3.3.5 \(\int \frac {(a+b x+c x^2)^{3/2} (d+e x+f x^2)}{(g+h x)^6} \, dx\) [205]

Optimal. Leaf size=1226 \[ -\frac {\left (128 c^4 f g^7-32 c^3 f g^5 h (11 b g-10 a h)+8 c^2 g h^2 \left (38 b^2 f g^4+2 a^2 h^2 \left (13 f g^2+3 d h^2\right )-a b g h \left (65 f g^2+3 d h^2\right )\right )-2 c h^3 \left (8 a^3 h^3 (2 f g-3 e h)-2 a b^2 g^2 h (34 f g+3 e h)+4 a^2 b h^2 \left (5 f g^2+6 e g h+3 d h^2\right )+b^3 \left (35 f g^4-3 d g^2 h^2\right )\right )-b h^4 (b g-2 a h) \left (16 a^2 f h^2-2 a b h (10 f g+3 e h)+b^2 \left (7 f g^2+3 h (e g+d h)\right )\right )+h \left (128 c f \left (c g^2-h (b g-a h)\right )^3+(2 c g-b h) \left (32 c^3 f g^5-8 c^2 g h \left (10 b f g^3-11 a f g^2 h+3 a d h^3\right )+2 c h^2 \left (4 a^2 h^2 (10 f g-3 e h)-6 a b h \left (11 f g^2-e g h-d h^2\right )+b^2 \left (29 f g^3+3 d g h^2\right )\right )-b h^3 \left (16 a^2 f h^2-2 a b h (10 f g+3 e h)+b^2 \left (7 f g^2+3 h (e g+d h)\right )\right )\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{128 h^5 \left (c g^2-b g h+a h^2\right )^3 (g+h x)^2}-\frac {\left (16 c^2 f g^5-2 c g h \left (13 b f g^3-10 a f g^2 h+3 b d g h^2-6 a d h^3\right )-h^2 \left (4 a^2 h^2 (2 f g-3 e h)-b^2 g \left (7 f g^2+3 h (e g+d h)\right )+2 a b h \left (f g^2+3 h (2 e g+d h)\right )\right )+h \left (4 c^2 \left (7 f g^4-3 d g^2 h^2\right )+2 c g h \left (2 a h (14 f g-3 e h)-b \left (28 f g^2-3 e g h-6 d h^2\right )\right )+h^2 \left (16 a^2 f h^2-2 a b h (22 f g-3 e h)+b^2 \left (25 f g^2-3 h (e g+d h)\right )\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{48 h^3 \left (c g^2-b g h+a h^2\right )^2 (g+h x)^4}-\frac {\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{5 h \left (c g^2-b g h+a h^2\right ) (g+h x)^5}+\frac {c^{3/2} f \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{h^6}-\frac {\left (256 c^5 f g^7-896 c^4 f g^5 h (b g-a h)+32 c^3 g h^2 \left (35 b^2 f g^4-70 a b f g^3 h+a^2 h^2 \left (35 f g^2-3 d h^2\right )\right )-16 c^2 h^3 \left (35 b^3 f g^4-6 a^3 h^3 (6 f g-e h)+3 a^2 b h^2 \left (35 f g^2-e g h-d h^2\right )-3 a b^2 g h \left (35 f g^2+d h^2\right )\right )+b^3 h^5 \left (16 a^2 f h^2-2 a b h (10 f g+3 e h)+b^2 \left (7 f g^2+3 h (e g+d h)\right )\right )-2 b c h^4 \left (96 a^3 f h^3-24 a^2 b h^2 (8 f g+e h)-b^3 \left (35 f g^3-3 d g h^2\right )+4 a b^2 h \left (35 f g^2+3 h (e g+d h)\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 )}{256 h^6 \left (c g^2-b g h+a h^2\right )^{7/2}} \]

[Out]

