3.197 \(\int (g+h x)^2 (a+b x+c x^2)^{3/2} (d+e x+f x^2) \, dx\)

Optimal. Leaf size=753 \[ \frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (16 c^2 \left (3 a^2 f h^2+12 a b h (e h+2 f g)+14 b^2 \left (d h^2+2 e g h+f g^2\right )\right )-72 b^2 c h (3 a f h+2 b e h+4 b f g)-128 c^3 \left (a \left (d h^2+2 e g h+f g^2\right )+3 b g (2 d h+e g)\right )+99 b^4 f h^2+768 c^4 d g^2\right )}{6144 c^5}-\frac{\left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2} \left (16 c^2 \left (3 a^2 f h^2+12 a b h (e h+2 f g)+14 b^2 \left (d h^2+2 e g h+f g^2\right )\right )-72 b^2 c h (3 a f h+2 b e h+4 b f g)-128 c^3 \left (a \left (d h^2+2 e g h+f g^2\right )+3 b g (2 d h+e g)\right )+99 b^4 f h^2+768 c^4 d g^2\right )}{16384 c^6}+\frac{\left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (16 c^2 \left (3 a^2 f h^2+12 a b h (e h+2 f g)+14 b^2 \left (d h^2+2 e g h+f g^2\right )\right )-72 b^2 c h (3 a f h+2 b e h+4 b f g)-128 c^3 \left (a \left (d h^2+2 e g h+f g^2\right )+3 b g (2 d h+e g)\right )+99 b^4 f h^2+768 c^4 d g^2\right )}{32768 c^{13/2}}-\frac{\left (a+b x+c x^2\right )^{5/2} \left (-10 c h x \left (-12 c h (7 a f h+2 b (6 e h+f g))+99 b^2 f h^2-8 c^2 \left (5 f g^2-4 h (7 d h+2 e g)\right )\right )+8 c^2 h \left (96 a h (e h+2 f g)+b \left (196 h (d h+2 e g)+31 f g^2\right )\right )-36 b c h^2 (31 a f h+28 b (e h+2 f g))+693 b^3 f h^3+96 c^3 g \left (5 f g^2-8 h (7 d h+e g)\right )\right )}{13440 c^4 h}-\frac{(g+h x)^2 \left (a+b x+c x^2\right )^{5/2} (11 b f h-16 c e h+10 c f g)}{112 c^2 h}+\frac{f (g+h x)^3 \left (a+b x+c x^2\right )^{5/2}}{8 c h} \]

[Out]

-((b^2 - 4*a*c)*(768*c^4*d*g^2 + 99*b^4*f*h^2 - 72*b^2*c*h*(4*b*f*g + 2*b*e*h + 3*a*f*h) - 128*c^3*(3*b*g*(e*g
+ 2*d*h) + a*(f*g^2 + 2*e*g*h + d*h^2)) + 16*c^2*(3*a^2*f*h^2 + 12*a*b*h*(2*f*g + e*h) + 14*b^2*(f*g^2 + 2*e*
g*h + d*h^2)))*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(16384*c^6) + ((768*c^4*d*g^2 + 99*b^4*f*h^2 - 72*b^2*c*h*(4
*b*f*g + 2*b*e*h + 3*a*f*h) - 128*c^3*(3*b*g*(e*g + 2*d*h) + a*(f*g^2 + 2*e*g*h + d*h^2)) + 16*c^2*(3*a^2*f*h^
2 + 12*a*b*h*(2*f*g + e*h) + 14*b^2*(f*g^2 + 2*e*g*h + d*h^2)))*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(6144*c^5
) - ((10*c*f*g - 16*c*e*h + 11*b*f*h)*(g + h*x)^2*(a + b*x + c*x^2)^(5/2))/(112*c^2*h) + (f*(g + h*x)^3*(a + b
*x + c*x^2)^(5/2))/(8*c*h) - ((693*b^3*f*h^3 + 96*c^3*g*(5*f*g^2 - 8*h*(e*g + 7*d*h)) - 36*b*c*h^2*(31*a*f*h +
28*b*(2*f*g + e*h)) + 8*c^2*h*(96*a*h*(2*f*g + e*h) + b*(31*f*g^2 + 196*h*(2*e*g + d*h))) - 10*c*h*(99*b^2*f*
h^2 - 8*c^2*(5*f*g^2 - 4*h*(2*e*g + 7*d*h)) - 12*c*h*(7*a*f*h + 2*b*(f*g + 6*e*h)))*x)*(a + b*x + c*x^2)^(5/2)
)/(13440*c^4*h) + ((b^2 - 4*a*c)^2*(768*c^4*d*g^2 + 99*b^4*f*h^2 - 72*b^2*c*h*(4*b*f*g + 2*b*e*h + 3*a*f*h) -
128*c^3*(3*b*g*(e*g + 2*d*h) + a*(f*g^2 + 2*e*g*h + d*h^2)) + 16*c^2*(3*a^2*f*h^2 + 12*a*b*h*(2*f*g + e*h) + 1
4*b^2*(f*g^2 + 2*e*g*h + d*h^2)))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(32768*c^(13/2))

________________________________________________________________________________________

Rubi [A]  time = 2.10369, antiderivative size = 749, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 6, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {1653, 832, 779, 612, 621, 206} \[ \frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (16 c^2 \left (3 a^2 f h^2+12 a b h (e h+2 f g)+14 b^2 \left (h (d h+2 e g)+f g^2\right )\right )-72 b^2 c h (3 a f h+2 b e h+4 b f g)-128 c^3 \left (a h (d h+2 e g)+a f g^2+3 b g (2 d h+e g)\right )+99 b^4 f h^2+768 c^4 d g^2\right )}{6144 c^5}-\frac{\left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2} \left (16 c^2 \left (3 a^2 f h^2+12 a b h (e h+2 f g)+14 b^2 \left (h (d h+2 e g)+f g^2\right )\right )-72 b^2 c h (3 a f h+2 b e h+4 b f g)-128 c^3 \left (a h (d h+2 e g)+a f g^2+3 b g (2 d h+e g)\right )+99 b^4 f h^2+768 c^4 d g^2\right )}{16384 c^6}+\frac{\left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (16 c^2 \left (3 a^2 f h^2+12 a b h (e h+2 f g)+14 b^2 \left (h (d h+2 e g)+f g^2\right )\right )-72 b^2 c h (3 a f h+2 b e h+4 b f g)-128 c^3 \left (a h (d h+2 e g)+a f g^2+3 b g (2 d h+e g)\right )+99 b^4 f h^2+768 c^4 d g^2\right )}{32768 c^{13/2}}-\frac{\left (a+b x+c x^2\right )^{5/2} \left (-10 c h x \left (-12 c h (7 a f h+2 b (6 e h+f g))+99 b^2 f h^2-8 c^2 \left (5 f g^2-4 h (7 d h+2 e g)\right )\right )+8 c^2 h \left (96 a h (e h+2 f g)+196 b h (d h+2 e g)+31 b f g^2\right )-36 b c h^2 (31 a f h+28 b (e h+2 f g))+693 b^3 f h^3+96 c^3 \left (5 f g^3-8 g h (7 d h+e g)\right )\right )}{13440 c^4 h}-\frac{(g+h x)^2 \left (a+b x+c x^2\right )^{5/2} (11 b f h-16 c e h+10 c f g)}{112 c^2 h}+\frac{f (g+h x)^3 \left (a+b x+c x^2\right )^{5/2}}{8 c h} \]

Antiderivative was successfully verified.

