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

Optimal. Leaf size=584 \[ -\frac {\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (8 c^2 \left (2 a^2 f h^2+6 a b h (e h+2 f g)+5 b^2 \left (d h^2+2 e g h+f g^2\right )\right )-28 b^2 c h (2 a f h+b e h+2 b f g)-32 c^3 \left (a \left (d h^2+2 e g h+f g^2\right )+2 b g (2 d h+e g)\right )+21 b^4 f h^2+128 c^4 d g^2\right )}{1024 c^{11/2}}+\frac {(b+2 c x) \sqrt {a+b x+c x^2} \left (8 c^2 \left (2 a^2 f h^2+6 a b h (e h+2 f g)+5 b^2 \left (d h^2+2 e g h+f g^2\right )\right )-28 b^2 c h (2 a f h+b e h+2 b f g)-32 c^3 \left (a \left (d h^2+2 e g h+f g^2\right )+2 b g (2 d h+e g)\right )+21 b^4 f h^2+128 c^4 d g^2\right )}{512 c^5}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-6 c h x \left (-4 c h (5 a f h+7 b e h+2 b f g)+21 b^2 f h^2-8 c^2 \left (f g^2-h (5 d h+2 e g)\right )\right )+8 c^2 h \left (16 a h (e h+2 f g)+b \left (25 h (d h+2 e g)+7 f g^2\right )\right )-28 b c h^2 (7 a f h+5 b (e h+2 f g))+105 b^3 f h^3+64 c^3 g \left (f g^2-2 h (5 d h+e g)\right )\right )}{960 c^4 h}-\frac {(g+h x)^2 \left (a+b x+c x^2\right )^{3/2} (3 b f h-4 c e h+2 c f g)}{20 c^2 h}+\frac {f (g+h x)^3 \left (a+b x+c x^2\right )^{3/2}}{6 c h} \]

[Out]

-1/20*(3*b*f*h-4*c*e*h+2*c*f*g)*(h*x+g)^2*(c*x^2+b*x+a)^(3/2)/c^2/h+1/6*f*(h*x+g)^3*(c*x^2+b*x+a)^(3/2)/c/h-1/
960*(105*b^3*f*h^3+64*c^3*g*(f*g^2-2*h*(5*d*h+e*g))-28*b*c*h^2*(7*a*f*h+5*b*(e*h+2*f*g))+8*c^2*h*(16*a*h*(e*h+
2*f*g)+b*(7*f*g^2+25*h*(d*h+2*e*g)))-6*c*h*(21*b^2*f*h^2-4*c*h*(5*a*f*h+7*b*e*h+2*b*f*g)-8*c^2*(f*g^2-h*(5*d*h
+2*e*g)))*x)*(c*x^2+b*x+a)^(3/2)/c^4/h-1/1024*(-4*a*c+b^2)*(128*c^4*d*g^2+21*b^4*f*h^2-28*b^2*c*h*(2*a*f*h+b*e
*h+2*b*f*g)-32*c^3*(2*b*g*(2*d*h+e*g)+a*(d*h^2+2*e*g*h+f*g^2))+8*c^2*(2*a^2*f*h^2+6*a*b*h*(e*h+2*f*g)+5*b^2*(d
*h^2+2*e*g*h+f*g^2)))*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(11/2)+1/512*(128*c^4*d*g^2+21*b^4*
f*h^2-28*b^2*c*h*(2*a*f*h+b*e*h+2*b*f*g)-32*c^3*(2*b*g*(2*d*h+e*g)+a*(d*h^2+2*e*g*h+f*g^2))+8*c^2*(2*a^2*f*h^2
+6*a*b*h*(e*h+2*f*g)+5*b^2*(d*h^2+2*e*g*h+f*g^2)))*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c^5

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Rubi [A]  time = 1.44, antiderivative size = 581, normalized size of antiderivative = 0.99, number of steps used = 6, 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} \[ \frac {(b+2 c x) \sqrt {a+b x+c x^2} \left (8 c^2 \left (2 a^2 f h^2+6 a b h (e h+2 f g)+5 b^2 \left (h (d h+2 e g)+f g^2\right )\right )-28 b^2 c h (2 a f h+b e h+2 b f g)-32 c^3 \left (a h (d h+2 e g)+a f g^2+2 b g (2 d h+e g)\right )+21 b^4 f h^2+128 c^4 d g^2\right )}{512 c^5}-\frac {\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (8 c^2 \left (2 a^2 f h^2+6 a b h (e h+2 f g)+5 b^2 \left (h (d h+2 e g)+f g^2\right )\right )-28 b^2 c h (2 a f h+b e h+2 b f g)-32 c^3 \left (a h (d h+2 e g)+a f g^2+2 b g (2 d h+e g)\right )+21 b^4 f h^2+128 c^4 d g^2\right )}{1024 c^{11/2}}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-6 c h x \left (-4 c h (5 a f h+7 b e h+2 b f g)+21 b^2 f h^2-8 c^2 \left (f g^2-h (5 d h+2 e g)\right )\right )+8 c^2 h \left (16 a h (e h+2 f g)+25 b h (d h+2 e g)+7 b f g^2\right )-28 b c h^2 (7 a f h+5 b (e h+2 f g))+105 b^3 f h^3+64 c^3 \left (f g^3-2 g h (5 d h+e g)\right )\right )}{960 c^4 h}-\frac {(g+h x)^2 \left (a+b x+c x^2\right )^{3/2} (3 b f h-4 c e h+2 c f g)}{20 c^2 h}+\frac {f (g+h x)^3 \left (a+b x+c x^2\right )^{3/2}}{6 c h} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((128*c^4*d*g^2 + 21*b^4*f*h^2 - 28*b^2*c*h*(2*b*f*g + b*e*h + 2*a*f*h) - 32*c^3*(a*f*g^2 + a*h*(2*e*g + d*h)
+ 2*b*g*(e*g + 2*d*h)) + 8*c^2*(2*a^2*f*h^2 + 6*a*b*h*(2*f*g + e*h) + 5*b^2*(f*g^2 + h*(2*e*g + d*h))))*(b + 2
*c*x)*Sqrt[a + b*x + c*x^2])/(512*c^5) - ((2*c*f*g - 4*c*e*h + 3*b*f*h)*(g + h*x)^2*(a + b*x + c*x^2)^(3/2))/(
20*c^2*h) + (f*(g + h*x)^3*(a + b*x + c*x^2)^(3/2))/(6*c*h) - ((105*b^3*f*h^3 + 64*c^3*(f*g^3 - 2*g*h*(e*g + 5
*d*h)) - 28*b*c*h^2*(7*a*f*h + 5*b*(2*f*g + e*h)) + 8*c^2*h*(7*b*f*g^2 + 25*b*h*(2*e*g + d*h) + 16*a*h*(2*f*g
+ e*h)) - 6*c*h*(21*b^2*f*h^2 - 4*c*h*(2*b*f*g + 7*b*e*h + 5*a*f*h) - 8*c^2*(f*g^2 - h*(2*e*g + 5*d*h)))*x)*(a
+ b*x + c*x^2)^(3/2))/(960*c^4*h) - ((b^2 - 4*a*c)*(128*c^4*d*g^2 + 21*b^4*f*h^2 - 28*b^2*c*h*(2*b*f*g + b*e*
h + 2*a*f*h) - 32*c^3*(a*f*g^2 + a*h*(2*e*g + d*h) + 2*b*g*(e*g + 2*d*h)) + 8*c^2*(2*a^2*f*h^2 + 6*a*b*h*(2*f*
g + e*h) + 5*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])])/(1024*c^(
11/2))

