### 3.200 $$\int \frac{(a+b x+c x^2)^{3/2} (d+e x+f x^2)}{g+h x} \, dx$$

Optimal. Leaf size=660 $\frac{\sqrt{a+b x+c x^2} \left (2 c h x \left (8 c h (2 c g-b h) (b f g-2 c d h)-\left (-4 c h (2 b g-3 a h)-3 b^2 h^2+16 c^2 g^2\right ) (b f h-2 c e h+2 c f g)\right )+6 b^2 c h^3 (-2 a f h-b e h+b f g)-8 b c^2 h^2 \left (3 a h (f g-e h)-2 b \left (d h^2-e g h+f g^2\right )\right )-32 c^3 h (5 b g-4 a h) \left (f g^2-h (e g-d h)\right )+3 b^4 f h^4+128 c^4 g^2 \left (f g^2-h (e g-d h)\right )\right )}{128 c^3 h^5}-\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (4 c h (2 c g-b h) \left (8 c h (b g-2 a h) (b f g-2 c d h)-g \left (-4 a c h-3 b^2 h+8 b c g\right ) (b f h-2 c e h+2 c f g)\right )-2 \left (-2 c h (b g-a h)-\frac{b^2 h^2}{2}+4 c^2 g^2\right ) \left (8 c h (2 c g-b h) (b f g-2 c d h)-\left (-4 c h (2 b g-3 a h)-3 b^2 h^2+16 c^2 g^2\right ) (b f h-2 c e h+2 c f g)\right )\right )}{256 c^{7/2} h^6}-\frac{\left (a+b x+c x^2\right )^{3/2} (8 c h (b f g-2 c d h)+6 c h x (b f h-2 c e h+2 c f g)-(8 c g-3 b h) (b f h-2 c e h+2 c f g))}{48 c^2 h^3}+\frac{\left (a h^2-b g h+c g^2\right )^{3/2} \left (f g^2-h (e g-d h)\right ) \tanh ^{-1}\left (\frac{-2 a h+x (2 c g-b h)+b g}{2 \sqrt{a+b x+c x^2} \sqrt{a h^2-b g h+c g^2}}\right )}{h^6}+\frac{f \left (a+b x+c x^2\right )^{5/2}}{5 c h}$

[Out]

((3*b^4*f*h^4 + 6*b^2*c*h^3*(b*f*g - b*e*h - 2*a*f*h) + 128*c^4*g^2*(f*g^2 - h*(e*g - d*h)) - 32*c^3*h*(5*b*g
- 4*a*h)*(f*g^2 - h*(e*g - d*h)) - 8*b*c^2*h^2*(3*a*h*(f*g - e*h) - 2*b*(f*g^2 - e*g*h + d*h^2)) + 2*c*h*(8*c*
h*(2*c*g - b*h)*(b*f*g - 2*c*d*h) - (2*c*f*g - 2*c*e*h + b*f*h)*(16*c^2*g^2 - 3*b^2*h^2 - 4*c*h*(2*b*g - 3*a*h
)))*x)*Sqrt[a + b*x + c*x^2])/(128*c^3*h^5) - ((8*c*h*(b*f*g - 2*c*d*h) - (8*c*g - 3*b*h)*(2*c*f*g - 2*c*e*h +
b*f*h) + 6*c*h*(2*c*f*g - 2*c*e*h + b*f*h)*x)*(a + b*x + c*x^2)^(3/2))/(48*c^2*h^3) + (f*(a + b*x + c*x^2)^(5
/2))/(5*c*h) - ((4*c*h*(2*c*g - b*h)*(8*c*h*(b*g - 2*a*h)*(b*f*g - 2*c*d*h) - g*(8*b*c*g - 3*b^2*h - 4*a*c*h)*
(2*c*f*g - 2*c*e*h + b*f*h)) - 2*(4*c^2*g^2 - (b^2*h^2)/2 - 2*c*h*(b*g - a*h))*(8*c*h*(2*c*g - b*h)*(b*f*g - 2
*c*d*h) - (2*c*f*g - 2*c*e*h + b*f*h)*(16*c^2*g^2 - 3*b^2*h^2 - 4*c*h*(2*b*g - 3*a*h))))*ArcTanh[(b + 2*c*x)/(
2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(256*c^(7/2)*h^6) + ((c*g^2 - b*g*h + a*h^2)^(3/2)*(f*g^2 - h*(e*g - d*h))*
ArcTanh[(b*g - 2*a*h + (2*c*g - b*h)*x)/(2*Sqrt[c*g^2 - b*g*h + a*h^2]*Sqrt[a + b*x + c*x^2])])/h^6