[In]

Int[(g + h*x)^2*(a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2),x]

[Out]

-((b^2 - 4*a*c)*(768*c^4*d*g^2 + 99*b^4*f*h^2 - 72*b^2*c*h*(4*b*f*g + 2*b*e*h + 3*a*f*h) - 128*c^3*(a*f*g^2 +
a*h*(2*e*g + d*h) + 3*b*g*(e*g + 2*d*h)) + 16*c^2*(3*a^2*f*h^2 + 12*a*b*h*(2*f*g + e*h) + 14*b^2*(f*g^2 + h*(2
*e*g + d*h))))*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(16384*c^6) + ((768*c^4*d*g^2 + 99*b^4*f*h^2 - 72*b^2*c*h*(4
*b*f*g + 2*b*e*h + 3*a*f*h) - 128*c^3*(a*f*g^2 + a*h*(2*e*g + d*h) + 3*b*g*(e*g + 2*d*h)) + 16*c^2*(3*a^2*f*h^
2 + 12*a*b*h*(2*f*g + e*h) + 14*b^2*(f*g^2 + h*(2*e*g + d*h))))*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(6144*c^5
) - ((10*c*f*g - 16*c*e*h + 11*b*f*h)*(g + h*x)^2*(a + b*x + c*x^2)^(5/2))/(112*c^2*h) + (f*(g + h*x)^3*(a + b
*x + c*x^2)^(5/2))/(8*c*h) - ((693*b^3*f*h^3 + 96*c^3*(5*f*g^3 - 8*g*h*(e*g + 7*d*h)) - 36*b*c*h^2*(31*a*f*h +
28*b*(2*f*g + e*h)) + 8*c^2*h*(31*b*f*g^2 + 196*b*h*(2*e*g + d*h) + 96*a*h*(2*f*g + e*h)) - 10*c*h*(99*b^2*f*
h^2 - 8*c^2*(5*f*g^2 - 4*h*(2*e*g + 7*d*h)) - 12*c*h*(7*a*f*h + 2*b*(f*g + 6*e*h)))*x)*(a + b*x + c*x^2)^(5/2)
)/(13440*c^4*h) + ((b^2 - 4*a*c)^2*(768*c^4*d*g^2 + 99*b^4*f*h^2 - 72*b^2*c*h*(4*b*f*g + 2*b*e*h + 3*a*f*h) -
128*c^3*(a*f*g^2 + a*h*(2*e*g + d*h) + 3*b*g*(e*g + 2*d*h)) + 16*c^2*(3*a^2*f*h^2 + 12*a*b*h*(2*f*g + e*h) + 1
4*b^2*(f*g^2 + h*(2*e*g + d*h))))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(32768*c^(13/2))

Rule 1653

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]]}, Simp[(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] + Dist[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] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2
, 0] &&  !(IGtQ[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))

Rule 832

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[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] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
&&  !(IGtQ[m, 0] && EqQ[f, 0])

Rule 779

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] + Dist[(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] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

