### 3.186 $$\int (g+h x)^3 \sqrt{a+b x+c x^2} (d+e x+f x^2) \, dx$$

Optimal. Leaf size=930 $\frac{f \left (c x^2+b x+a\right )^{3/2} (g+h x)^4}{7 c h}-\frac{(6 c f g-14 c e h+11 b f h) \left (c x^2+b x+a\right )^{3/2} (g+h x)^3}{84 c^2 h}+\frac{\left (-4 \left (3 f g^2-7 h (e g+2 d h)\right ) c^2-2 h (8 b f g+21 b e h+16 a f h) c+33 b^2 f h^2\right ) \left (c x^2+b x+a\right )^{3/2} (g+h x)^2}{280 c^3 h}+\frac{\left (-128 g^2 \left (3 f g^2-7 h (e g+12 d h)\right ) c^4-16 h \left (16 a h \left (15 f g^2+7 h (3 e g+d h)\right )+b g \left (17 f g^2+21 h (19 e g+25 d h)\right )\right ) c^3+8 h^2 \left (\left (537 f g^2+245 h (3 e g+d h)\right ) b^2+343 a h (3 f g+e h) b+128 a^2 f h^2\right ) c^2-42 b^2 h^3 (78 a f h+35 b (3 f g+e h)) c-6 h \left (16 g \left (3 f g^2-7 h (e g+7 d h)\right ) c^3+8 h \left (a h (41 f g+35 e h)+b \left (5 f g^2+7 h (9 e g+7 d h)\right )\right ) c^2-6 b h^2 (59 b f g+49 b e h+74 a f h) c+231 b^3 f h^3\right ) x c+1155 b^4 f h^4\right ) \left (c x^2+b x+a\right )^{3/2}}{13440 c^5 h}-\frac{\left (b^2-4 a c\right ) \left (-33 f h^3 b^5+6 c h^2 (20 a f h+7 b (3 f g+e h)) b^3-8 c^2 h \left (7 \left (3 f g^2+3 e h g+d h^2\right ) b^2+14 a h (3 f g+e h) b+10 a^2 f h^2\right ) b+256 c^5 d g^3-64 c^4 g \left (2 b g (e g+3 d h)+a \left (f g^2+3 h (e g+d h)\right )\right )+16 c^3 \left (5 g \left (f g^2+3 h (e g+d h)\right ) b^2+6 a h \left (3 f g^2+h (3 e g+d h)\right ) b+2 a^2 h^2 (3 f g+e h)\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right )}{2048 c^{13/2}}+\frac{\left (-33 f h^3 b^5+6 c h^2 (20 a f h+7 b (3 f g+e h)) b^3-8 c^2 h \left (7 \left (3 f g^2+3 e h g+d h^2\right ) b^2+14 a h (3 f g+e h) b+10 a^2 f h^2\right ) b+256 c^5 d g^3-64 c^4 g \left (2 b g (e g+3 d h)+a \left (f g^2+3 h (e g+d h)\right )\right )+16 c^3 \left (5 g \left (f g^2+3 h (e g+d h)\right ) b^2+6 a h \left (3 f g^2+h (3 e g+d h)\right ) b+2 a^2 h^2 (3 f g+e h)\right )\right ) (b+2 c x) \sqrt{c x^2+b x+a}}{1024 c^6}$

[Out]

((256*c^5*d*g^3 - 33*b^5*f*h^3 + 6*b^3*c*h^2*(20*a*f*h + 7*b*(3*f*g + e*h)) - 8*b*c^2*h*(10*a^2*f*h^2 + 14*a*b
*h*(3*f*g + e*h) + 7*b^2*(3*f*g^2 + 3*e*g*h + d*h^2)) - 64*c^4*g*(2*b*g*(e*g + 3*d*h) + a*(f*g^2 + 3*h*(e*g +
d*h))) + 16*c^3*(2*a^2*h^2*(3*f*g + e*h) + 5*b^2*g*(f*g^2 + 3*h*(e*g + d*h)) + 6*a*b*h*(3*f*g^2 + h*(3*e*g + d
*h))))*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(1024*c^6) + ((33*b^2*f*h^2 - 2*c*h*(8*b*f*g + 21*b*e*h + 16*a*f*h)
- 4*c^2*(3*f*g^2 - 7*h*(e*g + 2*d*h)))*(g + h*x)^2*(a + b*x + c*x^2)^(3/2))/(280*c^3*h) - ((6*c*f*g - 14*c*e*h
+ 11*b*f*h)*(g + h*x)^3*(a + b*x + c*x^2)^(3/2))/(84*c^2*h) + (f*(g + h*x)^4*(a + b*x + c*x^2)^(3/2))/(7*c*h)
+ ((1155*b^4*f*h^4 - 128*c^4*g^2*(3*f*g^2 - 7*h*(e*g + 12*d*h)) - 42*b^2*c*h^3*(78*a*f*h + 35*b*(3*f*g + e*h)
) + 8*c^2*h^2*(128*a^2*f*h^2 + 343*a*b*h*(3*f*g + e*h) + b^2*(537*f*g^2 + 245*h*(3*e*g + d*h))) - 16*c^3*h*(16
*a*h*(15*f*g^2 + 7*h*(3*e*g + d*h)) + b*g*(17*f*g^2 + 21*h*(19*e*g + 25*d*h))) - 6*c*h*(231*b^3*f*h^3 - 6*b*c*
h^2*(59*b*f*g + 49*b*e*h + 74*a*f*h) + 16*c^3*g*(3*f*g^2 - 7*h*(e*g + 7*d*h)) + 8*c^2*h*(a*h*(41*f*g + 35*e*h)
+ b*(5*f*g^2 + 7*h*(9*e*g + 7*d*h))))*x)*(a + b*x + c*x^2)^(3/2))/(13440*c^5*h) - ((b^2 - 4*a*c)*(256*c^5*d*g
^3 - 33*b^5*f*h^3 + 6*b^3*c*h^2*(20*a*f*h + 7*b*(3*f*g + e*h)) - 8*b*c^2*h*(10*a^2*f*h^2 + 14*a*b*h*(3*f*g + e
*h) + 7*b^2*(3*f*g^2 + 3*e*g*h + d*h^2)) - 64*c^4*g*(2*b*g*(e*g + 3*d*h) + a*(f*g^2 + 3*h*(e*g + d*h))) + 16*c
^3*(2*a^2*h^2*(3*f*g + e*h) + 5*b^2*g*(f*g^2 + 3*h*(e*g + d*h)) + 6*a*b*h*(3*f*g^2 + h*(3*e*g + d*h))))*ArcTan
h[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2048*c^(13/2))