________________________________________________________________________________________

Rubi [A]  time = 1.82506, antiderivative size = 660, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 32, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.188, Rules used = {1653, 814, 843, 621, 206, 724} $\frac{\sqrt{a+b x+c x^2} \left (2 c h x \left (8 c h (2 c g-b h) (b f g-2 c d h)-\left (-4 c h (2 b g-3 a h)-3 b^2 h^2+16 c^2 g^2\right ) (b f h-2 c e h+2 c f g)\right )+6 b^2 c h^3 (-2 a f h-b e h+b f g)-8 b c^2 h^2 \left (3 a h (f g-e h)-2 b \left (d h^2-e g h+f g^2\right )\right )-32 c^3 h (5 b g-4 a h) \left (f g^2-h (e g-d h)\right )+3 b^4 f h^4+128 c^4 \left (f g^4-g^2 h (e g-d h)\right )\right )}{128 c^3 h^5}-\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (4 c h (2 c g-b h) \left (8 c h (b g-2 a h) (b f g-2 c d h)-g \left (-4 a c h-3 b^2 h+8 b c g\right ) (b f h-2 c e h+2 c f g)\right )-2 \left (-2 c h (b g-a h)-\frac{b^2 h^2}{2}+4 c^2 g^2\right ) \left (8 c h (2 c g-b h) (b f g-2 c d h)-\left (-4 c h (2 b g-3 a h)-3 b^2 h^2+16 c^2 g^2\right ) (b f h-2 c e h+2 c f g)\right )\right )}{256 c^{7/2} h^6}-\frac{\left (a+b x+c x^2\right )^{3/2} (8 c h (b f g-2 c d h)+6 c h x (b f h-2 c e h+2 c f g)-(8 c g-3 b h) (b f h-2 c e h+2 c f g))}{48 c^2 h^3}+\frac{\left (a h^2-b g h+c g^2\right )^{3/2} \left (f g^2-h (e g-d h)\right ) \tanh ^{-1}\left (\frac{-2 a h+x (2 c g-b h)+b g}{2 \sqrt{a+b x+c x^2} \sqrt{a h^2-b g h+c g^2}}\right )}{h^6}+\frac{f \left (a+b x+c x^2\right )^{5/2}}{5 c h}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((3*b^4*f*h^4 + 6*b^2*c*h^3*(b*f*g - b*e*h - 2*a*f*h) - 32*c^3*h*(5*b*g - 4*a*h)*(f*g^2 - h*(e*g - d*h)) + 128
*c^4*(f*g^4 - g^2*h*(e*g - d*h)) - 8*b*c^2*h^2*(3*a*h*(f*g - e*h) - 2*b*(f*g^2 - e*g*h + d*h^2)) + 2*c*h*(8*c*
h*(2*c*g - b*h)*(b*f*g - 2*c*d*h) - (2*c*f*g - 2*c*e*h + b*f*h)*(16*c^2*g^2 - 3*b^2*h^2 - 4*c*h*(2*b*g - 3*a*h
)))*x)*Sqrt[a + b*x + c*x^2])/(128*c^3*h^5) - ((8*c*h*(b*f*g - 2*c*d*h) - (8*c*g - 3*b*h)*(2*c*f*g - 2*c*e*h +
b*f*h) + 6*c*h*(2*c*f*g - 2*c*e*h + b*f*h)*x)*(a + b*x + c*x^2)^(3/2))/(48*c^2*h^3) + (f*(a + b*x + c*x^2)^(5
/2))/(5*c*h) - ((4*c*h*(2*c*g - b*h)*(8*c*h*(b*g - 2*a*h)*(b*f*g - 2*c*d*h) - g*(8*b*c*g - 3*b^2*h - 4*a*c*h)*
(2*c*f*g - 2*c*e*h + b*f*h)) - 2*(4*c^2*g^2 - (b^2*h^2)/2 - 2*c*h*(b*g - a*h))*(8*c*h*(2*c*g - b*h)*(b*f*g - 2
*c*d*h) - (2*c*f*g - 2*c*e*h + b*f*h)*(16*c^2*g^2 - 3*b^2*h^2 - 4*c*h*(2*b*g - 3*a*h))))*ArcTanh[(b + 2*c*x)/(
2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(256*c^(7/2)*h^6) + ((c*g^2 - b*g*h + a*h^2)^(3/2)*(f*g^2 - h*(e*g - d*h))*
ArcTanh[(b*g - 2*a*h + (2*c*g - b*h)*x)/(2*Sqrt[c*g^2 - b*g*h + a*h^2]*Sqrt[a + b*x + c*x^2])])/h^6

