3.193 \(\int \frac {\sqrt {a+b x+c x^2} (d+e x+f x^2)}{(g+h x)^4} \, dx\)

Optimal. Leaf size=603 \[ -\frac {\sqrt {a+b x+c x^2} \left (h x \left (h^2 \left (8 a^2 f h^2-2 a b h (10 f g-e h)+b^2 \left (11 f g^2-h (d h+e g)\right )\right )+2 c g h \left (2 a h (6 f g-e h)-b \left (12 f g^2-h (2 d h+e g)\right )\right )+4 c^2 \left (3 f g^4-d g^2 h^2\right )\right )+h^2 \left (4 a^2 e h^3-2 a b h \left (d h^2+2 e g h+3 f g^2\right )+b^2 g \left (d h^2+e g h+5 f g^2\right )\right )-2 c g h \left (-2 a d h^3-6 a f g^2 h+b d g h^2+7 b f g^3\right )+8 c^2 f g^5\right )}{8 h^3 (g+h x)^2 \left (a h^2-b g h+c g^2\right )^2}-\frac {\tanh ^{-1}\left (\frac {-2 a h+x (2 c g-b h)+b g}{2 \sqrt {a+b x+c x^2} \sqrt {a h^2-b g h+c g^2}}\right ) \left (2 c h^2 \left (4 a^2 h^2 (4 f g-e h)-2 a b h \left (-d h^2-e g h+15 f g^2\right )+b^2 \left (d g h^2+15 f g^3\right )\right )-b h^3 \left (8 a^2 f h^2-2 a b h (e h+6 f g)+b^2 \left (d h^2+e g h+5 f g^2\right )\right )-8 c^2 g h \left (a d h^3-5 a f g^2 h+5 b f g^3\right )+16 c^3 f g^5\right )}{16 h^4 \left (a h^2-b g h+c g^2\right )^{5/2}}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (f g^2-h (e g-d h)\right )}{3 h (g+h x)^3 \left (a h^2-b g h+c g^2\right )}+\frac {\sqrt {c} f \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{h^4} \]

[Out]

-1/3*(f*g^2-h*(-d*h+e*g))*(c*x^2+b*x+a)^(3/2)/h/(a*h^2-b*g*h+c*g^2)/(h*x+g)^3-1/16*(16*c^3*f*g^5-8*c^2*g*h*(a*
d*h^3-5*a*f*g^2*h+5*b*f*g^3)-b*h^3*(8*a^2*f*h^2-2*a*b*h*(e*h+6*f*g)+b^2*(d*h^2+e*g*h+5*f*g^2))+2*c*h^2*(4*a^2*
h^2*(-e*h+4*f*g)-2*a*b*h*(-d*h^2-e*g*h+15*f*g^2)+b^2*(d*g*h^2+15*f*g^3)))*arctanh(1/2*(b*g-2*a*h+(-b*h+2*c*g)*
x)/(a*h^2-b*g*h+c*g^2)^(1/2)/(c*x^2+b*x+a)^(1/2))/h^4/(a*h^2-b*g*h+c*g^2)^(5/2)+f*arctanh(1/2*(2*c*x+b)/c^(1/2
)/(c*x^2+b*x+a)^(1/2))*c^(1/2)/h^4-1/8*(8*c^2*f*g^5-2*c*g*h*(-2*a*d*h^3-6*a*f*g^2*h+b*d*g*h^2+7*b*f*g^3)+h^2*(
4*a^2*e*h^3+b^2*g*(d*h^2+e*g*h+5*f*g^2)-2*a*b*h*(d*h^2+2*e*g*h+3*f*g^2))+h*(4*c^2*(-d*g^2*h^2+3*f*g^4)+h^2*(8*
a^2*f*h^2-2*a*b*h*(-e*h+10*f*g)+b^2*(11*f*g^2-h*(d*h+e*g)))+2*c*g*h*(2*a*h*(-e*h+6*f*g)-b*(12*f*g^2-h*(2*d*h+e
*g))))*x)*(c*x^2+b*x+a)^(1/2)/h^3/(a*h^2-b*g*h+c*g^2)^2/(h*x+g)^2