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])

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 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 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 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]))

Rubi steps

\begin {align*} \int (g+h x)^2 \sqrt {a+b x+c x^2} \left (d+e x+f x^2\right ) \, dx &=\frac {f (g+h x)^3 \left (a+b x+c x^2\right )^{3/2}}{6 c h}+\frac {\int (g+h x)^2 \left (-\frac {3}{2} h (b f g-4 c d h+2 a f h)-\frac {3}{2} h (2 c f g-4 c e h+3 b f h) x\right ) \sqrt {a+b x+c x^2} \, dx}{6 c h^2}\\ &=-\frac {(2 c f g-4 c e h+3 b f h) (g+h x)^2 \left (a+b x+c x^2\right )^{3/2}}{20 c^2 h}+\frac {f (g+h x)^3 \left (a+b x+c x^2\right )^{3/2}}{6 c h}+\frac {\int (g+h x) \left (\frac {3}{4} h \left (9 b^2 f g h+12 a b f h^2-4 b c g (f g+3 e h)+4 c h (10 c d g-3 a f g-4 a e h)\right )+\frac {3}{4} h \left (21 b^2 f h^2-4 c h (2 b f g+7 b e h+5 a f h)-8 c^2 \left (f g^2-h (2 e g+5 d h)\right )\right ) x\right ) \sqrt {a+b x+c x^2} \, dx}{30 c^2 h^2}\\ &=-\frac {(2 c f g-4 c e h+3 b f h) (g+h x)^2 \left (a+b x+c x^2\right )^{3/2}}{20 c^2 h}+\frac {f (g+h x)^3 \left (a+b x+c x^2\right )^{3/2}}{6 c h}-\frac {\left (105 b^3 f h^3+64 c^3 \left (f g^3-2 g h (e g+5 d h)\right )-28 b c h^2 (7 a f h+5 b (2 f g+e h))+8 c^2 h \left (7 b f g^2+25 b h (2 e g+d h)+16 a h (2 f g+e h)\right )-6 c h \left (21 b^2 f h^2-4 c h (2 b f g+7 b e h+5 a f h)-8 c^2 \left (f g^2-h (2 e g+5 d h)\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{960 c^4 h}+\frac {\left (128 c^4 d g^2+21 b^4 f h^2-28 b^2 c h (2 b f g+b e h+2 a f h)-32 c^3 \left (a f g^2+a h (2 e g+d h)+2 b g (e g+2 d h)\right )+8 c^2 \left (2 a^2 f h^2+6 a b h (2 f g+e h)+5 b^2 \left (f g^2+h (2 e g+d h)\right )\right )\right ) \int \sqrt {a+b x+c x^2} \, dx}{128 c^4}\\ &=\frac {\left (128 c^4 d g^2+21 b^4 f h^2-28 b^2 c h (2 b f g+b e h+2 a f h)-32 c^3 \left (a f g^2+a h (2 e g+d h)+2 b g (e g+2 d h)\right )+8 c^2 \left (2 a^2 f h^2+6 a b h (2 f g+e h)+5 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}}{512 c^5}-\frac {(2 c f g-4 c e h+3 b f h) (g+h x)^2 \left (a+b x+c x^2\right )^{3/2}}{20 c^2 h}+\frac {f (g+h x)^3 \left (a+b x+c x^2\right )^{3/2}}{6 c h}-\frac {\left (105 b^3 f h^3+64 c^3 \left (f g^3-2 g h (e g+5 d h)\right )-28 b c h^2 (7 a f h+5 b (2 f g+e h))+8 c^2 h \left (7 b f g^2+25 b h (2 e g+d h)+16 a h (2 f g+e h)\right )-6 c h \left (21 b^2 f h^2-4 c h (2 b f g+7 b e h+5 a f h)-8 c^2 \left (f g^2-h (2 e g+5 d h)\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{960 c^4 h}-\frac {\left (\left (b^2-4 a c\right ) \left (128 c^4 d g^2+21 b^4 f h^2-28 b^2 c h (2 b f g+b e h+2 a f h)-32 c^3 \left (a f g^2+a h (2 e g+d h)+2 b g (e g+2 d h)\right )+8 c^2 \left (2 a^2 f h^2+6 a b h (2 f g+e h)+5 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}{1024 c^5}\\ &=\frac {\left (128 c^4 d g^2+21 b^4 f h^2-28 b^2 c h (2 b f g+b e h+2 a f h)-32 c^3 \left (a f g^2+a h (2 e g+d h)+2 b g (e g+2 d h)\right )+8 c^2 \left (2 a^2 f h^2+6 a b h (2 f g+e h)+5 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}}{512 c^5}-\frac {(2 c f g-4 c e h+3 b f h) (g+h x)^2 \left (a+b x+c x^2\right )^{3/2}}{20 c^2 h}+\frac {f (g+h x)^3 \left (a+b x+c x^2\right )^{3/2}}{6 c h}-\frac {\left (105 b^3 f h^3+64 c^3 \left (f g^3-2 g h (e g+5 d h)\right )-28 b c h^2 (7 a f h+5 b (2 f g+e h))+8 c^2 h \left (7 b f g^2+25 b h (2 e g+d h)+16 a h (2 f g+e h)\right )-6 c h \left (21 b^2 f h^2-4 c h (2 b f g+7 b e h+5 a f h)-8 c^2 \left (f g^2-h (2 e g+5 d h)\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{960 c^4 h}-\frac {\left (\left (b^2-4 a c\right ) \left (128 c^4 d g^2+21 b^4 f h^2-28 b^2 c h (2 b f g+b e h+2 a f h)-32 c^3 \left (a f g^2+a h (2 e g+d h)+2 b g (e g+2 d h)\right )+8 c^2 \left (2 a^2 f h^2+6 a b h (2 f g+e h)+5 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 )}{512 c^5}\\ &=\frac {\left (128 c^4 d g^2+21 b^4 f h^2-28 b^2 c h (2 b f g+b e h+2 a f h)-32 c^3 \left (a f g^2+a h (2 e g+d h)+2 b g (e g+2 d h)\right )+8 c^2 \left (2 a^2 f h^2+6 a b h (2 f g+e h)+5 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}}{512 c^5}-\frac {(2 c f g-4 c e h+3 b f h) (g+h x)^2 \left (a+b x+c x^2\right )^{3/2}}{20 c^2 h}+\frac {f (g+h x)^3 \left (a+b x+c x^2\right )^{3/2}}{6 c h}-\frac {\left (105 b^3 f h^3+64 c^3 \left (f g^3-2 g h (e g+5 d h)\right )-28 b c h^2 (7 a f h+5 b (2 f g+e h))+8 c^2 h \left (7 b f g^2+25 b h (2 e g+d h)+16 a h (2 f g+e h)\right )-6 c h \left (21 b^2 f h^2-4 c h (2 b f g+7 b e h+5 a f h)-8 c^2 \left (f g^2-h (2 e g+5 d h)\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{960 c^4 h}-\frac {\left (b^2-4 a c\right ) \left (128 c^4 d g^2+21 b^4 f h^2-28 b^2 c h (2 b f g+b e h+2 a f h)-32 c^3 \left (a f g^2+a h (2 e g+d h)+2 b g (e g+2 d h)\right )+8 c^2 \left (2 a^2 f h^2+6 a b h (2 f g+e h)+5 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 )}{1024 c^{11/2}}\\ \end {align*}

