3.206 \(\int \frac{(a+b x+c x^2)^{3/2} (d+e x+f x^2)}{(g+h x)^7} \, dx\)

Optimal. Leaf size=657 \[ \frac{\left (a+b x+c x^2\right )^{3/2} (-2 a h+x (2 c g-b h)+b g) \left (24 a^2 f h^2-4 c \left (a \left (d h^2-7 e g h+f g^2\right )+3 b g (2 d h+e g)\right )-12 a b h (e h+2 f g)+b^2 \left (7 d h^2+5 e g h+7 f g^2\right )+24 c^2 d g^2\right )}{192 (g+h x)^4 \left (a h^2-b g h+c g^2\right )^3}-\frac{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (-2 a h+x (2 c g-b h)+b g) \left (24 a^2 f h^2-4 c \left (a \left (d h^2-7 e g h+f g^2\right )+3 b g (2 d h+e g)\right )-12 a b h (e h+2 f g)+b^2 \left (7 d h^2+5 e g h+7 f g^2\right )+24 c^2 d g^2\right )}{512 (g+h x)^2 \left (a h^2-b g h+c g^2\right )^4}+\frac{\left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{-2 a h+x (2 c g-b h)+b g}{2 \sqrt{a+b x+c x^2} \sqrt{a h^2-b g h+c g^2}}\right ) \left (24 a^2 f h^2-4 c \left (a \left (d h^2-7 e g h+f g^2\right )+3 b g (2 d h+e g)\right )-12 a b h (e h+2 f g)+b^2 \left (7 d h^2+5 e g h+7 f g^2\right )+24 c^2 d g^2\right )}{1024 \left (a h^2-b g h+c g^2\right )^{9/2}}+\frac{\left (a+b x+c x^2\right )^{5/2} \left (h \left (12 a h (2 f g-e h)-b \left (-7 d h^2-5 e g h+17 f g^2\right )\right )+2 c g \left (h (e g-7 d h)+5 f g^2\right )\right )}{60 h (g+h x)^5 \left (a h^2-b g h+c g^2\right )^2}-\frac{\left (a+b x+c x^2\right )^{5/2} \left (f g^2-h (e g-d h)\right )}{6 h (g+h x)^6 \left (a h^2-b g h+c g^2\right )} \]

[Out]

-((b^2 - 4*a*c)*(24*c^2*d*g^2 + 24*a^2*f*h^2 - 12*a*b*h*(2*f*g + e*h) + b^2*(7*f*g^2 + 5*e*g*h + 7*d*h^2) - 4*
c*(3*b*g*(e*g + 2*d*h) + a*(f*g^2 - 7*e*g*h + d*h^2)))*(b*g - 2*a*h + (2*c*g - b*h)*x)*Sqrt[a + b*x + c*x^2])/
(512*(c*g^2 - b*g*h + a*h^2)^4*(g + h*x)^2) + ((24*c^2*d*g^2 + 24*a^2*f*h^2 - 12*a*b*h*(2*f*g + e*h) + b^2*(7*
f*g^2 + 5*e*g*h + 7*d*h^2) - 4*c*(3*b*g*(e*g + 2*d*h) + a*(f*g^2 - 7*e*g*h + d*h^2)))*(b*g - 2*a*h + (2*c*g -
b*h)*x)*(a + b*x + c*x^2)^(3/2))/(192*(c*g^2 - b*g*h + a*h^2)^3*(g + h*x)^4) - ((f*g^2 - h*(e*g - d*h))*(a + b
*x + c*x^2)^(5/2))/(6*h*(c*g^2 - b*g*h + a*h^2)*(g + h*x)^6) + ((2*c*g*(5*f*g^2 + h*(e*g - 7*d*h)) + h*(12*a*h
*(2*f*g - e*h) - b*(17*f*g^2 - 5*e*g*h - 7*d*h^2)))*(a + b*x + c*x^2)^(5/2))/(60*h*(c*g^2 - b*g*h + a*h^2)^2*(
g + h*x)^5) + ((b^2 - 4*a*c)^2*(24*c^2*d*g^2 + 24*a^2*f*h^2 - 12*a*b*h*(2*f*g + e*h) + b^2*(7*f*g^2 + 5*e*g*h
+ 7*d*h^2) - 4*c*(3*b*g*(e*g + 2*d*h) + a*(f*g^2 - 7*e*g*h + d*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])])/(1024*(c*g^2 - b*g*h + a*h^2)^(9/2))