________________________________________________________________________________________

Rubi [A]  time = 3.01272, antiderivative size = 927, normalized size of antiderivative = 1., 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} $\frac{f \left (c x^2+b x+a\right )^{3/2} (g+h x)^4}{7 c h}-\frac{(6 c f g-14 c e h+11 b f h) \left (c x^2+b x+a\right )^{3/2} (g+h x)^3}{84 c^2 h}+\frac{\left (-4 \left (3 f g^2-7 h (e g+2 d h)\right ) c^2-2 h (8 b f g+21 b e h+16 a f h) c+33 b^2 f h^2\right ) \left (c x^2+b x+a\right )^{3/2} (g+h x)^2}{280 c^3 h}+\frac{\left (-128 \left (3 f g^4-7 g^2 h (e g+12 d h)\right ) c^4-16 h \left (16 a h \left (15 f g^2+7 h (3 e g+d h)\right )+b g \left (17 f g^2+21 h (19 e g+25 d h)\right )\right ) c^3+8 h^2 \left (\left (537 f g^2+245 h (3 e g+d h)\right ) b^2+343 a h (3 f g+e h) b+128 a^2 f h^2\right ) c^2-42 b^2 h^3 (78 a f h+35 b (3 f g+e h)) c-6 h \left (16 \left (3 f g^3-7 g h (e g+7 d h)\right ) c^3+8 h \left (5 b f g^2+7 b h (9 e g+7 d h)+a h (41 f g+35 e h)\right ) c^2-6 b h^2 (59 b f g+49 b e h+74 a f h) c+231 b^3 f h^3\right ) x c+1155 b^4 f h^4\right ) \left (c x^2+b x+a\right )^{3/2}}{13440 c^5 h}-\frac{\left (b^2-4 a c\right ) \left (-33 f h^3 b^5+6 c h^2 (20 a f h+7 b (3 f g+e h)) b^3-8 c^2 h \left (7 \left (3 f g^2+3 e h g+d h^2\right ) b^2+14 a h (3 f g+e h) b+10 a^2 f h^2\right ) b+256 c^5 d g^3-64 c^4 g \left (a f g^2+2 b (e g+3 d h) g+3 a h (e g+d h)\right )+16 c^3 \left (5 g \left (f g^2+3 h (e g+d h)\right ) b^2+6 a h \left (3 f g^2+h (3 e g+d h)\right ) b+2 a^2 h^2 (3 f g+e h)\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right )}{2048 c^{13/2}}+\frac{\left (-33 f h^3 b^5+6 c h^2 (20 a f h+7 b (3 f g+e h)) b^3-8 c^2 h \left (7 \left (3 f g^2+3 e h g+d h^2\right ) b^2+14 a h (3 f g+e h) b+10 a^2 f h^2\right ) b+256 c^5 d g^3-64 c^4 g \left (a f g^2+2 b (e g+3 d h) g+3 a h (e g+d h)\right )+16 c^3 \left (5 g \left (f g^2+3 h (e g+d h)\right ) b^2+6 a h \left (3 f g^2+h (3 e g+d h)\right ) b+2 a^2 h^2 (3 f g+e h)\right )\right ) (b+2 c x) \sqrt{c x^2+b x+a}}{1024 c^6}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((256*c^5*d*g^3 - 33*b^5*f*h^3 - 64*c^4*g*(a*f*g^2 + 3*a*h*(e*g + d*h) + 2*b*g*(e*g + 3*d*h)) + 6*b^3*c*h^2*(2
0*a*f*h + 7*b*(3*f*g + e*h)) - 8*b*c^2*h*(10*a^2*f*h^2 + 14*a*b*h*(3*f*g + e*h) + 7*b^2*(3*f*g^2 + 3*e*g*h + d
*h^2)) + 16*c^3*(2*a^2*h^2*(3*f*g + e*h) + 5*b^2*g*(f*g^2 + 3*h*(e*g + d*h)) + 6*a*b*h*(3*f*g^2 + h*(3*e*g + d
*h))))*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(1024*c^6) + ((33*b^2*f*h^2 - 2*c*h*(8*b*f*g + 21*b*e*h + 16*a*f*h)
- 4*c^2*(3*f*g^2 - 7*h*(e*g + 2*d*h)))*(g + h*x)^2*(a + b*x + c*x^2)^(3/2))/(280*c^3*h) - ((6*c*f*g - 14*c*e*h
+ 11*b*f*h)*(g + h*x)^3*(a + b*x + c*x^2)^(3/2))/(84*c^2*h) + (f*(g + h*x)^4*(a + b*x + c*x^2)^(3/2))/(7*c*h)
+ ((1155*b^4*f*h^4 - 128*c^4*(3*f*g^4 - 7*g^2*h*(e*g + 12*d*h)) - 42*b^2*c*h^3*(78*a*f*h + 35*b*(3*f*g + e*h)
) + 8*c^2*h^2*(128*a^2*f*h^2 + 343*a*b*h*(3*f*g + e*h) + b^2*(537*f*g^2 + 245*h*(3*e*g + d*h))) - 16*c^3*h*(16
*a*h*(15*f*g^2 + 7*h*(3*e*g + d*h)) + b*g*(17*f*g^2 + 21*h*(19*e*g + 25*d*h))) - 6*c*h*(231*b^3*f*h^3 - 6*b*c*
h^2*(59*b*f*g + 49*b*e*h + 74*a*f*h) + 16*c^3*(3*f*g^3 - 7*g*h*(e*g + 7*d*h)) + 8*c^2*h*(5*b*f*g^2 + 7*b*h*(9*
e*g + 7*d*h) + a*h*(41*f*g + 35*e*h)))*x)*(a + b*x + c*x^2)^(3/2))/(13440*c^5*h) - ((b^2 - 4*a*c)*(256*c^5*d*g
^3 - 33*b^5*f*h^3 - 64*c^4*g*(a*f*g^2 + 3*a*h*(e*g + d*h) + 2*b*g*(e*g + 3*d*h)) + 6*b^3*c*h^2*(20*a*f*h + 7*b
*(3*f*g + e*h)) - 8*b*c^2*h*(10*a^2*f*h^2 + 14*a*b*h*(3*f*g + e*h) + 7*b^2*(3*f*g^2 + 3*e*g*h + d*h^2)) + 16*c
^3*(2*a^2*h^2*(3*f*g + e*h) + 5*b^2*g*(f*g^2 + 3*h*(e*g + d*h)) + 6*a*b*h*(3*f*g^2 + h*(3*e*g + d*h))))*ArcTan