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Rubi [A]  time = 1.45, antiderivative size = 601, normalized size of antiderivative = 1.00, 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 = {1650, 810, 843, 621, 206, 724} \[ -\frac {\sqrt {a+b x+c x^2} \left (h x \left (8 a^2 f h^3-2 b \left (a h^2 (10 f g-e h)-c g h (2 d h+e g)+12 c f g^3\right )+4 a c g h (6 f g-e h)+b^2 h \left (11 f g^2-h (d h+e g)\right )+c^2 \left (\frac {12 f g^4}{h}-4 d g^2 h\right )\right )+4 a^2 e h^4-2 b \left (a h^2 \left (d h^2+2 e g h+3 f g^2\right )+c \left (d g^2 h^2+7 f g^4\right )\right )+4 a c g h \left (d h^2+3 f g^2\right )+b^2 g h \left (h (d h+e g)+5 f g^2\right )+\frac {8 c^2 f g^5}{h}\right )}{8 h^2 (g+h x)^2 \left (a h^2-b g h+c g^2\right )^2}-\frac {\tanh ^{-1}\left (\frac {-2 a h+x (2 c g-b h)+b g}{2 \sqrt {a+b x+c x^2} \sqrt {a h^2-b g h+c g^2}}\right ) \left (2 c h^2 \left (4 a^2 h^2 (4 f g-e h)-2 a b h \left (-d h^2-e g h+15 f g^2\right )+b^2 \left (d g h^2+15 f g^3\right )\right )-b h^3 \left (8 a^2 f h^2-2 a b h (e h+6 f g)+b^2 \left (d h^2+e g h+5 f g^2\right )\right )-8 c^2 g h \left (a d h^3-5 a f g^2 h+5 b f g^3\right )+16 c^3 f g^5\right )}{16 h^4 \left (a h^2-b g h+c g^2\right )^{5/2}}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (f g^2-h (e g-d h)\right )}{3 h (g+h x)^3 \left (a h^2-b g h+c g^2\right )}+\frac {\sqrt {c} f \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{h^4} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[a + b*x + c*x^2]*(d + e*x + f*x^2))/(g + h*x)^4,x]

[Out]

-(((8*c^2*f*g^5)/h + 4*a^2*e*h^4 + 4*a*c*g*h*(3*f*g^2 + d*h^2) + b^2*g*h*(5*f*g^2 + h*(e*g + d*h)) - 2*b*(a*h^
2*(3*f*g^2 + 2*e*g*h + d*h^2) + c*(7*f*g^4 + d*g^2*h^2)) + h*(8*a^2*f*h^3 + 4*a*c*g*h*(6*f*g - e*h) + c^2*((12
*f*g^4)/h - 4*d*g^2*h) + b^2*h*(11*f*g^2 - h*(e*g + d*h)) - 2*b*(12*c*f*g^3 - c*g*h*(e*g + 2*d*h) + a*h^2*(10*
f*g - e*h)))*x)*Sqrt[a + b*x + c*x^2])/(8*h^2*(c*g^2 - b*g*h + a*h^2)^2*(g + h*x)^2) - ((f*g^2 - h*(e*g - d*h)
)*(a + b*x + c*x^2)^(3/2))/(3*h*(c*g^2 - b*g*h + a*h^2)*(g + h*x)^3) + (Sqrt[c]*f*ArcTanh[(b + 2*c*x)/(2*Sqrt[
c]*Sqrt[a + b*x + c*x^2])])/h^4 - ((16*c^3*f*g^5 - 8*c^2*g*h*(5*b*f*g^3 - 5*a*f*g^2*h + a*d*h^3) - b*h^3*(8*a^
2*f*h^2 - 2*a*b*h*(6*f*g + e*h) + b^2*(5*f*g^2 + e*g*h + d*h^2)) + 2*c*h^2*(4*a^2*h^2*(4*f*g - e*h) - 2*a*b*h*
(15*f*g^2 - e*g*h - d*h^2) + b^2*(15*f*g^3 + d*g*h^2)))*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])])/(16*h^4*(c*g^2 - b*g*h + a*h^2)^(5/2))