________________________________________________________________________________________

Rubi [A]  time = 1.21644, antiderivative size = 660, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156, Rules used = {1650, 806, 720, 724, 206} \[ \frac{\left (a+b x+c x^2\right )^{3/2} (-2 a h+x (2 c g-b h)+b g) \left (24 a^2 f h^2-4 c \left (-a h (7 e g-d h)+a f g^2+3 b g (2 d h+e g)\right )-12 a b h (e h+2 f g)+b^2 \left (h (7 d h+5 e g)+7 f g^2\right )+24 c^2 d g^2\right )}{192 (g+h x)^4 \left (a h^2-b g h+c g^2\right )^3}-\frac{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (-2 a h+x (2 c g-b h)+b g) \left (24 a^2 f h^2-4 c \left (-a h (7 e g-d h)+a f g^2+3 b g (2 d h+e g)\right )-12 a b h (e h+2 f g)+b^2 \left (h (7 d h+5 e g)+7 f g^2\right )+24 c^2 d g^2\right )}{512 (g+h x)^2 \left (a h^2-b g h+c g^2\right )^4}+\frac{\left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{-2 a h+x (2 c g-b h)+b g}{2 \sqrt{a+b x+c x^2} \sqrt{a h^2-b g h+c g^2}}\right ) \left (24 a^2 f h^2-4 c \left (-a h (7 e g-d h)+a f g^2+3 b g (2 d h+e g)\right )-12 a b h (e h+2 f g)+b^2 \left (h (7 d h+5 e g)+7 f g^2\right )+24 c^2 d g^2\right )}{1024 \left (a h^2-b g h+c g^2\right )^{9/2}}+\frac{\left (a+b x+c x^2\right )^{5/2} \left (2 c \left (g h (e g-7 d h)+5 f g^3\right )-h \left (-12 a h (2 f g-e h)-b h (7 d h+5 e g)+17 b f g^2\right )\right )}{60 h (g+h x)^5 \left (a h^2-b g h+c g^2\right )^2}-\frac{\left (a+b x+c x^2\right )^{5/2} \left (f g^2-h (e g-d h)\right )}{6 h (g+h x)^6 \left (a h^2-b g h+c g^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2))/(g + h*x)^7,x]

[Out]

-((b^2 - 4*a*c)*(24*c^2*d*g^2 + 24*a^2*f*h^2 - 12*a*b*h*(2*f*g + e*h) - 4*c*(a*f*g^2 - a*h*(7*e*g - d*h) + 3*b
*g*(e*g + 2*d*h)) + b^2*(7*f*g^2 + h*(5*e*g + 7*d*h)))*(b*g - 2*a*h + (2*c*g - b*h)*x)*Sqrt[a + b*x + c*x^2])/
(512*(c*g^2 - b*g*h + a*h^2)^4*(g + h*x)^2) + ((24*c^2*d*g^2 + 24*a^2*f*h^2 - 12*a*b*h*(2*f*g + e*h) - 4*c*(a*
f*g^2 - a*h*(7*e*g - d*h) + 3*b*g*(e*g + 2*d*h)) + b^2*(7*f*g^2 + h*(5*e*g + 7*d*h)))*(b*g - 2*a*h + (2*c*g -
b*h)*x)*(a + b*x + c*x^2)^(3/2))/(192*(c*g^2 - b*g*h + a*h^2)^3*(g + h*x)^4) - ((f*g^2 - h*(e*g - d*h))*(a + b
*x + c*x^2)^(5/2))/(6*h*(c*g^2 - b*g*h + a*h^2)*(g + h*x)^6) + ((2*c*(5*f*g^3 + g*h*(e*g - 7*d*h)) - h*(17*b*f
*g^2 - b*h*(5*e*g + 7*d*h) - 12*a*h*(2*f*g - e*h)))*(a + b*x + c*x^2)^(5/2))/(60*h*(c*g^2 - b*g*h + a*h^2)^2*(
g + h*x)^5) + ((b^2 - 4*a*c)^2*(24*c^2*d*g^2 + 24*a^2*f*h^2 - 12*a*b*h*(2*f*g + e*h) - 4*c*(a*f*g^2 - a*h*(7*e
*g - d*h) + 3*b*g*(e*g + 2*d*h)) + b^2*(7*f*g^2 + h*(5*e*g + 7*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])])/(1024*(c*g^2 - b*g*h + a*h^2)^(9/2))