h[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2048*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)^3 \sqrt{a+b x+c x^2} \left (d+e x+f x^2\right ) \, dx &=\frac{f (g+h x)^4 \left (a+b x+c x^2\right )^{3/2}}{7 c h}+\frac{\int (g+h x)^3 \left (-\frac{1}{2} h (3 b f g-14 c d h+8 a f h)-\frac{1}{2} h (6 c f g-14 c e h+11 b f h) x\right ) \sqrt{a+b x+c x^2} \, dx}{7 c h^2}\\ &=-\frac{(6 c f g-14 c e h+11 b f h) (g+h x)^3 \left (a+b x+c x^2\right )^{3/2}}{84 c^2 h}+\frac{f (g+h x)^4 \left (a+b x+c x^2\right )^{3/2}}{7 c h}+\frac{\int (g+h x)^2 \left (\frac{3}{4} h \left (11 b^2 f g h+22 a b f h^2-2 b c g (3 f g+7 e h)+4 c h (14 c d g-5 a f g-7 a e h)\right )+\frac{3}{4} h \left (33 b^2 f h^2-2 c h (8 b f g+21 b e h+16 a f h)-4 c^2 \left (3 f g^2-7 h (e g+2 d h)\right )\right ) x\right ) \sqrt{a+b x+c x^2} \, dx}{42 c^2 h^2}\\ &=\frac{\left (33 b^2 f h^2-2 c h (8 b f g+21 b e h+16 a f h)-4 c^2 \left (3 f g^2-7 h (e g+2 d h)\right )\right ) (g+h x)^2 \left (a+b x+c x^2\right )^{3/2}}{280 c^3 h}-\frac{(6 c f g-14 c e h+11 b f h) (g+h x)^3 \left (a+b x+c x^2\right )^{3/2}}{84 c^2 h}+\frac{f (g+h x)^4 \left (a+b x+c x^2\right )^{3/2}}{7 c h}+\frac{\int (g+h x) \left (-\frac{3}{8} h \left (99 b^3 f g h^2+4 b c \left (6 c f g^3+14 c g h (4 e g+3 d h)-a h^2 (95 f g+42 e h)\right )+2 b^2 \left (66 a f h^3-c g h (79 f g+63 e h)\right )-8 c h \left (70 c^2 d g^2+16 a^2 f h^2-a c \left (19 f g^2+7 h (7 e g+4 d h)\right )\right )\right )-\frac{3}{8} h \left (231 b^3 f h^3-6 b c h^2 (59 b f g+49 b e h+74 a f h)+16 c^3 \left (3 f g^3-7 g h (e g+7 d h)\right )+8 c^2 h \left (5 b f g^2+7 b h (9 e g+7 d h)+a h (41 f g+35 e h)\right )\right ) x\right ) \sqrt{a+b x+c x^2} \, dx}{210 c^3 h^2}\\ &=\frac{\left (33 b^2 f h^2-2 c h (8 b f g+21 b e h+16 a f h)-4 c^2 \left (3 f g^2-7 h (e g+2 d h)\right )\right ) (g+h x)^2 \left (a+b x+c x^2\right )^{3/2}}{280 c^3 h}-\frac{(6 c f g-14 c e h+11 b f h) (g+h x)^3 \left (a+b x+c x^2\right )^{3/2}}{84 c^2 h}+\frac{f (g+h x)^4 \left (a+b x+c x^2\right )^{3/2}}{7 c h}+\frac{\left (1155 b^4 f h^4-128 c^4 \left (3 f g^4-7 g^2 h (e g+12 d h)\right )-42 b^2 c h^3 (78 a f h+35 b (3 f g+e h))+8 c^2 h^2 \left (128 a^2 f h^2+343 a b h (3 f g+e h)+b^2 \left (537 f g^2+245 h (3 e g+d h)\right )\right )-16 c^3 h \left (16 a h \left (15 f g^2+7 h (3 e g+d h)\right )+b g \left (17 f g^2+21 h (19 e g+25 d h)\right )\right )-6 c h \left (231 b^3 f h^3-6 b c h^2 (59 b f g+49 b e h+74 a f h)+16 c^3 \left (3 f g^3-7 g h (e g+7 d h)\right )+8 c^2 h \left (5 b f g^2+7 b h (9 e g+7 d h)+a h (41 f g+35 e h)\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{13440 c^5 h}+\frac{\left (256 c^5 d g^3-33 b^5 f h^3-64 c^4 g \left (a f g^2+3 a h (e g+d h)+2 b g (e g+3 d h)\right )+6 b^3 c h^2 (20 a f h+7 b (3 f g+e h))-8 b c^2 h \left (10 a^2 f h^2+14 a b h (3 f g+e h)+7 b^2 \left (3 f g^2+3 e g h+d h^2\right )\right )+16 c^3 \left (2 a^2 h^2 (3 f g+e h)+5 b^2 g \left (f g^2+3 h (e g+d h)\right )+6 a b h \left (3 f g^2+h (3 e g+d h)\right )\right )\right ) \int \sqrt{a+b x+c x^2} \, dx}{256 c^5}\\ &=\frac{\left (256 c^5 d g^3-33 b^5 f h^3-64 c^4 g \left (a f g^2+3 a h (e g+d h)+2 b g (e g+3 d h)\right )+6 b^3 c h^2 (20 a f h+7 b (3 f g+e h))-8 b c^2 h \left (10 a^2 f h^2+14 a b h (3 f g+e h)+7 b^2 \left (3 f g^2+3 e g h+d h^2\right )\right )+16 c^3 \left (2 a^2 h^2 (3 f g+e h)+5 b^2 g \left (f g^2+3 h (e g+d h)\right )+6 a b h \left (3 f g^2+h (3 e g+d h)\right )\right )\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{1024 c^6}+\frac{\left (33 b^2 f h^2-2 c h (8 b f g+21 b e h+16 a f h)-4 c^2 \left (3 f g^2-7 h (e g+2 d h)\right )\right ) (g+h x)^2 \left (a+b x+c x^2\right )^{3/2}}{280 c^3 h}-\frac{(6 c f g-14 c e h+11 b f h) (g+h x)^3 \left (a+b x+c x^2\right )^{3/2}}{84 c^2 h}+\frac{f (g+h x)^4 \left (a+b x+c x^2\right )^{3/2}}{7 c h}+\frac{\left (1155 b^4 f h^4-128 c^4 \left (3 f g^4-7 g^2 h (e g+12 d h)\right )-42 b^2 c h^3 (78 a f h+35 b (3 f g+e h))+8 c^2 h^2 \left (128 a^2 f h^2+343 a b h (3 f g+e h)+b^2 \left (537 f g^2+245 h (3 e g+d h)\right )\right )-16 c^3 h \left (16 a h \left (15 f g^2+7 h (3 e g+d h)\right )+b g \left (17 f g^2+21 h (19 e g+25 d h)\right )\right )-6 c h \left (231 b^3 f h^3-6 b c h^2 (59 b f g+49 b e h+74 a f h)+16 c^3 \left (3 f g^3-7 g h (e g+7 d h)\right )+8 c^2 h \left (5 b f g^2+7 b h (9 e g+7 d h)+a h (41 f g+35 e h)\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{13440 c^5 h}-\frac{\left (\left (b^2-4 a c\right ) \left (256 