Rule 612

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p +
1)), x] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int (g+h x)^2 \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right ) \, dx &=\frac{f (g+h x)^3 \left (a+b x+c x^2\right )^{5/2}}{8 c h}+\frac{\int (g+h x)^2 \left (-\frac{1}{2} h (5 b f g-16 c d h+6 a f h)-\frac{1}{2} h (10 c f g-16 c e h+11 b f h) x\right ) \left (a+b x+c x^2\right )^{3/2} \, dx}{8 c h^2}\\ &=-\frac{(10 c f g-16 c e h+11 b f h) (g+h x)^2 \left (a+b x+c x^2\right )^{5/2}}{112 c^2 h}+\frac{f (g+h x)^3 \left (a+b x+c x^2\right )^{5/2}}{8 c h}+\frac{\int (g+h x) \left (\frac{1}{4} h \left (55 b^2 f g h+44 a b f h^2-20 b c g (f g+4 e h)+4 c h (56 c d g-11 a f g-16 a e h)\right )+\frac{1}{4} h \left (99 b^2 f h^2-8 c^2 \left (5 f g^2-4 h (2 e g+7 d h)\right )-12 c h (7 a f h+2 b (f g+6 e h))\right ) x\right ) \left (a+b x+c x^2\right )^{3/2} \, dx}{56 c^2 h^2}\\ &=-\frac{(10 c f g-16 c e h+11 b f h) (g+h x)^2 \left (a+b x+c x^2\right )^{5/2}}{112 c^2 h}+\frac{f (g+h x)^3 \left (a+b x+c x^2\right )^{5/2}}{8 c h}-\frac{\left (693 b^3 f h^3+96 c^3 \left (5 f g^3-8 g h (e g+7 d h)\right )-36 b c h^2 (31 a f h+28 b (2 f g+e h))+8 c^2 h \left (31 b f g^2+196 b h (2 e g+d h)+96 a h (2 f g+e h)\right )-10 c h \left (99 b^2 f h^2-8 c^2 \left (5 f g^2-4 h (2 e g+7 d h)\right )-12 c h (7 a f h+2 b (f g+6 e h))\right ) x\right ) \left (a+b x+c x^2\right )^{5/2}}{13440 c^4 h}+\frac{\left (768 c^4 d g^2+99 b^4 f h^2-72 b^2 c h (4 b f g+2 b e h+3 a f h)-128 c^3 \left (a f g^2+a h (2 e g+d h)+3 b g (e g+2 d h)\right )+16 c^2 \left (3 a^2 f h^2+12 a b h (2 f g+e h)+14 b^2 \left (f g^2+h (2 e g+d h)\right )\right )\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{768 c^4}\\ &=\frac{\left (768 c^4 d g^2+99 b^4 f h^2-72 b^2 c h (4 b f g+2 b e h+3 a f h)-128 c^3 \left (a f g^2+a h (2 e g+d h)+3 b g (e g+2 d h)\right )+16 c^2 \left (3 a^2 f h^2+12 a b h (2 f g+e h)+14 b^2 \left (f g^2+h (2 e g+d h)\right )\right )\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{6144 c^5}-\frac{(10 c f g-16 c e h+11 b f h) (g+h x)^2 \left (a+b x+c x^2\right )^{5/2}}{112 c^2 h}+\frac{f (g+h x)^3 \left (a+b x+c x^2\right )^{5/2}}{8 c h}-\frac{\left (693 b^3 f h^3+96 c^3 \left (5 f g^3-8 g h (e g+7 d h)\right )-36 b c h^2 (31 a f h+28 b (2 f g+e h))+8 c^2 h \left (31 b f g^2+196 b h (2 e g+d h)+96 a h (2 f g+e h)\right )-10 c h \left (99 b^2 f h^2-8 c^2 \left (5 f g^2-4 h (2 e g+7 d h)\right )-12 c h (7 a f h+2 b (f g+6 e h))\right ) x\right ) \left (a+b x+c x^2\right )^{5/2}}{13440 c^4 h}-\frac{\left (\left (b^2-4 a c\right ) \left (768 c^4 d g^2+99 b^4 f h^2-72 b^2 c h (4 b f g+2 b e h+3 a f h)-128 c^3 \left (a f g^2+a h (2 e g+d h)+3 b g (e g+2 d h)\right )+16 c^2 \left (3 a^2 f h^2+12 a b h (2 f g+e h)+14 b^2 \left (f g^2+h (2 e g+d h)\right )\right )\right )\right ) \int \sqrt{a+b x+c x^2} \, dx}{4096 c^5}\\ &=-\frac{\left (b^2-4 a c\right ) \left (768 c^4 d g^2+99 b^4 f h^2-72 b^2 c h (4 b f g+2 b e h+3 a f h)-128 c^3 \left (a f g^2+a h (2 e g+d h)+3 b g (e g+2 d h)\right )+16 c^2 \left (3 a^2 f h^2+12 a b h (2 f g+e h)+14 b^2 \left (f g^2+h (2 e g+d h)\right )\right )\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{16384 c^6}+\frac{\left (768 c^4 d g^2+99 b^4 f h^2-72 b^2 c h (4 b f g+2 b e h+3 a f h)-128 c^3 \left (a f g^2+a h (2 e g+d h)+3 b g (e g+2 d h)\right )+16 c^2 \left (3 a^2 f h^2+12 a b h (2 f g+e h)+14 b^2 \left (f g^2+h (2 e g+d h)\right )\right )\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{6144 c^5}-\frac{(10 c f g-16 c e h+11 b f h) (g+h x)^2 \left (a+b x+c x^2\right )^{5/2}}{112 c^2 h}+\frac{f (g+h x)^3 \left (a+b x+c x^2\right )^{5/2}}{8 c h}-\frac{\left (693 b^3 f h^3+96 c^3 \left (5 f g^3-8 g h (e g+7 d h)\right )-36 b c h^2 (31 a f h+28 b (2 f g+e h))+8 c^2 h \left (31 b f g^2+196 b h (2 e g+d h)+96 a h (2 f g+e h)\right )-10 c h \left (99 b^2 f h^2-8 c^2 \left (5 f g^2-4 h (2 e g+7 d h)\right )-12 c h (7 a f h+2 b (f g+6 e h))\right ) x\right ) \left (a+b x+c x^2\right )^{5/2}}{13440 c^4 h}+\frac{\left (\left (b^2-4 a c\right )^2 \left (768 c^4 d g^2+99 b^4 f h^2-72 b^2 c h (4 b f g+2 b e h+3 a f h)-128 c^3 \left (a f g^2+a h (2 e g+d h)+3 b g (e g+2 d h)\right )+16 c^2 \left (3 a^2 f h^2+12 a b h (2 f g+e h)+14 b^2 \left (f g^2+h (2 e g+d h)\right )\right )\right )\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{32768 c^6}\\ &=-\frac{\left (b^2-4 a c\right ) \left (768 c^4 d g^2+99 b^4 f h^2-72 b^2 c h (4 b f g+2 b e h+3 a f h)-128 c^3 \left (a f g^2+a h (2 e g+d h)+3 b g (e g+2 d h)\right )+16 c^2 \left (3 a^2 f h^2+12 a b h (2 f g+e h)+14 b^2 \left (f g^2+h (2 e g+d h)\right )\right )\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{16384 c^6}+\frac{\left (768 c^4 d g^2+99 b^4 f h^2-72 b^2 c h (4 b f g+2 b e h+3 a f h)-128 c^3 \left (a f g^2+a h (2 e g+d h)+3 b g (e g+2 d h)\right )+16 c^2 \left (3 a^2 f h^2+12 a b h (2 f g+e h)+14 b^2 \left (f g^2+h (2 e g+d h)\right )\right )\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{6144 c^5}-\frac{(10 c f g-16 c e h+11 b f h) (g+h x)^2 \left (a+b x+c x^2\right )^{5/2}}{112 c^2 h}+\frac{f (g+h x)^3 \left (a+b x+c x^2\right )^{5/2}}{8 c h}-\frac{\left (693 b^3 f h^3+96 c^3 \left (5 f g^3-8 g h (e g+7 d h)\right )-36 b c h^2 (31 a f h+28 b (2 f g+e h))+8 c^2 h \left (31 b f g^2+196 b h (2 e g+d h)+96 a h (2 f g+e h)\right )-10 c h \left (99 b^2 f h^2-8 c^2 \left (5 f g^2-4 h (2 e g+7 d h)\right )-12 c h (7 a f h+2 b (f g+6 e h))\right ) x\right ) \left (a+b x+c x^2\right )^{5/2}}{13440 c^4 h}+\frac{\left (\left (b^2-4 a c\right )^2 \left (768 c^4 d g^2+99 b^4 f h^2-72 b^2 c h (4 b f g+2 b e h+3 a f h)-128 c^3 \left (a f g^2+a h (2 e g+d h)+3 b g (e g+2 d h)\right )+16 c^2 \left (3 a^2 f h^2+12 a b h (2 f g+e h)+14 b^2 \left (f g^2+h (2 e g+d h)\right )\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x}{\sqrt{a+b x+c x^2}}\right )}{16384 c^6}\\ &=-\frac{\left (b^2-4 a c\right ) \left (768 c^4 d g^2+99 b^4 f h^2-72 b^2 c h (4 b f g+2 b e h+3 a f h)-128 c^3 \left (a f g^2+a h (2 e g+d h)+3 b g (e g+2 d h)\right )+16 c^2 \left (3 a^2 f h^2+12 a b h (2 f g+e h)+14 b^2 \left (f g^2+h (2 e g+d h)\right )\right )\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{16384 c^6}+\frac{\left (768 c^4 d g^2+99 b^4 f h^2-72 b^2 c h (4 b f g+2 b e h+3 a f h)-128 c^3 \left (a f g^2+a h (2 e g+d h)+3 b g (e g+2 d h)\right )+16 c^2 \left (3 a^2 f h^2+12 a b h (2 f g+e h)+14 b^2 \left (f g^2+h (2 e g+d h)\right )\right )\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{6144 c^5}-\frac{(10 c f g-16 c e h+11 b f h) (g+h x)^2 \left (a+b x+c x^2\right )^{5/2}}{112 c^2 h}+\frac{f (g+h x)^3 \left (a+b x+c x^2\right )^{5/2}}{8 c h}-\frac{\left (693 b^3 f h^3+96 c^3 \left (5 f g^3-8 g h (e g+7 d h)\right )-36 b c h^2 (31 a f h+28 b (2 f g+e h))+8 c^2 h \left (31 b f g^2+196 b h (2 e g+d h)+96 a h (2 f g+e h)\right )-10 c h \left (99 b^2 f h^2-8 c^2 \left (5 f g^2-4 h (2 e g+7 d h)\right )-12 c h (7 a f h+2 b (f g+6 e h))\right ) x\right ) \left (a+b x+c x^2\right )^{5/2}}{13440 c^4 h}+\frac{\left (b^2-4 a c\right )^2 \left (768 c^4 d g^2+99 b^4 f h^2-72 b^2 c h (4 b f g+2 b e h+3 a f h)-128 c^3 \left (a f g^2+a h (2 e g+d h)+3 b g (e g+2 d h)\right )+16 c^2 \left (3 a^2 f h^2+12 a b h (2 f g+e h)+14 b^2 \left (f g^2+h (2 e g+d h)\right )\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{32768 c^{13/2}}\\ \end{align*}