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Mathematica [A]  time = 0.97, size = 436, normalized size = 0.75 \[ \frac {\frac {3 h \left (2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)}-\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right ) \left (8 c^2 \left (2 a^2 f h^2+6 a b h (e h+2 f g)+5 b^2 \left (h (d h+2 e g)+f g^2\right )\right )-28 b^2 c h (2 a f h+b e h+2 b f g)-32 c^3 \left (a h (d h+2 e g)+a f g^2+2 b g (2 d h+e g)\right )+21 b^4 f h^2+128 c^4 d g^2\right )}{512 c^{9/2}}-\frac {(a+x (b+c x))^{3/2} \left (8 c^2 h (a h (16 e h+32 f g+15 f h x)+b h (25 d h+50 e g+21 e h x)+b f g (7 g+6 h x))-14 b c h^2 (14 a f h+b (10 e h+20 f g+9 f h x))+105 b^3 f h^3+16 c^3 \left (f g^2 (4 g+3 h x)-h (5 d h (8 g+3 h x)+2 e g (4 g+3 h x))\right )\right )}{160 c^3}-\frac {3 (g+h x)^2 (a+x (b+c x))^{3/2} (3 b f h-4 c e h+2 c f g)}{10 c}+f (g+h x)^3 (a+x (b+c x))^{3/2}}{6 c h} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-3*(2*c*f*g - 4*c*e*h + 3*b*f*h)*(g + h*x)^2*(a + x*(b + c*x))^(3/2))/(10*c) + f*(g + h*x)^3*(a + x*(b + c*x
))^(3/2) - ((a + x*(b + c*x))^(3/2)*(105*b^3*f*h^3 - 14*b*c*h^2*(14*a*f*h + b*(20*f*g + 10*e*h + 9*f*h*x)) + 8
*c^2*h*(b*f*g*(7*g + 6*h*x) + b*h*(50*e*g + 25*d*h + 21*e*h*x) + a*h*(32*f*g + 16*e*h + 15*f*h*x)) + 16*c^3*(f
*g^2*(4*g + 3*h*x) - h*(2*e*g*(4*g + 3*h*x) + 5*d*h*(8*g + 3*h*x)))))/(160*c^3) + (3*h*(128*c^4*d*g^2 + 21*b^4
*f*h^2 - 28*b^2*c*h*(2*b*f*g + b*e*h + 2*a*f*h) - 32*c^3*(a*f*g^2 + a*h*(2*e*g + d*h) + 2*b*g*(e*g + 2*d*h)) +
8*c^2*(2*a^2*f*h^2 + 6*a*b*h*(2*f*g + e*h) + 5*b^2*(f*g^2 + h*(2*e*g + d*h))))*(2*Sqrt[c]*(b + 2*c*x)*Sqrt[a
+ x*(b + c*x)] - (b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))/(512*c^(9/2)))/(6*c*h)

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fricas [A]  time = 1.75, size = 1791, normalized size = 3.07 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)^2*(f*x^2+e*x+d)*(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")

[Out]