-1/48*(16*c^2*f*g^5-2*c*g*h*(-6*a*d*h^3-10*a*f*g^2*h+3*b*d*g*h^2+13*b*f*g^3)-h^2*(4*a^2*h^2*(-3*e*h+2*f*g)-b^2
*g*(7*f*g^2+3*h*(d*h+e*g))+2*a*b*h*(f*g^2+3*h*(d*h+2*e*g)))+h*(4*c^2*(-3*d*g^2*h^2+7*f*g^4)+2*c*g*h*(2*a*h*(-3
*e*h+14*f*g)-b*(-6*d*h^2-3*e*g*h+28*f*g^2))+h^2*(16*a^2*f*h^2-2*a*b*h*(-3*e*h+22*f*g)+b^2*(25*f*g^2-3*h*(d*h+e
*g))))*x)*(c*x^2+b*x+a)^(3/2)/h^3/(a*h^2-b*g*h+c*g^2)^2/(h*x+g)^4-1/5*(f*g^2-h*(-d*h+e*g))*(c*x^2+b*x+a)^(5/2)
/h/(a*h^2-b*g*h+c*g^2)/(h*x+g)^5+c^(3/2)*f*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/h^6-1/256*(256*c
^5*f*g^7-896*c^4*f*g^5*h*(-a*h+b*g)+32*c^3*g*h^2*(35*b^2*f*g^4-70*a*b*f*g^3*h+a^2*h^2*(-3*d*h^2+35*f*g^2))-16*
c^2*h^3*(35*b^3*f*g^4-6*a^3*h^3*(-e*h+6*f*g)+3*a^2*b*h^2*(-d*h^2-e*g*h+35*f*g^2)-3*a*b^2*g*h*(d*h^2+35*f*g^2))
+b^3*h^5*(16*a^2*f*h^2-2*a*b*h*(3*e*h+10*f*g)+b^2*(7*f*g^2+3*h*(d*h+e*g)))-2*b*c*h^4*(96*a^3*f*h^3-24*a^2*b*h^
2*(e*h+8*f*g)-b^3*(-3*d*g*h^2+35*f*g^3)+4*a*b^2*h*(35*f*g^2+3*h*(d*h+e*g))))*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^6/(a*h^2-b*g*h+c*g^2)^(7/2)-1/128*(128*c^4*f*g^7-32*c^3
*f*g^5*h*(-10*a*h+11*b*g)+8*c^2*g*h^2*(38*b^2*f*g^4+2*a^2*h^2*(3*d*h^2+13*f*g^2)-a*b*g*h*(3*d*h^2+65*f*g^2))-2
*c*h^3*(8*a^3*h^3*(-3*e*h+2*f*g)-2*a*b^2*g^2*h*(3*e*h+34*f*g)+4*a^2*b*h^2*(3*d*h^2+6*e*g*h+5*f*g^2)+b^3*(-3*d*
g^2*h^2+35*f*g^4))-b*h^4*(-2*a*h+b*g)*(16*a^2*f*h^2-2*a*b*h*(3*e*h+10*f*g)+b^2*(7*f*g^2+3*h*(d*h+e*g)))+h*(128
*c*f*(c*g^2-h*(-a*h+b*g))^3+(-b*h+2*c*g)*(32*c^3*f*g^5-8*c^2*g*h*(3*a*d*h^3-11*a*f*g^2*h+10*b*f*g^3)+2*c*h^2*(
4*a^2*h^2*(-3*e*h+10*f*g)-6*a*b*h*(-d*h^2-e*g*h+11*f*g^2)+b^2*(3*d*g*h^2+29*f*g^3))-b*h^3*(16*a^2*f*h^2-2*a*b*
h*(3*e*h+10*f*g)+b^2*(7*f*g^2+3*h*(d*h+e*g)))))*x)*(c*x^2+b*x+a)^(1/2)/h^5/(a*h^2-b*g*h+c*g^2)^3/(h*x+g)^2

________________________________________________________________________________________

Rubi [A]
time = 2.46, antiderivative size = 1223, normalized size of antiderivative = 1.00, 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 = {1664, 824, 857, 635, 212, 738} \begin {gather*} -\frac {\left (f g^2-h (e g-d h)\right ) \left (c x^2+b x+a\right )^{5/2}}{5 h \left (c g^2-b h g+a h^2\right ) (g+h x)^5}-\frac {\left (16 c^2 f g^5-2 c h \left (13 b f g^3-10 a f h g^2+3 b d h^2 g-6 a d h^3\right ) g-h^2 \left (-g \left (7 f g^2+3 h (e g+d h)\right ) b^2+2 a h \left (f g^2+3 h (2 e g+d h)\right ) b+4 a^2 h^2 (2 f g-3 e h)\right )+h^2 \left (16 a^2 f h^3+4 a c g (14 f g-3 e h) h+b^2 \left (25 f g^2-3 h (e g+d h)\right ) h+c^2 \left (\frac {28 f g^4}{h}-12 d g^2 h\right )-b \left (56 c f g^3-6 c h (e g+2 d h) g+2 a h^2 (22 f g-3 e h)\right )\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{48 h^3 \left (c g^2-b h g+a h^2\right )^2 (g+h x)^4}-\frac {\left (\frac {128 c^4 f g^7}{h}-32 c^3 f (11 b g-10 a h) g^5+8 c^2 h \left (38 b^2 f g^4-a b h \left (65 f g^2+3 d h^2\right ) g+2 a^2 h^2 \left (13 f g^2+3 d h^2\right )\right ) g-b h^3 (b g-2 a h) \left (\left (7 f g^2+3 h (e g+d h)\right ) b^2-2 a h (10 f g+3 e h) b+16 a^2 f h^2\right )-2 c h^2 \left (\left (35 f g^4-3 d g^2 h^2\right ) b^3-2 a g^2 h (34 f g+3 e h) b^2+4 a^2 h^2 \left (5 f g^2+3 h (2 e g+d h)\right ) b+8 a^3 h^3 (2 f g-3 e h)\right )+\left (128 c f \left (c g^2-h (b g-a h)\right )^3+(2 c g-b h) \left (32 c^3 f g^5-8 c^2 h \left (10 b f g^3-11 a f h g^2+3 a d h^3\right ) g+2 c h^2 \left (\left (29 f g^3+3 d h^2 g\right ) b^2-6 a h \left (11 f g^2-e h g-d h^2\right ) b+4 a^2 h^2 (10 f g-3 e h)\right )-b h^3 \left (\left (7 f g^2+3 h (e g+d h)\right ) b^2-2 a h (10 f g+3 e h) b+16 a^2 f h^2\right )\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{128 h^4 \left (c g^2-b h g+a h^2\right )^3 (g+h x)^2}+\frac {c^{3/2} f \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right )}{h^6}-\frac {\left (256 c^5 f g^7-896 c^4 f h (b g-a h) g^5+32 c^3 h^2 \left (35 b^2 f g^4-70 a b f h g^3+a^2 h^2 \left (35 f g^2-3 d h^2\right )\right ) g-16 c^2 h^3 \left (35 b^3 f g^4-3 a b^2 h \left (35 f g^2+d h^2\right ) g-6 a^3 h^3 (6 f g-e h)+3 a^2 b h^2 \left (35 f g^2-e h g-d h^2\right )\right )+b^3 h^5 \left (\left (7 f g^2+3 h (e g+d h)\right ) b^2-2 a h (10 f g+3 e h) b+16 a^2 f h^2\right )-2 b c h^4 \left (-\left (\left (35 f g^3-3 d g h^2\right ) b^3\right )+4 a h \left (35 f g^2+3 h (e g+d h)\right ) b^2-24 a^2 h^2 (8 f g+e h) b+96 a^3 f h^3\right )\right ) \tanh ^{-1}\left (\frac {b g-2 a h+(2 c g-b h) x}{2 \sqrt {c g^2-b h g+a h^2} \sqrt {c x^2+b x+a}}\right )}{256 h^6 \left (c g^2-b h g+a h^2\right )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2))/(g + h*x)^6,x]