Mathematica [A]  time = 1.67799, size = 468, normalized size = 0.62 \[ \frac{\frac{h \left (2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)} \left (4 c \left (5 a+2 c x^2\right )-3 b^2+8 b c x\right )+3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )\right ) \left (16 c^2 \left (3 a^2 f h^2+12 a b h (e h+2 f g)+14 b^2 \left (h (d h+2 e g)+f g^2\right )\right )-72 b^2 c h (3 a f h+2 b e h+4 b f g)-128 c^3 \left (a h (d h+2 e g)+a f g^2+3 b g (2 d h+e g)\right )+99 b^4 f h^2+768 c^4 d g^2\right )}{12288 c^{11/2}}-\frac{(a+x (b+c x))^{5/2} \left (8 c^2 h (3 a h (32 e h+64 f g+35 f h x)+4 b h (49 d h+98 e g+45 e h x)+b f g (31 g+30 h x))-18 b c h^2 (62 a f h+b (56 e h+112 f g+55 f h x))+693 b^3 f h^3+16 c^3 \left (5 f g^2 (6 g+5 h x)-4 h (7 d h (12 g+5 h x)+2 e g (6 g+5 h x))\right )\right )}{1680 c^3}-\frac{(g+h x)^2 (a+x (b+c x))^{5/2} (11 b f h+2 c (5 f g-8 e h))}{14 c}+f (g+h x)^3 (a+x (b+c x))^{5/2}}{8 c h} \]

Antiderivative was successfully verified.

[In]

Integrate[(g + h*x)^2*(a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2),x]

[Out]

(-((11*b*f*h + 2*c*(5*f*g - 8*e*h))*(g + h*x)^2*(a + x*(b + c*x))^(5/2))/(14*c) + f*(g + h*x)^3*(a + x*(b + c*
x))^(5/2) - ((a + x*(b + c*x))^(5/2)*(693*b^3*f*h^3 + 8*c^2*h*(b*f*g*(31*g + 30*h*x) + 4*b*h*(98*e*g + 49*d*h
+ 45*e*h*x) + 3*a*h*(64*f*g + 32*e*h + 35*f*h*x)) - 18*b*c*h^2*(62*a*f*h + b*(112*f*g + 56*e*h + 55*f*h*x)) +
16*c^3*(5*f*g^2*(6*g + 5*h*x) - 4*h*(2*e*g*(6*g + 5*h*x) + 7*d*h*(12*g + 5*h*x)))))/(1680*c^3) + (h*(768*c^4*d
*g^2 + 99*b^4*f*h^2 - 72*b^2*c*h*(4*b*f*g + 2*b*e*h + 3*a*f*h) - 128*c^3*(a*f*g^2 + a*h*(2*e*g + d*h) + 3*b*g*
(e*g + 2*d*h)) + 16*c^2*(3*a^2*f*h^2 + 12*a*b*h*(2*f*g + e*h) + 14*b^2*(f*g^2 + h*(2*e*g + d*h))))*(2*Sqrt[c]*
(b + 2*c*x)*Sqrt[a + x*(b + c*x)]*(-3*b^2 + 8*b*c*x + 4*c*(5*a + 2*c*x^2)) + 3*(b^2 - 4*a*c)^2*ArcTanh[(b + 2*
c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))/(12288*c^(11/2)))/(8*c*h)

________________________________________________________________________________________

Maple [B]  time = 0.065, size = 3769, normalized size = 5. \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((h*x+g)^2*(c*x^2+b*x+a)^(3/2)*(f*x^2+e*x+d),x)

[Out]