[-1/30720*(15*(8*(16*(b^2*c^4 - 4*a*c^5)*d - 8*(b^3*c^3 - 4*a*b*c^4)*e + (5*b^4*c^2 - 24*a*b^2*c^3 + 16*a^2*c^
4)*f)*g^2 - 8*(16*(b^3*c^3 - 4*a*b*c^4)*d - 2*(5*b^4*c^2 - 24*a*b^2*c^3 + 16*a^2*c^4)*e + (7*b^5*c - 40*a*b^3*
c^2 + 48*a^2*b*c^3)*f)*g*h + (8*(5*b^4*c^2 - 24*a*b^2*c^3 + 16*a^2*c^4)*d - 4*(7*b^5*c - 40*a*b^3*c^2 + 48*a^2
*b*c^3)*e + (21*b^6 - 140*a*b^4*c + 240*a^2*b^2*c^2 - 64*a^3*c^3)*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*(1280*c^6*f*h^2*x^5 + 128*(24*c^6*f*g*h + (12*c^
6*e + b*c^5*f)*h^2)*x^4 + 16*(120*c^6*f*g^2 + 24*(10*c^6*e + b*c^5*f)*g*h + (120*c^6*d + 12*b*c^5*e - (9*b^2*c
^4 - 20*a*c^5)*f)*h^2)*x^3 + 40*(48*b*c^5*d - 8*(3*b^2*c^4 - 8*a*c^5)*e + (15*b^3*c^3 - 52*a*b*c^4)*f)*g^2 - 8
*(80*(3*b^2*c^4 - 8*a*c^5)*d - 10*(15*b^3*c^3 - 52*a*b*c^4)*e + (105*b^4*c^2 - 460*a*b^2*c^3 + 256*a^2*c^4)*f)
*g*h + (40*(15*b^3*c^3 - 52*a*b*c^4)*d - 4*(105*b^4*c^2 - 460*a*b^2*c^3 + 256*a^2*c^4)*e + (315*b^5*c - 1680*a
*b^3*c^2 + 1808*a^2*b*c^3)*f)*h^2 + 8*(40*(8*c^6*e + b*c^5*f)*g^2 + 8*(80*c^6*d + 10*b*c^5*e - (7*b^2*c^4 - 16
*a*c^5)*f)*g*h + (40*b*c^5*d - 4*(7*b^2*c^4 - 16*a*c^5)*e + (21*b^3*c^3 - 68*a*b*c^4)*f)*h^2)*x^2 + 2*(40*(48*
c^6*d + 8*b*c^5*e - (5*b^2*c^4 - 12*a*c^5)*f)*g^2 + 8*(80*b*c^5*d - 10*(5*b^2*c^4 - 12*a*c^5)*e + (35*b^3*c^3
- 116*a*b*c^4)*f)*g*h - (40*(5*b^2*c^4 - 12*a*c^5)*d - 4*(35*b^3*c^3 - 116*a*b*c^4)*e + (105*b^4*c^2 - 448*a*b
^2*c^3 + 240*a^2*c^4)*f)*h^2)*x)*sqrt(c*x^2 + b*x + a))/c^6, 1/15360*(15*(8*(16*(b^2*c^4 - 4*a*c^5)*d - 8*(b^3
*c^3 - 4*a*b*c^4)*e + (5*b^4*c^2 - 24*a*b^2*c^3 + 16*a^2*c^4)*f)*g^2 - 8*(16*(b^3*c^3 - 4*a*b*c^4)*d - 2*(5*b^
4*c^2 - 24*a*b^2*c^3 + 16*a^2*c^4)*e + (7*b^5*c - 40*a*b^3*c^2 + 48*a^2*b*c^3)*f)*g*h + (8*(5*b^4*c^2 - 24*a*b
^2*c^3 + 16*a^2*c^4)*d - 4*(7*b^5*c - 40*a*b^3*c^2 + 48*a^2*b*c^3)*e + (21*b^6 - 140*a*b^4*c + 240*a^2*b^2*c^2
- 64*a^3*c^3)*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*(1280*c^6*f*h^2*x^5 + 128*(24*c^6*f*g*h + (12*c^6*e + b*c^5*f)*h^2)*x^4 + 16*(120*c^6*f*g^2 + 24*(10*c^6*
e + b*c^5*f)*g*h + (120*c^6*d + 12*b*c^5*e - (9*b^2*c^4 - 20*a*c^5)*f)*h^2)*x^3 + 40*(48*b*c^5*d - 8*(3*b^2*c^
4 - 8*a*c^5)*e + (15*b^3*c^3 - 52*a*b*c^4)*f)*g^2 - 8*(80*(3*b^2*c^4 - 8*a*c^5)*d - 10*(15*b^3*c^3 - 52*a*b*c^
4)*e + (105*b^4*c^2 - 460*a*b^2*c^3 + 256*a^2*c^4)*f)*g*h + (40*(15*b^3*c^3 - 52*a*b*c^4)*d - 4*(105*b^4*c^2 -
460*a*b^2*c^3 + 256*a^2*c^4)*e + (315*b^5*c - 1680*a*b^3*c^2 + 1808*a^2*b*c^3)*f)*h^2 + 8*(40*(8*c^6*e + b*c^
5*f)*g^2 + 8*(80*c^6*d + 10*b*c^5*e - (7*b^2*c^4 - 16*a*c^5)*f)*g*h + (40*b*c^5*d - 4*(7*b^2*c^4 - 16*a*c^5)*e
+ (21*b^3*c^3 - 68*a*b*c^4)*f)*h^2)*x^2 + 2*(40*(48*c^6*d + 8*b*c^5*e - (5*b^2*c^4 - 12*a*c^5)*f)*g^2 + 8*(80
*b*c^5*d - 10*(5*b^2*c^4 - 12*a*c^5)*e + (35*b^3*c^3 - 116*a*b*c^4)*f)*g*h - (40*(5*b^2*c^4 - 12*a*c^5)*d - 4*
(35*b^3*c^3 - 116*a*b*c^4)*e + (105*b^4*c^2 - 448*a*b^2*c^3 + 240*a^2*c^4)*f)*h^2)*x)*sqrt(c*x^2 + b*x + a))/c
^6]

