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 \[ \frac{(b+2 c x) \sqrt{a+b x+c x^2} \left (8 c^2 \left (2 a^2 f h^2+6 a b h (e h+2 f g)+5 b^2 \left (d h^2+2 e g h+f g^2\right )\right )-28 b^2 c h (2 a f h+b e h+2 b f g)-32 c^3 \left (a \left (d h^2+2 e g h+f g^2\right )+2 b g (2 d h+e g)\right )+21 b^4 f h^2+128 c^4 d g^2\right )}{512 c^5}-\frac{\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (8 c^2 \left (2 a^2 f h^2+6 a b h (e h+2 f g)+5 b^2 \left (d h^2+2 e g h+f g^2\right )\right )-28 b^2 c h (2 a f h+b e h+2 b f g)-32 c^3 \left (a \left (d h^2+2 e g h+f g^2\right )+2 b g (2 d h+e g)\right )+21 b^4 f h^2+128 c^4 d g^2\right )}{1024 c^{11/2}}-\frac{\left (a+b x+c x^2\right )^{3/2} \left (-6 c h x \left (-4 c h (5 a f h+7 b e h+2 b f g)+21 b^2 f h^2-8 c^2 \left (f g^2-h (5 d h+2 e g)\right )\right )+8 c^2 h \left (16 a h (e h+2 f g)+b \left (25 h (d h+2 e g)+7 f g^2\right )\right )-28 b c h^2 (7 a f h+5 b (e h+2 f g))+105 b^3 f h^3+64 c^3 g \left (f g^2-2 h (5 d h+e g)\right )\right )}{960 c^4 h}-\frac{(g+h x)^2 \left (a+b x+c x^2\right )^{3/2} (3 b f h-4 c e h+2 c f g)}{20 c^2 h}+\frac{f (g+h x)^3 \left (a+b x+c x^2\right )^{3/2}}{6 c h} \]
[Out]
((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*(2*b*g*(e*g + 2*d*h) + a*(f*g
^2 + 2*e*g*h + d*h^2)) + 8*c^2*(2*a^2*f*h^2 + 6*a*b*h*(2*f*g + e*h) + 5*b^2*(f*g^2 + 2*e*g*h + d*h^2)))*(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*g*(f*g^2 - 2*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*(16*a*h*(2*f*g + e*h) + b*(7*f*g^2 + 25*h*(2*e*g +
d*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*(2*b*g*(e*g + 2*d*h) + a*(f*g^2 + 2*e*g*h + d*h^2)) + 8*c^2*(2*a^2*f*h^2 + 6*a*b*h*(2*f*
g + e*h) + 5*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])])/(1024*c^(
11/2))
________________________________________________________________________________________
Rubi [A] time = 1.44001, 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 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 \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*}
Mathematica [A] time = 0.96637, 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)
________________________________________________________________________________________
Maple [B] time = 0.059, size = 2179, normalized size = 3.7 \begin{align*} \text{result too large to display} \end{align*}
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]
-7/32*f*h^2*b^2/c^3*a*x*(c*x^2+b*x+a)^(1/2)-1/2*b/c*x*(c*x^2+b*x+a)^(1/2)*d*g*h+1/6*f*h^2*x^3*(c*x^2+b*x+a)^(3
/2)/c-7/64*b^3/c^3*x*(c*x^2+b*x+a)^(1/2)*e*h^2-7/64*b^4/c^4*(c*x^2+b*x+a)^(1/2)*f*g*h-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*e*h^2+7/128*b^5/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*f*g
*h+3/32*b^2/c^3*a*(c*x^2+b*x+a)^(1/2)*e*h^2+7/24*b^2/c^3*(c*x^2+b*x+a)^(3/2)*f*g*h+3/8*b/c^2*a*x*(c*x^2+b*x+a)
^(1/2)*f*g*h+5/64*b^3/c^3*(c*x^2+b*x+a)^(1/2)*d*h^2+5/64*b^3/c^3*(c*x^2+b*x+a)^(1/2)*f*g^2-5/128*b^4/c^(7/2)*l
n((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d*h^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^2-1/8*a^2/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d*h^2-1/8*a^2/c^(3/2)*ln((1/2*b+c*x)/c^(1/2
)+(c*x^2+b*x+a)^(1/2))*f*g^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^2-4/15*a/c^2*(c
*x^2+b*x+a)^(3/2)*f*g*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^2+1/8*b^3/c^(5/2)*ln((
1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d*g*h-3/20*f*h^2*b/c^2*x^2*(c*x^2+b*x+a)^(3/2)+21/160*f*h^2*b^2/c^3*x*