1/16*b^2/c^3*a*(c*x^2+b*x+a)^(3/2)*f*g*h-3/14*b/c^2*x*(c*x^2+b*x+a)^(5/2)*f*g*h-3/32*b^3/c^3*(c*x^2+b*x+a)^(1/
2)*x*a*e*h^2-15/64*b^3/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2*f*g*h+3/32*b^2/c^3*a^2*(c*x^2+b
*x+a)^(1/2)*f*g*h+21/256*b^5/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*f*g*h-33/640*f*h^2*b^3/c^4*
(c*x^2+b*x+a)^(5/2)-99/16384*f*h^2*b^7/c^6*(c*x^2+b*x+a)^(1/2)-3/64*d*g^2/c^2*(c*x^2+b*x+a)^(1/2)*b^3+3/8*d*g^
2/c^(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2+3/128*d*g^2/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*
x+a)^(1/2))*b^4+1/8*d*g^2/c*(c*x^2+b*x+a)^(3/2)*b+3/8*d*g^2*(c*x^2+b*x+a)^(1/2)*x*a+3/128*f*h^2*a^4/c^(5/2)*ln
((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+33/2048*f*h^2*b^5/c^5*(c*x^2+b*x+a)^(3/2)+99/32768*f*h^2*b^8/c^(13/2
)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+1/8*f*h^2*x^3*(c*x^2+b*x+a)^(5/2)/c+1/6*x*(c*x^2+b*x+a)^(5/2)/c*
f*g^2-7/60*b/c^2*(c*x^2+b*x+a)^(5/2)*d*h^2-7/60*b/c^2*(c*x^2+b*x+a)^(5/2)*f*g^2+2/5*(c*x^2+b*x+a)^(5/2)/c*d*g*
h-1/16*b^2/c^2*(c*x^2+b*x+a)^(3/2)*e*g^2+7/192*b^3/c^3*(c*x^2+b*x+a)^(3/2)*d*h^2+7/192*b^3/c^3*(c*x^2+b*x+a)^(
3/2)*f*g^2-7/512*b^5/c^4*(c*x^2+b*x+a)^(1/2)*d*h^2-7/512*b^5/c^4*(c*x^2+b*x+a)^(1/2)*f*g^2+7/1024*b^6/c^(9/2)*
ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d*h^2+7/1024*b^6/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2
))*f*g^2-1/16*a^3/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d*h^2+9/1024*b^6/c^5*(c*x^2+b*x+a)^(1/2)
*e*h^2-3/32*b^4/c^4*(c*x^2+b*x+a)^(1/2)*a*f*g*h+3/16*b/c^(5/2)*a^3*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))
*f*g*h+1/8*b^3/c^3*(c*x^2+b*x+a)^(1/2)*a*e*g*h+9/32*b^2/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^
2*e*g*h-15/128*b^4/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*e*g*h-1/12*a/c*x*(c*x^2+b*x+a)^(3/2)*
e*g*h+1/8*b^2/c^2*(c*x^2+b*x+a)^(1/2)*x*a*d*h^2+1/8*b^2/c^2*(c*x^2+b*x+a)^(1/2)*x*a*f*g^2-1/4*b/c*x*(c*x^2+b*x
+a)^(3/2)*d*g*h-3/16*b/c*(c*x^2+b*x+a)^(1/2)*x*a*e*g^2+3/32*b^3/c^2*(c*x^2+b*x+a)^(1/2)*x*d*g*h-3/16*b^2/c^2*(
c*x^2+b*x+a)^(1/2)*a*d*g*h-3/8*b/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2*d*g*h+3/16*b^3/c^(5/2
)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*d*g*h-1/16*a^3/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1
/2))*f*g^2+1/6*x*(c*x^2+b*x+a)^(5/2)/c*d*h^2-2/35*a/c^2*(c*x^2+b*x+a)^(5/2)*e*h^2+1/7*x^2*(c*x^2+b*x+a)^(5/2)/
c*e*h^2+3/40*b^2/c^3*(c*x^2+b*x+a)^(5/2)*e*h^2-3/128*b^4/c^4*(c*x^2+b*x+a)^(3/2)*e*h^2-9/2048*b^7/c^(11/2)*ln(
(1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*e*h^2+1/8*b/c^2*a*x*(c*x^2+b*x+a)^(3/2)*f*g*h+3/16*b/c^2*a^2*(c*x^2+b
*x+a)^(1/2)*x*f*g*h-3/8*b/c*(c*x^2+b*x+a)^(1/2)*x*a*d*g*h+1/4*b^2/c^2*(c*x^2+b*x+a)^(1/2)*x*a*e*g*h-3/16*b^3/c
^3*(c*x^2+b*x+a)^(1/2)*x*a*f*g*h+3/128*b^4/c^3*(c*x^2+b*x+a)^(1/2)*e*g^2-3/256*b^5/c^(7/2)*ln((1/2*b+c*x)/c^(1
/2)+(c*x^2+b*x+a)^(1/2))*e*g^2+1/4*d*g^2*x*(c*x^2+b*x+a)^(3/2)+1/5*(c*x^2+b*x+a)^(5/2)/c*e*g^2+1/16*b/c^2*a*x*
(c*x^2+b*x+a)^(3/2)*e*h^2+3/32*b/c^2*a^2*(c*x^2+b*x+a)^(1/2)*x*e*h^2-57/512*f*h^2*b^2/c^3*a^2*(c*x^2+b*x+a)^(1
/2)*x+153/2048*f*h^2*b^4/c^4*(c*x^2+b*x+a)^(1/2)*x*a-9/128*f*h^2*b^2/c^3*a*x*(c*x^2+b*x+a)^(3/2)-1/24*a/c^2*(c
*x^2+b*x+a)^(3/2)*b*e*g*h-1/8*a^2/c*(c*x^2+b*x+a)^(1/2)*x*e*g*h-1/16*a^2/c^2*(c*x^2+b*x+a)^(1/2)*b*e*g*h+7/48*
b^2/c^2*x*(c*x^2+b*x+a)^(3/2)*e*g*h-7/128*b^4/c^3*(c*x^2+b*x+a)^(1/2)*x*e*g*h-3/32*b^3/c^3*x*(c*x^2+b*x+a)^(3/