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]

Rule 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)), x] - Dist[(b
*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(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] && EqQ[Sim
plify[m + 2*p + 3], 0]

Rule 720

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*
(d*b - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^p)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[(p*(b^2 -
4*a*c))/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[
{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m +
2*p + 2, 0] && GtQ[p, 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 206

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

Rubi steps

\begin{align*} \int \frac{\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{(g+h x)^7} \, dx &=-\frac{\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{6 h \left (c g^2-b g h+a h^2\right ) (g+h x)^6}-\frac{\int \frac{\left (\frac{1}{2} \left (-12 c d g+5 b e g+12 a f g-\frac{5 b f g^2}{h}+7 b d h-12 a e h\right )-\left (c e g-6 b f g+\frac{5 c f g^2}{h}-c d h+6 a f h\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{(g+h x)^6} \, dx}{6 \left (c g^2-b g h+a h^2\right )}\\ &=-\frac{\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{6 h \left (c g^2-b g h+a h^2\right ) (g+h x)^6}+\frac{\left (2 c \left (5 f g^3+g h (e g-7 d h)\right )-h \left (17 b f g^2-b h (5 e g+7 d h)-12 a h (2 f g-e h)\right )\right ) \left (a+b x+c x^2\right )^{5/2}}{60 h \left (c g^2-b g h+a h^2\right )^2 (g+h x)^5}+\frac{\left (24 c^2 d g^2+24 a^2 f h^2-12 a b h (2 f g+e h)-4 c \left (a f g^2-a h (7 e g-d h)+3 b g (e g+2 d h)\right )+b^2 \left (7 f g^2+h (5 e g+7 d h)\right )\right ) \int \frac{\left (a+b x+c x^2\right )^{3/2}}{(g+h x)^5} \, dx}{24 \left (c g^2-b g h+a h^2\right )^2}\\ &=\frac{\left (24 c^2 d g^2+24 a^2 f h^2-12 a b h (2 f g+e h)-4 c \left (a f g^2-a h (7 e g-d h)+3 b g (e g+2 d h)\right )+b^2 \left (7 f g^2+h (5 e g+7 d h)\right )\right ) (b g-2 a h+(2 c g-b h) x) \left (a+b x+c x^2\right )^{3/2}}{192 \left (c g^2-b g h+a h^2\right )^3 (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}}{6 h \left (c g^2-b g h+a h^2\right ) (g+h x)^6}+\frac{\left (2 c \left (5 f g^3+g h (e g-7 d h)\right )-h \left (17 b f g^2-b h (5 e g+7 d h)-12 a h (2 f g-e h)\right )\right ) \left (a+b x+c x^2\right )^{5/2}}{60 h \left (c g^2-b g h+a h^2\right )^2 (g+h x)^5}-\frac{\left (\left (b^2-4 a c\right ) \left (24 c^2 d g^2+24 a^2 f h^2-12 a b h (2 f g+e h)-4 c \left (a f g^2-a h (7 e g-d h)+3 b g (e g+2 d h)\right )+b^2 \left (7 f g^2+h (5 e g+7 d h)\right )\right )\right ) \int \frac{\sqrt{a+b x+c x^2}}{(g+h x)^3} \, dx}{128 \left (c g^2-b g h+a h^2\right )^3}\\ &=-\frac{\left (b^2-4 a c\right ) \left (24 c^2 d g^2+24 a^2 f h^2-12 a b h (2 f g+e h)-4 c \left (a f g^2-a h (7 e g-d h)+3 b g (e g+2 d h)\right )+b^2 \left (7 f g^2+h (5 e g+7 d h)\right )\right ) (b g-2 a h+(2 c g-b h) x) \sqrt{a+b x+c x^2}}{512 \left (c g^2-b g h+a h^2\right )^4 (g+h x)^2}+\frac{\left (24 c^2 d g^2+24 a^2 f h^2-12 a b h (2 f g+e h)-4 c \left (a f g^2-a h (7 e g-d h)+3 b g (e g+2 d h)\right )+b^2 \left (7 f g^2+h (5 e g+7 d h)\right )\right ) (b g-2 a h+(2 c g-b h) x) \left (a+b x+c x^2\right )^{3/2}}{192 \left (c g^2-b g h+a h^2\right )^3 (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}}{6 h \left (c g^2-b g h+a h^2\right ) (g+h x)^6}+\frac{\left (2 c \left (5 f g^3+g h (e g-7 d h)\right )-h \left (17 b f g^2-b h (5 e g+7 d h)-12 a h (2 f g-e h)\right )\right ) \left (a+b x+c x^2\right )^{5/2}}{60 h \left (c g^2-b g h+a h^2\right )^2 (g+h x)^5}+\frac{\left (\left (b^2-4 a c\right )^2 \left (24 c^2 d g^2+24 a^2 f h^2-12 a b h (2 f g+e h)-4 c \left (a f g^2-a h (7 e g-d h)+3 b g (e g+2 d h)\right )+b^2 \left (7 f g^2+h (5 e g+7 d h)\right )\right )\right ) \int \frac{1}{(g+h x) \sqrt{a+b x+c x^2}} \, dx}{1024 \left (c g^2-b g h+a h^2\right )^4}\\ &=-\frac{\left (b^2-4 a c\right ) \left (24 c^2 d g^2+24 a^2 f h^2-12 a b h (2 f g+e h)-4 c \left (a f g^2-a h (7 e g-d h)+3 b g (e g+2 d h)\right )+b^2 \left (7 f g^2+h (5 e g+7 d h)\right )\right ) (b g-2 a h+(2 c g-b h) x) \sqrt{a+b x+c x^2}}{512 \left (c g^2-b g h+a h^2\right )^4 (g+h x)^2}+\frac{\left (24 c^2 d g^2+24 a^2 f h^2-12 a b h (2 f g+e h)-4 c \left (a f g^2-a h (7 e g-d h)+3 b g (e g+2 d h)\right )+b^2 \left (7 f g^2+h (5 e g+7 d h)\right )\right ) (b g-2 a h+(2 c g-b h) x) \left (a+b x+c x^2\right )^{3/2}}{192 \left (c g^2-b g h+a h^2\right )^3 (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}}{6 h \left (c g^2-b g h+a h^2\right ) (g+h x)^6}+\frac{\left (2 c \left (5 f g^3+g h (e g-7 d h)\right )-h \left (17 b f g^2-b h (5 e g+7 d h)-12 a h (2 f g-e h)\right )\right ) \left (a+b x+c x^2\right )^{5/2}}{60 h \left (c g^2-b g h+a h^2\right )^2 (g+h x)^5}-\frac{\left (\left (b^2-4 a c\right )^2 \left (24 c^2 d g^2+24 a^2 f h^2-12 a b h (2 f g+e h)-4 c \left (a f g^2-a h (7 e g-d h)+3 b g (e g+2 d h)\right )+b^2 \left (7 f g^2+h (5 e g+7 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 )}{512 \left (c g^2-b g h+a h^2\right )^4}\\ &=-\frac{\left (b^2-4 a c\right ) \left (24 c^2 d g^2+24 a^2 f h^2-12 a b h (2 f g+e h)-4 c \left (a f g^2-a h (7 e g-d h)+3 b g (e g+2 d h)\right )+b^2 \left (7 f g^2+h (5 e g+7 d h)\right )\right ) (b g-2 a h+(2 c g-b h) x) \sqrt{a+b x+c x^2}}{512 \left (c g^2-b g h+a h^2\right )^4 (g+h x)^2}+\frac{\left (24 c^2 d g^2+24 a^2 f h^2-12 a b h (2 f g+e h)-4 c \left (a f g^2-a h (7 e g-d h)+3 b g (e g+2 d h)\right )+b^2 \left (7 f g^2+h (5 e g+7 d h)\right )\right ) (b g-2 a h+(2 c g-b h) x) \left (a+b x+c x^2\right )^{3/2}}{192 \left (c g^2-b g h+a h^2\right )^3 (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}}{6 h \left (c g^2-b g h+a h^2\right ) (g+h x)^6}+\frac{\left (2 c \left (5 f g^3+g h (e g-7 d h)\right )-h \left (17 b f g^2-b h (5 e g+7 d h)-12 a h (2 f g-e h)\right )\right ) \left (a+b x+c x^2\right )^{5/2}}{60 h \left (c g^2-b g h+a h^2\right )^2 (g+h x)^5}+\frac{\left (b^2-4 a c\right )^2 \left (24 c^2 d g^2+24 a^2 f h^2-12 a b h (2 f g+e h)-4 c \left (a f g^2-a h (7 e g-d h)+3 b g (e g+2 d h)\right )+b^2 \left (7 f g^2+h (5 e g+7 d h)\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 )}{1024 \left (c g^2-b g h+a h^2\right )^{9/2}}\\ \end{align*}