c^5 d g^3-33 b^5 f h^3-64 c^4 g \left (a f g^2+3 a h (e g+d h)+2 b g (e g+3 d h)\right )+6 b^3 c h^2 (20 a f h+7 b (3 f g+e h))-8 b c^2 h \left (10 a^2 f h^2+14 a b h (3 f g+e h)+7 b^2 \left (3 f g^2+3 e g h+d h^2\right )\right )+16 c^3 \left (2 a^2 h^2 (3 f g+e h)+5 b^2 g \left (f g^2+3 h (e g+d h)\right )+6 a b h \left (3 f g^2+h (3 e g+d h)\right )\right )\right )\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{2048 c^6}\\ &=\frac{\left (256 c^5 d g^3-33 b^5 f h^3-64 c^4 g \left (a f g^2+3 a h (e g+d h)+2 b g (e g+3 d h)\right )+6 b^3 c h^2 (20 a f h+7 b (3 f g+e h))-8 b c^2 h \left (10 a^2 f h^2+14 a b h (3 f g+e h)+7 b^2 \left (3 f g^2+3 e g h+d h^2\right )\right )+16 c^3 \left (2 a^2 h^2 (3 f g+e h)+5 b^2 g \left (f g^2+3 h (e g+d h)\right )+6 a b h \left (3 f g^2+h (3 e g+d h)\right )\right )\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{1024 c^6}+\frac{\left (33 b^2 f h^2-2 c h (8 b f g+21 b e h+16 a f h)-4 c^2 \left (3 f g^2-7 h (e g+2 d h)\right )\right ) (g+h x)^2 \left (a+b x+c x^2\right )^{3/2}}{280 c^3 h}-\frac{(6 c f g-14 c e h+11 b f h) (g+h x)^3 \left (a+b x+c x^2\right )^{3/2}}{84 c^2 h}+\frac{f (g+h x)^4 \left (a+b x+c x^2\right )^{3/2}}{7 c h}+\frac{\left (1155 b^4 f h^4-128 c^4 \left (3 f g^4-7 g^2 h (e g+12 d h)\right )-42 b^2 c h^3 (78 a f h+35 b (3 f g+e h))+8 c^2 h^2 \left (128 a^2 f h^2+343 a b h (3 f g+e h)+b^2 \left (537 f g^2+245 h (3 e g+d h)\right )\right )-16 c^3 h \left (16 a h \left (15 f g^2+7 h (3 e g+d h)\right )+b g \left (17 f g^2+21 h (19 e g+25 d h)\right )\right )-6 c h \left (231 b^3 f h^3-6 b c h^2 (59 b f g+49 b e h+74 a f h)+16 c^3 \left (3 f g^3-7 g h (e g+7 d h)\right )+8 c^2 h \left (5 b f g^2+7 b h (9 e g+7 d h)+a h (41 f g+35 e h)\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{13440 c^5 h}-\frac{\left (\left (b^2-4 a c\right ) \left (256 c^5 d g^3-33 b^5 f h^3-64 c^4 g \left (a f g^2+3 a h (e g+d h)+2 b g (e g+3 d h)\right )+6 b^3 c h^2 (20 a f h+7 b (3 f g+e h))-8 b c^2 h \left (10 a^2 f h^2+14 a b h (3 f g+e h)+7 b^2 \left (3 f g^2+3 e g h+d h^2\right )\right )+16 c^3 \left (2 a^2 h^2 (3 f g+e h)+5 b^2 g \left (f g^2+3 h (e g+d h)\right )+6 a b h \left (3 f g^2+h (3 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 )}{1024 c^6}\\ &=\frac{\left (256 c^5 d g^3-33 b^5 f h^3-64 c^4 g \left (a f g^2+3 a h (e g+d h)+2 b g (e g+3 d h)\right )+6 b^3 c h^2 (20 a f h+7 b (3 f g+e h))-8 b c^2 h \left (10 a^2 f h^2+14 a b h (3 f g+e h)+7 b^2 \left (3 f g^2+3 e g h+d h^2\right )\right )+16 c^3 \left (2 a^2 h^2 (3 f g+e h)+5 b^2 g \left (f g^2+3 h (e g+d h)\right )+6 a b h \left (3 f g^2+h (3 e g+d h)\right )\right )\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{1024 c^6}+\frac{\left (33 b^2 f h^2-2 c h (8 b f g+21 b e h+16 a f h)-4 c^2 \left (3 f g^2-7 h (e g+2 d h)\right )\right ) (g+h x)^2 \left (a+b x+c x^2\right )^{3/2}}{280 c^3 h}-\frac{(6 c f g-14 c e h+11 b f h) (g+h x)^3 \left (a+b x+c x^2\right )^{3/2}}{84 c^2 h}+\frac{f (g+h x)^4 \left (a+b x+c x^2\right )^{3/2}}{7 c h}+\frac{\left (1155 b^4 f h^4-128 c^4 \left (3 f g^4-7 g^2 h (e g+12 d h)\right )-42 b^2 c h^3 (78 a f h+35 b (3 f g+e h))+8 c^2 h^2 \left (128 a^2 f h^2+343 a b h (3 f g+e h)+b^2 \left (537 f g^2+245 h (3 e g+d h)\right )\right )-16 c^3 h \left (16 a h \left (15 f g^2+7 h (3 e g+d h)\right )+b g \left (17 f g^2+21 h (19 e g+25 d h)\right )\right )-6 c h \left (231 b^3 f h^3-6 b c h^2 (59 b f g+49 b e h+74 a f h)+16 c^3 \left (3 f g^3-7 g h (e g+7 d h)\right )+8 c^2 h \left (5 b f g^2+7 b h (9 e g+7 d h)+a h (41 f g+35 e h)\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{13440 c^5 h}-\frac{\left (b^2-4 a c\right ) \left (256 c^5 d g^3-33 b^5 f h^3-64 c^4 g \left (a f g^2+3 a h (e g+d h)+2 b g (e g+3 d h)\right )+6 b^3 c h^2 (20 a f h+7 b (3 f g+e h))-8 b c^2 h \left (10 a^2 f h^2+14 a b h (3 f g+e h)+7 b^2 \left (3 f g^2+3 e g h+d h^2\right )\right )+16 c^3 \left (2 a^2 h^2 (3 f g+e h)+5 b^2 g \left (f g^2+3 h (e g+d h)\right )+6 a b h \left (3 f g^2+h (3 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 )}{2048 c^{13/2}}\\ \end{align*}

