3.20 \(\int \frac {(A+B x) (a+b x+c x^2)^{3/2}}{d+e x+f x^2} \, dx\)

Optimal. Leaf size=1092 \[ \frac {B \left (c x^2+b x+a\right )^{3/2}}{3 f}-\frac {\left (2 A c f (4 c e-5 b f)-B \left (8 \left (e^2-d f\right ) c^2-2 f (5 b e-4 a f) c+b^2 f^2\right )+2 c f (2 B c e-b B f-2 A c f) x\right ) \sqrt {c x^2+b x+a}}{8 c f^3}+\frac {\left (2 A c f \left (8 \left (e^2-d f\right ) c^2-12 f (b e-a f) c+3 b^2 f^2\right )-B \left (16 \left (e^3-2 d e f\right ) c^3-24 f \left (b e^2-a f e-b d f\right ) c^2+6 b f^2 (b e-2 a f) c+b^3 f^3\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right )}{16 c^{3/2} f^4}-\frac {\left (2 c f \left (B d (c e-b f) \left (c e^2-b f e+2 a f^2-2 c d f\right )+A f \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right )-c \left (e-\sqrt {e^2-4 d f}\right ) \left (A f (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+B \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )\right )\right ) \tanh ^{-1}\left (\frac {4 a f-b \left (e-\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-b f e+2 a f^2-2 c d f-(c e-b f) \sqrt {e^2-4 d f}} \sqrt {c x^2+b x+a}}\right )}{\sqrt {2} c f^4 \sqrt {e^2-4 d f} \sqrt {c e^2-b f e+2 a f^2-2 c d f-(c e-b f) \sqrt {e^2-4 d f}}}+\frac {\left (2 f \left (B d (c e-b f) \left (c e^2-b f e+2 a f^2-2 c d f\right )+A f \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right )-\left (e+\sqrt {e^2-4 d f}\right ) \left (A f (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+B \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )\right )\right ) \tanh ^{-1}\left (\frac {4 a f-b \left (e+\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e+\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-b f e+2 a f^2-2 c d f+(c e-b f) \sqrt {e^2-4 d f}} \sqrt {c x^2+b x+a}}\right )}{\sqrt {2} f^4 \sqrt {e^2-4 d f} \sqrt {c e^2-b f e+2 a f^2-2 c d f+(c e-b f) \sqrt {e^2-4 d f}}} \]

[Out]