[Out]

-1/128*(((128*c^4*f*g^7)/h - 32*c^3*f*g^5*(11*b*g - 10*a*h) + 8*c^2*g*h*(38*b^2*f*g^4 + 2*a^2*h^2*(13*f*g^2 +
3*d*h^2) - a*b*g*h*(65*f*g^2 + 3*d*h^2)) - b*h^3*(b*g - 2*a*h)*(16*a^2*f*h^2 - 2*a*b*h*(10*f*g + 3*e*h) + b^2*
(7*f*g^2 + 3*h*(e*g + d*h))) - 2*c*h^2*(8*a^3*h^3*(2*f*g - 3*e*h) - 2*a*b^2*g^2*h*(34*f*g + 3*e*h) + b^3*(35*f
*g^4 - 3*d*g^2*h^2) + 4*a^2*b*h^2*(5*f*g^2 + 3*h*(2*e*g + d*h))) + (128*c*f*(c*g^2 - h*(b*g - a*h))^3 + (2*c*g
- b*h)*(32*c^3*f*g^5 - 8*c^2*g*h*(10*b*f*g^3 - 11*a*f*g^2*h + 3*a*d*h^3) + 2*c*h^2*(4*a^2*h^2*(10*f*g - 3*e*h
) - 6*a*b*h*(11*f*g^2 - e*g*h - d*h^2) + b^2*(29*f*g^3 + 3*d*g*h^2)) - b*h^3*(16*a^2*f*h^2 - 2*a*b*h*(10*f*g +
3*e*h) + b^2*(7*f*g^2 + 3*h*(e*g + d*h)))))*x)*Sqrt[a + b*x + c*x^2])/(h^4*(c*g^2 - b*g*h + a*h^2)^3*(g + h*x
)^2) - ((16*c^2*f*g^5 - 2*c*g*h*(13*b*f*g^3 - 10*a*f*g^2*h + 3*b*d*g*h^2 - 6*a*d*h^3) - h^2*(4*a^2*h^2*(2*f*g
- 3*e*h) - b^2*g*(7*f*g^2 + 3*h*(e*g + d*h)) + 2*a*b*h*(f*g^2 + 3*h*(2*e*g + d*h))) + h^2*(16*a^2*f*h^3 + 4*a*
c*g*h*(14*f*g - 3*e*h) + c^2*((28*f*g^4)/h - 12*d*g^2*h) + b^2*h*(25*f*g^2 - 3*h*(e*g + d*h)) - b*(56*c*f*g^3
- 6*c*g*h*(e*g + 2*d*h) + 2*a*h^2*(22*f*g - 3*e*h)))*x)*(a + b*x + c*x^2)^(3/2))/(48*h^3*(c*g^2 - b*g*h + a*h^
2)^2*(g + h*x)^4) - ((f*g^2 - h*(e*g - d*h))*(a + b*x + c*x^2)^(5/2))/(5*h*(c*g^2 - b*g*h + a*h^2)*(g + h*x)^5
) + (c^(3/2)*f*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/h^6 - ((256*c^5*f*g^7 - 896*c^4*f*g^5*h
*(b*g - a*h) + 32*c^3*g*h^2*(35*b^2*f*g^4 - 70*a*b*f*g^3*h + a^2*h^2*(35*f*g^2 - 3*d*h^2)) - 16*c^2*h^3*(35*b^
3*f*g^4 - 6*a^3*h^3*(6*f*g - e*h) + 3*a^2*b*h^2*(35*f*g^2 - e*g*h - d*h^2) - 3*a*b^2*g*h*(35*f*g^2 + d*h^2)) +
b^3*h^5*(16*a^2*f*h^2 - 2*a*b*h*(10*f*g + 3*e*h) + b^2*(7*f*g^2 + 3*h*(e*g + d*h))) - 2*b*c*h^4*(96*a^3*f*h^3
- 24*a^2*b*h^2*(8*f*g + e*h) - b^3*(35*f*g^3 - 3*d*g*h^2) + 4*a*b^2*h*(35*f*g^2 + 3*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])])/(256*h^6*(c*g^2 - b*g*
h + a*h^2)^(7/2))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 635