Mathematica [A]  time = 2.44677, size = 1093, normalized size = 1.18 $\frac{2 \sqrt{c} \sqrt{a+x (b+c x)} \left (-3465 f h^3 b^6+210 c h^2 (63 f g+21 e h+11 f h x) b^5-84 c h \left (-260 a f h^2+35 c (6 e g+2 d h+e h x) h+c f \left (210 g^2+105 h x g+22 h^2 x^2\right )\right ) b^4+16 c^2 \left (c \left (f \left (525 g^3+735 h x g^2+441 h^2 x^2 g+99 h^3 x^3\right )+7 h \left (5 d h (45 g+7 h x)+3 e \left (75 g^2+35 h x g+7 h^2 x^2\right )\right )\right )-42 a h^2 (35 e h+3 f (35 g+6 h x))\right ) b^3-16 c^2 \left (2163 a^2 f h^3-2 a c \left (7 h (345 e g+115 d h+56 e h x)+3 f \left (805 g^2+392 h x g+81 h^2 x^2\right )\right ) h+2 c^2 \left (7 d h \left (180 g^2+75 h x g+14 h^2 x^2\right )+21 e \left (20 g^3+25 h x g^2+14 h^2 x^2 g+3 h^3 x^3\right )+f x \left (175 g^3+294 h x g^2+189 h^2 x^2 g+44 h^3 x^3\right )\right )\right ) b^2+32 c^3 \left (4 \left (21 d \left (10 g^3+10 h x g^2+5 h^2 x^2 g+h^3 x^3\right )+x \left (7 e \left (10 g^3+15 h x g^2+9 h^2 x^2 g+2 h^3 x^3\right )+f x \left (35 g^3+63 h x g^2+42 h^2 x^2 g+10 h^3 x^3\right )\right )\right ) c^2-2 a \left (f \left (455 g^3+609 h x g^2+357 h^2 x^2 g+79 h^3 x^3\right )+7 h \left (d h (195 g+29 h x)+e \left (195 g^2+87 h x g+17 h^2 x^2\right )\right )\right ) c+a^2 h^2 (2373 f g+791 e h+397 f h x)\right ) b+64 c^3 \left (4 x \left (21 d \left (10 g^3+20 h x g^2+15 h^2 x^2 g+4 h^3 x^3\right )+x \left (7 e \left (20 g^3+45 h x g^2+36 h^2 x^2 g+10 h^3 x^3\right )+3 f x \left (35 g^3+84 h x g^2+70 h^2 x^2 g+20 h^3 x^3\right )\right )\right ) c^3+2 a \left (7 d h \left (120 g^2+45 h x g+8 h^2 x^2\right )+7 e \left (40 g^3+45 h x g^2+24 h^2 x^2 g+5 h^3 x^3\right )+3 f x \left (35 g^3+56 h x g^2+35 h^2 x^2 g+8 h^3 x^3\right )\right ) c^2-a^2 h \left (7 h (96 e g+32 d h+15 e h x)+f \left (672 g^2+315 h x g+64 h^2 x^2\right )\right ) c+128 a^3 f h^3\right )\right )+105 \left (b^2-4 a c\right ) \left (33 f h^3 b^5-6 c h^2 (20 a f h+7 b (3 f g+e h)) b^3+8 c^2 h \left (7 \left (3 f g^2+3 e h g+d h^2\right ) b^2+14 a h (3 f g+e h) b+10 a^2 f h^2\right ) b-256 c^5 d g^3+64 c^4 g \left (a f g^2+2 b (e g+3 d h) g+3 a h (e g+d h)\right )-16 c^3 \left (5 g \left (f g^2+3 h (e g+d h)\right ) b^2+6 a h \left (3 f g^2+h (3 e g+d h)\right ) b+2 a^2 h^2 (3 f g+e h)\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )}{215040 c^{13/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(-3465*b^6*f*h^3 + 210*b^5*c*h^2*(63*f*g + 21*e*h + 11*f*h*x) - 84*b^4*c*h*(-
260*a*f*h^2 + 35*c*h*(6*e*g + 2*d*h + e*h*x) + c*f*(210*g^2 + 105*g*h*x + 22*h^2*x^2)) - 16*b^2*c^2*(2163*a^2*
f*h^3 - 2*a*c*h*(7*h*(345*e*g + 115*d*h + 56*e*h*x) + 3*f*(805*g^2 + 392*g*h*x + 81*h^2*x^2)) + 2*c^2*(7*d*h*(
180*g^2 + 75*g*h*x + 14*h^2*x^2) + 21*e*(20*g^3 + 25*g^2*h*x + 14*g*h^2*x^2 + 3*h^3*x^3) + f*x*(175*g^3 + 294*
g^2*h*x + 189*g*h^2*x^2 + 44*h^3*x^3))) + 16*b^3*c^2*(-42*a*h^2*(35*e*h + 3*f*(35*g + 6*h*x)) + c*(f*(525*g^3
+ 735*g^2*h*x + 441*g*h^2*x^2 + 99*h^3*x^3) + 7*h*(5*d*h*(45*g + 7*h*x) + 3*e*(75*g^2 + 35*g*h*x + 7*h^2*x^2))
)) + 32*b*c^3*(a^2*h^2*(2373*f*g + 791*e*h + 397*f*h*x) - 2*a*c*(f*(455*g^3 + 609*g^2*h*x + 357*g*h^2*x^2 + 79
*h^3*x^3) + 7*h*(d*h*(195*g + 29*h*x) + e*(195*g^2 + 87*g*h*x + 17*h^2*x^2))) + 4*c^2*(21*d*(10*g^3 + 10*g^2*h
*x + 5*g*h^2*x^2 + h^3*x^3) + x*(7*e*(10*g^3 + 15*g^2*h*x + 9*g*h^2*x^2 + 2*h^3*x^3) + f*x*(35*g^3 + 63*g^2*h*
x + 42*g*h^2*x^2 + 10*h^3*x^3)))) + 64*c^3*(128*a^3*f*h^3 - a^2*c*h*(7*h*(96*e*g + 32*d*h + 15*e*h*x) + f*(672
*g^2 + 315*g*h*x + 64*h^2*x^2)) + 2*a*c^2*(7*d*h*(120*g^2 + 45*g*h*x + 8*h^2*x^2) + 7*e*(40*g^3 + 45*g^2*h*x +
24*g*h^2*x^2 + 5*h^3*x^3) + 3*f*x*(35*g^3 + 56*g^2*h*x + 35*g*h^2*x^2 + 8*h^3*x^3)) + 4*c^3*x*(21*d*(10*g^3 +
20*g^2*h*x + 15*g*h^2*x^2 + 4*h^3*x^3) + x*(7*e*(20*g^3 + 45*g^2*h*x + 36*g*h^2*x^2 + 10*h^3*x^3) + 3*f*x*(35
*g^3 + 84*g^2*h*x + 70*g*h^2*x^2 + 20*h^3*x^3))))) + 105*(b^2 - 4*a*c)*(-256*c^5*d*g^3 + 33*b^5*f*h^3 + 64*c^4
*g*(a*f*g^2 + 3*a*h*(e*g + d*h) + 2*b*g*(e*g + 3*d*h)) - 6*b^3*c*h^2*(20*a*f*h + 7*b*(3*f*g + e*h)) + 8*b*c^2*
h*(10*a^2*f*h^2 + 14*a*b*h*(3*f*g + e*h) + 7*b^2*(3*f*g^2 + 3*e*g*h + d*h^2)) - 16*c^3*(2*a^2*h^2*(3*f*g + e*h
) + 5*b^2*g*(f*g^2 + 3*h*(e*g + d*h)) + 6*a*b*h*(3*f*g^2 + h*(3*e*g + d*h))))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*S
qrt[a + x*(b + c*x)])])/(215040*c^(13/2))