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 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 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]

Rule 810

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*((d*g - e*f*(m + 2))*(c*d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*
f - d*g) - e*(g*(m + 1)*(c*d^2 - b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x))/(e^2*(m + 1)*(m + 2)*(c*d^2
- b*d*e + a*e^2)), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(a + b*x
+ c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + p + 2)) + b*(a*e^2*g*(m + 1)
- c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c*(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1
) - b*(d*g*(m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*
c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 0]

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 1650

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomia
lQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + b*x + c*
x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^
(m + 1)*(a + b*x + c*x^2)^p*ExpandToSum[(m + 1)*(c*d^2 - b*d*e + a*e^2)*Q + c*d*R*(m + 1) - b*e*R*(m + p + 2)
- c*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, e, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&
NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1]

Rubi steps

\begin {align*} \int \frac {\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^4} \, dx &=-\frac {\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{3/2}}{3 h \left (c g^2-b g h+a h^2\right ) (g+h x)^3}-\frac {\int \frac {\left (-\frac {3}{2} \left (2 c d g-b e g-2 a f g+\frac {b f g^2}{h}-b d h+2 a e h\right )+3 f \left (b g-\frac {c g^2}{h}-a h\right ) x\right ) \sqrt {a+b x+c x^2}}{(g+h x)^3} \, dx}{3 \left (c g^2-b g h+a h^2\right )}\\ &=-\frac {\left (\frac {8 c^2 f g^5}{h}+4 a^2 e h^4+4 a c g h \left (3 f g^2+d h^2\right )+b^2 g h \left (5 f g^2+h (e g+d h)\right )-2 b \left (a h^2 \left (3 f g^2+2 e g h+d h^2\right )+c \left (7 f g^4+d g^2 h^2\right )\right )+h \left (8 a^2 f h^3+4 a c g h (6 f g-e h)+c^2 \left (\frac {12 f g^4}{h}-4 d g^2 h\right )+b^2 h \left (11 f g^2-h (e g+d h)\right )-2 b \left (12 c f g^3-c g h (e g+2 d h)+a h^2 (10 f g-e h)\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{8 h^2 \left (c g^2-b g h+a h^2\right )^2 (g+h x)^2}-\frac {\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{3/2}}{3 h \left (c g^2-b g h+a h^2\right ) (g+h x)^3}+\frac {\int \frac {\frac {3 \left (b^3 h^2 \left (5 f g^2+h (e g+d h)\right )+4 b \left (2 c^2 f g^4+2 a^2 f h^4+a c h^2 \left (7 f g^2-e g h-d h^2\right )\right )-8 a c h \left (a h^2 (2 f g-e h)+c \left (f g^3-d g h^2\right )\right )-2 b^2 \left (a h^3 (6 f g+e h)+c \left (7 f g^3 h+d g h^3\right )\right )\right )}{4 h}+\frac {12 c f \left (c g^2-b g h+a h^2\right )^2 x}{h}}{(g+h x) \sqrt {a+b x+c x^2}} \, dx}{12 h^2 \left (c g^2-b g h+a h^2\right )^2}\\ &=-\frac {\left (\frac {8 c^2 f g^5}{h}+4 a^2 e h^4+4 a c g h \left (3 f g^2+d h^2\right )+b^2 g h \left (5 f g^2+h (e g+d h)\right )-2 b \left (a h^2 \left (3 f g^2+2 e g h+d h^2\right )+c \left (7 f g^4+d g^2 h^2\right )\right )+h \left (8 a^2 f h^3+4 a c g h (6 f g-e h)+c^2 \left (\frac {12 f g^4}{h}-4 d g^2 h\right )+b^2 h \left (11 f g^2-h (e g+d h)\right )-2 b \left (12 c f g^3-c g h (e g+2 d h)+a h^2 (10 f g-e h)\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{8 h^2 \left (c g^2-b g h+a h^2\right )^2 (g+h x)^2}-\frac {\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{3/2}}{3 h \left (c g^2-b g h+a h^2\right ) (g+h x)^3}+\frac {(c f) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{h^4}-\frac {\left (16 c^3 f g^5-8 c^2 g h \left (5 b f g^3-5 a f g^2 h+a d h^3\right )-b h^3 \left (8 a^2 f h^2-2 a b h (6 f g+e h)+b^2 \left (5 f g^2+e g h+d h^2\right )\right )+2 c h^2 \left (4 a^2 h^2 (4 f g-e h)-2 a b h \left (15 f g^2-e g h-d h^2\right )+b^2 \left (15 f g^3+d g h^2\right )\right )\right ) \int \frac {1}{(g+h x) \sqrt {a+b x+c x^2}} \, dx}{16 h^4 \left (c g^2-b g h+a h^2\right )^2}\\ &=-\frac {\left (\frac {8 c^2 f g^5}{h}+4 a^2 e h^4+4 a c g h \left (3 f g^2+d h^2\right )+b^2 g h \left (5 f g^2+h (e g+d h)\right )-2 b \left (a h^2 \left (3 f g^2+2 e g h+d h^2\right )+c \left (7 f g^4+d g^2 h^2\right )\right )+h \left (8 a^2 f h^3+4 a c g h (6 f g-e h)+c^2 \left (\frac {12 f g^4}{h}-4 d g^2 h\right )+b^2 h \left (11 f g^2-h (e g+d h)\right )-2 b \left (12 c f g^3-c g h (e g+2 d h)+a h^2 (10 f g-e h)\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{8 h^2 \left (c g^2-b g h+a h^2\right )^2 (g+h x)^2}-\frac {\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{3/2}}{3 h \left (c g^2-b g h+a h^2\right ) (g+h x)^3}+\frac {(2 c f) \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 )}{h^4}+\frac {\left (16 c^3 f g^5-8 c^2 g h \left (5 b f g^3-5 a f g^2 h+a d h^3\right )-b h^3 \left (8 a^2 f h^2-2 a b h (6 f g+e h)+b^2 \left (5 f g^2+e g h+d h^2\right )\right )+2 c h^2 \left (4 a^2 h^2 (4 f g-e h)-2 a b h \left (15 f g^2-e g h-d h^2\right )+b^2 \left (15 f g^3+d g h^2\right )\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 )}{8 h^4 \left (c g^2-b g h+a h^2\right )^2}\\ &=-\frac {\left (\frac {8 c^2 f g^5}{h}+4 a^2 e h^4+4 a c g h \left (3 f g^2+d h^2\right )+b^2 g h \left (5 f g^2+h (e g+d h)\right )-2 b \left (a h^2 \left (3 f g^2+2 e g h+d h^2\right )+c \left (7 f g^4+d g^2 h^2\right )\right )+h \left (8 a^2 f h^3+4 a c g h (6 f g-e h)+c^2 \left (\frac {12 f g^4}{h}-4 d g^2 h\right )+b^2 h \left (11 f g^2-h (e g+d h)\right )-2 b \left (12 c f g^3-c g h (e g+2 d h)+a h^2 (10 f g-e h)\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{8 h^2 \left (c g^2-b g h+a h^2\right )^2 (g+h x)^2}-\frac {\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{3/2}}{3 h \left (c g^2-b g h+a h^2\right ) (g+h x)^3}+\frac {\sqrt {c} f \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{h^4}-\frac {\left (16 c^3 f g^5-8 c^2 g h \left (5 b f g^3-5 a f g^2 h+a d h^3\right )-b h^3 \left (8 a^2 f h^2-2 a b h (6 f g+e h)+b^2 \left (5 f g^2+e g h+d h^2\right )\right )+2 c h^2 \left (4 a^2 h^2 (4 f g-e h)-2 a b h \left (15 f g^2-e g h-d h^2\right )+b^2 \left (15 f g^3+d g h^2\right )\right )\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 )}{16 h^4 \left (c g^2-b g h+a h^2\right )^{5/2}}\\ \end {align*}