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giac [A]  time = 0.31, size = 1012, normalized size = 1.73 \[ \frac {1}{7680} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, f h^{2} x + \frac {24 \, c^{5} f g h + b c^{4} f h^{2} + 12 \, c^{5} h^{2} e}{c^{5}}\right )} x + \frac {120 \, c^{5} f g^{2} + 24 \, b c^{4} f g h + 120 \, c^{5} d h^{2} - 9 \, b^{2} c^{3} f h^{2} + 20 \, a c^{4} f h^{2} + 240 \, c^{5} g h e + 12 \, b c^{4} h^{2} e}{c^{5}}\right )} x + \frac {40 \, b c^{4} f g^{2} + 640 \, c^{5} d g h - 56 \, b^{2} c^{3} f g h + 128 \, a c^{4} f g h + 40 \, b c^{4} d h^{2} + 21 \, b^{3} c^{2} f h^{2} - 68 \, a b c^{3} f h^{2} + 320 \, c^{5} g^{2} e + 80 \, b c^{4} g h e - 28 \, b^{2} c^{3} h^{2} e + 64 \, a c^{4} h^{2} e}{c^{5}}\right )} x + \frac {1920 \, c^{5} d g^{2} - 200 \, b^{2} c^{3} f g^{2} + 480 \, a c^{4} f g^{2} + 640 \, b c^{4} d g h + 280 \, b^{3} c^{2} f g h - 928 \, a b c^{3} f g h - 200 \, b^{2} c^{3} d h^{2} + 480 \, a c^{4} d h^{2} - 105 \, b^{4} c f h^{2} + 448 \, a b^{2} c^{2} f h^{2} - 240 \, a^{2} c^{3} f h^{2} + 320 \, b c^{4} g^{2} e - 400 \, b^{2} c^{3} g h e + 960 \, a c^{4} g h e + 140 \, b^{3} c^{2} h^{2} e - 464 \, a b c^{3} h^{2} e}{c^{5}}\right )} x + \frac {1920 \, b c^{4} d g^{2} + 600 \, b^{3} c^{2} f g^{2} - 2080 \, a b c^{3} f g^{2} - 1920 \, b^{2} c^{3} d g h + 5120 \, a c^{4} d g h - 840 \, b^{4} c f g h + 3680 \, a b^{2} c^{2} f g h - 2048 \, a^{2} c^{3} f g h + 600 \, b^{3} c^{2} d h^{2} - 2080 \, a b c^{3} d h^{2} + 315 \, b^{5} f h^{2} - 1680 \, a b^{3} c f h^{2} + 1808 \, a^{2} b c^{2} f h^{2} - 960 \, b^{2} c^{3} g^{2} e + 2560 \, a c^{4} g^{2} e + 1200 \, b^{3} c^{2} g h e - 4160 \, a b c^{3} g h e - 420 \, b^{4} c h^{2} e + 1840 \, a b^{2} c^{2} h^{2} e - 1024 \, a^{2} c^{3} h^{2} e}{c^{5}}\right )} + \frac {{\left (128 \, b^{2} c^{4} d g^{2} - 512 \, a c^{5} d g^{2} + 40 \, b^{4} c^{2} f g^{2} - 192 \, a b^{2} c^{3} f g^{2} + 128 \, a^{2} c^{4} f g^{2} - 128 \, b^{3} c^{3} d g h + 512 \, a b c^{4} d g h - 56 \, b^{5} c f g h + 320 \, a b^{3} c^{2} f g h - 384 \, a^{2} b c^{3} f g h + 40 \, b^{4} c^{2} d h^{2} - 192 \, a b^{2} c^{3} d h^{2} + 128 \, a^{2} c^{4} d h^{2} + 21 \, b^{6} f h^{2} - 140 \, a b^{4} c f h^{2} + 240 \, a^{2} b^{2} c^{2} f h^{2} - 64 \, a^{3} c^{3} f h^{2} - 64 \, b^{3} c^{3} g^{2} e + 256 \, a b c^{4} g^{2} e + 80 \, b^{4} c^{2} g h e - 384 \, a b^{2} c^{3} g h e + 256 \, a^{2} c^{4} g h e - 28 \, b^{5} c h^{2} e + 160 \, a b^{3} c^{2} h^{2} e - 192 \, a^{2} b c^{3} h^{2} e\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{1024 \, c^{\frac {11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)^2*(f*x^2+e*x+d)*(c*x^2+b*x+a)^(1/2),x, algorithm="giac")

[Out]