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 814

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

Rule 843

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

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])

Rule 724

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

Rubi steps

\begin{align*} \int \frac{\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{g+h x} \, dx &=\frac{f \left (a+b x+c x^2\right )^{5/2}}{5 c h}+\frac{\int \frac{\left (-\frac{5}{2} h (b f g-2 c d h)-\frac{5}{2} h (2 c f g-2 c e h+b f h) x\right ) \left (a+b x+c x^2\right )^{3/2}}{g+h x} \, dx}{5 c h^2}\\ &=-\frac{(8 c h (b f g-2 c d h)-(8 c g-3 b h) (2 c f g-2 c e h+b f h)+6 c h (2 c f g-2 c e h+b f h) x) \left (a+b x+c x^2\right )^{3/2}}{48 c^2 h^3}+\frac{f \left (a+b x+c x^2\right )^{5/2}}{5 c h}-\frac{\int \frac{\left (-\frac{5}{4} h \left (8 c h (b g-2 a h) (b f g-2 c d h)-g \left (8 b c g-3 b^2 h-4 a c h\right ) (2 c f g-2 c e h+b f h)\right )-\frac{5}{4} h \left (8 c h (2 c g-b h) (b f g-2 c d h)-2 (2 c f g-2 c e h+b f h) \left (8 c^2 g^2-\frac{3 b^2 h^2}{2}-2 c h (2 b g-3 a h)\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{g+h x} \, dx}{40 c^2 h^4}\\ &=\frac{\left (3 b^4 f h^4+6 b^2 c h^3 (b f g-b e h-2 a f h)-32 c^3 h (5 b g-4 a h) \left (f g^2-h (e g-d h)\right )+128 c^4 \left (f g^4-g^2 h (e g-d h)\right )-8 b c^2 h^2 \left (3 a h (f g-e h)-2 b \left (f g^2-e g h+d h^2\right )\right )+2 c h \left (8 c h (2 c g-b h) (b f g-2 c d h)-(2 c f g-2 c e h+b f h) \left (16 c^2 g^2-3 b^2 h^2-4 c h (2 b g-3 a h)\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{128 c^3 h^5}-\frac{(8 c h (b f g-2 c d h)-(8 c g-3 b h) (2 c f g-2 c e h+b f h)+6 c h (2 c f g-2 c e h+b f h) x) \left (a+b x+c x^2\right )^{3/2}}{48 c^2 h^3}+\frac{f \left (a+b x+c x^2\right )^{5/2}}{5 c h}+\frac{\int \frac{-\frac{5}{8} h \left (4 c h (b g-2 a h) \left (8 c h (b g-2 a h) (b f g-2 c d h)-g \left (8 b c g-3 b^2 h-4 a c h\right ) (2 c f g-2 c e h+b f h)\right )-g \left (4 b c g-b^2 h-4 a c h\right ) \left (8 c h (2 c g-b h) (b f g-2 c d h)-(2 