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Mathematica [A]  time = 1.93, size = 439, normalized size = 0.73 \[ \frac {\frac {\left (\frac {\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {2 a h-b g+b h x-2 c g x}{2 \sqrt {a+x (b+c x)} \sqrt {h (a h-b g)+c g^2}}\right )}{8 \left (h (a h-b g)+c g^2\right )^{3/2}}+\frac {\sqrt {a+x (b+c x)} (-2 a h+b (g-h x)+2 c g x)}{4 (g+h x)^2 \left (h (a h-b g)+c g^2\right )}\right ) \left (2 a h^2 (e h-2 f g)-b h \left (h (d h+e g)-3 f g^2\right )+c \left (2 d g h^2-2 f g^3\right )\right )}{2 \left (h (a h-b g)+c g^2\right )}-\frac {h (a+x (b+c x))^{3/2} \left (h (d h-e g)+f g^2\right )}{3 (g+h x)^3 \left (h (a h-b g)+c g^2\right )}+\frac {f \left (\frac {(2 c g-b h) \tanh ^{-1}\left (\frac {2 a h-b g+b h x-2 c g x}{2 \sqrt {a+x (b+c x)} \sqrt {h (a h-b g)+c g^2}}\right )}{2 \sqrt {h (a h-b g)+c g^2}}-\frac {h \sqrt {a+x (b+c x)}}{g+h x}+\sqrt {c} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )}{h^2}}{h^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[a + b*x + c*x^2]*(d + e*x + f*x^2))/(g + h*x)^4,x]

[Out]

(-1/3*(h*(f*g^2 + h*(-(e*g) + d*h))*(a + x*(b + c*x))^(3/2))/((c*g^2 + h*(-(b*g) + a*h))*(g + h*x)^3) + ((2*a*
h^2*(-2*f*g + e*h) + c*(-2*f*g^3 + 2*d*g*h^2) - b*h*(-3*f*g^2 + h*(e*g + d*h)))*((Sqrt[a + x*(b + c*x)]*(-2*a*
h + 2*c*g*x + b*(g - h*x)))/(4*(c*g^2 + h*(-(b*g) + a*h))*(g + h*x)^2) + ((b^2 - 4*a*c)*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)])])/(8*(c*g^2 + h*(-(b*g) + a*h))^
(3/2))))/(2*(c*g^2 + h*(-(b*g) + a*h))) + (f*(-((h*Sqrt[a + x*(b + c*x)])/(g + h*x)) + Sqrt[c]*ArcTanh[(b + 2*
c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] + ((2*c*g - b*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)])])/(2*Sqrt[c*g^2 + h*(-(b*g) + a*h)])))/h^2)/h^2