1/3*B*(c*x^2+b*x+a)^(3/2)/f+1/16*(2*A*c*f*(3*b^2*f^2-12*c*f*(-a*f+b*e)+8*c^2*(-d*f+e^2))-B*(b^3*f^3+6*b*c*f^2*
(-2*a*f+b*e)-24*c^2*f*(-a*e*f-b*d*f+b*e^2)+16*c^3*(-2*d*e*f+e^3)))*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)
^(1/2))/c^(3/2)/f^4-1/8*(2*A*c*f*(-5*b*f+4*c*e)-B*(b^2*f^2-2*c*f*(-4*a*f+5*b*e)+8*c^2*(-d*f+e^2))+2*c*f*(-2*A*
c*f-B*b*f+2*B*c*e)*x)*(c*x^2+b*x+a)^(1/2)/c/f^3-1/2*arctanh(1/4*(4*a*f+2*x*(b*f-c*(e-(-4*d*f+e^2)^(1/2)))-b*(e
-(-4*d*f+e^2)^(1/2)))*2^(1/2)/(c*x^2+b*x+a)^(1/2)/(c*e^2-2*c*d*f-b*e*f+2*a*f^2-(-b*f+c*e)*(-4*d*f+e^2)^(1/2))^
(1/2))*(2*c*f*(B*d*(-b*f+c*e)*(2*a*f^2-b*e*f-2*c*d*f+c*e^2)+A*f*(2*c*d*f*(-a*f+b*e)-f^2*(-a^2*f+b^2*d)-c^2*d*(
-d*f+e^2)))-c*(A*f*(-b*f+c*e)*(f*(-2*a*f+b*e)-c*(-2*d*f+e^2))+B*(c^2*(d^2*f^2-3*d*e^2*f+e^4)-f^2*(2*a*b*e*f-a^
2*f^2-b^2*(-d*f+e^2))+2*c*f*(a*f*(-d*f+e^2)-b*(-2*d*e*f+e^3))))*(e-(-4*d*f+e^2)^(1/2)))/c/f^4*2^(1/2)/(-4*d*f+
e^2)^(1/2)/(c*e^2-2*c*d*f-b*e*f+2*a*f^2-(-b*f+c*e)*(-4*d*f+e^2)^(1/2))^(1/2)+1/2*arctanh(1/4*(4*a*f-b*(e+(-4*d
*f+e^2)^(1/2))+2*x*(b*f-c*(e+(-4*d*f+e^2)^(1/2))))*2^(1/2)/(c*x^2+b*x+a)^(1/2)/(c*e^2-2*c*d*f-b*e*f+2*a*f^2+(-
b*f+c*e)*(-4*d*f+e^2)^(1/2))^(1/2))*(2*f*(B*d*(-b*f+c*e)*(2*a*f^2-b*e*f-2*c*d*f+c*e^2)+A*f*(2*c*d*f*(-a*f+b*e)
-f^2*(-a^2*f+b^2*d)-c^2*d*(-d*f+e^2)))-(A*f*(-b*f+c*e)*(f*(-2*a*f+b*e)-c*(-2*d*f+e^2))+B*(c^2*(d^2*f^2-3*d*e^2
*f+e^4)-f^2*(2*a*b*e*f-a^2*f^2-b^2*(-d*f+e^2))+2*c*f*(a*f*(-d*f+e^2)-b*(-2*d*e*f+e^3))))*(e+(-4*d*f+e^2)^(1/2)
))/f^4*2^(1/2)/(-4*d*f+e^2)^(1/2)/(c*e^2-2*c*d*f-b*e*f+2*a*f^2+(-b*f+c*e)*(-4*d*f+e^2)^(1/2))^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 18.87, antiderivative size = 1092, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {1019, 1066, 1076, 621, 206, 1032, 724} \[ \frac {B \left (c x^2+b x+a\right )^{3/2}}{3 f}-\frac {\left (2 A c f (4 c e-5 b f)-B \left (8 \left (e^2-d f\right ) c^2-2 f (5 b e-4 a f) c+b^2 f^2\right )+2 c f (2 B c e-b B f-2 A c f) x\right ) \sqrt {c x^2+b x+a}}{8 c f^3}+\frac {\left (2 A c f \left (8 \left (e^2-d f\right ) c^2-12 f (b e-a f) c+3 b^2 f^2\right )-B \left (16 \left (e^3-2 d e f\right ) c^3-24 f \left (b e^2-a f e-b d f\right ) c^2+6 b f^2 (b e-2 a f) c+b^3 f^3\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right )}{16 c^{3/2} f^4}-\frac {\left (2 c f \left (B d (c e-b f) \left (c e^2-b f e+2 a f^2-2 c d f\right )+A f \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right )-c \left (e-\sqrt {e^2-4 d f}\right ) \left (A f (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+B \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (e^2-d f\right ) b^2+2 a e f b-a^2 f^2\right )\right )\right )\right ) \tanh ^{-1}\left (\frac {4 a f-b \left (e-\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-b f e+2 a f^2-2 c d f-(c e-b f) \sqrt {e^2-4 d f}} \sqrt {c x^2+b x+a}}\right )}{\sqrt {2} c f^4 \sqrt {e^2-4 d f} \sqrt {c e^2-b f e+2 a f^2-2 c d f-(c e-b f) \sqrt {e^2-4 d f}}}+\frac {\left (2 f \left (B d (c e-b f) \left (c e^2-b f e+2 a f^2-2 c d f\right )+A f \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right )-\left (e+\sqrt {e^2-4 d f}\right ) \left (A f (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+B \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (e^2-d f\right ) b^2+2 a e f b-a^2 f^2\right )\right )\right )\right ) \tanh ^{-1}\left (\frac {4 a f-b \left (e+\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e+\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-b f e+2 a f^2-2 c d f+(c e-b f) \sqrt {e^2-4 d f}} \sqrt {c x^2+b x+a}}\right )}{\sqrt {2} f^4 \sqrt {e^2-4 d f} \sqrt {c e^2-b f e+2 a f^2-2 c d f+(c e-b f) \sqrt {e^2-4 d f}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((2*A*c*f*(4*c*e - 5*b*f) - B*(b^2*f^2 - 2*c*f*(5*b*e - 4*a*f) + 8*c^2*(e^2 - d*f)) + 2*c*f*(2*B*c*e - b*B*f
- 2*A*c*f)*x)*Sqrt[a + b*x + c*x^2])/(8*c*f^3) + (B*(a + b*x + c*x^2)^(3/2))/(3*f) + ((2*A*c*f*(3*b^2*f^2 - 12
*c*f*(b*e - a*f) + 8*c^2*(e^2 - d*f)) - B*(b^3*f^3 + 6*b*c*f^2*(b*e - 2*a*f) - 24*c^2*f*(b*e^2 - b*d*f - a*e*f
) + 16*c^3*(e^3 - 2*d*e*f)))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16*c^(3/2)*f^4) - ((2*c*
f*(B*d*(c*e - b*f)*(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2) + A*f*(2*c*d*f*(b*e - a*f) - f^2*(b^2*d - a^2*f) - c^2*
d*(e^2 - d*f))) - c*(e - Sqrt[e^2 - 4*d*f])*(A*f*(c*e - b*f)*(f*(b*e - 2*a*f) - c*(e^2 - 2*d*f)) + B*(c^2*(e^4
- 3*d*e^2*f + d^2*f^2) - f^2*(2*a*b*e*f - a^2*f^2 - b^2*(e^2 - d*f)) + 2*c*f*(a*f*(e^2 - d*f) - b*(e^3 - 2*d*
e*f)))))*ArcTanh[(4*a*f - b*(e - Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e - Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c
*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*c*f^4*Sqrt
[e^2 - 4*d*f]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]) + ((2*f*(B*d*(c*e - b*f
)*(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2) + A*f*(2*c*d*f*(b*e - a*f) - f^2*(b^2*d - a^2*f) - c^2*d*(e^2 - d*f))) -
(e + Sqrt[e^2 - 4*d*f])*(A*f*(c*e - b*f)*(f*(b*e - 2*a*f) - c*(e^2 - 2*d*f)) + B*(c^2*(e^4 - 3*d*e^2*f + d^2*
f^2) - f^2*(2*a*b*e*f - a^2*f^2 - b^2*(e^2 - d*f)) + 2*c*f*(a*f*(e^2 - d*f) - b*(e^3 - 2*d*e*f)))))*ArcTanh[(4
*a*f - b*(e + Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e + Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*
e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*f^4*Sqrt[e^2 - 4*d*f]*Sqrt[c*
e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]])