2)*f*g*h+9/256*b^5/c^4*(c*x^2+b*x+a)^(1/2)*x*f*g*h+7/96*b^2/c^2*x*(c*x^2+b*x+a)^(3/2)*d*h^2-7/30*b/c^2*(c*x^2+
b*x+a)^(5/2)*e*g*h+9/64*b^2/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2*f*g^2-15/256*b^4/c^(7/2)*l
n((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*d*h^2-15/256*b^4/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/
2))*a*f*g^2+7/512*b^6/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*e*g*h-1/24*a/c*x*(c*x^2+b*x+a)^(3/2)
*d*h^2-1/24*a/c*x*(c*x^2+b*x+a)^(3/2)*f*g^2-1/48*a/c^2*(c*x^2+b*x+a)^(3/2)*b*d*h^2+7/96*b^2/c^2*x*(c*x^2+b*x+a
)^(3/2)*f*g^2+1/3*x*(c*x^2+b*x+a)^(5/2)/c*e*g*h+1/16*b^3/c^3*(c*x^2+b*x+a)^(1/2)*a*f*g^2-7/256*b^5/c^4*(c*x^2+
b*x+a)^(1/2)*e*g*h+9/64*b^2/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2*d*h^2-1/16*a^2/c*(c*x^2+b*
x+a)^(1/2)*x*f*g^2-1/32*a^2/c^2*(c*x^2+b*x+a)^(1/2)*b*d*h^2-1/32*a^2/c^2*(c*x^2+b*x+a)^(1/2)*b*f*g^2-1/8*a^3/c
^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*e*g*h+7/96*b^3/c^3*(c*x^2+b*x+a)^(3/2)*e*g*h-7/256*b^4/c^3*
(c*x^2+b*x+a)^(1/2)*x*d*h^2-7/256*b^4/c^3*(c*x^2+b*x+a)^(1/2)*x*f*g^2+1/16*b^3/c^3*(c*x^2+b*x+a)^(1/2)*a*d*h^2
-1/48*a/c^2*(c*x^2+b*x+a)^(3/2)*b*f*g^2+3/128*f*h^2*a^3/c^2*(c*x^2+b*x+a)^(1/2)*x+33/1024*f*h^2*b^4/c^4*x*(c*x
^2+b*x+a)^(3/2)-11/112*f*h^2*b/c^2*x^2*(c*x^2+b*x+a)^(5/2)-3/28*b/c^2*x*(c*x^2+b*x+a)^(5/2)*e*h^2+3/20*b^2/c^3
*(c*x^2+b*x+a)^(5/2)*f*g*h-3/64*b^4/c^4*(c*x^2+b*x+a)^(3/2)*f*g*h-3/64*b^3/c^3*x*(c*x^2+b*x+a)^(3/2)*e*h^2+3/3
2*b/c^(5/2)*a^3*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*e*h^2-4/35*a/c^2*(c*x^2+b*x+a)^(5/2)*f*g*h+2/7*x^2
*(c*x^2+b*x+a)^(5/2)/c*f*g*h-1/8*b/c*x*(c*x^2+b*x+a)^(3/2)*e*g^2-1/8*b^2/c^2*(c*x^2+b*x+a)^(3/2)*d*g*h+3/64*b^
3/c^2*(c*x^2+b*x+a)^(1/2)*x*e*g^2-3/32*b^2/c^2*(c*x^2+b*x+a)^(1/2)*a*e*g^2+3/64*b^4/c^3*(c*x^2+b*x+a)^(1/2)*d*
g*h-3/16*b/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2*e*g^2+3/32*b^3/c^(5/2)*ln((1/2*b+c*x)/c^(1/
2)+(c*x^2+b*x+a)^(1/2))*a*e*g^2-3/128*b^5/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d*g*h-3/32*d*g^2
/c*(c*x^2+b*x+a)^(1/2)*x*b^2+3/16*d*g^2/c*(c*x^2+b*x+a)^(1/2)*b*a-3/16*d*g^2/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c
*x^2+b*x+a)^(1/2))*b^2*a-99/8192*f*h^2*b^6/c^5*(c*x^2+b*x+a)^(1/2)*x+153/4096*f*h^2*b^5/c^5*(c*x^2+b*x+a)^(1/2
)*a+33/448*f*h^2*b^2/c^3*x*(c*x^2+b*x+a)^(5/2)+105/1024*f*h^2*b^4/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)
^(1/2))*a^2-57/1024*f*h^2*b^3/c^4*a^2*(c*x^2+b*x+a)^(1/2)-63/2048*f*h^2*b^6/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c
*x^2+b*x+a)^(1/2))*a-15/128*f*h^2*b^2/c^(7/2)*a^3*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+93/1120*f*h^2*b/
c^3*a*(c*x^2+b*x+a)^(5/2)+3/256*f*h^2*a^3/c^3*(c*x^2+b*x+a)^(1/2)*b+1/64*f*h^2*a^2/c^2*x*(c*x^2+b*x+a)^(3/2)+1
/128*f*h^2*a^2/c^3*(c*x^2+b*x+a)^(3/2)*b-1/16*f*h^2*a/c^2*x*(c*x^2+b*x+a)^(5/2)-9/256*f*h^2*b^3/c^4*a*(c*x^2+b
*x+a)^(3/2)+1/32*b^2/c^3*a*(c*x^2+b*x+a)^(3/2)*e*h^2-1/16*a^2/c*(c*x^2+b*x+a)^(1/2)*x*d*h^2+3/64*b^2/c^3*a^2*(
c*x^2+b*x+a)^(1/2)*e*h^2+9/512*b^6/c^5*(c*x^2+b*x+a)^(1/2)*f*g*h+21/512*b^5/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*
x^2+b*x+a)^(1/2))*a*e*h^2-15/128*b^3/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2*e*h^2+9/512*b^5/c
^4*(c*x^2+b*x+a)^(1/2)*x*e*h^2-3/64*b^4/c^4*(c*x^2+b*x+a)^(1/2)*a*e*h^2-9/1024*b^7/c^(11/2)*ln((1/2*b+c*x)/c^(
1/2)+(c*x^2+b*x+a)^(1/2))*f*g*h