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^2+e*x+d)*(c*x^2+b*x+a)^(1/2)/(h*x+g)^4,x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^2+e*x+d)*(c*x^2+b*x+a)^(1/2)/(h*x+g)^4,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Eval
uation time: 42.01Unable to divide, perhaps due to rounding error%%%{%%{[-1,0]:[1,0,%%%{-1,[1]%%%}]%%},[8,0,0,
0,0,0,8,0]%%%}+%%%{%%%{8,[1]%%%},[7,0,0,0,0,0,7,1]%%%}+%%%{%%{[-4,0]:[1,0,%%%{-1,[1]%%%}]%%},[6,0,1,0,0,0,7,1]
%%%}+%%%{%%{[4,0]:[1,0,%%%{-1,[1]%%%}]%%},[6,0,0,0,1,0,8,0]%%%}+%%%{%%{[%%%{-24,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}
]%%},[6,0,0,0,0,0,6,2]%%%}+%%%{%%%{24,[1]%%%},[5,0,1,0,0,0,6,2]%%%}+%%%{%%%{-24,[1]%%%},[5,0,0,0,1,0,7,1]%%%}+
%%%{%%%{32,[2]%%%},[5,0,0,0,0,0,5,3]%%%}+%%%{%%{[-6,0]:[1,0,%%%{-1,[1]%%%}]%%},[4,0,2,0,0,0,6,2]%%%}+%%%{%%{[1
2,0]:[1,0,%%%{-1,[1]%%%}]%%},[4,0,1,0,1,0,7,1]%%%}+%%%{%%{[%%%{-48,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[4,0,1,0
,0,0,5,3]%%%}+%%%{%%{[-6,0]:[1,0,%%%{-1,[1]%%%}]%%},[4,0,0,0,2,0,8,0]%%%}+%%%{%%{[%%%{48,[1]%%%},0]:[1,0,%%%{-
1,[1]%%%}]%%},[4,0,0,0,1,0,6,2]%%%}+%%%{%%{[%%%{-16,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[4,0,0,0,0,0,4,4]%%%}+%
%%{%%%{24,[1]%%%},[3,0,2,0,0,0,5,3]%%%}+%%%{%%%{-48,[1]%%%},[3,0,1,0,1,0,6,2]%%%}+%%%{%%%{32,[2]%%%},[3,0,1,0,
0,0,4,4]%%%}+%%%{%%%{24,[1]%%%},[3,0,0,0,2,0,7,1]%%%}+%%%{%%%{-32,[2]%%%},[3,0,0,0,1,0,5,3]%%%}+%%%{%%{[-4,0]:
[1,0,%%%{-1,[1]%%%}]%%},[2,0,3,0,0,0,5,3]%%%}+%%%{%%{[12,0]:[1,0,%%%{-1,[1]%%%}]%%},[2,0,2,0,1,0,6,2]%%%}+%%%{
%%{[%%%{-24,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,0,2,0,0,0,4,4]%%%}+%%%{%%{[-12,0]:[1,0,%%%{-1,[1]%%%}]%%},[2
,0,1,0,2,0,7,1]%%%}+%%%{%%{[%%%{48,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,0,1,0,1,0,5,3]%%%}+%%%{%%{[4,0]:[1,0,
%%%{-1,[1]%%%}]%%},[2,0,0,0,3,0,8,0]%%%}+%%%{%%{[%%%{-24,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,0,0,0,2,0,6,2]%
%%}+%%%{%%%{8,[1]%%%},[1,0,3,0,0,0,4,4]%%%}+%%%{%%%{-24,[1]%%%},[1,0,2,0,1,0,5,3]%%%}+%%%{%%%{24,[1]%%%},[1,0,
1,0,2,0,6,2]%%%}+%%%{%%%{-8,[1]%%%},[1,0,0,0,3,0,7,1]%%%}+%%%{%%{[-1,0]:[1,0,%%%{-1,[1]%%%}]%%},[0,0,4,0,0,0,4
,4]%%%}+%%%{%%{[4,0]:[1,0,%%%{-1,[1]%%%}]%%},[0,0,3,0,1,0,5,3]%%%}+%%%{%%{[-6,0]:[1,0,%%%{-1,[1]%%%}]%%},[0,0,
2,0,2,0,6,2]%%%}+%%%{%%{[4,0]:[1,0,%%%{-1,[1]%%%}]%%},[0,0,1,0,3,0,7,1]%%%}+%%%{%%{[-1,0]:[1,0,%%%{-1,[1]%%%}]
%%},[0,0,0,0,4,0,8,0]%%%} / %%%{%%%{1,[2]%%%},[8,0,0,0,0,0,4,0]%%%}+%%%{%%{poly1[%%%{-8,[2]%%%},0]:[1,0,%%%{-1
,[1]%%%}]%%},[7,0,0,0,0,0,3,1]%%%}+%%%{%%%{4,[2]%%%},[6,0,1,0,0,0,3,1]%%%}+%%%{%%%{-4,[2]%%%},[6,0,0,0,1,0,4,0
]%%%}+%%%{%%%{24,[3]%%%},[6,0,0,0,0,0,2,2]%%%}+%%%{%%{[%%%{-24,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,0,1,0,0,0
,2,2]%%%}+%%%{%%{poly1[%%%{24,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,0,0,0,1,0,3,1]%%%}+%%%{%%{poly1[%%%{-32,[3