1/7680*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*f*h^2*x + (24*c^5*f*g*h + b*c^4*f*h^2 + 12*c^5*h^2*e)/c^5)*x + (1
20*c^5*f*g^2 + 24*b*c^4*f*g*h + 120*c^5*d*h^2 - 9*b^2*c^3*f*h^2 + 20*a*c^4*f*h^2 + 240*c^5*g*h*e + 12*b*c^4*h^
2*e)/c^5)*x + (40*b*c^4*f*g^2 + 640*c^5*d*g*h - 56*b^2*c^3*f*g*h + 128*a*c^4*f*g*h + 40*b*c^4*d*h^2 + 21*b^3*c
^2*f*h^2 - 68*a*b*c^3*f*h^2 + 320*c^5*g^2*e + 80*b*c^4*g*h*e - 28*b^2*c^3*h^2*e + 64*a*c^4*h^2*e)/c^5)*x + (19
20*c^5*d*g^2 - 200*b^2*c^3*f*g^2 + 480*a*c^4*f*g^2 + 640*b*c^4*d*g*h + 280*b^3*c^2*f*g*h - 928*a*b*c^3*f*g*h -
200*b^2*c^3*d*h^2 + 480*a*c^4*d*h^2 - 105*b^4*c*f*h^2 + 448*a*b^2*c^2*f*h^2 - 240*a^2*c^3*f*h^2 + 320*b*c^4*g
^2*e - 400*b^2*c^3*g*h*e + 960*a*c^4*g*h*e + 140*b^3*c^2*h^2*e - 464*a*b*c^3*h^2*e)/c^5)*x + (1920*b*c^4*d*g^2
+ 600*b^3*c^2*f*g^2 - 2080*a*b*c^3*f*g^2 - 1920*b^2*c^3*d*g*h + 5120*a*c^4*d*g*h - 840*b^4*c*f*g*h + 3680*a*b
^2*c^2*f*g*h - 2048*a^2*c^3*f*g*h + 600*b^3*c^2*d*h^2 - 2080*a*b*c^3*d*h^2 + 315*b^5*f*h^2 - 1680*a*b^3*c*f*h^
2 + 1808*a^2*b*c^2*f*h^2 - 960*b^2*c^3*g^2*e + 2560*a*c^4*g^2*e + 1200*b^3*c^2*g*h*e - 4160*a*b*c^3*g*h*e - 42
0*b^4*c*h^2*e + 1840*a*b^2*c^2*h^2*e - 1024*a^2*c^3*h^2*e)/c^5) + 1/1024*(128*b^2*c^4*d*g^2 - 512*a*c^5*d*g^2
+ 40*b^4*c^2*f*g^2 - 192*a*b^2*c^3*f*g^2 + 128*a^2*c^4*f*g^2 - 128*b^3*c^3*d*g*h + 512*a*b*c^4*d*g*h - 56*b^5*
c*f*g*h + 320*a*b^3*c^2*f*g*h - 384*a^2*b*c^3*f*g*h + 40*b^4*c^2*d*h^2 - 192*a*b^2*c^3*d*h^2 + 128*a^2*c^4*d*h
^2 + 21*b^6*f*h^2 - 140*a*b^4*c*f*h^2 + 240*a^2*b^2*c^2*f*h^2 - 64*a^3*c^3*f*h^2 - 64*b^3*c^3*g^2*e + 256*a*b*
c^4*g^2*e + 80*b^4*c^2*g*h*e - 384*a*b^2*c^3*g*h*e + 256*a^2*c^4*g*h*e - 28*b^5*c*h^2*e + 160*a*b^3*c^2*h^2*e
- 192*a^2*b*c^3*h^2*e)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(11/2)

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maple [B]  time = 0.02, size = 2179, normalized size = 3.73 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((h*x+g)^2*(f*x^2+e*x+d)*(c*x^2+b*x+a)^(1/2),x)

[Out]

-1/2/c^(3/2)*b*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*d*g*h-1/4*a/c*(c*x^2+b*x+a)^(1/2)*x*e*g*h-1/8*a/c
^2*(c*x^2+b*x+a)^(1/2)*b*e*g*h-7/32/c^3*b^3*(c*x^2+b*x+a)^(1/2)*x*f*g*h-5/16/c^(7/2)*b^3*ln((c*x+1/2*b)/c^(1/2
)+(c*x^2+b*x+a)^(1/2))*a*f*g*h+1/3*(c*x^2+b*x+a)^(3/2)/c*e*g^2+1/2*d*g^2*(c*x^2+b*x+a)^(1/2)*x-1/4*a^2/c^(3/2)
*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*e*g*h+1/16*f*h^2*a^2/c^2*(c*x^2+b*x+a)^(1/2)*x+1/32*f*h^2*a^2/c^3
*(c*x^2+b*x+a)^(1/2)*b+21/256*f*h^2/c^4*b^4*(c*x^2+b*x+a)^(1/2)*x+35/256*f*h^2/c^(9/2)*b^4*ln((c*x+1/2*b)/c^(1
/2)+(c*x^2+b*x+a)^(1/2))*a-7/64*f*h^2/c^4*b^3*a*(c*x^2+b*x+a)^(1/2)-15/64*f*h^2/c^(7/2)*b^2*a^2*ln((c*x+1/2*b)
/c^(1/2)+(c*x^2+b*x+a)^(1/2))+49/240*f*h^2/c^3*b*a*(c*x^2+b*x+a)^(3/2)-1/8*f*h^2*a/c^2*x*(c*x^2+b*x+a)^(3/2)+5
/32/c^2*b^2*(c*x^2+b*x+a)^(1/2)*x*f*g^2+5/32/c^3*b^3*(c*x^2+b*x+a)^(1/2)*e*g*h+3/16/c^(5/2)*b^2*ln((c*x+1/2*b)
/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*d*h^2+2/5*x^2*(c*x^2+b*x+a)^(3/2)/c*f*g*h-7/40/c^2*b*x*(c*x^2+b*x+a)^(3/2)*e*h