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 1019

Int[((g_.) + (h_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Sy
mbol] :> Simp[(h*(a + b*x + c*x^2)^p*(d + e*x + f*x^2)^(q + 1))/(2*f*(p + q + 1)), x] - Dist[1/(2*f*(p + q + 1
)), Int[(a + b*x + c*x^2)^(p - 1)*(d + e*x + f*x^2)^q*Simp[h*p*(b*d - a*e) + a*(h*e - 2*g*f)*(p + q + 1) + (2*
h*p*(c*d - a*f) + b*(h*e - 2*g*f)*(p + q + 1))*x + (h*p*(c*e - b*f) + c*(h*e - 2*g*f)*(p + q + 1))*x^2, x], x]
, x] /; FreeQ[{a, b, c, d, e, f, g, h, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && GtQ[p, 0] && Ne
Q[p + q + 1, 0]

Rule 1032

Int[((g_.) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbo
l] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(2*c*g - h*(b - q))/q, Int[1/((b - q + 2*c*x)*Sqrt[d + e*x + f*x^2])
, x], x] - Dist[(2*c*g - h*(b + q))/q, Int[1/((b + q + 2*c*x)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, b,
c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && PosQ[b^2 - 4*a*c]

Rule 1066

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)*((d_) + (e_.)*(x_) + (f_.)*(x_
)^2)^(q_), x_Symbol] :> Simp[((B*c*f*(2*p + 2*q + 3) + C*(b*f*p - c*e*(2*p + q + 2)) + 2*c*C*f*(p + q + 1)*x)*
(a + b*x + c*x^2)^p*(d + e*x + f*x^2)^(q + 1))/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 3)), x] - Dist[1/(2*c*f^2*(p
+ q + 1)*(2*p + 2*q + 3)), Int[(a + b*x + c*x^2)^(p - 1)*(d + e*x + f*x^2)^q*Simp[p*(b*d - a*e)*(C*(c*e - b*f)
*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(b^2*C*d*f*p + a*c*(C*(2*d*f - e^2*(2*p + q + 2)) + f*
(B*e - 2*A*f)*(2*p + 2*q + 3))) + (2*p*(c*d - a*f)*(C*(c*e - b*f)*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (
p + q + 1)*(C*e*f*p*(b^2 - 4*a*c) - b*c*(C*(e^2 - 4*d*f)*(2*p + q + 2) + f*(2*C*d - B*e + 2*A*f)*(2*p + 2*q +
3))))*x + (p*(c*e - b*f)*(C*(c*e - b*f)*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(C*f^2*p*(b^2 -
4*a*c) - c^2*(C*(e^2 - 4*d*f)*(2*p + q + 2) + f*(2*C*d - B*e + 2*A*f)*(2*p + 2*q + 3))))*x^2, x], x], x] /; F
reeQ[{a, b, c, d, e, f, A, B, C, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && GtQ[p, 0] && NeQ[p +
q + 1, 0] && NeQ[2*p + 2*q + 3, 0] &&  !IGtQ[p, 0] &&  !IGtQ[q, 0]