________________________________________________________________________________________

Maple [B]  time = 0.067, size = 3543, normalized size = 3.8 \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)^3*(f*x^2+e*x+d)*(c*x^2+b*x+a)^(1/2),x)

[Out]

7/16*b^2/c^3*(c*x^2+b*x+a)^(3/2)*f*g^2*h+3/16*b/c^(5/2)*a^2*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d*h^3-
1/8*d*g^3/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b^2-5/128*b^4/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*
x^2+b*x+a)^(1/2))*f*g^3-1/8*a^2/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*f*g^3+5/64*b^3/c^3*(c*x^2+
b*x+a)^(1/2)*f*g^3+(c*x^2+b*x+a)^(3/2)/c*d*g^2*h-1/8*b^2/c^2*(c*x^2+b*x+a)^(1/2)*e*g^3+1/16*b^3/c^(5/2)*ln((1/
2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*e*g^3+1/4*x*(c*x^2+b*x+a)^(3/2)/c*f*g^3-5/24*b/c^2*(c*x^2+b*x+a)^(3/2)*f
*g^3+21/512*b^5/c^5*(c*x^2+b*x+a)^(1/2)*e*h^3-21/1024*b^6/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))
*e*h^3+1/2*x^3*(c*x^2+b*x+a)^(3/2)/c*f*g*h^2+9/16*b/c^2*a*x*(c*x^2+b*x+a)^(1/2)*f*g^2*h+9/16*b/c^2*a*x*(c*x^2+
b*x+a)^(1/2)*e*g*h^2-21/32*b^2/c^3*a*x*(c*x^2+b*x+a)^(1/2)*f*g*h^2-1/8*a/c^2*x*(c*x^2+b*x+a)^(3/2)*e*h^3+1/4*d
*g^3/c*(c*x^2+b*x+a)^(1/2)*b+9/16*b/c^(5/2)*a^2*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*e*g*h^2+3/16*b/c^2
*a*x*(c*x^2+b*x+a)^(1/2)*d*h^3-21/64*b^3/c^3*x*(c*x^2+b*x+a)^(1/2)*e*g*h^2+9/32*b^2/c^3*a*(c*x^2+b*x+a)^(1/2)*
f*g^2*h+15/32*b^2/c^2*x*(c*x^2+b*x+a)^(1/2)*d*g*h^2+15/32*b^2/c^2*x*(c*x^2+b*x+a)^(1/2)*e*g^2*h+9/32*b^2/c^3*a
*(c*x^2+b*x+a)^(1/2)*e*g*h^2-21/64*b^3/c^3*x*(c*x^2+b*x+a)^(1/2)*f*g^2*h-15/32*b^3/c^(7/2)*ln((1/2*b+c*x)/c^(1
/2)+(c*x^2+b*x+a)^(1/2))*a*e*g*h^2-15/32*b^3/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*f*g^2*h-3/1
6*a/c^2*(c*x^2+b*x+a)^(1/2)*b*e*g^2*h-3/8*a/c*x*(c*x^2+b*x+a)^(1/2)*e*g^2*h+9/16*b^2/c^(5/2)*ln((1/2*b+c*x)/c^
(1/2)+(c*x^2+b*x+a)^(1/2))*a*e*g^2*h-3/8*a/c*x*(c*x^2+b*x+a)^(1/2)*d*g*h^2-11/84*f*h^3*b/c^2*x^3*(c*x^2+b*x+a)
^(3/2)-33/320*f*h^3*b^3/c^4*x*(c*x^2+b*x+a)^(3/2)-5/32*f*h^3*b/c^(7/2)*a^3*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a
)^(1/2))+35/128*f*h^3*b^3/c^(9/2)*a^2*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+3/5*x^2*(c*x^2+b*x+a)^(3/2)/
c*e*g*h^2+21/256*b^5/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*e*g*h^2-63/512*f*h^3*b^5/c^(11/2)*ln(
(1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a-4/35*f*h^3*a/c^2*x^2*(c*x^2+b*x+a)^(3/2)-5/32*b^3/c^(7/2)*ln((1/2*b
+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*d*h^3+21/160*b^2/c^3*x*(c*x^2+b*x+a)^(3/2)*e*h^3+35/256*b^4/c^(9/2)*ln((1
/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*e*h^3-7/40*b/c^2*x*(c*x^2+b*x+a)^(3/2)*d*h^3+21/256*b^5/c^(9/2)*ln((1
/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*f*g^2*h+7/16*b^2/c^3*(c*x^2+b*x+a)^(3/2)*e*g*h^2-21/128*b^4/c^4*(c*x^2+
b*x+a)^(1/2)*e*g*h^2-21/128*b^4/c^4*(c*x^2+b*x+a)^(1/2)*f*g^2*h+49/240*b/c^3*a*(c*x^2+b*x+a)^(3/2)*e*h^3+21/25
6*b^4/c^4*x*(c*x^2+b*x+a)^(1/2)*e*h^3+63/512*b^5/c^5*(c*x^2+b*x+a)^(1/2)*f*g*h^2+3/32*b^2/c^3*a*(c*x^2+b*x+a)^
(1/2)*d*h^3+3/5*x^2*(c*x^2+b*x+a)^(3/2)/c*f*g^2*h-5/8*b/c^2*(c*x^2+b*x+a)^(3/2)*e*g^2*h+1/2*d*g^3/c^(1/2)*ln((
1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a+3/4*x*(c*x^2+b*x+a)^(3/2)/c*d*g*h^2+3/4*x*(c*x^2+b*x+a)^(3/2)/c*e*g^
2*h+15/64*b^3/c^3*(c*x^2+b*x+a)^(1/2)*e*g^2*h+5/32*b^2/c^2*x*(c*x^2+b*x+a)^(1/2)*f*g^3+15/64*b^3/c^3*(c*x^2+b*
x+a)^(1/2)*d*g*h^2-3/8*a^2/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*e*g^2*h+1/16*a^2/c^2*x*(c*x^2+b
*x+a)^(1/2)*e*h^3+1/32*a^2/c^3*(c*x^2+b*x+a)^(1/2)*b*e*h^3-7/64*b^3/c^3*x*(c*x^2+b*x+a)^(1/2)*d*h^3+9/16*b^2/c
^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*d*g*h^2-2/5*a/c^2*(c*x^2+b*x+a)^(3/2)*e*g*h^2-2/5*a/c^2*(
c*x^2+b*x+a)^(3/2)*f*g^2*h+33/280*f*h^3*b^2/c^3*x^2*(c*x^2+b*x+a)^(3/2)-33/512*f*h^3*b^5/c^5*x*(c*x^2+b*x+a)^(
1/2)+15/128*f*h^3*b^4/c^5*a*(c*x^2+b*x+a)^(1/2)-39/160*f*h^3*b^2/c^4*a*(c*x^2+b*x+a)^(3/2)-5/64*f*h^3*b^2/c^4*
a^2*(c*x^2+b*x+a)^(1/2)-33/1024*f*h^3*b^6/c^6*(c*x^2+b*x+a)^(1/2)+33/2048*f*h^3*b^7/c^(13/2)*ln((1/2*b+c*x)/c^
(1/2)+(c*x^2+b*x+a)^(1/2))+8/105*f*h^3*a^2/c^3*(c*x^2+b*x+a)^(3/2)+7/256*b^5/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c
*x^2+b*x+a)^(1/2))*d*h^3-2/15*a/c^2*(c*x^2+b*x+a)^(3/2)*d*h^3+1/3*(c*x^2+b*x+a)^(3/2)/c*e*g^3+1/2*d*g^3*x*(c*x
^2+b*x+a)^(1/2)+11/128*f*h^3*b^4/c^5*(c*x^2+b*x+a)^(3/2)-1/4*b/c*x*(c*x^2+b*x+a)^(1/2)*e*g^3-3/8*b^2/c^2*(c*x^
2+b*x+a)^(1/2)*d*g^2*h-1/4*b/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*e*g^3+3/16*b^3/c^(5/2)*ln((
1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d*g^2*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))
*e*g^2*h-5/8*b/c^2*(c*x^2+b*x+a)^(3/2)*d*g*h^2-3/20*b/c^2*x^2*(c*x^2+b*x+a)^(3/2)*e*h^3-21/64*b^3/c^4*(c*x^2+b
*x+a)^(3/2)*f*g*h^2-63/1024*b^6/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*f*g*h^2-7/64*b^3/c^4*a*(c
*x^2+b*x+a)^(1/2)*e*h^3-15/64*b^2/c^(7/2)*a^2*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*e*h^3-7/128*b^4/c^4*
(c*x^2+b*x+a)^(1/2)*d*h^3-7/64*b^3/c^4*(c*x^2+b*x+a)^(3/2)*e*h^3+1/7*f*h^3*x^4*(c*x^2+b*x+a)^(3/2)/c-3/8*a^2/c
^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d*g*h^2-1/8*a/c*x*(c*x^2+b*x+a)^(1/2)*f*g^3-1/16*a/c^2*(c*x
^2+b*x+a)^(1/2)*b*f*g^3+3/16*a^3/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*f*g*h^2+3/16*b^2/c^(5/2)*
ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*f*g^3-15/128*b^4/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1
/2))*d*g*h^2+1/16*a^3/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*e*h^3+1/6*x^3*(c*x^2+b*x+a)^(3/2)/c*
e*h^3+1/5*x^2*(c*x^2+b*x+a)^(3/2)/c*d*h^3+7/48*b^2/c^3*(c*x^2+b*x+a)^(3/2)*d*h^3-21/40*b/c^2*x*(c*x^2+b*x+a)^(
3/2)*e*g*h^2-3/16*a/c^2*(c*x^2+b*x+a)^(1/2)*b*d*g*h^2-3/4*b/c*x*(c*x^2+b*x+a)^(1/2)*d*g^2*h-3/4*b/c^(3/2)*ln((
1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*d*g^2*h+63/160*b^2/c^3*x*(c*x^2+b*x+a)^(3/2)*f*g*h^2-45/64*b^2/c^(7/
2)*a^2*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*f*g*h^2+49/80*b/c^3*a*(c*x^2+b*x+a)^(3/2)*f*g*h^2-9/20*b/c^
2*x^2*(c*x^2+b*x+a)^(3/2)*f*g*h^2-3/8*a/c^2*x*(c*x^2+b*x+a)^(3/2)*f*g*h^2+3/16*a^2/c^2*x*(c*x^2+b*x+a)^(1/2)*f
*g*h^2+3/32*a^2/c^3*(c*x^2+b*x+a)^(1/2)*b*f*g*h^2+63/256*b^4/c^4*x*(c*x^2+b*x+a)^(1/2)*f*g*h^2+105/256*b^4/c^(
9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*f*g*h^2-7/32*b^2/c^3*a*x*(c*x^2+b*x+a)^(1/2)*e*h^3-21/40*b/
c^2*x*(c*x^2+b*x+a)^(3/2)*f*g^2*h-21/64*b^3/c^4*a*(c*x^2+b*x+a)^(1/2)*f*g*h^2-5/32*f*h^3*b/c^3*a^2*x*(c*x^2+b*
x+a)^(1/2)+15/64*f*h^3*b^3/c^4*a*x*(c*x^2+b*x+a)^(1/2)+111/560*f*h^3*b/c^3*a*x*(c*x^2+b*x+a)^(3/2)+9/16*b/c^(5
/2)*a^2*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*f*g^2*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)^3*(f*x^2+e*x+d)*(c*x^2+b*x+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 12.1682, size = 6407, normalized size = 6.89 \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)^3*(f*x^2+e*x+d)*(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")

[Out]