Mathematica [A]  time = 6.24459, size = 765, normalized size = 1.16 \[ \frac{(a+x (b+c x))^{3/2} \left (-\frac{-\frac{\left (\frac{\left (a+b x+c x^2\right )^{3/2} (-2 a h+x (2 c g-b h)+b g)}{8 (g+h x)^4 \left (a h^2-b g h+c g^2\right )}-\frac{3 \left (b^2-4 a c\right ) \left (\frac{\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a h-x (2 c g-b h)-b g}{2 \sqrt{a+b x+c x^2} \sqrt{a h^2-b g h+c g^2}}\right )}{2 \sqrt{a h^2-b g h+c g^2} \left (4 a h^2-4 b g h+4 c g^2\right )}+\frac{\sqrt{a+b x+c x^2} (-2 a h+x (2 c g-b h)+b g)}{4 (g+h x)^2 \left (a h^2-b g h+c g^2\right )}\right )}{16 \left (a h^2-b g h+c g^2\right )}\right ) \left (b \left (-c g \left (-6 f h (b g-a h)+c h (e g-d h)+5 c f g^2\right )-\frac{1}{2} c h \left (12 h (a e h-a f g+c d g)-b h (7 d h+5 e g)+5 b f g^2\right )\right )-2 \left (-\frac{1}{2} c^2 g \left (12 h (a e h-a f g+c d g)-b h (7 d h+5 e g)+5 b f g^2\right )-a c h \left (-6 f h (b g-a h)+c h (e g-d h)+5 c f g^2\right )\right )\right )}{2 \left (a h^2-b g h+c g^2\right )}-\frac{\left (a+b x+c x^2\right )^{5/2} \left (c g \left (-6 f h (b g-a h)+c h (e g-d h)+5 c f g^2\right )-\frac{1}{2} c h \left (12 h (a e h-a f g+c d g)-b h (7 d h+5 e g)+5 b f g^2\right )\right )}{5 (g+h x)^5 \left (a h^2-b g h+c g^2\right )}}{6 \left (a h^2-b g h+c g^2\right )}-\frac{\left (a+b x+c x^2\right )^{5/2} \left (\frac{1}{2} h (-12 a f h+5 b f g+2 c d h)-\frac{1}{2} g (2 c (e h+5 f g)-7 b f h)\right )}{6 (g+h x)^6 \left (a h^2-b g h+c g^2\right )}\right )}{c h \left (a+b x+c x^2\right )^{3/2}}-\frac{f \left (a+b x+c x^2\right ) (a+x (b+c x))^{3/2}}{c h (g+h x)^6} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2))/(g + h*x)^7,x]

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.322, size = 100754, normalized size = 153.4 \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)^7,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)^7,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)^7,x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

Sympy [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)**7,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [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)^7,x, algorithm="giac")

[Out]

Timed out