]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,0,0,0,0,0,1,3]%%%}+%%%{%%%{6,[2]%%%},[4,0,2,0,0,0,2,2]%%%}+%%%{%%%{-12,[2]
%%%},[4,0,1,0,1,0,3,1]%%%}+%%%{%%%{48,[3]%%%},[4,0,1,0,0,0,1,3]%%%}+%%%{%%%{6,[2]%%%},[4,0,0,0,2,0,4,0]%%%}+%%
%{%%%{-48,[3]%%%},[4,0,0,0,1,0,2,2]%%%}+%%%{%%%{16,[4]%%%},[4,0,0,0,0,0,0,4]%%%}+%%%{%%{poly1[%%%{-24,[2]%%%},
0]:[1,0,%%%{-1,[1]%%%}]%%},[3,0,2,0,0,0,1,3]%%%}+%%%{%%{[%%%{48,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,0,1,0,1,
0,2,2]%%%}+%%%{%%{[%%%{-32,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,0,1,0,0,0,0,4]%%%}+%%%{%%{poly1[%%%{-24,[2]%%
%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,0,0,0,2,0,3,1]%%%}+%%%{%%{poly1[%%%{32,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,
0,0,0,1,0,1,3]%%%}+%%%{%%%{4,[2]%%%},[2,0,3,0,0,0,1,3]%%%}+%%%{%%%{-12,[2]%%%},[2,0,2,0,1,0,2,2]%%%}+%%%{%%%{2
4,[3]%%%},[2,0,2,0,0,0,0,4]%%%}+%%%{%%%{12,[2]%%%},[2,0,1,0,2,0,3,1]%%%}+%%%{%%%{-48,[3]%%%},[2,0,1,0,1,0,1,3]
%%%}+%%%{%%%{-4,[2]%%%},[2,0,0,0,3,0,4,0]%%%}+%%%{%%%{24,[3]%%%},[2,0,0,0,2,0,2,2]%%%}+%%%{%%{[%%%{-8,[2]%%%},
0]:[1,0,%%%{-1,[1]%%%}]%%},[1,0,3,0,0,0,0,4]%%%}+%%%{%%{poly1[%%%{24,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,0,2
,0,1,0,1,3]%%%}+%%%{%%{[%%%{-24,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,0,1,0,2,0,2,2]%%%}+%%%{%%{poly1[%%%{8,[2
]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,0,0,0,3,0,3,1]%%%}+%%%{%%%{1,[2]%%%},[0,0,4,0,0,0,0,4]%%%}+%%%{%%%{-4,[2]%
%%},[0,0,3,0,1,0,1,3]%%%}+%%%{%%%{6,[2]%%%},[0,0,2,0,2,0,2,2]%%%}+%%%{%%%{-4,[2]%%%},[0,0,1,0,3,0,3,1]%%%}+%%%
{%%%{1,[2]%%%},[0,0,0,0,4,0,4,0]%%%} Error: Bad Argument Value

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maple [B]  time = 0.02, size = 19321, normalized size = 32.04 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x^2+e*x+d)*(c*x^2+b*x+a)^(1/2)/(h*x+g)^4,x)

[Out]

result too large to display

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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((f*x^2+e*x+d)*(c*x^2+b*x+a)^(1/2)/(h*x+g)^4,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(a*h^2-b*g*h>0)', see `assume?`
for more details)Is a*h^2-b*g*h                            +c*g^2 zero or nonzero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {c\,x^2+b\,x+a}\,\left (f\,x^2+e\,x+d\right )}{{\left (g+h\,x\right )}^4} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x + c*x^2)^(1/2)*(d + e*x + f*x^2))/(g + h*x)^4,x)

[Out]

int(((a + b*x + c*x^2)^(1/2)*(d + e*x + f*x^2))/(g + h*x)^4, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a + b x + c x^{2}} \left (d + e x + f x^{2}\right )}{\left (g + h x\right )^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x**2+e*x+d)*(c*x**2+b*x+a)**(1/2)/(h*x+g)**4,x)

[Out]

Integral(sqrt(a + b*x + c*x**2)*(d + e*x + f*x**2)/(g + h*x)**4, x)

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