(c*x^2+b*x+a)^(3/2)+21/256*f*h^2*b^4/c^4*x*(c*x^2+b*x+a)^(1/2)+35/256*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+7/48*b^2/c^3*(c*x^2+b*x+a)^(3/2)*e*h^2-7/128*b^4/c^4*(c*x^2+b*x+a)^(1/2)*e*h^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))*e*h^2+1/16*f*h^2*a^3/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*
x^2+b*x+a)^(1/2))-21/1024*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))+1/2*d*g^2/c^(1/2)*ln(
(1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a-2/15*a/c^2*(c*x^2+b*x+a)^(3/2)*e*h^2+1/5*x^2*(c*x^2+b*x+a)^(3/2)/c*
e*h^2-1/8*d*g^2/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b^2-5/24*b/c^2*(c*x^2+b*x+a)^(3/2)*f*g^2+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))*e*h^2-1/8*a/c^2*(c*x^2+b*x+a)^(1/2)*b*e*g*h+5/16
*b^2/c^2*x*(c*x^2+b*x+a)^(1/2)*e*g*h+3/8*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*h-1/4*a
/c*x*(c*x^2+b*x+a)^(1/2)*e*g*h+3/16*b^2/c^3*a*(c*x^2+b*x+a)^(1/2)*f*g*h+3/8*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*h-5/16*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*h+3/16*b/c^2*
a*x*(c*x^2+b*x+a)^(1/2)*e*h^2-7/20*b/c^2*x*(c*x^2+b*x+a)^(3/2)*f*g*h-7/32*b^3/c^3*x*(c*x^2+b*x+a)^(1/2)*f*g*h-
1/2*b/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*d*g*h-5/24*b/c^2*(c*x^2+b*x+a)^(3/2)*d*h^2+2/3*(c*
x^2+b*x+a)^(3/2)/c*d*g*h-1/8*b^2/c^2*(c*x^2+b*x+a)^(1/2)*e*g^2+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^2+1/4*d*g^2/c*(c*x^2+b*x+a)^(1/2)*b+1/3*(c*x^2+b*x+a)^(3/2)/c*e*g^2+1/2*d*g^2*x*(c*x^2+b*x+a
)^(1/2)+21/512*f*h^2*b^5/c^5*(c*x^2+b*x+a)^(1/2)-7/64*f*h^2*b^3/c^4*(c*x^2+b*x+a)^(3/2)-7/40*b/c^2*x*(c*x^2+b*
x+a)^(3/2)*e*h^2-7/64*f*h^2*b^3/c^4*a*(c*x^2+b*x+a)^(1/2)-15/64*f*h^2*b^2/c^(7/2)*a^2*ln((1/2*b+c*x)/c^(1/2)+(
c*x^2+b*x+a)^(1/2))+49/240*f*h^2*b/c^3*a*(c*x^2+b*x+a)^(3/2)-1/16*a/c^2*(c*x^2+b*x+a)^(1/2)*b*f*g^2-1/4*a^2/c^
(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*e*g*h+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*d*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^2-5/64*b^4/c^(7/2)*ln((1/2*b
+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*e*g*h-1/8*a/c*x*(c*x^2+b*x+a)^(1/2)*d*h^2+5/32*b^3/c^3*(c*x^2+b*x+a)^(1/2)*
e*g*h+2/5*x^2*(c*x^2+b*x+a)^(3/2)/c*f*g*h-1/4*b^2/c^2*(c*x^2+b*x+a)^(1/2)*d*g*h+1/2*x*(c*x^2+b*x+a)^(3/2)/c*e*
g*h-1/4*b/c*x*(c*x^2+b*x+a)^(1/2)*e*g^2-1/8*a/c*x*(c*x^2+b*x+a)^(1/2)*f*g^2-1/16*a/c^2*(c*x^2+b*x+a)^(1/2)*b*d
*h^2+5/32*b^2/c^2*x*(c*x^2+b*x+a)^(1/2)*d*h^2+5/32*b^2/c^2*x*(c*x^2+b*x+a)^(1/2)*f*g^2-1/8*f*h^2*a/c^2*x*(c*x^
2+b*x+a)^(3/2)+1/16*f*h^2*a^2/c^2*x*(c*x^2+b*x+a)^(1/2)+1/32*f*h^2*a^2/c^3*(c*x^2+b*x+a)^(1/2)*b-5/12*b/c^2*(c
*x^2+b*x+a)^(3/2)*e*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*(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 = 6.63003, size = 4019, normalized size = 6.88 \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*(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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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (g + h x\right )^{2} \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)**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)
________________________________________________________________________________________
Giac [A] time = 1.27079, size = 1366, normalized size = 2.34 \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*(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)