c f g-2 c e h+b f h) \left (16 c^2 g^2-3 b^2 h^2-4 c h (2 b g-3 a h)\right )\right )\right )-\frac{5}{8} h \left (4 c h (2 c g-b h) \left (8 c h (b g-2 a h) (b f g-2 c d h)-g \left (8 b c g-3 b^2 h-4 a c h\right ) (2 c f g-2 c e h+b f h)\right )-2 \left (4 c^2 g^2-\frac{b^2 h^2}{2}-2 c h (b g-a h)\right ) \left (8 c h (2 c g-b h) (b f g-2 c d h)-(2 c f g-2 c e h+b f h) \left (16 c^2 g^2-3 b^2 h^2-4 c h (2 b g-3 a h)\right )\right )\right ) x}{(g+h x) \sqrt{a+b x+c x^2}} \, dx}{160 c^3 h^6}\\ &=\frac{\left (3 b^4 f h^4+6 b^2 c h^3 (b f g-b e h-2 a f h)-32 c^3 h (5 b g-4 a h) \left (f g^2-h (e g-d h)\right )+128 c^4 \left (f g^4-g^2 h (e g-d h)\right )-8 b c^2 h^2 \left (3 a h (f g-e h)-2 b \left (f g^2-e g h+d h^2\right )\right )+2 c h \left (8 c h (2 c g-b h) (b f g-2 c d h)-(2 c f g-2 c e h+b f h) \left (16 c^2 g^2-3 b^2 h^2-4 c h (2 b g-3 a h)\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{128 c^3 h^5}-\frac{(8 c h (b f g-2 c d h)-(8 c g-3 b h) (2 c f g-2 c e h+b f h)+6 c h (2 c f g-2 c e h+b f h) x) \left (a+b x+c x^2\right )^{3/2}}{48 c^2 h^3}+\frac{f \left (a+b x+c x^2\right )^{5/2}}{5 c h}+\frac{\left (\left (c g^2-b g h+a h^2\right )^2 \left (f g^2-h (e g-d h)\right )\right ) \int \frac{1}{(g+h x) \sqrt{a+b x+c x^2}} \, dx}{h^6}-\frac{\left (4 c h (2 c g-b h) \left (8 c h (b g-2 a h) (b f g-2 c d h)-g \left (8 b c g-3 b^2 h-4 a c h\right ) (2 c f g-2 c e h+b f h)\right )-2 \left (4 c^2 g^2-\frac{b^2 h^2}{2}-2 c h (b g-a h)\right ) \left (8 c h (2 c g-b h) (b f g-2 c d h)-(2 c f g-2 c e h+b f h) \left (16 c^2 g^2-3 b^2 h^2-4 c h (2 b g-3 a h)\right )\right )\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{256 c^3 h^6}\\ &=\frac{\left (3 b^4 f h^4+6 b^2 c h^3 (b f g-b e h-2 a f h)-32 c^3 h (5 b g-4 a h) \left (f g^2-h (e g-d h)\right )+128 c^4 \left (f g^4-g^2 h (e g-d h)\right )-8 b c^2 h^2 \left (3 a h (f g-e h)-2 b \left (f g^2-e g h+d h^2\right )\right )+2 c h \left (8 c h (2 c g-b h) (b f g-2 c d h)-(2 c f g-2 c e h+b f h) \left (16 c^2 g^2-3 b^2 h^2-4 c h (2 b g-3 a h)\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{128 