Rule 1076

Int[((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x
_)^2]), x_Symbol] :> Dist[C/c, Int[1/Sqrt[d + e*x + f*x^2], x], x] + Dist[1/c, Int[(A*c - a*C + (B*c - b*C)*x)
/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b^2 - 4*a*c
, 0] && NeQ[e^2 - 4*d*f, 0]

Rubi steps

\begin {align*} \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{d+e x+f x^2} \, dx &=\frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}-\frac {\int \frac {\sqrt {a+b x+c x^2} \left (\frac {3}{2} (b B d-2 a A f)-\frac {3}{2} (2 A b f-B (2 c d+b e-2 a f)) x+\frac {3}{2} (2 B c e-b B f-2 A c f) x^2\right )}{d+e x+f x^2} \, dx}{3 f}\\ &=-\frac {\left (2 A c f (4 c e-5 b f)-B \left (b^2 f^2-2 c f (5 b e-4 a f)+8 c^2 \left (e^2-d f\right )\right )+2 c f (2 B c e-b B f-2 A c f) x\right ) \sqrt {a+b x+c x^2}}{8 c f^3}+\frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}+\frac {\int \frac {-\frac {3}{8} \left (b^3 B d f^2-10 b^2 c d f (B e-A f)-8 a c f (B c d e-A f (c d-2 a f))-4 b c d \left (2 A c e f-B \left (2 c e^2-2 c d f+5 a f^2\right )\right )\right )+\frac {3}{8} \left (2 A c f \left (8 c^2 d e-4 a c e f-b f (5 b e-16 a f)+4 b c \left (e^2-4 d f\right )\right )-B \left (b^3 e f^2+16 c^3 d \left (e^2-d f\right )+2 c f \left (10 a b e f-8 a^2 f^2-b^2 \left (5 e^2-8 d f\right )\right )-8 c^2 \left (a f \left (e^2-4 d f\right )-b \left (e^3-5 d e f\right )\right )\right )\right ) x+\frac {3}{8} \left (2 A c f \left (3 b^2 f^2-12 c f (b e-a f)+8 c^2 \left (e^2-d f\right )\right )-B \left (b^3 f^3+6 b c f^2 (b e-2 a f)-24 c^2 f \left (b e^2-b d f-a e f\right )+16 c^3 \left (e^3-2 d e f\right )\right )\right ) x^2}{\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )} \, dx}{6 c f^3}\\ &=-\frac {\left (2 A c f (4 c e-5 b f)-B \left (b^2 f^2-2 c f (5 b e-4 a f)+8 c^2 \left (e^2-d f\right )\right )+2 c f (2 B c e-b B f-2 A c f) x\right ) \sqrt {a+b x+c x^2}}{8 c f^3}+\frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}+\frac {\int \frac {-\frac {3}{8} d \left (2 A c f \left (3 b^2 f^2-12 c f (b e-a f)+8 c^2 \left (e^2-d f\right )\right )-B \left (b^3 f^3+6 b c f^2 (b e-2 a f)-24 c^2 f \left (b e^2-b d f-a e f\right )+16 c^3 \left (e^3-2 d e f\right )\right )\right )-\frac {3}{8} f \left (b^3 B d f^2-10 b^2 c d f (B e-A f)-8 a c f (B c d e-A f (c d-2 a f))-4 b c d \left (2 A c e f-B \left (2 c e^2-2 c d f+5 a f^2\right )\right )\right )+\left (-\frac {3}{8} e \left (2 A c f \left (3 b^2 f^2-12 c f (b e-a f)+8 c^2 \left (e^2-d f\right )\right )-B \left (b^3 f^3+6 b c f^2 (b e-2 a f)-24 c^2 f \left (b e^2-b d f-a e f\right )+16 c^3 \left (e^3-2 d e f\right )\right )\right )+\frac {3}{8} f \left (2 A c f \left (8 c^2 d e-4 a c e f-b f (5 b e-16 a f)+4 b c \left (e^2-4 d f\right )\right )-B \left (b^3 e f^2+16 c^3 d \left (e^2-d f\right )+2 c f \left (10 a b e f-8 a^2 f^2-b^2 \left (5 e^2-8 d f\right )\right )-8 c^2 \left (a f \left (e^2-4 d f\right )-b \left (e^3-5 d e f\right )\right )\right )\right )\right ) x}{\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )} \, dx}{6 c f^4}+\frac {\left (2 A c f \left (3 b^2 f^2-12 c f (b e-a f)+8 c^2 \left (e^2-d f\right )\right )-B \left (b^3 f^3+6 b c f^2 (b e-2 a f)-24 c^2 f \left (b e^2-b d f-a e f\right )+16 c^3 \left (e^3-2 d e f\right )\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{16 c f^4}\\ &=-\frac {\left (2 A c f (4 c e-5 b f)-B \left (b^2 f^2-2 c f (5 b e-4 a f)+8 c^2 \left (e^2-d f\right )\right )+2 c f (2 B c e-b B f-2 A c f) x\right ) \sqrt {a+b x+c x^2}}{8 c f^3}+\frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}+\frac {\left (2 A c f \left (3 b^2 f^2-12 c f (b e-a f)+8 c^2 \left (e^2-d f\right )\right )-B \left (b^3 f^3+6 b c f^2 (b e-2 a f)-24 c^2 f \left (b e^2-b d f-a e f\right )+16 c^3 \left (e^3-2 d e f\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{8 c f^4}-\frac {\left (A f \left (c^2 \left (e^4-4 d e^2 f+2 d^2 f^2+e^3 \sqrt {e^2-4 d f}-2 d e f \sqrt {e^2-4 