[1/430080*(105*(16*(16*(b^2*c^5 - 4*a*c^6)*d - 8*(b^3*c^4 - 4*a*b*c^5)*e + (5*b^4*c^3 - 24*a*b^2*c^4 + 16*a^2*
c^5)*f)*g^3 - 24*(16*(b^3*c^4 - 4*a*b*c^5)*d - 2*(5*b^4*c^3 - 24*a*b^2*c^4 + 16*a^2*c^5)*e + (7*b^5*c^2 - 40*a
*b^3*c^3 + 48*a^2*b*c^4)*f)*g^2*h + 6*(8*(5*b^4*c^3 - 24*a*b^2*c^4 + 16*a^2*c^5)*d - 4*(7*b^5*c^2 - 40*a*b^3*c
^3 + 48*a^2*b*c^4)*e + (21*b^6*c - 140*a*b^4*c^2 + 240*a^2*b^2*c^3 - 64*a^3*c^4)*f)*g*h^2 - (8*(7*b^5*c^2 - 40
*a*b^3*c^3 + 48*a^2*b*c^4)*d - 2*(21*b^6*c - 140*a*b^4*c^2 + 240*a^2*b^2*c^3 - 64*a^3*c^4)*e + (33*b^7 - 252*a
*b^5*c + 560*a^2*b^3*c^2 - 320*a^3*b*c^3)*f)*h^3)*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*(15360*c^7*f*h^3*x^6 + 1280*(42*c^7*f*g*h^2 + (14*c^7*e + b*c^6*f)*h^3)*
x^5 + 128*(504*c^7*f*g^2*h + 42*(12*c^7*e + b*c^6*f)*g*h^2 + (168*c^7*d + 14*b*c^6*e - (11*b^2*c^5 - 24*a*c^6)
*f)*h^3)*x^4 + 560*(48*b*c^6*d - 8*(3*b^2*c^5 - 8*a*c^6)*e + (15*b^3*c^4 - 52*a*b*c^5)*f)*g^3 - 168*(80*(3*b^2
*c^5 - 8*a*c^6)*d - 10*(15*b^3*c^4 - 52*a*b*c^5)*e + (105*b^4*c^3 - 460*a*b^2*c^4 + 256*a^2*c^5)*f)*g^2*h + 42
*(40*(15*b^3*c^4 - 52*a*b*c^5)*d - 4*(105*b^4*c^3 - 460*a*b^2*c^4 + 256*a^2*c^5)*e + (315*b^5*c^2 - 1680*a*b^3
*c^3 + 1808*a^2*b*c^4)*f)*g*h^2 - (56*(105*b^4*c^3 - 460*a*b^2*c^4 + 256*a^2*c^5)*d - 14*(315*b^5*c^2 - 1680*a
*b^3*c^3 + 1808*a^2*b*c^4)*e + (3465*b^6*c - 21840*a*b^4*c^2 + 34608*a^2*b^2*c^3 - 8192*a^3*c^4)*f)*h^3 + 16*(
1680*c^7*f*g^3 + 504*(10*c^7*e + b*c^6*f)*g^2*h + 42*(120*c^7*d + 12*b*c^6*e - (9*b^2*c^5 - 20*a*c^6)*f)*g*h^2
+ (168*b*c^6*d - 14*(9*b^2*c^5 - 20*a*c^6)*e + (99*b^3*c^4 - 316*a*b*c^5)*f)*h^3)*x^3 + 8*(560*(8*c^7*e + b*c
^6*f)*g^3 + 168*(80*c^7*d + 10*b*c^6*e - (7*b^2*c^5 - 16*a*c^6)*f)*g^2*h + 42*(40*b*c^6*d - 4*(7*b^2*c^5 - 16*
a*c^6)*e + (21*b^3*c^4 - 68*a*b*c^5)*f)*g*h^2 - (56*(7*b^2*c^5 - 16*a*c^6)*d - 14*(21*b^3*c^4 - 68*a*b*c^5)*e
+ (231*b^4*c^3 - 972*a*b^2*c^4 + 512*a^2*c^5)*f)*h^3)*x^2 + 2*(560*(48*c^7*d + 8*b*c^6*e - (5*b^2*c^5 - 12*a*c
^6)*f)*g^3 + 168*(80*b*c^6*d - 10*(5*b^2*c^5 - 12*a*c^6)*e + (35*b^3*c^4 - 116*a*b*c^5)*f)*g^2*h - 42*(40*(5*b
^2*c^5 - 12*a*c^6)*d - 4*(35*b^3*c^4 - 116*a*b*c^5)*e + (105*b^4*c^3 - 448*a*b^2*c^4 + 240*a^2*c^5)*f)*g*h^2 +
(56*(35*b^3*c^4 - 116*a*b*c^5)*d - 14*(105*b^4*c^3 - 448*a*b^2*c^4 + 240*a^2*c^5)*e + (1155*b^5*c^2 - 6048*a*
b^3*c^3 + 6352*a^2*b*c^4)*f)*h^3)*x)*sqrt(c*x^2 + b*x + a))/c^7, 1/215040*(105*(16*(16*(b^2*c^5 - 4*a*c^6)*d -
8*(b^3*c^4 - 4*a*b*c^5)*e + (5*b^4*c^3 - 24*a*b^2*c^4 + 16*a^2*c^5)*f)*g^3 - 24*(16*(b^3*c^4 - 4*a*b*c^5)*d -
2*(5*b^4*c^3 - 24*a*b^2*c^4 + 16*a^2*c^5)*e + (7*b^5*c^2 - 40*a*b^3*c^3 + 48*a^2*b*c^4)*f)*g^2*h + 6*(8*(5*b^
4*c^3 - 24*a*b^2*c^4 + 16*a^2*c^5)*d - 4*(7*b^5*c^2 - 40*a*b^3*c^3 + 48*a^2*b*c^4)*e + (21*b^6*c - 140*a*b^4*c
^2 + 240*a^2*b^2*c^3 - 64*a^3*c^4)*f)*g*h^2 - (8*(7*b^5*c^2 - 40*a*b^3*c^3 + 48*a^2*b*c^4)*d - 2*(21*b^6*c - 1
40*a*b^4*c^2 + 240*a^2*b^2*c^3 - 64*a^3*c^4)*e + (33*b^7 - 252*a*b^5*c + 560*a^2*b^3*c^2 - 320*a^3*b*c^3)*f)*h
^3)*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*(15360*c^7*f*h
^3*x^6 + 1280*(42*c^7*f*g*h^2 + (14*c^7*e + b*c^6*f)*h^3)*x^5 + 128*(504*c^7*f*g^2*h + 42*(12*c^7*e + b*c^6*f)
*g*h^2 + (168*c^7*d + 14*b*c^6*e - (11*b^2*c^5 - 24*a*c^6)*f)*h^3)*x^4 + 560*(48*b*c^6*d - 8*(3*b^2*c^5 - 8*a*
c^6)*e + (15*b^3*c^4 - 52*a*b*c^5)*f)*g^3 - 168*(80*(3*b^2*c^5 - 8*a*c^6)*d - 10*(15*b^3*c^4 - 52*a*b*c^5)*e +
(105*b^4*c^3 - 460*a*b^2*c^4 + 256*a^2*c^5)*f)*g^2*h + 42*(40*(15*b^3*c^4 - 52*a*b*c^5)*d - 4*(105*b^4*c^3 -
460*a*b^2*c^4 + 256*a^2*c^5)*e + (315*b^5*c^2 - 1680*a*b^3*c^3 + 1808*a^2*b*c^4)*f)*g*h^2 - (56*(105*b^4*c^3 -
460*a*b^2*c^4 + 256*a^2*c^5)*d - 14*(315*b^5*c^2 - 1680*a*b^3*c^3 + 1808*a^2*b*c^4)*e + (3465*b^6*c - 21840*a
*b^4*c^2 + 34608*a^2*b^2*c^3 - 8192*a^3*c^4)*f)*h^3 + 16*(1680*c^7*f*g^3 + 504*(10*c^7*e + b*c^6*f)*g^2*h + 42
*(120*c^7*d + 12*b*c^6*e - (9*b^2*c^5 - 20*a*c^6)*f)*g*h^2 + (168*b*c^6*d - 14*(9*b^2*c^5 - 20*a*c^6)*e + (99*
b^3*c^4 - 316*a*b*c^5)*f)*h^3)*x^3 + 8*(560*(8*c^7*e + b*c^6*f)*g^3 + 168*(80*c^7*d + 10*b*c^6*e - (7*b^2*c^5
- 16*a*c^6)*f)*g^2*h + 42*(40*b*c^6*d - 4*(7*b^2*c^5 - 16*a*c^6)*e + (21*b^3*c^4 - 68*a*b*c^5)*f)*g*h^2 - (56*
(7*b^2*c^5 - 16*a*c^6)*d - 14*(21*b^3*c^4 - 68*a*b*c^5)*e + (231*b^4*c^3 - 972*a*b^2*c^4 + 512*a^2*c^5)*f)*h^3
)*x^2 + 2*(560*(48*c^7*d + 8*b*c^6*e - (5*b^2*c^5 - 12*a*c^6)*f)*g^3 + 168*(80*b*c^6*d - 10*(5*b^2*c^5 - 12*a*
c^6)*e + (35*b^3*c^4 - 116*a*b*c^5)*f)*g^2*h - 42*(40*(5*b^2*c^5 - 12*a*c^6)*d - 4*(35*b^3*c^4 - 116*a*b*c^5)*
e + (105*b^4*c^3 - 448*a*b^2*c^4 + 240*a^2*c^5)*f)*g*h^2 + (56*(35*b^3*c^4 - 116*a*b*c^5)*d - 14*(105*b^4*c^3
- 448*a*b^2*c^4 + 240*a^2*c^5)*e + (1155*b^5*c^2 - 6048*a*b^3*c^3 + 6352*a^2*b*c^4)*f)*h^3)*x)*sqrt(c*x^2 + b*
x + a))/c^7]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (g + h x\right )^{3} \sqrt{a + b x + c x^{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)**3*(f*x**2+e*x+d)*(c*x**2+b*x+a)**(1/2),x)