c^3 h^5}-\frac{(8 c h (b f g-2 c d h)-(8 c g-3 b h) (2 c f g-2 c e h+b f h)+6 c h (2 c f g-2 c e h+b f h) x) \left (a+b x+c x^2\right )^{3/2}}{48 c^2 h^3}+\frac{f \left (a+b x+c x^2\right )^{5/2}}{5 c h}-\frac{\left (2 \left (c g^2-b g h+a h^2\right )^2 \left (f g^2-h (e g-d h)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 c g^2-4 b g h+4 a h^2-x^2} \, dx,x,\frac{-b g+2 a h-(2 c g-b h) x}{\sqrt{a+b x+c x^2}}\right )}{h^6}-\frac{\left (4 c h (2 c g-b h) \left (8 c h (b g-2 a h) (b f g-2 c d h)-g \left (8 b c g-3 b^2 h-4 a c h\right ) (2 c f g-2 c e h+b f h)\right )-2 \left (4 c^2 g^2-\frac{b^2 h^2}{2}-2 c h (b g-a h)\right ) \left (8 c h (2 c g-b h) (b f g-2 c d h)-(2 c f g-2 c e h+b f h) \left (16 c^2 g^2-3 b^2 h^2-4 c h (2 b g-3 a h)\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 )}{128 c^3 h^6}\\ &=\frac{\left (3 b^4 f h^4+6 b^2 c h^3 (b f g-b e h-2 a f h)-32 c^3 h (5 b g-4 a h) \left (f g^2-h (e g-d h)\right )+128 c^4 \left (f g^4-g^2 h (e g-d h)\right )-8 b c^2 h^2 \left (3 a h (f g-e h)-2 b \left (f g^2-e g h+d h^2\right )\right )+2 c h \left (8 c h (2 c g-b h) (b f g-2 c d h)-(2 c f g-2 c e h+b f h) \left (16 c^2 g^2-3 b^2 h^2-4 c h (2 b g-3 a h)\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{128 c^3 h^5}-\frac{(8 c h (b f g-2 c d h)-(8 c g-3 b h) (2 c f g-2 c e h+b f h)+6 c h (2 c f g-2 c e h+b f h) x) \left (a+b x+c x^2\right )^{3/2}}{48 c^2 h^3}+\frac{f \left (a+b x+c x^2\right )^{5/2}}{5 c h}-\frac{\left (4 c h (2 c g-b h) \left (8 c h (b g-2 a h) (b f g-2 c d h)-g \left (8 b c g-3 b^2 h-4 a c h\right ) (2 c f g-2 c e h+b f h)\right )-2 \left (4 c^2 g^2-\frac{b^2 h^2}{2}-2 c h (b g-a h)\right ) \left (8 c h (2 c g-b h) (b f g-2 c d h)-(2 c f g-2 c e h+b f h) \left (16 c^2 g^2-3 b^2 h^2-4 c h (2 b g-3 a h)\right )\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{256 c^{7/2} h^6}+\frac{\left (c g^2-b g h+a h^2\right )^{3/2} \left (f g^2-h (e g-d h)\right ) \tanh ^{-1}\left (\frac{b g-2 a h+(2 c g-b h) x}{2 \sqrt{c g^2-b g h+a h^2} \sqrt{a+b x+c x^2}}\right )}{h^6}\\ \end{align*}