d f}\right )+f^2 \left (2 a^2 f^2-2 a b f \left (e+\sqrt {e^2-4 d f}\right )+b^2 \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )\right )+2 c f \left (a f \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )-b \left (e^3-3 d e f+e^2 \sqrt {e^2-4 d f}-d f \sqrt {e^2-4 d f}\right )\right )\right )-B \left (c^2 \left (e^5-5 d e^3 f+5 d^2 e f^2+e^4 \sqrt {e^2-4 d f}-3 d e^2 f \sqrt {e^2-4 d f}+d^2 f^2 \sqrt {e^2-4 d f}\right )+f^2 \left (a^2 f^2 \left (e+\sqrt {e^2-4 d f}\right )-2 a b f \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )+b^2 \left (e^3-3 d e f+e^2 \sqrt {e^2-4 d f}-d f \sqrt {e^2-4 d f}\right )\right )+2 c f \left (a f \left (e^3-3 d e f+e^2 \sqrt {e^2-4 d f}-d f \sqrt {e^2-4 d f}\right )-b \left (e^4-4 d e^2 f+2 d^2 f^2+e^3 \sqrt {e^2-4 d f}-2 d e f \sqrt {e^2-4 d f}\right )\right )\right )\right ) \int \frac {1}{\left (e+\sqrt {e^2-4 d f}+2 f x\right ) \sqrt {a+b x+c x^2}} \, dx}{f^4 \sqrt {e^2-4 d f}}--\frac {\left (2 f \left (-\frac {3}{8} d \left (2 A c f \left (3 b^2 f^2-12 c f (b e-a f)+8 c^2 \left (e^2-d f\right )\right )-B \left (b^3 f^3+6 b c f^2 (b e-2 a f)-24 c^2 f \left (b e^2-b d f-a e f\right )+16 c^3 \left (e^3-2 d e f\right )\right )\right )-\frac {3}{8} f \left (b^3 B d f^2-10 b^2 c d f (B e-A f)-8 a c f (B c d e-A f (c d-2 a f))-4 b c d \left (2 A c e f-B \left (2 c e^2-2 c d f+5 a f^2\right )\right )\right )\right )-\left (e-\sqrt {e^2-4 d f}\right ) \left (-\frac {3}{8} e \left (2 A c f \left (3 b^2 f^2-12 c f (b e-a f)+8 c^2 \left (e^2-d f\right )\right )-B \left (b^3 f^3+6 b c f^2 (b e-2 a f)-24 c^2 f \left (b e^2-b d f-a e f\right )+16 c^3 \left (e^3-2 d e f\right )\right )\right )+\frac {3}{8} f \left (2 A c f \left (8 c^2 d e-4 a c e f-b f (5 b e-16 a f)+4 b c \left (e^2-4 d f\right )\right )-B \left (b^3 e f^2+16 c^3 d \left (e^2-d f\right )+2 c f \left (10 a b e f-8 a^2 f^2-b^2 \left (5 e^2-8 d f\right )\right )-8 c^2 \left (a f \left (e^2-4 d f\right )-b \left (e^3-5 d e f\right )\right )\right )\right )\right )\right ) \int \frac {1}{\left (e-\sqrt {e^2-4 d f}+2 f x\right ) \sqrt {a+b x+c x^2}} \, dx}{6 c f^4 \sqrt {e^2-4 d f}}\\ &=-\frac {\left (2 A c f (4 c e-5 b f)-B \left (b^2 f^2-2 c f (5 b e-4 a f)+8 c^2 \left (e^2-d f\right )\right )+2 c f (2 B c e-b B f-2 A c f) x\right ) \sqrt {a+b x+c x^2}}{8 c f^3}+\frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}+\frac {\left (2 A c f \left (3 b^2 f^2-12 c f (b e-a f)+8 c^2 \left (e^2-d f\right )\right )-B \left (b^3 f^3+6 b c f^2 (b e-2 a f)-24 c^2 f \left (b e^2-b d f-a e f\right )+16 c^3 \left (e^3-2 d e f\right )\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} f^4}+\frac {\left (2 \left (A f \left (c^2 \left (e^4-4 d e^2 f+2 d^2 f^2+e^3 \sqrt {e^2-4 d f}-2 d e f \sqrt {e^2-4 d f}\right )+f^2 \left (2 a^2 f^2-2 a b f \left (e+\sqrt {e^2-4 d f}\right )+b^2 \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )\right )+2 c f \left (a f \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )-b \left (e^3-3 d e f+e^2 \sqrt {e^2-4 d f}-d f \sqrt {e^2-4 d f}\right )\right )\right )-B \left (c^2 \left (e^5-5 d e^3 f+5 d^2 e f^2+e^4 \sqrt {e^2-4 d f}-3 d e^2 f \sqrt {e^2-4 d f}+d^2 f^2 \sqrt {e^2-4 d f}\right )+f^2 \left (a^2 f^2 \left (e+\sqrt {e^2-4 d f}\right )-2 a b f \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )+b^2 \left (e^3-3 d e f+e^2 \sqrt {e^2-4 d f}-d f \sqrt {e^2-4 d f}\right )\right )+2 c f \left (a f \left (e^3-3 d e f+e^2 \sqrt {e^2-4 d f}-d f \sqrt {e^2-4 d f}\right )-b \left (e^4-4 d e^2 f+2 d^2 f^2+e^3 \sqrt {e^2-4 d f}-2 d e f \sqrt {e^2-4 d f}\right )\right )\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{16 a f^2-8 b f \left (e+\sqrt {e^2-4 d f}\right )+4 c \left (e+\sqrt {e^2-4 d f}\right )^2-x^2} \, dx,x,\frac {4 a f-b \left (e+\sqrt {e^2-4 d f}\right )-\left (-2 b f+2 c \left (e+\sqrt {e^2-4 d f}\right )\right ) x}{\sqrt {a+b x+c x^2}}\right )}{f^4 \sqrt {e^2-4 d f}}+-\frac {\left (2 f \left (-\frac {3}{8} d \left (2 A c f \left (3 b^2 f^2-12 c f (b e-a f)+8 c^2 \left (e^2-d f\right )\right )-B \left (b^3 f^3+6 b c f^2 (b e-2 a f)-24 c^2 f \left (b e^2-b d f-a e f\right )+16 c^3 \left (e^3-2 d e f\right )\right )\right )-\frac {3}{8} f \left (b^3 B d f^2-10 b^2 c d f (B e-A f)-8 a c f (B c d e-A f (c d-2 a f))-4 b c d \left (2 A c e f-B \left (2 c e^2-2 c d f+5 a f^2\right )\right )\right )\right )-\left (e-\sqrt {e^2-4 d f}\right ) \left (-\frac {3}{8} e \left (2 A c f \left (3 b^2 f^2-12 c f (b e-a f)+8 c^2 \left (e^2-d f\right )\right )-B \left (b^3 f^3+6 b c f^2 (b e-2 a f)-24 c^2 f \left (b e^2-b d f-a e f\right )+16 c^3 \left (e^3-2 d e f\right )\right )\right )+\frac {3}{8} f \left (2 A c f \left (8 c^2 d e-4 a c e f-b f (5 b e-16 a f)+4 b c \left (e^2-4 d f\right )\right )-B \left (b^3 e f^2+16 c^3 d \left (e^2-d f\right )+2 c f \left (10 a b e f-8 a^2 f^2-b^2 \left (5 e^2-8 d f\right )\right )-8 c^2 \left (a f \left (e^2-4 d f\right )-b \left (e^3-5 d e f\right )\right )\right )\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{16 a f^2-8 b f \left (e-\sqrt {e^2-4 d f}\right )+4 c \left (e-\sqrt {e^2-4 d f}\right )^2-x^2} \, dx,x,\frac {4 a f-b \left (e-\sqrt {e^2-4 d f}\right )-\left (-2 b f+2 c \left (e-\sqrt {e^2-4 d f}\right )\right ) x}{\sqrt {a+b x+c x^2}}\right )}{3 c f^4 \sqrt {e^2-4 d f}}\\ &=-\frac {\left (2 A c f (4 c e-5 b f)-B \left (b^2 f^2-2 c f (5 b e-4 a f)+8 c^2 \left (e^2-d f\right )\right )+2 c f (2 B c e-b B f-2 A c f) x\right ) \sqrt {a+b x+c x^2}}{8 c f^3}+\frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}+\frac {\left (2 A c f \left (3 b^2 f^2-12 c f (b e-a f)+8 c^2 \left (e^2-d f\right )\right )-B \left (b^3 f^3+6 b c f^2 (b e-2 a f)-24 c^2 f \left (b e^2-b d f-a e f\right )+16 c^3 \left (e^3-2 d e f\right )\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} f^4}+\frac {\left (2 c f \left (B d (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-A f \left (2 c d f (b e-a f)-f^2 \left (b^2 d-a^2 f\right )-c^2 d \left (e^2-d f\right )\right )\right )+c \left (e-\sqrt {e^2-4 d f}\right ) \left (A f (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+B \left (c^2 \left (e^4-3 d e^2 f+d^2 f^2\right )-f^2 \left (2 a b e f-a^2 f^2-b^2 \left (e^2-d f\right )\right )+2 c f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right )\right )\right )\right ) \tanh ^{-1}\left (\frac {4 a f-b \left (e-\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-2 c d f-b e f+2 a f^2-(c e-b f) \sqrt {e^2-4 d f}} \sqrt {a+b x+c x^2}}\right )}{\sqrt {2} c f^4 \sqrt {e^2-4 d f} \sqrt {c e^2-2 c d f-b e f+2 a f^2-(c e-b f) \sqrt {e^2-4 d f}}}+\frac {\left (A f \left (c^2 \left (e^4-4 d e^2 f+2 d^2 f^2+e^3 \sqrt {e^2-4 d f}-2 d e f \sqrt {e^2-4 d f}\right )+f^2 \left (2 a^2 f^2-2 a b f \left (e+\sqrt {e^2-4 d f}\right )+b^2 \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )\right )+2 c f \left (a f \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )-b \left (e^3-3 d e f+e^2 \sqrt {e^2-4 d f}-d f \sqrt {e^2-4 d f}\right )\right )\right )-B \left (c^2 \left (e^5-5 d e^3 f+5 d^2 e f^2+e^4 \sqrt {e^2-4 d f}-3 d e^2 f \sqrt {e^2-4 d f}+d^2 f^2 \sqrt {e^2-4 d f}\right )+f^2 \left (a^2 f^2 \left (e+\sqrt {e^2-4 d f}\right )-2 a b f \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )+b^2 \left (e^3-3 d e f+e^2 \sqrt {e^2-4 d f}-d f \sqrt {e^2-4 d f}\right )\right )+2 c f \left (a f \left (e^3-3 d e f+e^2 \sqrt {e^2-4 d f}-d f \sqrt {e^2-4 d f}\right )-b \left (e^4-4 d e^2 f+2 d^2 f^2+e^3 \sqrt {e^2-4 d f}-2 d e f \sqrt {e^2-4 d f}\right )\right )\right )\right ) \tanh ^{-1}\left (\frac {4 a f-b \left (e+\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e+\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-2 c d f-b e f+2 a f^2+(c e-b f) \sqrt {e^2-4 d f}} \sqrt {a+b x+c x^2}}\right )}{\sqrt {2} f^4 \sqrt {e^2-4 d f} \sqrt {c e^2-2 c d f-b e f+2 a f^2+(c e-b f) \sqrt {e^2-4 d f}}}\\ \end {align*}