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)^2*(c*x^2+b*x+a)^(3/2)*(f*x^2+e*x+d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 13.6364, size = 7302, normalized size = 9.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)^2*(c*x^2+b*x+a)^(3/2)*(f*x^2+e*x+d),x, algorithm="fricas")

[Out]

[1/6881280*(105*(32*(24*(b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*d - 12*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*e +
(7*b^6*c^2 - 60*a*b^4*c^3 + 144*a^2*b^2*c^4 - 64*a^3*c^5)*f)*g^2 - 32*(24*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c
^5)*d - 2*(7*b^6*c^2 - 60*a*b^4*c^3 + 144*a^2*b^2*c^4 - 64*a^3*c^5)*e + 3*(3*b^7*c - 28*a*b^5*c^2 + 80*a^2*b^3
*c^3 - 64*a^3*b*c^4)*f)*g*h + (32*(7*b^6*c^2 - 60*a*b^4*c^3 + 144*a^2*b^2*c^4 - 64*a^3*c^5)*d - 48*(3*b^7*c -
28*a*b^5*c^2 + 80*a^2*b^3*c^3 - 64*a^3*b*c^4)*e + 3*(33*b^8 - 336*a*b^6*c + 1120*a^2*b^4*c^2 - 1280*a^3*b^2*c^
3 + 256*a^4*c^4)*f)*h^2)*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*(215040*c^8*f*h^2*x^7 + 15360*(32*c^8*f*g*h + (16*c^8*e + 17*b*c^7*f)*h^2)*x^6 + 1280*(224*c^8*f*
g^2 + 32*(14*c^8*e + 15*b*c^7*f)*g*h + (224*c^8*d + 240*b*c^7*e + 3*(b^2*c^6 + 84*a*c^7)*f)*h^2)*x^5 + 128*(22
4*(12*c^8*e + 13*b*c^7*f)*g^2 + 32*(168*c^8*d + 182*b*c^7*e + 3*(b^2*c^6 + 64*a*c^7)*f)*g*h + (2912*b*c^7*d +
48*(b^2*c^6 + 64*a*c^7)*e - 3*(11*b^3*c^5 - 52*a*b*c^6)*f)*h^2)*x^4 + 16*(224*(120*c^8*d + 132*b*c^7*e + (3*b^
2*c^6 + 140*a*c^7)*f)*g^2 + 32*(1848*b*c^7*d + 14*(3*b^2*c^6 + 140*a*c^7)*e - 3*(9*b^3*c^5 - 44*a*b*c^6)*f)*g*
h + (224*(3*b^2*c^6 + 140*a*c^7)*d - 48*(9*b^3*c^5 - 44*a*b*c^6)*e + 3*(99*b^4*c^4 - 568*a*b^2*c^5 + 560*a^2*c
^6)*f)*h^2)*x^3 - 224*(120*(3*b^3*c^5 - 20*a*b*c^6)*d - 12*(15*b^4*c^4 - 100*a*b^2*c^5 + 128*a^2*c^6)*e + (105
*b^5*c^3 - 760*a*b^3*c^4 + 1296*a^2*b*c^5)*f)*g^2 + 32*(168*(15*b^4*c^4 - 100*a*b^2*c^5 + 128*a^2*c^6)*d - 14*
(105*b^5*c^3 - 760*a*b^3*c^4 + 1296*a^2*b*c^5)*e + 3*(315*b^6*c^2 - 2520*a*b^4*c^3 + 5488*a^2*b^2*c^4 - 2048*a
^3*c^5)*f)*g*h - (224*(105*b^5*c^3 - 760*a*b^3*c^4 + 1296*a^2*b*c^5)*d - 48*(315*b^6*c^2 - 2520*a*b^4*c^3 + 54
88*a^2*b^2*c^4 - 2048*a^3*c^5)*e + 3*(3465*b^7*c - 30660*a*b^5*c^2 + 81648*a^2*b^3*c^3 - 58816*a^3*b*c^4)*f)*h
^2 + 8*(224*(360*b*c^7*d + 12*(b^2*c^6 + 32*a*c^7)*e - (7*b^3*c^5 - 36*a*b*c^6)*f)*g^2 + 32*(168*(b^2*c^6 + 32
*a*c^7)*d - 14*(7*b^3*c^5 - 36*a*b*c^6)*e + 3*(21*b^4*c^4 - 124*a*b^2*c^5 + 128*a^2*c^6)*f)*g*h - (224*(7*b^3*
c^5 - 36*a*b*c^6)*d - 48*(21*b^4*c^4 - 124*a*b^2*c^5 + 128*a^2*c^6)*e + 3*(231*b^5*c^3 - 1560*a*b^3*c^4 + 2416
*a^2*b*c^5)*f)*h^2)*x^2 + 2*(224*(120*(b^2*c^6 + 20*a*c^7)*d - 12*(5*b^3*c^5 - 28*a*b*c^6)*e + (35*b^4*c^4 - 2
16*a*b^2*c^5 + 240*a^2*c^6)*f)*g^2 - 32*(168*(5*b^3*c^5 - 28*a*b*c^6)*d - 14*(35*b^4*c^4 - 216*a*b^2*c^5 + 240
*a^2*c^6)*e + 3*(105*b^5*c^3 - 728*a*b^3*c^4 + 1168*a^2*b*c^5)*f)*g*h + (224*(35*b^4*c^4 - 216*a*b^2*c^5 + 240
*a^2*c^6)*d - 48*(105*b^5*c^3 - 728*a*b^3*c^4 + 1168*a^2*b*c^5)*e + 3*(1155*b^6*c^2 - 8988*a*b^4*c^3 + 18896*a
^2*b^2*c^4 - 6720*a^3*c^5)*f)*h^2)*x)*sqrt(c*x^2 + b*x + a))/c^7, -1/3440640*(105*(32*(24*(b^4*c^4 - 8*a*b^2*c
^5 + 16*a^2*c^6)*d - 12*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*e + (7*b^6*c^2 - 60*a*b^4*c^3 + 144*a^2*b^2*c^4
- 64*a^3*c^5)*f)*g^2 - 32*(24*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*d - 2*(7*b^6*c^2 - 60*a*b^4*c^3 + 144*a^
2*b^2*c^4 - 64*a^3*c^5)*e + 3*(3*b^7*c - 28*a*b^5*c^2 + 80*a^2*b^3*c^3 - 64*a^3*b*c^4)*f)*g*h + (32*(7*b^6*c^2
- 60*a*b^4*c^3 + 144*a^2*b^2*c^4 - 64*a^3*c^5)*d - 48*(3*b^7*c - 28*a*b^5*c^2 + 80*a^2*b^3*c^3 - 64*a^3*b*c^4
)*e + 3*(33*b^8 - 336*a*b^6*c + 1120*a^2*b^4*c^2 - 1280*a^3*b^2*c^3 + 256*a^4*c^4)*f)*h^2)*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*(215040*c^8*f*h^2*x^7 + 15360*(32*c^8
*f*g*h + (16*c^8*e + 17*b*c^7*f)*h^2)*x^6 + 1280*(224*c^8*f*g^2 + 32*(14*c^8*e + 15*b*c^7*f)*g*h + (224*c^8*d
+ 240*b*c^7*e + 3*(b^2*c^6 + 84*a*c^7)*f)*h^2)*x^5 + 128*(224*(12*c^8*e + 13*b*c^7*f)*g^2 + 32*(168*c^8*d + 18
2*b*c^7*e + 3*(b^2*c^6 + 64*a*c^7)*f)*g*h + (2912*b*c^7*d + 48*(b^2*c^6 + 64*a*c^7)*e - 3*(11*b^3*c^5 - 52*a*b
*c^6)*f)*h^2)*x^4 + 16*(224*(120*c^8*d + 132*b*c^7*e + (3*b^2*c^6 + 140*a*c^7)*f)*g^2 + 32*(1848*b*c^7*d + 14*
(3*b^2*c^6 + 140*a*c^7)*e - 3*(9*b^3*c^5 - 44*a*b*c^6)*f)*g*h + (224*(3*b^2*c^6 + 140*a*c^7)*d - 48*(9*b^3*c^5
- 44*a*b*c^6)*e + 3*(99*b^4*c^4 - 568*a*b^2*c^5 + 560*a^2*c^6)*f)*h^2)*x^3 - 224*(120*(3*b^3*c^5 - 20*a*b*c^6
)*d - 12*(15*b^4*c^4 - 100*a*b^2*c^5 + 128*a^2*c^6)*e + (105*b^5*c^3 - 760*a*b^3*c^4 + 1296*a^2*b*c^5)*f)*g^2
+ 32*(168*(15*b^4*c^4 - 100*a*b^2*c^5 + 128*a^2*c^6)*d - 14*(105*b^5*c^3 - 760*a*b^3*c^4 + 1296*a^2*b*c^5)*e +
3*(315*b^6*c^2 - 2520*a*b^4*c^3 + 5488*a^2*b^2*c^4 - 2048*a^3*c^5)*f)*g*h - (224*(105*b^5*c^3 - 760*a*b^3*c^4
+ 1296*a^2*b*c^5)*d - 48*(315*b^6*c^2 - 2520*a*b^4*c^3 + 5488*a^2*b^2*c^4 - 2048*a^3*c^5)*e + 3*(3465*b^7*c -
30660*a*b^5*c^2 + 81648*a^2*b^3*c^3 - 58816*a^3*b*c^4)*f)*h^2 + 8*(224*(360*b*c^7*d + 12*(b^2*c^6 + 32*a*c^7)
*e - (7*b^3*c^5 - 36*a*b*c^6)*f)*g^2 + 32*(168*(b^2*c^6 + 32*a*c^7)*d - 14*(7*b^3*c^5 - 36*a*b*c^6)*e + 3*(21*
b^4*c^4 - 124*a*b^2*c^5 + 128*a^2*c^6)*f)*g*h - (224*(7*b^3*c^5 - 36*a*b*c^6)*d - 48*(21*b^4*c^4 - 124*a*b^2*c
^5 + 128*a^2*c^6)*e + 3*(231*b^5*c^3 - 1560*a*b^3*c^4 + 2416*a^2*b*c^5)*f)*h^2)*x^2 + 2*(224*(120*(b^2*c^6 + 2
0*a*c^7)*d - 12*(5*b^3*c^5 - 28*a*b*c^6)*e + (35*b^4*c^4 - 216*a*b^2*c^5 + 240*a^2*c^6)*f)*g^2 - 32*(168*(5*b^
3*c^5 - 28*a*b*c^6)*d - 14*(35*b^4*c^4 - 216*a*b^2*c^5 + 240*a^2*c^6)*e + 3*(105*b^5*c^3 - 728*a*b^3*c^4 + 116
8*a^2*b*c^5)*f)*g*h + (224*(35*b^4*c^4 - 216*a*b^2*c^5 + 240*a^2*c^6)*d - 48*(105*b^5*c^3 - 728*a*b^3*c^4 + 11
68*a^2*b*c^5)*e + 3*(1155*b^6*c^2 - 8988*a*b^4*c^3 + 18896*a^2*b^2*c^4 - 6720*a^3*c^5)*f)*h^2)*x)*sqrt(c*x^2 +
b*x + a))/c^7]

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (g + h x\right )^{2} \left (a + b x + c x^{2}\right )^{\frac{3}{2}} \left (d + e x + f x^{2}\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)**2*(c*x**2+b*x+a)**(3/2)*(f*x**2+e*x+d),x)

[Out]

Integral((g + h*x)**2*(a + b*x + c*x**2)**(3/2)*(d + e*x + f*x**2), x)

________________________________________________________________________________________

Giac [B]  time = 1.20415, size = 2500, normalized size = 3.32 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)^2*(c*x^2+b*x+a)^(3/2)*(f*x^2+e*x+d),x, algorithm="giac")

[Out]