^2+7/24/c^3*b^2*(c*x^2+b*x+a)^(3/2)*f*g*h-7/64/c^3*b^3*(c*x^2+b*x+a)^(1/2)*x*e*h^2-7/64/c^4*b^4*(c*x^2+b*x+a)^
(1/2)*f*g*h-5/32/c^(7/2)*b^3*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*e*h^2+7/128/c^(9/2)*b^5*ln((c*x+1/2
*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*f*g*h+3/32/c^3*b^2*a*(c*x^2+b*x+a)^(1/2)*e*h^2+3/16/c^(5/2)*b*a^2*ln((c*x+1/2
*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*e*h^2-4/15*a/c^2*(c*x^2+b*x+a)^(3/2)*f*g*h-5/12/c^2*b*(c*x^2+b*x+a)^(3/2)*e*g
*h+5/32/c^2*b^2*(c*x^2+b*x+a)^(1/2)*x*d*h^2-3/20*f*h^2/c^2*b*x^2*(c*x^2+b*x+a)^(3/2)+21/160*f*h^2/c^3*b^2*x*(c
*x^2+b*x+a)^(3/2)-1/8*a/c*(c*x^2+b*x+a)^(1/2)*x*d*h^2-1/8*a/c*(c*x^2+b*x+a)^(1/2)*x*f*g^2-1/16*a/c^2*(c*x^2+b*
x+a)^(1/2)*b*d*h^2-1/16*a/c^2*(c*x^2+b*x+a)^(1/2)*b*f*g^2-1/8*d*g^2/c^(3/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+
a)^(1/2))*b^2+1/5*x^2*(c*x^2+b*x+a)^(3/2)/c*e*h^2+7/48/c^3*b^2*(c*x^2+b*x+a)^(3/2)*e*h^2-7/128/c^4*b^4*(c*x^2+
b*x+a)^(1/2)*e*h^2+7/256/c^(9/2)*b^5*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*e*h^2-2/15*a/c^2*(c*x^2+b*x+a
)^(3/2)*e*h^2+1/6*f*h^2*x^3*(c*x^2+b*x+a)^(3/2)/c-7/64*f*h^2/c^4*b^3*(c*x^2+b*x+a)^(3/2)+21/512*f*h^2/c^5*b^5*
(c*x^2+b*x+a)^(1/2)-21/1024*f*h^2/c^(11/2)*b^6*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+1/16*f*h^2*a^3/c^(5
/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-5/24/c^2*b*(c*x^2+b*x+a)^(3/2)*f*g^2+3/16/c^2*b*a*(c*x^2+b*x+a
)^(1/2)*x*e*h^2+3/16/c^3*b^2*a*(c*x^2+b*x+a)^(1/2)*f*g*h+3/8/c^(5/2)*b*a^2*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a
)^(1/2))*f*g*h+5/16/c^2*b^2*(c*x^2+b*x+a)^(1/2)*x*e*g*h+3/8/c^(5/2)*b^2*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(
1/2))*a*e*g*h-1/2/c*b*(c*x^2+b*x+a)^(1/2)*x*d*g*h-7/32*f*h^2/c^3*b^2*a*(c*x^2+b*x+a)^(1/2)*x-7/20/c^2*b*x*(c*x
^2+b*x+a)^(3/2)*f*g*h+1/2*d*g^2/c^(1/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a+1/4*d*g^2/c*(c*x^2+b*x+a
)^(1/2)*b+5/64/c^3*b^3*(c*x^2+b*x+a)^(1/2)*d*h^2+5/64/c^3*b^3*(c*x^2+b*x+a)^(1/2)*f*g^2-5/128/c^(7/2)*b^4*ln((
c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d*h^2-5/128/c^(7/2)*b^4*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*f*
g^2-1/8*a^2/c^(3/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d*h^2-1/8*a^2/c^(3/2)*ln((c*x+1/2*b)/c^(1/2)+(
c*x^2+b*x+a)^(1/2))*f*g^2+2/3*(c*x^2+b*x+a)^(3/2)/c*d*g*h-1/8/c^2*b^2*(c*x^2+b*x+a)^(1/2)*e*g^2+1/16/c^(5/2)*b
^3*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*e*g^2+1/4*x*(c*x^2+b*x+a)^(3/2)/c*d*h^2+1/4*x*(c*x^2+b*x+a)^(3/
2)/c*f*g^2-5/24/c^2*b*(c*x^2+b*x+a)^(3/2)*d*h^2+3/8/c^2*b*a*(c*x^2+b*x+a)^(1/2)*x*f*g*h+3/16/c^(5/2)*b^2*ln((c
*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*f*g^2-5/64/c^(7/2)*b^4*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*e*
g*h+1/2*x*(c*x^2+b*x+a)^(3/2)/c*e*g*h-1/4/c*b*(c*x^2+b*x+a)^(1/2)*x*e*g^2-1/4/c^2*b^2*(c*x^2+b*x+a)^(1/2)*d*g*
h-1/4/c^(3/2)*b*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*e*g^2+1/8/c^(5/2)*b^3*ln((c*x+1/2*b)/c^(1/2)+(c*
x^2+b*x+a)^(1/2))*d*g*h