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 738

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 824

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(d + e*x)^(m + 1))*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((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), 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 857

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 1664

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 {\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{(g+h x)^6} \, dx &=-\frac {\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{5 h \left (c g^2-b g h+a h^2\right ) (g+h x)^5}-\frac {\int \frac {\left (-\frac {5}{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 )+5 f \left (b g-\frac {c g^2}{h}-a h\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{(g+h x)^5} \, dx}{5 \left (c g^2-b g h+a h^2\right )}\\ &=-\frac {\left (16 c^2 f g^5-2 c g h \left (13 b f g^3-10 a f g^2 h+3 b d g h^2-6 a d h^3\right )-h^2 \left (4 a^2 h^2 (2 f g-3 e h)-b^2 g \left (7 f g^2+3 h (e g+d h)\right )+2 a b h \left (f g^2+3 h (2 e g+d h)\right )\right )+h^2 \left (16 a^2 f h^3+4 a c g h (14 f g-3 e h)+c^2 \left (\frac {28 f g^4}{h}-12 d g^2 h\right )+b^2 h \left (25 f g^2-3 h (e g+d h)\right )-b \left (56 c f g^3-6 c g h (e g+2 d h)+2 a h^2 (22 f g-3 e h)\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{48 h^3 \left (c g^2-b g h+a h^2\right )^2 (g+h x)^4}-\frac {\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{5 h \left (c g^2-b g h+a h^2\right ) (g+h x)^5}+\frac {\int \frac {\left (\frac {5 \left (b^3 h^2 \left (7 f g^2+3 h (e g+d h)\right )-24 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 (10 f g+3 e h)+c \left (13 f g^3 h+3 d g h^3\right )\right )+4 b \left (4 c^2 f g^4+4 a^2 f h^4+a c h^2 \left (17 f g^2-3 h (e g+d h)\right )\right )\right )}{4 h}+\frac {40 c f \left (c g^2-b g h+a h^2\right )^2 x}{h}\right ) \sqrt {a+b x+c x^2}}{(g+h x)^3} \, dx}{40 h^2 \left (c g^2-b g h+a h^2\right )^2}\\ &=-\frac {\left (\frac {128 c^4 f g^7}{h}-32 c^3 f g^5 (11 b g-10 a h)+8 c^2 g h \left (38 b^2 f g^4+2 a^2 h^2 \left (13 f g^2+3 d h^2\right )-a b g h \left (65 f g^2+3 d h^2\right )\right )-b h^3 (b g-2 a h) \left (16 a^2 f h^2-2 a b h (10 f g+3 e h)+b^2 \left (7 f g^2+3 h (e g+d h)\right )\right )-2 c h^2 \left (8 a^3 h^3 (2 f g-3 e h)-2 a b^2 g^2 h (34 f g+3 e h)+b^3 \left (35 f g^4-3 d g^2 h^2\right )+4 a^2 b h^2 \left (5 f g^2+3 h (2 e g+d h)\right )\right )+\left (128 c f \left (c g^2-h (b g-a h)\right )^3+(2 c g-b h) \left (32 c^3 f g^5-8 c^2 g h \left (10 b f g^3-11 a f g^2 h+3 a d h^3\right )+2 c h^2 \left (4 a^2 h^2 (10 f g-3 e h)-6 a b h \left (11 f g^2-e g h-d h^2\right )+b^2 \left (29 f g^3+3 d g h^2\right )\right )-b h^3 \left (16 a^2 f h^2-2 a b h (10 f g+3 e h)+b^2 \left (7 f g^2+3 h (e g+d h)\right )\right )\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{128 h^4 \left (c g^2-b g h+a h^2\right )^3 (g+h x)^2}-\frac {\left (16 c^2 f g^5-2 c g h \left (13 b f g^3-10 a f g^2 h+3 b d g h^2-6 a d h^3\right )-h^2 \left (4 a^2 h^2 (2 f g-3 e h)-b^2 g \left (7 f g^2+3 h (e g+d h)\right )+2 a b h \left (f g^2+3 h (2 e g+d h)\right )\right )+h^2 \left (16 a^2 f h^3+4 a c g h (14 f g-3 e h)+c^2 \left (\frac {28 f g^4}{h}-12 d g^2 h\right )+b^2 h \left (25 f g^2-3 h (e g+d h)\right )-b \left (56 c f g^3-6 c g h (e g+2 d h)+2 a h^2 (22 f g-3 e h)\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{48 h^3 \left (c g^2-b g h+a h^2\right )^2 (g+h x)^4}-\frac {\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{5 h \left (c g^2-b g h+a h^2\right ) (g+h x)^5}-\frac {\int \frac {\frac {5 \left (b^4 \left (70 c f g^3 h^3-6 c d g h^5-20 a f g h^5-6 a e h^6\right )+b^5 h^4 \left (7 f g^2+3 h (e g+d h)\right )-16 b c \left (8 c^3 f g^6+44 a c^2 f g^4 h^2+12 a^3 f h^6+3 a^2 c h^4 \left (19 f g^2-e g h-d h^2\right )\right )+16 b^2 c h \left (22 c^2 f g^5+3 a^2 h^4 (8 f g+e h)+3 a c g h^2 \left (19 f g^2+d h^2\right )\right )+32 a c^2 h \left (4 c^2 f g^5+a^2 h^4 (10 f g-3 e h)+a c \left (11 f g^3 h^2-3 d g h^4\right )\right )-8 b^3 \left (38 c^2 f g^4 h^2-2 a^2 f h^6+a c h^4 \left (35 f g^2+3 h (e g+d h)\right )\right )\right )}{8 h}-\frac {160 c^2 f \left (c g^2-b g h+a h^2\right )^3 x}{h}}{(g+h x) \sqrt {a+b x+c x^2}} \, dx}{160 h^4 \left (c g^2-b g h+a h^2\right )^3}\\ &=-\frac {\left (\frac {128 c^4 f g^7}{h}-32 c^3 f g^5 (11 b g-10 a h)+8 c^2 g h \left (38 b^2 f g^4+2 a^2 h^2 \left (13 f g^2+3 d h^2\right )-a b g h \left (65 f g^2+3 d h^2\right )\right )-b h^3 (b g-2 a h) \left (16 a^2 f h^2-2 a b h (10 f g+3 e h)+b^2 \left (7 f g^2+3 h (e g+d h)\right )\right )-2 c h^2 \left (8 a^3 h^3 (2 f g-3 e h)-2 a b^2 g^2 h (34 f g+3 e h)+b^3 \left (35 f g^4-3 d g^2 h^2\right )+4 a^2 b h^2 \left (5 f g^2+3 h (2 e g+d h)\right )\right )+\left (128 c f \left (c g^2-h (b g-a h)\right )^3+(2 c g-b h) \left (32 c^3 f g^5-8 c^2 g h \left (10 b f g^3-11 a f g^2 h+3 a d h^3\right )+2 c h^2 \left (4 a^2 h^2 (10 f g-3 e h)-6 a b h \left (11 f g^2-e g h-d h^2\right )+b^2 \left (29 f g^3+3 d g h^2\right )\right )-b h^3 \left (16 a^2 f h^2-2 a b h (10 f g+3 e h)+b^2 \left (7 f g^2+3 h (e g+d h)\right )\right )\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{128 h^4 \left (c g^2-b g h+a h^2\right )^3 (g+h x)^2}-\frac {\left (16 c^2 f g^5-2 c g h \left (13 b f g^3-10 a f g^2 h+3 b d g h^2-6 a d h^3\right )-h^2 \left (4 a^2 h^2 (2 f g-3 e h)-b^2 g \left (7 f g^2+3 h (e g+d h)\right )+2 a b h \left (f g^2+3 h (2 e g+d h)\right )\right )+h^2 \left (16 a^2 f h^3+4 a c g h (14 f g-3 e h)+c^2 \left (\frac {28 f g^4}{h}-12 d g^2 h\right )+b^2 h \left (25 f g^2-3 h (e g+d h)\right )-b \left (56 c f g^3-6 c g h (e g+2 d h)+2 a h^2 (22 f g-3 e h)\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{48 h^3 \left (c g^2-b g h+a h^2\right )^2 (g+h x)^4}-\frac {\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{5 h \left (c g^2-b g h+a h^2\right ) (g+h x)^5}+\frac {\left (c^2 f\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{h^6}-\frac {\left (256 c^5 f g^7-896 c^4 f g^5 h (b g-a h)+32 c^3 g h^2 \left (35 b^2 f g^4-70 a b f g^3 h+a^2 h^2 \left (35 f g^2-3 d h^2\right )\right )-16 c^2 h^3 \left (35 b^3 f g^4-6 a^3 h^3 (6 f g-e h)+3 a^2 b h^2 \left (35 f g^2-e g h-d h^2\right )-3 a b^2 g h \left (35 f g^2+d h^2\right )\right )+b^3 h^5 \left (16 a^2 f h^2-2 a b h (10 f g+3 e h)+b^2 \left (7 f g^2+3 h (e g+d h)\right )\right )-2 b c h^4 \left (96 a^3 f h^3-24 a^2 b h^2 (8 f g+e h)-b^3 \left (35 f g^3-3 d g h^2\right )+4 a b^2 h \left (35 f g^2+3 h (e g+d h)\right )\right )\right ) \int \frac {1}{(g+h x) \sqrt {a+b x+c x^2}} \, dx}{256 h^6 \left (c g^2-b g h+a h^2\right )^3}\\ &=-\frac {\left (\frac {128 c^4 f g^7}{h}-32 c^3 f g^5 (11 b g-10 a h)+8 c^2 g h \left (38 b^2 f g^4+2 a^2 h^2 \left (13 f g^2+3 d h^2\right )-a b g h \left (65 f g^2+3 d h^2\right )\right )-b h^3 (b g-2 a h) \left (16 a^2 f h^2-2 a b h (10 f g+3 e h)+b^2 \left (7 f g^2+3 h (e g+d h)\right )\right )-2 c h^2 \left (8 a^3 h^3 (2 f g-3 e h)-2 a b^2 g^2 h (34 f g+3 e h)+b^3 \left (35 f g^4-3 d g^2 h^2\right )+4 a^2 b h^2 \left (5 f g^2+3 h (2 e g+d h)\right )\right )+\left (128 c f \left (c g^2-h (b g-a h)\right )^3+(2 c g-b h) \left (32 c^3 f g^5-8 c^2 g h \left (10 b f g^3-11 a f g^2 h+3 a d h^3\right )+2 c h^2 \left (4 a^2 h^2 (10 f g-3 e h)-6 a b h \left (11 f g^2-e g h-d h^2\right )+b^2 \left (29 f g^3+3 d g h^2\right )\right )-b h^3 \left (16 a^2 f h^2-2 a b h (10 f g+3 e h)+b^2 \left (7 f g^2+3 h (e g+d h)\right )\right )\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{128 h^4 \left (c g^2-b g h+a h^2\right )^3 (g+h x)^2}-\frac {\left (16 c^2 f g^5-2 c g h \left (13 b f g^3-10 a f g^2 h+3 b d g h^2-6 a d h^3\right )-h^2 \left (4 a^2 h^2 (2 f g-3 e h)-b^2 g \left (7 f g^2+3 h (e g+d h)\right )+2 a b h \left (f g^2+3 h (2 e g+d h)\right )\right )+h^2 \left (16 a^2 f h^3+4 a c g h (14 f g-3 e h)+c^2 \left (\frac {28 f g^4}{h}-12 d g^2 h\right )+b^2 h \left (25 f g^2-3 h (e g+d h)\right )-b \left (56 c f g^3-6 c g h (e g+2 d h)+2 a h^2 (22 f g-3 e h)\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{48 h^3 \left (c g^2-b g h+a h^2\right )^2 (g+h x)^4}-\frac {\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{5 h \left (c g^2-b g h+a h^2\right ) (g+h x)^5}+\frac {\left (2 c^2 f\right ) \text {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^6}+\frac {\left (256 c^5 f g^7-896 c^4 f g^5 h (b g-a h)+32 c^3 g h^2 \left (35 b^2 f g^4-70 a b f g^3 h+a^2 h^2 \left (35 f g^2-3 d h^2\right )\right )-16 c^2 h^3 \left (35 b^3 f g^4-6 a^3 h^3 (6 f g-e h)+3 a^2 b h^2 \left (35 f g^2-e g h-d h^2\right )-3 a b^2 g h \left (35 f g^2+d h^2\right )\right )+b^3 h^5 \left (16 a^2 f h^2-2 a b h (10 f g+3 e h)+b^2 \left (7 f g^2+3 h (e g+d h)\right )\right )-2 b c h^4 \left (96 a^3 f h^3-24 a^2 b h^2 (8 f g+e h)-b^3 \left (35 f g^3-3 d g h^2\right )+4 a b^2 h \left (35 f g^2+3 h (e g+d h)\right )\right )\right ) \text {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 )}{128 h^6 \left (c g^2-b g h+a h^2\right )^3}\\ &=-\frac {\left (\frac {128 c^4 f g^7}{h}-32 c^3 f g^5 (11 b g-10 a h)+8 c^2 g h \left (38 b^2 f g^4+2 a^2 h^2 \left (13 f g^2+3 d h^2\right )-a b g h \left (65 f g^2+3 d h^2\right )\right )-b h^3 (b g-2 a h) \left (16 a^2 f h^2-2 a b h (10 f g+3 e h)+b^2 \left (7 f g^2+3 h (e g+d h)\right )\right )-2 c h^2 \left (8 a^3 h^3 (2 f g-3 e h)-2 a b^2 g^2 h (34 f g+3 e h)+b^3 \left (35 f g^4-3 d g^2 h^2\right )+4 a^2 b h^2 \left (5 f g^2+3 h (2 e g+d h)\right )\right )+\left (128 c f \left (c g^2-h (b g-a h)\right )^3+(2 c g-b h) \left (32 c^3 f g^5-8 c^2 g h \left (10 b f g^3-11 a f g^2 h+3 a d h^3\right )+2 c h^2 \left (4 a^2 h^2 (10 f g-3 e h)-6 a b h \left (11 f g^2-e g h-d h^2\right )+b^2 \left (29 f g^3+3 d g h^2\right )\right )-b h^3 \left (16 a^2 f h^2-2 a b h (10 f g+3 e h)+b^2 \left (7 f g^2+3 h (e g+d h)\right )\right )\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{128 h^4 \left (c g^2-b g h+a h^2\right )^3 (g+h x)^2}-\frac {\left (16 c^2 f g^5-2 c g h \left (13 b f g^3-10 a f g^2 h+3 b d g h^2-6 a d h^3\right )-h^2 \left (4 a^2 h^2 (2 f g-3 e h)-b^2 g \left (7 f g^2+3 h (e g+d h)\right )+2 a b h \left (f g^2+3 h (2 e g+d h)\right )\right )+h^2 \left (16 a^2 f h^3+4 a c g h (14 f g-3 e h)+c^2 \left (\frac {28 f g^4}{h}-12 d g^2 h\right )+b^2 h \left (25 f g^2-3 h (e g+d h)\right )-b \left (56 c f g^3-6 c g h (e g+2 d h)+2 a h^2 (22 f g-3 e h)\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{48 h^3 \left (c g^2-b g h+a h^2\right )^2 (g+h x)^4}-\frac {\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{5 h \left (c g^2-b g h+a h^2\right ) (g+h x)^5}+\frac {c^{3/2} f \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{h^6}-\frac {\left (256 c^5 f g^7-896 c^4 f g^5 h (b g-a h)+32 c^3 g h^2 \left (35 b^2 f g^4-70 a b f g^3 h+a^2 h^2 \left (35 f g^2-3 d h^2\right )\right )-16 c^2 h^3 \left (35 b^3 f g^4-6 a^3 h^3 (6 f g-e h)+3 a^2 b h^2 \left (35 f g^2-e g h-d h^2\right )-3 a b^2 g h \left (35 f g^2+d h^2\right )\right )+b^3 h^5 \left (16 a^2 f h^2-2 a b h (10 f g+3 e h)+b^2 \left (7 f g^2+3 h (e g+d h)\right )\right )-2 b c h^4 \left (96 a^3 f h^3-24 a^2 b h^2 (8 f g+e h)-b^3 \left (35 f g^3-3 d g h^2\right )+4 a b^2 h \left (35 f g^2+3 h (e g+d h)\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 )}{256 h^6 \left (c g^2-b g h+a h^2\right )^{7/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 16.74, size = 2229, normalized size = 1.82 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2))/(g + h*x)^6,x]

[Out]

((a + x*(b + c*x))^(3/2)*(-1/5*((c*g^2 - b*g*h + a*h^2)*(f*g^2 - e*g*h + d*h^2))/(h^5*(g + h*x)^5) + (42*c*f*g
^3 - 32*c*e*g^2*h - 31*b*f*g^2*h + 22*c*d*g*h^2 + 21*b*e*g*h^2 + 20*a*f*g*h^2 - 11*b*d*h^3 - 10*a*e*h^3)/(40*h
^5*(g + h*x)^4) + (-548*c^2*f*g^4 + 288*c^2*e*g^3*h + 808*b*c*f*g^3*h - 108*c^2*d*g^2*h^2 - 378*b*c*e*g^2*h^2
- 263*b^2*f*g^2*h^2 - 616*a*c*f*g^2*h^2 + 108*b*c*d*g*h^3 + 93*b^2*e*g*h^3 + 276*a*c*e*g*h^3 + 340*a*b*f*g*h^3
- 3*b^2*d*h^4 - 96*a*c*d*h^4 - 90*a*b*e*h^4 - 80*a^2*f*h^4)/(240*h^5*(c*g^2 - b*g*h + a*h^2)*(g + h*x)^3) + (
2608*c^3*f*g^5 - 768*c^3*e*g^4*h - 5752*b*c^2*f*g^4*h + 48*c^3*d*g^3*h^2 + 1512*b*c^2*e*g^3*h^2 + 3734*b^2*c*f
*g^3*h^2 + 5048*a*c^2*f*g^3*h^2 - 72*b*c^2*d*g^2*h^3 - 744*b^2*c*e*g^2*h^3 - 1488*a*c^2*e*g^2*h^3 - 605*b^3*f*
g^2*h^3 - 6084*a*b*c*f*g^2*h^3 - 6*b^2*c*d*g*h^4 + 168*a*c^2*d*g*h^4 + 15*b^3*e*g*h^4 + 1404*a*b*c*e*g*h^4 + 1
180*a*b^2*f*g*h^4 + 2320*a^2*c*f*g*h^4 + 15*b^3*d*h^5 - 84*a*b*c*d*h^5 - 30*a*b^2*e*h^5 - 600*a^2*c*e*h^5 - 56
0*a^2*b*f*h^5)/(960*h^5*(c*g^2 - b*g*h + a*h^2)^2*(g + h*x)^2) + (-4384*c^4*f*g^6 + 384*c^4*e*g^5*h + 12768*b*
c^3*f*g^5*h + 96*c^4*d*g^4*h^2 - 1008*b*c^3*e*g^4*h^2 - 12324*b^2*c^2*f*g^4*h^2 - 12528*a*c^3*f*g^4*h^2 - 192*
b*c^3*d*g^3*h^3 + 744*b^2*c^2*e*g^3*h^3 + 1248*a*c^3*e*g^3*h^3 + 4000*b^3*c*f*g^3*h^3 + 23808*a*b*c^2*f*g^3*h^
3 + 36*b^2*c^2*d*g^2*h^4 + 432*a*c^3*d*g^2*h^4 - 30*b^3*c*e*g^2*h^4 - 2088*a*b*c^2*e*g^2*h^4 - 105*b^4*f*g^2*h
^4 - 11100*a*b^2*c*f*g^2*h^4 - 11424*a^2*c^2*f*g^2*h^4 + 60*b^3*c*d*g*h^5 - 432*a*b*c^2*d*g*h^5 - 45*b^4*e*g*h
^5 + 360*a*b^2*c*e*g*h^5 + 1584*a^2*c^2*e*g*h^5 + 300*a*b^3*f*g*h^5 + 9840*a^2*b*c*f*g*h^5 - 45*b^4*d*h^6 + 30
0*a*b^2*c*d*h^6 - 384*a^2*c^2*d*h^6 + 90*a*b^3*e*h^6 - 600*a^2*b*c*e*h^6 - 240*a^2*b^2*f*h^6 - 2560*a^3*c*f*h^
6)/(1920*h^5*(c*g^2 - b*g*h + a*h^2)^3*(g + h*x))))/(a + b*x + c*x^2) - ((256*c^5*f*g^7 - 896*b*c^4*f*g^6*h +
1120*b^2*c^3*f*g^5*h^2 + 896*a*c^4*f*g^5*h^2 - 560*b^3*c^2*f*g^4*h^3 - 2240*a*b*c^3*f*g^4*h^3 + 70*b^4*c*f*g^3
*h^4 + 1680*a*b^2*c^2*f*g^3*h^4 + 1120*a^2*c^3*f*g^3*h^4 + 7*b^5*f*g^2*h^5 - 280*a*b^3*c*f*g^2*h^5 - 1680*a^2*
b*c^2*f*g^2*h^5 - 6*b^4*c*d*g*h^6 + 48*a*b^2*c^2*d*g*h^6 - 96*a^2*c^3*d*g*h^6 + 3*b^5*e*g*h^6 - 24*a*b^3*c*e*g
*h^6 + 48*a^2*b*c^2*e*g*h^6 - 20*a*b^4*f*g*h^6 + 384*a^2*b^2*c*f*g*h^6 + 576*a^3*c^2*f*g*h^6 + 3*b^5*d*h^7 - 2
4*a*b^3*c*d*h^7 + 48*a^2*b*c^2*d*h^7 - 6*a*b^4*e*h^7 + 48*a^2*b^2*c*e*h^7 - 96*a^3*c^2*e*h^7 + 16*a^2*b^3*f*h^
7 - 192*a^3*b*c*f*h^7)*(a + x*(b + c*x))^(3/2)*Log[g + h*x])/(256*h^6*(c*g^2 - b*g*h + a*h^2)^(7/2)*(a + b*x +
c*x^2)^(3/2)) + (((c^5*f*g^6)/(h^6*(c*g^2 - b*g*h + a*h^2)^3) - (3*b*c^4*f*g^5)/(h^5*(c*g^2 - b*g*h + a*h^2)^
3) + (3*c^3*(b^2 + a*c)*f*g^4)/(h^4*(c*g^2 - b*g*h + a*h^2)^3) - (b*c^2*(b^2 + 6*a*c)*f*g^3)/(h^3*(c*g^2 - b*g
*h + a*h^2)^3) + (3*a*c^2*(b^2 + a*c)*f*g^2)/(h^2*(c*g^2 - b*g*h + a*h^2)^3) + (a^2*c^2*f*(-3*b*g + a*h))/(h*(
c*g^2 - b*g*h + a*h^2)^3))*(a + x*(b + c*x))^(3/2)*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + b*x + c*x^2]])/(Sqrt[c]*
(a + b*x + c*x^2)^(3/2)) + ((256*c^5*f*g^7 - 896*b*c^4*f*g^6*h + 1120*b^2*c^3*f*g^5*h^2 + 896*a*c^4*f*g^5*h^2
- 560*b^3*c^2*f*g^4*h^3 - 2240*a*b*c^3*f*g^4*h^3 + 70*b^4*c*f*g^3*h^4 + 1680*a*b^2*c^2*f*g^3*h^4 + 1120*a^2*c^
3*f*g^3*h^4 + 7*b^5*f*g^2*h^5 - 280*a*b^3*c*f*g^2*h^5 - 1680*a^2*b*c^2*f*g^2*h^5 - 6*b^4*c*d*g*h^6 + 48*a*b^2*
c^2*d*g*h^6 - 96*a^2*c^3*d*g*h^6 + 3*b^5*e*g*h^6 - 24*a*b^3*c*e*g*h^6 + 48*a^2*b*c^2*e*g*h^6 - 20*a*b^4*f*g*h^
6 + 384*a^2*b^2*c*f*g*h^6 + 576*a^3*c^2*f*g*h^6 + 3*b^5*d*h^7 - 24*a*b^3*c*d*h^7 + 48*a^2*b*c^2*d*h^7 - 6*a*b^
4*e*h^7 + 48*a^2*b^2*c*e*h^7 - 96*a^3*c^2*e*h^7 + 16*a^2*b^3*f*h^7 - 192*a^3*b*c*f*h^7)*(a + x*(b + c*x))^(3/2
)*Log[-(b*g) + 2*a*h - 2*c*g*x + b*h*x + 2*Sqrt[c*g^2 - b*g*h + a*h^2]*Sqrt[a + b*x + c*x^2]])/(256*h^6*(c*g^2
- b*g*h + a*h^2)^(7/2)*(a + b*x + c*x^2)^(3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(13371\) vs. \(2(1196)=2392\).
time = 0.14, size = 13372, normalized size = 10.91

method	result	size
default	\(\text {Expression too large to display}\)	\(13372\)

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)^6,x,method=_RETURNVERBOSE)

[Out]

result too large to display

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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)^6,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 de

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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)^6,x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x + c x^{2}\right )^{\frac {3}{2}} \left (d + e x + f x^{2}\right )}{\left (g + h x\right )^{6}}\, dx \end {gather*}

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)**6,x)

[Out]

Integral((a + b*x + c*x**2)**(3/2)*(d + e*x + f*x**2)/(g + h*x)**6, x)

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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)^6,x, algorithm="giac")

[Out]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}\,\left (f\,x^2+e\,x+d\right )}{{\left (g+h\,x\right )}^6} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2))/(g + h*x)^6,x)

[Out]

int(((a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2))/(g + h*x)^6, x)

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