1/1720320*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*(12*(14*c*f*h^2*x + (32*c^8*f*g*h + 17*b*c^7*f*h^2 + 16*c^8*h^
2*e)/c^7)*x + (224*c^8*f*g^2 + 480*b*c^7*f*g*h + 224*c^8*d*h^2 + 3*b^2*c^6*f*h^2 + 252*a*c^7*f*h^2 + 448*c^8*g
*h*e + 240*b*c^7*h^2*e)/c^7)*x + (2912*b*c^7*f*g^2 + 5376*c^8*d*g*h + 96*b^2*c^6*f*g*h + 6144*a*c^7*f*g*h + 29
12*b*c^7*d*h^2 - 33*b^3*c^5*f*h^2 + 156*a*b*c^6*f*h^2 + 2688*c^8*g^2*e + 5824*b*c^7*g*h*e + 48*b^2*c^6*h^2*e +
3072*a*c^7*h^2*e)/c^7)*x + (26880*c^8*d*g^2 + 672*b^2*c^6*f*g^2 + 31360*a*c^7*f*g^2 + 59136*b*c^7*d*g*h - 864
*b^3*c^5*f*g*h + 4224*a*b*c^6*f*g*h + 672*b^2*c^6*d*h^2 + 31360*a*c^7*d*h^2 + 297*b^4*c^4*f*h^2 - 1704*a*b^2*c
^5*f*h^2 + 1680*a^2*c^6*f*h^2 + 29568*b*c^7*g^2*e + 1344*b^2*c^6*g*h*e + 62720*a*c^7*g*h*e - 432*b^3*c^5*h^2*e
+ 2112*a*b*c^6*h^2*e)/c^7)*x + (80640*b*c^7*d*g^2 - 1568*b^3*c^5*f*g^2 + 8064*a*b*c^6*f*g^2 + 5376*b^2*c^6*d*
g*h + 172032*a*c^7*d*g*h + 2016*b^4*c^4*f*g*h - 11904*a*b^2*c^5*f*g*h + 12288*a^2*c^6*f*g*h - 1568*b^3*c^5*d*h
^2 + 8064*a*b*c^6*d*h^2 - 693*b^5*c^3*f*h^2 + 4680*a*b^3*c^4*f*h^2 - 7248*a^2*b*c^5*f*h^2 + 2688*b^2*c^6*g^2*e
+ 86016*a*c^7*g^2*e - 3136*b^3*c^5*g*h*e + 16128*a*b*c^6*g*h*e + 1008*b^4*c^4*h^2*e - 5952*a*b^2*c^5*h^2*e +
6144*a^2*c^6*h^2*e)/c^7)*x + (26880*b^2*c^6*d*g^2 + 537600*a*c^7*d*g^2 + 7840*b^4*c^4*f*g^2 - 48384*a*b^2*c^5*
f*g^2 + 53760*a^2*c^6*f*g^2 - 26880*b^3*c^5*d*g*h + 150528*a*b*c^6*d*g*h - 10080*b^5*c^3*f*g*h + 69888*a*b^3*c
^4*f*g*h - 112128*a^2*b*c^5*f*g*h + 7840*b^4*c^4*d*h^2 - 48384*a*b^2*c^5*d*h^2 + 53760*a^2*c^6*d*h^2 + 3465*b^
6*c^2*f*h^2 - 26964*a*b^4*c^3*f*h^2 + 56688*a^2*b^2*c^4*f*h^2 - 20160*a^3*c^5*f*h^2 - 13440*b^3*c^5*g^2*e + 75
264*a*b*c^6*g^2*e + 15680*b^4*c^4*g*h*e - 96768*a*b^2*c^5*g*h*e + 107520*a^2*c^6*g*h*e - 5040*b^5*c^3*h^2*e +
34944*a*b^3*c^4*h^2*e - 56064*a^2*b*c^5*h^2*e)/c^7)*x - (80640*b^3*c^5*d*g^2 - 537600*a*b*c^6*d*g^2 + 23520*b^
5*c^3*f*g^2 - 170240*a*b^3*c^4*f*g^2 + 290304*a^2*b*c^5*f*g^2 - 80640*b^4*c^4*d*g*h + 537600*a*b^2*c^5*d*g*h -
688128*a^2*c^6*d*g*h - 30240*b^6*c^2*f*g*h + 241920*a*b^4*c^3*f*g*h - 526848*a^2*b^2*c^4*f*g*h + 196608*a^3*c
^5*f*g*h + 23520*b^5*c^3*d*h^2 - 170240*a*b^3*c^4*d*h^2 + 290304*a^2*b*c^5*d*h^2 + 10395*b^7*c*f*h^2 - 91980*a
*b^5*c^2*f*h^2 + 244944*a^2*b^3*c^3*f*h^2 - 176448*a^3*b*c^4*f*h^2 - 40320*b^4*c^4*g^2*e + 268800*a*b^2*c^5*g^
2*e - 344064*a^2*c^6*g^2*e + 47040*b^5*c^3*g*h*e - 340480*a*b^3*c^4*g*h*e + 580608*a^2*b*c^5*g*h*e - 15120*b^6
*c^2*h^2*e + 120960*a*b^4*c^3*h^2*e - 263424*a^2*b^2*c^4*h^2*e + 98304*a^3*c^5*h^2*e)/c^7) - 1/32768*(768*b^4*
c^4*d*g^2 - 6144*a*b^2*c^5*d*g^2 + 12288*a^2*c^6*d*g^2 + 224*b^6*c^2*f*g^2 - 1920*a*b^4*c^3*f*g^2 + 4608*a^2*b
^2*c^4*f*g^2 - 2048*a^3*c^5*f*g^2 - 768*b^5*c^3*d*g*h + 6144*a*b^3*c^4*d*g*h - 12288*a^2*b*c^5*d*g*h - 288*b^7
*c*f*g*h + 2688*a*b^5*c^2*f*g*h - 7680*a^2*b^3*c^3*f*g*h + 6144*a^3*b*c^4*f*g*h + 224*b^6*c^2*d*h^2 - 1920*a*b
^4*c^3*d*h^2 + 4608*a^2*b^2*c^4*d*h^2 - 2048*a^3*c^5*d*h^2 + 99*b^8*f*h^2 - 1008*a*b^6*c*f*h^2 + 3360*a^2*b^4*
c^2*f*h^2 - 3840*a^3*b^2*c^3*f*h^2 + 768*a^4*c^4*f*h^2 - 384*b^5*c^3*g^2*e + 3072*a*b^3*c^4*g^2*e - 6144*a^2*b
*c^5*g^2*e + 448*b^6*c^2*g*h*e - 3840*a*b^4*c^3*g*h*e + 9216*a^2*b^2*c^4*g*h*e - 4096*a^3*c^5*g*h*e - 144*b^7*
c*h^2*e + 1344*a*b^5*c^2*h^2*e - 3840*a^2*b^3*c^3*h^2*e + 3072*a^3*b*c^4*h^2*e)*log(abs(-2*(sqrt(c)*x - sqrt(c
*x^2 + b*x + a))*sqrt(c) - b))/c^(13/2)