Mathematica [A]  time = 2.40673, size = 635, normalized size = 0.96 $\frac{\sqrt{c} h \sqrt{a+x (b+c x)} \left (6 b^2 c h^3 (-2 a f h-b e h+b f (g+h x))-16 c^3 h \left (a h (h (-8 d h+8 e g-3 e h x)+f g (3 h x-8 g))+2 b (5 g-h x) \left (h (d h-e g)+f g^2\right )\right )+4 b c^2 h^2 \left (6 a e h^2-6 a f h (g+h x)+b h (4 d h-4 e g-3 e h x)+b f g (4 g+3 h x)\right )+3 b^4 f h^4+64 c^4 g (2 g-h x) \left (h (d h-e g)+f g^2\right )\right )-\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right ) \left (\left (2 c h (b g-a h)+\frac{b^2 h^2}{2}-4 c^2 g^2\right ) \left (8 c h (2 c g-b h) (b f g-2 c d h)-\left (4 c h (3 a h-2 b g)-3 b^2 h^2+16 c^2 g^2\right ) (b f h-2 c e h+2 c f g)\right )+2 c h (2 c g-b h) \left (8 c h (b g-2 a h) (b f g-2 c d h)-g \left (-4 a c h-3 b^2 h+8 b c g\right ) (b f h-2 c e h+2 c f g)\right )\right )-128 c^{7/2} \left (h (a h-b g)+c g^2\right )^{3/2} \left (h (d h-e g)+f g^2\right ) \tanh ^{-1}\left (\frac{2 a h-b g+b h x-2 c g x}{2 \sqrt{a+x (b+c x)} \sqrt{h (a h-b g)+c g^2}}\right )}{128 c^{7/2} h^6}-\frac{(a+x (b+c x))^{3/2} \left (3 b^2 f h^2+6 b c h (f (g+h x)-e h)-4 c^2 (h (4 d h-4 e g+3 e h x)+f g (4 g-3 h x))\right )}{48 c^2 h^3}+\frac{f (a+x (b+c x))^{5/2}}{5 c h}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(f*(a + x*(b + c*x))^(5/2))/(5*c*h) - ((a + x*(b + c*x))^(3/2)*(3*b^2*f*h^2 + 6*b*c*h*(-(e*h) + f*(g + h*x)) -
4*c^2*(f*g*(4*g - 3*h*x) + h*(-4*e*g + 4*d*h + 3*e*h*x))))/(48*c^2*h^3) + (Sqrt[c]*h*Sqrt[a + x*(b + c*x)]*(3
*b^4*f*h^4 + 64*c^4*g*(f*g^2 + h*(-(e*g) + d*h))*(2*g - h*x) + 6*b^2*c*h^3*(-(b*e*h) - 2*a*f*h + b*f*(g + h*x)
) + 4*b*c^2*h^2*(6*a*e*h^2 - 6*a*f*h*(g + h*x) + b*f*g*(4*g + 3*h*x) + b*h*(-4*e*g + 4*d*h - 3*e*h*x)) - 16*c^
3*h*(2*b*(f*g^2 + h*(-(e*g) + d*h))*(5*g - h*x) + a*h*(f*g*(-8*g + 3*h*x) + h*(8*e*g - 8*d*h - 3*e*h*x)))) - (
2*c*h*(2*c*g - b*h)*(8*c*h*(b*g - 2*a*h)*(b*f*g - 2*c*d*h) - g*(8*b*c*g - 3*b^2*h - 4*a*c*h)*(2*c*f*g - 2*c*e*
h + b*f*h)) + (-4*c^2*g^2 + (b^2*h^2)/2 + 2*c*h*(b*g - a*h))*(8*c*h*(2*c*g - b*h)*(b*f*g - 2*c*d*h) - (2*c*f*g
- 2*c*e*h + b*f*h)*(16*c^2*g^2 - 3*b^2*h^2 + 4*c*h*(-2*b*g + 3*a*h))))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a
+ x*(b + c*x)])] - 128*c^(7/2)*(c*g^2 + h*(-(b*g) + a*h))^(3/2)*(f*g^2 + h*(-(e*g) + d*h))*ArcTanh[(-(b*g) + 2
*a*h - 2*c*g*x + b*h*x)/(2*Sqrt[c*g^2 + h*(-(b*g) + a*h)]*Sqrt[a + x*(b + c*x)])])/(128*c^(7/2)*h^6)

________________________________________________________________________________________

Maple [B]  time = 0.263, size = 6715, normalized size = 10.2 \begin{align*} \text{output too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError