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

Optimal. Leaf size=824 \[ -\frac {\left (f g^2-h (e g-d h)\right ) \left (c x^2+b x+a\right )^{5/2}}{2 h \left (c g^2-b h g+a h^2\right ) (g+h x)^2}-\frac {\left (4 c g \left (-\frac {10 f g^2}{h}+6 e g-3 d h\right )-4 a h (7 f g-3 e h)+b \left (31 f g^2-3 h (5 e g-d h)\right )+2 h \left (-\frac {5 c f g^2}{h}+3 c e g+2 b f g-3 c d h-2 a f h\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{12 h^2 \left (c g^2-b h g+a h^2\right ) (g+h x)}-\frac {\left (-8 g^2 \left (10 f g^2-3 h (2 e g-d h)\right ) c^3-2 h \left (2 a h \left (19 f g^2-9 e h g+3 d h^2\right )-3 b g \left (22 f g^2-12 e h g+5 d h^2\right )\right ) c^2-h^2 \left (\left (53 f g^2-6 h (4 e g-d h)\right ) b^2-18 a h (3 f g-e h) b+8 a^2 f h^2\right ) c+2 h \left (2 g \left (10 f g^2-3 h (2 e g-d h)\right ) c^2+h \left (2 a h (7 f g-3 e h)-3 b \left (6 f g^2-3 e h g+d h^2\right )\right ) c+b f h^2 (b g-a h)\right ) x c+b^2 f h^3 (b g-a h)\right ) \sqrt {c x^2+b x+a}}{8 c h^5 \left (c g^2-b h g+a h^2\right )}-\frac {\left (16 g \left (10 f g^2-3 h (2 e g-d h)\right ) c^3+24 h \left (a h (3 f g-e h)-b \left (6 f g^2-3 e h g+d h^2\right )\right ) c^2+6 b h^2 (3 b f g-b e h-2 a f h) c+b^3 f h^3\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right )}{16 c^{3/2} h^6}+\frac {\left (8 c^2 \left (10 f g^2-3 h (2 e g-d h)\right ) g^2+4 c h \left (a h \left (19 f g^2-9 e h g+3 d h^2\right )-b g \left (28 f g^2-15 e h g+6 d h^2\right )\right )+h^2 \left (\left (35 f g^2-3 h (5 e g-d h)\right ) b^2-4 a h (10 f g-3 e h) b+8 a^2 f h^2\right )\right ) \tanh ^{-1}\left (\frac {b g-2 a h+(2 c g-b h) x}{2 \sqrt {c g^2-b h g+a h^2} \sqrt {c x^2+b x+a}}\right )}{8 h^6 \sqrt {c g^2-b h g+a h^2}} \]

[Out]

-1/12*(4*c*g*(6*e*g-10*f*g^2/h-3*d*h)-4*a*h*(-3*e*h+7*f*g)+b*(31*f*g^2-3*h*(-d*h+5*e*g))+2*h*(3*c*e*g+2*b*f*g-
5*c*f*g^2/h-3*c*d*h-2*a*f*h)*x)*(c*x^2+b*x+a)^(3/2)/h^2/(a*h^2-b*g*h+c*g^2)/(h*x+g)-1/2*(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)^2-1/16*(b^3*f*h^3+6*b*c*h^2*(-2*a*f*h-b*e*h+3*b*f*g)+16*c^3*g
*(10*f*g^2-3*h*(-d*h+2*e*g))+24*c^2*h*(a*h*(-e*h+3*f*g)-b*(d*h^2-3*e*g*h+6*f*g^2)))*arctanh(1/2*(2*c*x+b)/c^(1
/2)/(c*x^2+b*x+a)^(1/2))/c^(3/2)/h^6+1/8*(8*c^2*g^2*(10*f*g^2-3*h*(-d*h+2*e*g))+4*c*h*(a*h*(3*d*h^2-9*e*g*h+19
*f*g^2)-b*g*(6*d*h^2-15*e*g*h+28*f*g^2))+h^2*(8*a^2*f*h^2-4*a*b*h*(-3*e*h+10*f*g)+b^2*(35*f*g^2-3*h*(-d*h+5*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)^(1/2)-1/8*(b^2*f*h^3*(-a*h+b*g)-8*c^3*g^2*(10*f*g^2-3*h*(-d*h+2*e*g))-2*c^2*h*(2*a*h*(3*d*h^2-9*e*g*h+19
*f*g^2)-3*b*g*(5*d*h^2-12*e*g*h+22*f*g^2))-c*h^2*(8*a^2*f*h^2-18*a*b*h*(-e*h+3*f*g)+b^2*(53*f*g^2-6*h*(-d*h+4*
e*g)))+2*c*h*(b*f*h^2*(-a*h+b*g)+2*c^2*g*(10*f*g^2-3*h*(-d*h+2*e*g))+c*h*(2*a*h*(-3*e*h+7*f*g)-3*b*(d*h^2-3*e*
g*h+6*f*g^2)))*x)*(c*x^2+b*x+a)^(1/2)/c/h^5/(a*h^2-b*g*h+c*g^2)

________________________________________________________________________________________

Rubi [A]  time = 2.14, antiderivative size = 819, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 7, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {1650, 812, 814, 843, 621, 206, 724} \[ -\frac {\left (f g^2-h (e g-d h)\right ) \left (c x^2+b x+a\right )^{5/2}}{2 h \left (c g^2-b h g+a h^2\right ) (g+h x)^2}-\frac {\left (31 b f g^2+4 c \left (-\frac {10 f g^2}{h}+6 e g-3 d h\right ) g-3 b h (5 e g-d h)-4 a h (7 f g-3 e h)+2 h \left (-\frac {5 c f g^2}{h}+3 c e g+2 b f g-3 c d h-2 a f h\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{12 h^2 \left (c g^2-b h g+a h^2\right ) (g+h x)}-\frac {\left (8 g^2 \left (-\frac {10 f g^2}{h}+6 e g-3 d h\right ) c^3-2 \left (2 a h \left (19 f g^2-9 e h g+3 d h^2\right )-3 b g \left (22 f g^2-12 e h g+5 d h^2\right )\right ) c^2-h \left (\left (53 f g^2-6 h (4 e g-d h)\right ) b^2-18 a h (3 f g-e h) b+8 a^2 f h^2\right ) c+2 \left (2 \left (10 f g^3-3 g h (2 e g-d h)\right ) c^2+h \left (2 a h (7 f g-3 e h)-3 b \left (6 f g^2-3 e h g+d h^2\right )\right ) c+b f h^2 (b g-a h)\right ) x c+b^2 f h^2 (b g-a h)\right ) \sqrt {c x^2+b x+a}}{8 c h^4 \left (c g^2-b h g+a h^2\right )}-\frac {\left (16 \left (10 f g^3-3 g h (2 e g-d h)\right ) c^3-24 h \left (6 b f g^2-b h (3 e g-d h)-a h (3 f g-e h)\right ) c^2+6 b h^2 (3 b f g-b e h-2 a f h) c+b^3 f h^3\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right )}{16 c^{3/2} h^6}+\frac {\left (8 \left (10 f g^4-3 g^2 h (2 e g-d h)\right ) c^2-4 h \left (28 b f g^3-3 b h (5 e g-2 d h) g-a h \left (19 f g^2-9 e h g+3 d h^2\right )\right ) c+h^2 \left (\left (35 f g^2-3 h (5 e g-d h)\right ) b^2-4 a h (10 f g-3 e h) b+8 a^2 f h^2\right )\right ) \tanh ^{-1}\left (\frac {b g-2 a h+(2 c g-b h) x}{2 \sqrt {c g^2-b h g+a h^2} \sqrt {c x^2+b x+a}}\right )}{8 h^6 \sqrt {c g^2-b h g+a h^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b^2*f*h^2*(b*g - a*h) + 8*c^3*g^2*(6*e*g - (10*f*g^2)/h - 3*d*h) - 2*c^2*(2*a*h*(19*f*g^2 - 9*e*g*h + 3*d*h
^2) - 3*b*g*(22*f*g^2 - 12*e*g*h + 5*d*h^2)) - c*h*(8*a^2*f*h^2 - 18*a*b*h*(3*f*g - e*h) + b^2*(53*f*g^2 - 6*h
*(4*e*g - d*h))) + 2*c*(b*f*h^2*(b*g - a*h) + 2*c^2*(10*f*g^3 - 3*g*h*(2*e*g - d*h)) + c*h*(2*a*h*(7*f*g - 3*e
*h) - 3*b*(6*f*g^2 - 3*e*g*h + d*h^2)))*x)*Sqrt[a + b*x + c*x^2])/(8*c*h^4*(c*g^2 - b*g*h + a*h^2)) - ((31*b*f
*g^2 + 4*c*g*(6*e*g - (10*f*g^2)/h - 3*d*h) - 3*b*h*(5*e*g - d*h) - 4*a*h*(7*f*g - 3*e*h) + 2*h*(3*c*e*g + 2*b
*f*g - (5*c*f*g^2)/h - 3*c*d*h - 2*a*f*h)*x)*(a + b*x + c*x^2)^(3/2))/(12*h^2*(c*g^2 - b*g*h + a*h^2)*(g + h*x
)) - ((f*g^2 - h*(e*g - d*h))*(a + b*x + c*x^2)^(5/2))/(2*h*(c*g^2 - b*g*h + a*h^2)*(g + h*x)^2) - ((b^3*f*h^3
+ 6*b*c*h^2*(3*b*f*g - b*e*h - 2*a*f*h) + 16*c^3*(10*f*g^3 - 3*g*h*(2*e*g - d*h)) - 24*c^2*h*(6*b*f*g^2 - b*h
*(3*e*g - d*h) - a*h*(3*f*g - e*h)))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16*c^(3/2)*h^6)
+ ((8*c^2*(10*f*g^4 - 3*g^2*h*(2*e*g - d*h)) - 4*c*h*(28*b*f*g^3 - 3*b*g*h*(5*e*g - 2*d*h) - a*h*(19*f*g^2 - 9
*e*g*h + 3*d*h^2)) + h^2*(8*a^2*f*h^2 - 4*a*b*h*(10*f*g - 3*e*h) + b^2*(35*f*g^2 - 3*h*(5*e*g - d*h))))*ArcTan
h[(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])])/(8*h^6*Sqrt[c*g^2 -
b*g*h + a*h^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 812

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

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 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 {\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{(g+h x)^3} \, dx &=-\frac {\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{2 h \left (c g^2-b g h+a h^2\right ) (g+h x)^2}-\frac {\int \frac {\left (\frac {1}{2} \left (-4 c d g+5 b e g+4 a f g-\frac {5 b f g^2}{h}-b d h-4 a e h\right )+\left (3 c e g+2 b f g-\frac {5 c f g^2}{h}-3 c d h-2 a f h\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{(g+h x)^2} \, dx}{2 \left (c g^2-b g h+a h^2\right )}\\ &=-\frac {\left (31 b f g^2+4 c g \left (6 e g-\frac {10 f g^2}{h}-3 d h\right )-3 b h (5 e g-d h)-4 a h (7 f g-3 e h)+2 h \left (3 c e g+2 b f g-\frac {5 c f g^2}{h}-3 c d h-2 a f h\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{12 h^2 \left (c g^2-b g h+a h^2\right ) (g+h x)}-\frac {\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{2 h \left (c g^2-b g h+a h^2\right ) (g+h x)^2}+\frac {\int \frac {\left (\frac {1}{2} \left (2 (4 b g-2 a h) \left (3 c e g+2 b f g-\frac {5 c f g^2}{h}-3 c d h-2 a f h\right )+3 b \left (5 b f g^2-b h (5 e g-d h)+4 h (c d g-a f g+a e h)\right )\right )-\frac {2 \left (b f h^2 (b g-a h)+2 c^2 \left (10 f g^3-3 g h (2 e g-d h)\right )+c h \left (2 a h (7 f g-3 e h)-3 b \left (6 f g^2-3 e g h+d h^2\right )\right )\right ) x}{h}\right ) \sqrt {a+b x+c x^2}}{g+h x} \, dx}{4 h^2 \left (c g^2-b g h+a h^2\right )}\\ &=-\frac {\left (b^2 f h^2 (b g-a h)+8 c^3 g^2 \left (6 e g-\frac {10 f g^2}{h}-3 d h\right )-2 c^2 \left (2 a h \left (19 f g^2-9 e g h+3 d h^2\right )-3 b g \left (22 f g^2-12 e g h+5 d h^2\right )\right )-c h \left (8 a^2 f h^2-18 a b h (3 f g-e h)+b^2 \left (53 f g^2-6 h (4 e g-d h)\right )\right )+2 c \left (b f h^2 (b g-a h)+2 c^2 \left (10 f g^3-3 g h (2 e g-d h)\right )+c h \left (2 a h (7 f g-3 e h)-3 b \left (6 f g^2-3 e g h+d h^2\right )\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{8 c h^4 \left (c g^2-b g h+a h^2\right )}-\frac {\left (31 b f g^2+4 c g \left (6 e g-\frac {10 f g^2}{h}-3 d h\right )-3 b h (5 e g-d h)-4 a h (7 f g-3 e h)+2 h \left (3 c e g+2 b f g-\frac {5 c f g^2}{h}-3 c d h-2 a f h\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{12 h^2 \left (c g^2-b g h+a h^2\right ) (g+h x)}-\frac {\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{2 h \left (c g^2-b g h+a h^2\right ) (g+h x)^2}-\frac {\int \frac {\frac {\left (c g^2-b g h+a h^2\right ) \left (b^3 f g h^2-8 a c h \left (10 c f g^2+2 a f h^2-3 c h (2 e g-d h)\right )-2 b^2 c h \left (26 f g^2-3 h (4 e g-d h)\right )+4 b c \left (20 c f g^3-6 c g h (2 e g-d h)+a h^2 (17 f g-6 e h)\right )\right )}{h}+\frac {\left (c g^2-b g h+a h^2\right ) \left (b^3 f h^3+6 b c h^2 (3 b f g-b e h-2 a f h)+16 c^3 \left (10 f g^3-3 g h (2 e g-d h)\right )-24 c^2 h \left (6 b f g^2-b h (3 e g-d h)-a h (3 f g-e h)\right )\right ) x}{h}}{(g+h x) \sqrt {a+b x+c x^2}} \, dx}{16 c h^4 \left (c g^2-b g h+a h^2\right )}\\ &=-\frac {\left (b^2 f h^2 (b g-a h)+8 c^3 g^2 \left (6 e g-\frac {10 f g^2}{h}-3 d h\right )-2 c^2 \left (2 a h \left (19 f g^2-9 e g h+3 d h^2\right )-3 b g \left (22 f g^2-12 e g h+5 d h^2\right )\right )-c h \left (8 a^2 f h^2-18 a b h (3 f g-e h)+b^2 \left (53 f g^2-6 h (4 e g-d h)\right )\right )+2 c \left (b f h^2 (b g-a h)+2 c^2 \left (10 f g^3-3 g h (2 e g-d h)\right )+c h \left (2 a h (7 f g-3 e h)-3 b \left (6 f g^2-3 e g h+d h^2\right )\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{8 c h^4 \left (c g^2-b g h+a h^2\right )}-\frac {\left (31 b f g^2+4 c g \left (6 e g-\frac {10 f g^2}{h}-3 d h\right )-3 b h (5 e g-d h)-4 a h (7 f g-3 e h)+2 h \left (3 c e g+2 b f g-\frac {5 c f g^2}{h}-3 c d h-2 a f h\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{12 h^2 \left (c g^2-b g h+a h^2\right ) (g+h x)}-\frac {\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{2 h \left (c g^2-b g h+a h^2\right ) (g+h x)^2}-\frac {\left (b^3 f h^3+6 b c h^2 (3 b f g-b e h-2 a f h)+16 c^3 \left (10 f g^3-3 g h (2 e g-d h)\right )-24 c^2 h \left (6 b f g^2-b h (3 e g-d h)-a h (3 f g-e h)\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{16 c h^6}+\frac {\left (8 c^2 \left (10 f g^4-3 g^2 h (2 e g-d h)\right )-4 c h \left (28 b f g^3-3 b g h (5 e g-2 d h)-a h \left (19 f g^2-9 e g h+3 d h^2\right )\right )+h^2 \left (8 a^2 f h^2-4 a b h (10 f g-3 e h)+b^2 \left (35 f g^2-3 h (5 e g-d h)\right )\right )\right ) \int \frac {1}{(g+h x) \sqrt {a+b x+c x^2}} \, dx}{8 h^6}\\ &=-\frac {\left (b^2 f h^2 (b g-a h)+8 c^3 g^2 \left (6 e g-\frac {10 f g^2}{h}-3 d h\right )-2 c^2 \left (2 a h \left (19 f g^2-9 e g h+3 d h^2\right )-3 b g \left (22 f g^2-12 e g h+5 d h^2\right )\right )-c h \left (8 a^2 f h^2-18 a b h (3 f g-e h)+b^2 \left (53 f g^2-6 h (4 e g-d h)\right )\right )+2 c \left (b f h^2 (b g-a h)+2 c^2 \left (10 f g^3-3 g h (2 e g-d h)\right )+c h \left (2 a h (7 f g-3 e h)-3 b \left (6 f g^2-3 e g h+d h^2\right )\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{8 c h^4 \left (c g^2-b g h+a h^2\right )}-\frac {\left (31 b f g^2+4 c g \left (6 e g-\frac {10 f g^2}{h}-3 d h\right )-3 b h (5 e g-d h)-4 a h (7 f g-3 e h)+2 h \left (3 c e g+2 b f g-\frac {5 c f g^2}{h}-3 c d h-2 a f h\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{12 h^2 \left (c g^2-b g h+a h^2\right ) (g+h x)}-\frac {\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{2 h \left (c g^2-b g h+a h^2\right ) (g+h x)^2}-\frac {\left (b^3 f h^3+6 b c h^2 (3 b f g-b e h-2 a f h)+16 c^3 \left (10 f g^3-3 g h (2 e g-d h)\right )-24 c^2 h \left (6 b f g^2-b h (3 e g-d h)-a h (3 f g-e h)\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 )}{8 c h^6}-\frac {\left (8 c^2 \left (10 f g^4-3 g^2 h (2 e g-d h)\right )-4 c h \left (28 b f g^3-3 b g h (5 e g-2 d h)-a h \left (19 f g^2-9 e g h+3 d h^2\right )\right )+h^2 \left (8 a^2 f h^2-4 a b h (10 f g-3 e h)+b^2 \left (35 f g^2-3 h (5 e g-d h)\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 )}{4 h^6}\\ &=-\frac {\left (b^2 f h^2 (b g-a h)+8 c^3 g^2 \left (6 e g-\frac {10 f g^2}{h}-3 d h\right )-2 c^2 \left (2 a h \left (19 f g^2-9 e g h+3 d h^2\right )-3 b g \left (22 f g^2-12 e g h+5 d h^2\right )\right )-c h \left (8 a^2 f h^2-18 a b h (3 f g-e h)+b^2 \left (53 f g^2-6 h (4 e g-d h)\right )\right )+2 c \left (b f h^2 (b g-a h)+2 c^2 \left (10 f g^3-3 g h (2 e g-d h)\right )+c h \left (2 a h (7 f g-3 e h)-3 b \left (6 f g^2-3 e g h+d h^2\right )\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{8 c h^4 \left (c g^2-b g h+a h^2\right )}-\frac {\left (31 b f g^2+4 c g \left (6 e g-\frac {10 f g^2}{h}-3 d h\right )-3 b h (5 e g-d h)-4 a h (7 f g-3 e h)+2 h \left (3 c e g+2 b f g-\frac {5 c f g^2}{h}-3 c d h-2 a f h\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{12 h^2 \left (c g^2-b g h+a h^2\right ) (g+h x)}-\frac {\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{2 h \left (c g^2-b g h+a h^2\right ) (g+h x)^2}-\frac {\left (b^3 f h^3+6 b c h^2 (3 b f g-b e h-2 a f h)+16 c^3 \left (10 f g^3-3 g h (2 e g-d h)\right )-24 c^2 h \left (6 b f g^2-b h (3 e g-d h)-a h (3 f g-e h)\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{3/2} h^6}+\frac {\left (8 c^2 \left (10 f g^4-3 g^2 h (2 e g-d h)\right )-4 c h \left (28 b f g^3-3 b g h (5 e g-2 d h)-a h \left (19 f g^2-9 e g h+3 d h^2\right )\right )+h^2 \left (8 a^2 f h^2-4 a b h (10 f g-3 e h)+b^2 \left (35 f g^2-3 h (5 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 )}{8 h^6 \sqrt {c g^2-b g h+a h^2}}\\ \end {align*}

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Mathematica [B]  time = 6.27, size = 4162, normalized size = 5.05 \[ \text {Result too large to show} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(f*(a + b*x + c*x^2)*(a + x*(b + c*x))^(3/2))/(3*c*h*(g + h*x)^2) - ((a + x*(b + c*x))^(3/2)*(-1/2*(((h*(5*b*f
*g - 6*c*d*h - 4*a*f*h))/2 - (g*(10*c*f*g - 6*c*e*h + b*f*h))/2)*(a + b*x + c*x^2)^(5/2))/((c*g^2 - b*g*h + a*
h^2)*(g + h*x)^2) - (((-3*c*g*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) + (3*c*h*(5*b*f*g^2 - b*h*(5
*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2)*(a + b*x + c*x^2)^(5/2))/((-(c*g^2) + b*g*h - a*h^2)*(g + h*x))
+ (((4*c*(4*c*g - (3*b*h)/2)*(-3*c*g*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) + (3*c*h*(5*b*f*g^2
- b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2) + 4*c*h*(-3*a*c*h*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*
h*(e*g - d*h)) - (3*c^2*g*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2 + (3*b*c*h*(5*b*f*g
^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2 + (5*b*(3*c*g*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*
(e*g - d*h)) - (3*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2))/2) - 12*c^2*h*(-3*c*g
*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) + (3*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*
f*g + a*e*h)))/2)*x)*(a + b*x + c*x^2)^(3/2))/(12*c*h^2) - (((2*c*h*(-4*c*(2*a*c*g*h + b*g*(-4*c*g + (3*b*h)/2
))*(-3*c*g*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) + (3*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(
c*d*g - a*f*g + a*e*h)))/2) + 4*c*h*(b*g - 2*a*h)*(-3*a*c*h*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)
) - (3*c^2*g*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2 + (3*b*c*h*(5*b*f*g^2 - b*h*(5*e
*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2 + (5*b*(3*c*g*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h))
- (3*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2))/2)) - (2*c*g - (b*h)/2)*(-4*c*(-8*
c^2*g^2 + (3*b^2*h^2)/2 - c*h*(-4*b*g + 6*a*h))*(-3*c*g*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) +
(3*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2) + 4*c*h*(2*c*g - b*h)*(-3*a*c*h*(5*c*
f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) - (3*c^2*g*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g
+ a*e*h)))/2 + (3*b*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2 + (5*b*(3*c*g*(5*c*f*
g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) - (3*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*
e*h)))/2))/2)) + c*h*(-4*c*(-8*c^2*g^2 + (3*b^2*h^2)/2 - c*h*(-4*b*g + 6*a*h))*(-3*c*g*(5*c*f*g^2 - 2*f*h*(b*g
- a*h) - 3*c*h*(e*g - d*h)) + (3*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2) + 4*c*
h*(2*c*g - b*h)*(-3*a*c*h*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) - (3*c^2*g*(5*b*f*g^2 - b*h*(5*e
*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2 + (3*b*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a
*e*h)))/2 + (5*b*(3*c*g*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) - (3*c*h*(5*b*f*g^2 - b*h*(5*e*g -
d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2))/2))*x)*Sqrt[a + b*x + c*x^2])/(2*c*h^2) - (((2*c*h*(2*c*g - b*h)*(-4
*c*(2*a*c*g*h + b*g*(-4*c*g + (3*b*h)/2))*(-3*c*g*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) + (3*c*h
*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2) + 4*c*h*(b*g - 2*a*h)*(-3*a*c*h*(5*c*f*g^2
- 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) - (3*c^2*g*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*
h)))/2 + (3*b*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2 + (5*b*(3*c*g*(5*c*f*g^2 -
2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) - (3*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))
/2))/2)) + (-4*c^2*g^2 + (b^2*h^2)/2 - c*h*(-2*b*g + 2*a*h))*(-4*c*(-8*c^2*g^2 + (3*b^2*h^2)/2 - c*h*(-4*b*g +
6*a*h))*(-3*c*g*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) + (3*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) +
4*h*(c*d*g - a*f*g + a*e*h)))/2) + 4*c*h*(2*c*g - b*h)*(-3*a*c*h*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g
- d*h)) - (3*c^2*g*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2 + (3*b*c*h*(5*b*f*g^2 - b*
h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2 + (5*b*(3*c*g*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g -
d*h)) - (3*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2))/2)))*ArcTanh[(b + 2*c*x)/(2*
Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(Sqrt[c]*h) - (4*Sqrt[c*g^2 - b*g*h + a*h^2]*(-(g*(2*c*h*(2*c*g - b*h)*(-4*c*
(2*a*c*g*h + b*g*(-4*c*g + (3*b*h)/2))*(-3*c*g*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) + (3*c*h*(5
*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2) + 4*c*h*(b*g - 2*a*h)*(-3*a*c*h*(5*c*f*g^2 - 2
*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) - (3*c^2*g*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h))
)/2 + (3*b*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2 + (5*b*(3*c*g*(5*c*f*g^2 - 2*f
*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) - (3*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2)
)/2)) + (-4*c^2*g^2 + (b^2*h^2)/2 - c*h*(-2*b*g + 2*a*h))*(-4*c*(-8*c^2*g^2 + (3*b^2*h^2)/2 - c*h*(-4*b*g + 6*
a*h))*(-3*c*g*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) + (3*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*
h*(c*d*g - a*f*g + a*e*h)))/2) + 4*c*h*(2*c*g - b*h)*(-3*a*c*h*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d
*h)) - (3*c^2*g*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2 + (3*b*c*h*(5*b*f*g^2 - b*h*(
5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2 + (5*b*(3*c*g*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h
)) - (3*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2))/2)))) + h*(2*c*h*(b*g - 2*a*h)*
(-4*c*(2*a*c*g*h + b*g*(-4*c*g + (3*b*h)/2))*(-3*c*g*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) + (3*
c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2) + 4*c*h*(b*g - 2*a*h)*(-3*a*c*h*(5*c*f*g
^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) - (3*c^2*g*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a
*e*h)))/2 + (3*b*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2 + (5*b*(3*c*g*(5*c*f*g^2
- 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) - (3*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h
)))/2))/2)) + (2*a*c*g*h + b*g*(-2*c*g + (b*h)/2))*(-4*c*(-8*c^2*g^2 + (3*b^2*h^2)/2 - c*h*(-4*b*g + 6*a*h))*(
-3*c*g*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) + (3*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*
g - a*f*g + a*e*h)))/2) + 4*c*h*(2*c*g - b*h)*(-3*a*c*h*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) -
(3*c^2*g*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2 + (3*b*c*h*(5*b*f*g^2 - b*h*(5*e*g -
d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2 + (5*b*(3*c*g*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) - (3
*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2))/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])])/(h*(4*c*g^2 - 4*b*g*h + 4*a*h^2)))/(4*c*h^2
))/(8*c*h^2))/(-(c*g^2) + b*g*h - a*h^2))/(2*(c*g^2 - b*g*h + a*h^2))))/(3*c*h*(a + b*x + c*x^2)^(3/2))

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Eval
uation time: 1.47Unable to divide, perhaps due to rounding error%%%{1,[6,0,0,0,9,0,0,0]%%%}+%%%{%%{[-6,0]:[1,0
,%%%{-1,[1]%%%}]%%},[5,0,0,0,8,1,0,0]%%%}+%%%{-3,[4,1,0,0,9,0,0,0]%%%}+%%%{3,[4,0,1,0,8,1,0,0]%%%}+%%%{%%%{12,
[1]%%%},[4,0,0,0,7,2,0,0]%%%}+%%%{%%{[12,0]:[1,0,%%%{-1,[1]%%%}]%%},[3,1,0,0,8,1,0,0]%%%}+%%%{%%{[-12,0]:[1,0,
%%%{-1,[1]%%%}]%%},[3,0,1,0,7,2,0,0]%%%}+%%%{%%{[%%%{-8,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,0,0,0,6,3,0,0]%%
%}+%%%{3,[2,2,0,0,9,0,0,0]%%%}+%%%{-6,[2,1,1,0,8,1,0,0]%%%}+%%%{%%%{-12,[1]%%%},[2,1,0,0,7,2,0,0]%%%}+%%%{3,[2
,0,2,0,7,2,0,0]%%%}+%%%{%%%{12,[1]%%%},[2,0,1,0,6,3,0,0]%%%}+%%%{%%{[-6,0]:[1,0,%%%{-1,[1]%%%}]%%},[1,2,0,0,8,
1,0,0]%%%}+%%%{%%{[12,0]:[1,0,%%%{-1,[1]%%%}]%%},[1,1,1,0,7,2,0,0]%%%}+%%%{%%{[-6,0]:[1,0,%%%{-1,[1]%%%}]%%},[
1,0,2,0,6,3,0,0]%%%}+%%%{-1,[0,3,0,0,9,0,0,0]%%%}+%%%{3,[0,2,1,0,8,1,0,0]%%%}+%%%{-3,[0,1,2,0,7,2,0,0]%%%}+%%%
{1,[0,0,3,0,6,3,0,0]%%%} / %%%{%%{poly1[%%%{1,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[6,0,0,0,3,0,0,0]%%%}+%%%{%%%
{-6,[2]%%%},[5,0,0,0,2,1,0,0]%%%}+%%%{%%{[%%%{-3,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[4,1,0,0,3,0,0,0]%%%}+%%%{
%%{poly1[%%%{3,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[4,0,1,0,2,1,0,0]%%%}+%%%{%%{poly1[%%%{12,[2]%%%},0]:[1,0,%%
%{-1,[1]%%%}]%%},[4,0,0,0,1,2,0,0]%%%}+%%%{%%%{12,[2]%%%},[3,1,0,0,2,1,0,0]%%%}+%%%{%%%{-12,[2]%%%},[3,0,1,0,1
,2,0,0]%%%}+%%%{%%%{-8,[3]%%%},[3,0,0,0,0,3,0,0]%%%}+%%%{%%{[%%%{3,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,2,0,0
,3,0,0,0]%%%}+%%%{%%{[%%%{-6,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,1,1,0,2,1,0,0]%%%}+%%%{%%{[%%%{-12,[2]%%%},
0]:[1,0,%%%{-1,[1]%%%}]%%},[2,1,0,0,1,2,0,0]%%%}+%%%{%%{poly1[%%%{3,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,0,2,
0,1,2,0,0]%%%}+%%%{%%{poly1[%%%{12,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,0,1,0,0,3,0,0]%%%}+%%%{%%%{-6,[2]%%%}
,[1,2,0,0,2,1,0,0]%%%}+%%%{%%%{12,[2]%%%},[1,1,1,0,1,2,0,0]%%%}+%%%{%%%{-6,[2]%%%},[1,0,2,0,0,3,0,0]%%%}+%%%{%
%{[%%%{-1,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,3,0,0,3,0,0,0]%%%}+%%%{%%{[%%%{3,[1]%%%},0]:[1,0,%%%{-1,[1]%%%
}]%%},[0,2,1,0,2,1,0,0]%%%}+%%%{%%{[%%%{-3,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,1,2,0,1,2,0,0]%%%}+%%%{%%{pol
y1[%%%{1,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,0,3,0,0,3,0,0]%%%} Error: Bad Argument Value

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maple [B]  time = 0.02, size = 26596, normalized size = 32.28 \[ \text {output too large to display} \]

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)^3,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((c*x^2+b*x+a)^(3/2)*(f*x^2+e*x+d)/(h*x+g)^3,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 \[ \int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}\,\left (f\,x^2+e\,x+d\right )}{{\left (g+h\,x\right )}^3} \,d x \]

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)^3,x)

[Out]

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

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

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)**3,x)

[Out]

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

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