + b x\right )^{\frac {9}{2}} \left (- 5 a g + b f\right )}{9 b^{3}} + \frac {\left (a + b x\right )^{\frac {7}{2}} \cdot \left (10 a^{2} g - 4 a b f + b^{2} e\right )}{7 b^{3}} + \frac {\left (a + b x\right )^{\frac {5}{2}} \left (- 10 a^{3} g + 6 a^{2} b f - 3 a b^{2} e + b^{3} d\right )}{5 b^{3}} + \frac {\left (a + b x\right )^{\frac {3}{2}} \cdot \left (5 a^{4} g - 4 a^{3} b f + 3 a^{2} b^{2} e - 2 a b^{3} d\right )}{3 b^{3}} + \frac {\sqrt {a + b x} \left (- a^{5} g + a^{4} b f - a^{3} b^{2} e + a^{2} b^{3} d\right )}{b^{3}}\right )}{b^{3}} & \text {for}\: b \neq 0 \\\frac {\frac {d x^{3}}{3} + \frac {e x^{4}}{4} + \frac {f x^{5}}{5} + \frac {g x^{6}}{6}}{\sqrt {a}} & \text {otherwise} \end {cases} \]
Piecewise(((-a*(-3*a*(-9*b*g/(10*c) + f)/(4*c) - 5*b*(-4*a*g/(5*c) - 7*b*( -9*b*g/(10*c) + f)/(8*c) + e)/(6*c) + d)/(2*c) - b*(-2*a*(-4*a*g/(5*c) - 7 *b*(-9*b*g/(10*c) + f)/(8*c) + e)/(3*c) - 3*b*(-3*a*(-9*b*g/(10*c) + f)/(4 *c) - 5*b*(-4*a*g/(5*c) - 7*b*(-9*b*g/(10*c) + f)/(8*c) + e)/(6*c) + d)/(4 *c))/(2*c))*Piecewise((log(b + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + 2*c*x)/s qrt(c), Ne(a - b**2/(4*c), 0)), ((b/(2*c) + x)*log(b/(2*c) + x)/sqrt(c*(b/ (2*c) + x)**2), True)) + sqrt(a + b*x + c*x**2)*(g*x**4/(5*c) + x**3*(-9*b *g/(10*c) + f)/(4*c) + x**2*(-4*a*g/(5*c) - 7*b*(-9*b*g/(10*c) + f)/(8*c) + e)/(3*c) + x*(-3*a*(-9*b*g/(10*c) + f)/(4*c) - 5*b*(-4*a*g/(5*c) - 7*b*( -9*b*g/(10*c) + f)/(8*c) + e)/(6*c) + d)/(2*c) + (-2*a*(-4*a*g/(5*c) - 7*b *(-9*b*g/(10*c) + f)/(8*c) + e)/(3*c) - 3*b*(-3*a*(-9*b*g/(10*c) + f)/(4*c ) - 5*b*(-4*a*g/(5*c) - 7*b*(-9*b*g/(10*c) + f)/(8*c) + e)/(6*c) + d)/(4*c ))/c), Ne(c, 0)), (2*(g*(a + b*x)**(11/2)/(11*b**3) + (a + b*x)**(9/2)*(-5 *a*g + b*f)/(9*b**3) + (a + b*x)**(7/2)*(10*a**2*g - 4*a*b*f + b**2*e)/(7* b**3) + (a + b*x)**(5/2)*(-10*a**3*g + 6*a**2*b*f - 3*a*b**2*e + b**3*d)/( 5*b**3) + (a + b*x)**(3/2)*(5*a**4*g - 4*a**3*b*f + 3*a**2*b**2*e - 2*a*b* *3*d)/(3*b**3) + sqrt(a + b*x)*(-a**5*g + a**4*b*f - a**3*b**2*e + a**2*b* *3*d)/b**3)/b**3, Ne(b, 0)), ((d*x**3/3 + e*x**4/4 + f*x**5/5 + g*x**6/6)/ sqrt(a), True))
Exception generated. \[ \int \frac {x^2 \left (d+e x+f x^2+g x^3\right )}{\sqrt {a+b x+c x^2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
Time = 0.32 (sec) , antiderivative size = 321, normalized size of antiderivative = 0.93 \[ \int \frac {x^2 \left (d+e x+f x^2+g x^3\right )}{\sqrt {a+b x+c x^2}} \, dx=\frac {1}{1920} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (6 \, {\left (\frac {8 \, g x}{c} + \frac {10 \, c^{4} f - 9 \, b c^{3} g}{c^{5}}\right )} x + \frac {80 \, c^{4} e - 70 \, b c^{3} f + 63 \, b^{2} c^{2} g - 64 \, a c^{3} g}{c^{5}}\right )} x + \frac {480 \, c^{4} d - 400 \, b c^{3} e + 350 \, b^{2} c^{2} f - 360 \, a c^{3} f - 315 \, b^{3} c g + 644 \, a b c^{2} g}{c^{5}}\right )} x - \frac {1440 \, b c^{3} d - 1200 \, b^{2} c^{2} e + 1280 \, a c^{3} e + 1050 \, b^{3} c f - 2200 \, a b c^{2} f - 945 \, b^{4} g + 2940 \, a b^{2} c g - 1024 \, a^{2} c^{2} g}{c^{5}}\right )} - \frac {{\left (96 \, b^{2} c^{3} d - 128 \, a c^{4} d - 80 \, b^{3} c^{2} e + 192 \, a b c^{3} e + 70 \, b^{4} c f - 240 \, a b^{2} c^{2} f + 96 \, a^{2} c^{3} f - 63 \, b^{5} g + 280 \, a b^{3} c g - 240 \, a^{2} b c^{2} g\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{256 \, c^{\frac {11}{2}}} \]
1/1920*sqrt(c*x^2 + b*x + a)*(2*(4*(6*(8*g*x/c + (10*c^4*f - 9*b*c^3*g)/c^ 5)*x + (80*c^4*e - 70*b*c^3*f + 63*b^2*c^2*g - 64*a*c^3*g)/c^5)*x + (480*c ^4*d - 400*b*c^3*e + 350*b^2*c^2*f - 360*a*c^3*f - 315*b^3*c*g + 644*a*b*c ^2*g)/c^5)*x - (1440*b*c^3*d - 1200*b^2*c^2*e + 1280*a*c^3*e + 1050*b^3*c* f - 2200*a*b*c^2*f - 945*b^4*g + 2940*a*b^2*c*g - 1024*a^2*c^2*g)/c^5) - 1 /256*(96*b^2*c^3*d - 128*a*c^4*d - 80*b^3*c^2*e + 192*a*b*c^3*e + 70*b^4*c *f - 240*a*b^2*c^2*f + 96*a^2*c^3*f - 63*b^5*g + 280*a*b^3*c*g - 240*a^2*b *c^2*g)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/c^(11/ 2)
Timed out. \[ \int \frac {x^2 \left (d+e x+f x^2+g x^3\right )}{\sqrt {a+b x+c x^2}} \, dx=\int \frac {x^2\,\left (g\,x^3+f\,x^2+e\,x+d\right )}{\sqrt {c\,x^2+b\,x+a}} \,d x \]