[Out]

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

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Giac [A]  time = 1.32431, size = 2298, normalized size = 2.47 \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)^3*(f*x^2+e*x+d)*(c*x^2+b*x+a)^(1/2),x, algorithm="giac")

[Out]

1/107520*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*(12*f*h^3*x + (42*c^6*f*g*h^2 + b*c^5*f*h^3 + 14*c^6*h^3*e)/c^6
)*x + (504*c^6*f*g^2*h + 42*b*c^5*f*g*h^2 + 168*c^6*d*h^3 - 11*b^2*c^4*f*h^3 + 24*a*c^5*f*h^3 + 504*c^6*g*h^2*
e + 14*b*c^5*h^3*e)/c^6)*x + (1680*c^6*f*g^3 + 504*b*c^5*f*g^2*h + 5040*c^6*d*g*h^2 - 378*b^2*c^4*f*g*h^2 + 84
0*a*c^5*f*g*h^2 + 168*b*c^5*d*h^3 + 99*b^3*c^3*f*h^3 - 316*a*b*c^4*f*h^3 + 5040*c^6*g^2*h*e + 504*b*c^5*g*h^2*
e - 126*b^2*c^4*h^3*e + 280*a*c^5*h^3*e)/c^6)*x + (560*b*c^5*f*g^3 + 13440*c^6*d*g^2*h - 1176*b^2*c^4*f*g^2*h
+ 2688*a*c^5*f*g^2*h + 1680*b*c^5*d*g*h^2 + 882*b^3*c^3*f*g*h^2 - 2856*a*b*c^4*f*g*h^2 - 392*b^2*c^4*d*h^3 + 8
96*a*c^5*d*h^3 - 231*b^4*c^2*f*h^3 + 972*a*b^2*c^3*f*h^3 - 512*a^2*c^4*f*h^3 + 4480*c^6*g^3*e + 1680*b*c^5*g^2
*h*e - 1176*b^2*c^4*g*h^2*e + 2688*a*c^5*g*h^2*e + 294*b^3*c^3*h^3*e - 952*a*b*c^4*h^3*e)/c^6)*x + (26880*c^6*
d*g^3 - 2800*b^2*c^4*f*g^3 + 6720*a*c^5*f*g^3 + 13440*b*c^5*d*g^2*h + 5880*b^3*c^3*f*g^2*h - 19488*a*b*c^4*f*g
^2*h - 8400*b^2*c^4*d*g*h^2 + 20160*a*c^5*d*g*h^2 - 4410*b^4*c^2*f*g*h^2 + 18816*a*b^2*c^3*f*g*h^2 - 10080*a^2
*c^4*f*g*h^2 + 1960*b^3*c^3*d*h^3 - 6496*a*b*c^4*d*h^3 + 1155*b^5*c*f*h^3 - 6048*a*b^3*c^2*f*h^3 + 6352*a^2*b*
c^3*f*h^3 + 4480*b*c^5*g^3*e - 8400*b^2*c^4*g^2*h*e + 20160*a*c^5*g^2*h*e + 5880*b^3*c^3*g*h^2*e - 19488*a*b*c
^4*g*h^2*e - 1470*b^4*c^2*h^3*e + 6272*a*b^2*c^3*h^3*e - 3360*a^2*c^4*h^3*e)/c^6)*x + (26880*b*c^5*d*g^3 + 840
0*b^3*c^3*f*g^3 - 29120*a*b*c^4*f*g^3 - 40320*b^2*c^4*d*g^2*h + 107520*a*c^5*d*g^2*h - 17640*b^4*c^2*f*g^2*h +
77280*a*b^2*c^3*f*g^2*h - 43008*a^2*c^4*f*g^2*h + 25200*b^3*c^3*d*g*h^2 - 87360*a*b*c^4*d*g*h^2 + 13230*b^5*c
*f*g*h^2 - 70560*a*b^3*c^2*f*g*h^2 + 75936*a^2*b*c^3*f*g*h^2 - 5880*b^4*c^2*d*h^3 + 25760*a*b^2*c^3*d*h^3 - 14
336*a^2*c^4*d*h^3 - 3465*b^6*f*h^3 + 21840*a*b^4*c*f*h^3 - 34608*a^2*b^2*c^2*f*h^3 + 8192*a^3*c^3*f*h^3 - 1344
0*b^2*c^4*g^3*e + 35840*a*c^5*g^3*e + 25200*b^3*c^3*g^2*h*e - 87360*a*b*c^4*g^2*h*e - 17640*b^4*c^2*g*h^2*e +
77280*a*b^2*c^3*g*h^2*e - 43008*a^2*c^4*g*h^2*e + 4410*b^5*c*h^3*e - 23520*a*b^3*c^2*h^3*e + 25312*a^2*b*c^3*h
^3*e)/c^6) + 1/2048*(256*b^2*c^5*d*g^3 - 1024*a*c^6*d*g^3 + 80*b^4*c^3*f*g^3 - 384*a*b^2*c^4*f*g^3 + 256*a^2*c
^5*f*g^3 - 384*b^3*c^4*d*g^2*h + 1536*a*b*c^5*d*g^2*h - 168*b^5*c^2*f*g^2*h + 960*a*b^3*c^3*f*g^2*h - 1152*a^2
*b*c^4*f*g^2*h + 240*b^4*c^3*d*g*h^2 - 1152*a*b^2*c^4*d*g*h^2 + 768*a^2*c^5*d*g*h^2 + 126*b^6*c*f*g*h^2 - 840*
a*b^4*c^2*f*g*h^2 + 1440*a^2*b^2*c^3*f*g*h^2 - 384*a^3*c^4*f*g*h^2 - 56*b^5*c^2*d*h^3 + 320*a*b^3*c^3*d*h^3 -
384*a^2*b*c^4*d*h^3 - 33*b^7*f*h^3 + 252*a*b^5*c*f*h^3 - 560*a^2*b^3*c^2*f*h^3 + 320*a^3*b*c^3*f*h^3 - 128*b^3
*c^4*g^3*e + 512*a*b*c^5*g^3*e + 240*b^4*c^3*g^2*h*e - 1152*a*b^2*c^4*g^2*h*e + 768*a^2*c^5*g^2*h*e - 168*b^5*
c^2*g*h^2*e + 960*a*b^3*c^3*g*h^2*e - 1152*a^2*b*c^4*g*h^2*e + 42*b^6*c*h^3*e - 280*a*b^4*c^2*h^3*e + 480*a^2*
b^2*c^3*h^3*e - 128*a^3*c^4*h^3*e)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(13/2)