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)^2*(f*x^2+e*x+d)*(c*x^2+b*x+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more details)Is 4*a*c-b^2 positive, negative or zero?

________________________________________________________________________________________

mupad [B]  time = 7.91, size = 1881, normalized size = 3.22 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g + h*x)^2*(a + b*x + c*x^2)^(1/2)*(d + e*x + f*x^2),x)

[Out]

d*g^2*(x/2 + b/(4*c))*(a + b*x + c*x^2)^(1/2) + (e*h^2*x^2*(a + b*x + c*x^2)^(3/2))/(5*c) + (f*h^2*x^3*(a + b*
x + c*x^2)^(3/2))/(6*c) - (a*d*h^2*((x/2 + b/(4*c))*(a + b*x + c*x^2)^(1/2) + (log((b/2 + c*x)/c^(1/2) + (a +
b*x + c*x^2)^(1/2))*(a*c - b^2/4))/(2*c^(3/2))))/(4*c) - (a*f*g^2*((x/2 + b/(4*c))*(a + b*x + c*x^2)^(1/2) + (
log((b/2 + c*x)/c^(1/2) + (a + b*x + c*x^2)^(1/2))*(a*c - b^2/4))/(2*c^(3/2))))/(4*c) + (d*g^2*log((b/2 + c*x)
/c^(1/2) + (a + b*x + c*x^2)^(1/2))*(a*c - b^2/4))/(2*c^(3/2)) + (e*g^2*log((b + 2*c*x)/c^(1/2) + 2*(a + b*x +
c*x^2)^(1/2))*(b^3 - 4*a*b*c))/(16*c^(5/2)) - (2*a*e*h^2*((log((b + 2*c*x)/c^(1/2) + 2*(a + b*x + c*x^2)^(1/2
))*(b^3 - 4*a*b*c))/(16*c^(5/2)) + ((8*c*(a + c*x^2) - 3*b^2 + 2*b*c*x)*(a + b*x + c*x^2)^(1/2))/(24*c^2)))/(5
*c) - (5*b*d*h^2*((log((b + 2*c*x)/c^(1/2) + 2*(a + b*x + c*x^2)^(1/2))*(b^3 - 4*a*b*c))/(16*c^(5/2)) + ((8*c*
(a + c*x^2) - 3*b^2 + 2*b*c*x)*(a + b*x + c*x^2)^(1/2))/(24*c^2)))/(8*c) - (5*b*f*g^2*((log((b + 2*c*x)/c^(1/2
) + 2*(a + b*x + c*x^2)^(1/2))*(b^3 - 4*a*b*c))/(16*c^(5/2)) + ((8*c*(a + c*x^2) - 3*b^2 + 2*b*c*x)*(a + b*x +
c*x^2)^(1/2))/(24*c^2)))/(8*c) + (e*g^2*(8*c*(a + c*x^2) - 3*b^2 + 2*b*c*x)*(a + b*x + c*x^2)^(1/2))/(24*c^2)
+ (d*h^2*x*(a + b*x + c*x^2)^(3/2))/(4*c) + (f*g^2*x*(a + b*x + c*x^2)^(3/2))/(4*c) + (a*f*h^2*((5*b*((log((b
+ 2*c*x)/c^(1/2) + 2*(a + b*x + c*x^2)^(1/2))*(b^3 - 4*a*b*c))/(16*c^(5/2)) + ((8*c*(a + c*x^2) - 3*b^2 + 2*b
*c*x)*(a + b*x + c*x^2)^(1/2))/(24*c^2)))/(8*c) - (x*(a + b*x + c*x^2)^(3/2))/(4*c) + (a*((x/2 + b/(4*c))*(a +
b*x + c*x^2)^(1/2) + (log((b/2 + c*x)/c^(1/2) + (a + b*x + c*x^2)^(1/2))*(a*c - b^2/4))/(2*c^(3/2))))/(4*c)))
/(2*c) + (7*b*e*h^2*((5*b*((log((b + 2*c*x)/c^(1/2) + 2*(a + b*x + c*x^2)^(1/2))*(b^3 - 4*a*b*c))/(16*c^(5/2))
+ ((8*c*(a + c*x^2) - 3*b^2 + 2*b*c*x)*(a + b*x + c*x^2)^(1/2))/(24*c^2)))/(8*c) - (x*(a + b*x + c*x^2)^(3/2)
)/(4*c) + (a*((x/2 + b/(4*c))*(a + b*x + c*x^2)^(1/2) + (log((b/2 + c*x)/c^(1/2) + (a + b*x + c*x^2)^(1/2))*(a
*c - b^2/4))/(2*c^(3/2))))/(4*c)))/(10*c) - (3*b*f*h^2*((7*b*((5*b*((log((b + 2*c*x)/c^(1/2) + 2*(a + b*x + c*
x^2)^(1/2))*(b^3 - 4*a*b*c))/(16*c^(5/2)) + ((8*c*(a + c*x^2) - 3*b^2 + 2*b*c*x)*(a + b*x + c*x^2)^(1/2))/(24*
c^2)))/(8*c) - (x*(a + b*x + c*x^2)^(3/2))/(4*c) + (a*((x/2 + b/(4*c))*(a + b*x + c*x^2)^(1/2) + (log((b/2 + c
*x)/c^(1/2) + (a + b*x + c*x^2)^(1/2))*(a*c - b^2/4))/(2*c^(3/2))))/(4*c)))/(10*c) - (2*a*((log((b + 2*c*x)/c^
(1/2) + 2*(a + b*x + c*x^2)^(1/2))*(b^3 - 4*a*b*c))/(16*c^(5/2)) + ((8*c*(a + c*x^2) - 3*b^2 + 2*b*c*x)*(a + b
*x + c*x^2)^(1/2))/(24*c^2)))/(5*c) + (x^2*(a + b*x + c*x^2)^(3/2))/(5*c)))/(4*c) + (2*f*g*h*x^2*(a + b*x + c*
x^2)^(3/2))/(5*c) - (a*e*g*h*((x/2 + b/(4*c))*(a + b*x + c*x^2)^(1/2) + (log((b/2 + c*x)/c^(1/2) + (a + b*x +
c*x^2)^(1/2))*(a*c - b^2/4))/(2*c^(3/2))))/(2*c) + (d*g*h*log((b + 2*c*x)/c^(1/2) + 2*(a + b*x + c*x^2)^(1/2))
*(b^3 - 4*a*b*c))/(8*c^(5/2)) - (4*a*f*g*h*((log((b + 2*c*x)/c^(1/2) + 2*(a + b*x + c*x^2)^(1/2))*(b^3 - 4*a*b
*c))/(16*c^(5/2)) + ((8*c*(a + c*x^2) - 3*b^2 + 2*b*c*x)*(a + b*x + c*x^2)^(1/2))/(24*c^2)))/(5*c) - (5*b*e*g*
h*((log((b + 2*c*x)/c^(1/2) + 2*(a + b*x + c*x^2)^(1/2))*(b^3 - 4*a*b*c))/(16*c^(5/2)) + ((8*c*(a + c*x^2) - 3
*b^2 + 2*b*c*x)*(a + b*x + c*x^2)^(1/2))/(24*c^2)))/(4*c) + (d*g*h*(8*c*(a + c*x^2) - 3*b^2 + 2*b*c*x)*(a + b*
x + c*x^2)^(1/2))/(12*c^2) + (e*g*h*x*(a + b*x + c*x^2)^(3/2))/(2*c) + (7*b*f*g*h*((5*b*((log((b + 2*c*x)/c^(1
/2) + 2*(a + b*x + c*x^2)^(1/2))*(b^3 - 4*a*b*c))/(16*c^(5/2)) + ((8*c*(a + c*x^2) - 3*b^2 + 2*b*c*x)*(a + b*x
+ c*x^2)^(1/2))/(24*c^2)))/(8*c) - (x*(a + b*x + c*x^2)^(3/2))/(4*c) + (a*((x/2 + b/(4*c))*(a + b*x + c*x^2)^
(1/2) + (log((b/2 + c*x)/c^(1/2) + (a + b*x + c*x^2)^(1/2))*(a*c - b^2/4))/(2*c^(3/2))))/(4*c)))/(5*c)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (g + h x\right )^{2} \sqrt {a + b x + c x^{2}} \left (d + e x + f x^{2}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)**2*(f*x**2+e*x+d)*(c*x**2+b*x+a)**(1/2),x)

[Out]

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

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