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Mathematica [A]  time = 6.56, size = 1627, normalized size = 1.49 \[ \text {result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((B - (B*e - 2*A*f)/Sqrt[e^2 - 4*d*f])*(a + x*(b + c*x))^(3/2))/(6*f) + ((B + (B*e - 2*A*f)/Sqrt[e^2 - 4*d*f])
*(a + x*(b + c*x))^(3/2))/(6*f) - ((B + (-(B*e) + 2*A*f)/Sqrt[e^2 - 4*d*f])*(a + x*(b + c*x))^(3/2)*(((4*c*f*(
-4*a*f + b*(e - Sqrt[e^2 - 4*d*f])) + 2*(b*f - c*(e - Sqrt[e^2 - 4*d*f]))*(-(b*f) + 2*c*(e - Sqrt[e^2 - 4*d*f]
)) - 4*c*f*(b*f - c*(e - Sqrt[e^2 - 4*d*f]))*x)*Sqrt[a + b*x + c*x^2])/(8*c*f^2) - ((-2*(b*f - c*(e - Sqrt[e^2
- 4*d*f]))*(b^2*f^2 - 4*c^2*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]) - 4*c*f*(3*a*f - b*(e - Sqrt[e^2 - 4*d*f])))*
ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(Sqrt[c]*f) - (2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f
+ 2*a*f^2 - c*e*Sqrt[e^2 - 4*d*f] + b*f*Sqrt[e^2 - 4*d*f]]*(4*(e - Sqrt[e^2 - 4*d*f])*(b*f - c*(e - Sqrt[e^2 -
4*d*f]))*(b^2*f^2 - 4*c^2*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]) - 4*c*f*(3*a*f - b*(e - Sqrt[e^2 - 4*d*f]))) +
4*f*(2*c*f*(4*a*f - b*(e - Sqrt[e^2 - 4*d*f]))^2 - (e - Sqrt[e^2 - 4*d*f])*(b*f - c*(e - Sqrt[e^2 - 4*d*f]))*(
b^2*f + 4*a*c*f - 2*b*c*(e - Sqrt[e^2 - 4*d*f]))))*ArcTanh[(4*a*f - b*(e - Sqrt[e^2 - 4*d*f]) - (-2*b*f + 2*c*
(e - Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - c*e*Sqrt[e^2 - 4*d*f] + b*f*Sq
rt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(f*(16*a*f^2 - 8*b*f*(e - Sqrt[e^2 - 4*d*f]) + 4*c*(e - Sqrt[e^2 - 4
*d*f])^2)))/(16*c*f^2)))/(4*f*(a + b*x + c*x^2)^(3/2)) - ((B - (-(B*e) + 2*A*f)/Sqrt[e^2 - 4*d*f])*(a + x*(b +
c*x))^(3/2)*(((4*c*f*(-4*a*f + b*(e + Sqrt[e^2 - 4*d*f])) + 2*(b*f - c*(e + Sqrt[e^2 - 4*d*f]))*(-(b*f) + 2*c
*(e + Sqrt[e^2 - 4*d*f])) - 4*c*f*(b*f - c*(e + Sqrt[e^2 - 4*d*f]))*x)*Sqrt[a + b*x + c*x^2])/(8*c*f^2) - ((-2
*(b*f - c*(e + Sqrt[e^2 - 4*d*f]))*(b^2*f^2 - 4*c^2*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]) - 4*c*f*(3*a*f - b*(e
+ Sqrt[e^2 - 4*d*f])))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(Sqrt[c]*f) - (2*Sqrt[2]*Sqrt[c
*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + c*e*Sqrt[e^2 - 4*d*f] - b*f*Sqrt[e^2 - 4*d*f]]*(4*(e + Sqrt[e^2 - 4*d*f])*(
b*f - c*(e + Sqrt[e^2 - 4*d*f]))*(b^2*f^2 - 4*c^2*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]) - 4*c*f*(3*a*f - b*(e +
Sqrt[e^2 - 4*d*f]))) + 4*f*(2*c*f*(4*a*f - b*(e + Sqrt[e^2 - 4*d*f]))^2 - (e + Sqrt[e^2 - 4*d*f])*(b*f - c*(e
+ Sqrt[e^2 - 4*d*f]))*(b^2*f + 4*a*c*f - 2*b*c*(e + Sqrt[e^2 - 4*d*f]))))*ArcTanh[(4*a*f - b*(e + Sqrt[e^2 - 4
*d*f]) - (-2*b*f + 2*c*(e + Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + c*e*Sqr
t[e^2 - 4*d*f] - b*f*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(f*(16*a*f^2 - 8*b*f*(e + Sqrt[e^2 - 4*d*f])
+ 4*c*(e + Sqrt[e^2 - 4*d*f])^2)))/(16*c*f^2)))/(4*f*(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((B*x+A)*(c*x^2+b*x+a)^(3/2)/(f*x^2+e*x+d),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((B*x+A)*(c*x^2+b*x+a)^(3/2)/(f*x^2+e*x+d),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Erro
r: Bad Argument Type

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maple [B]  time = 0.03, size = 59465, normalized size = 54.46 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x+a)^(3/2)/(f*x^2+e*x+d),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((B*x+A)*(c*x^2+b*x+a)^(3/2)/(f*x^2+e*x+d),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(4*d*f-e^2>0)', see `assume?` f
or more details)Is 4*d*f-e^2 positive, negative or zero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (A+B\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{f\,x^2+e\,x+d} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [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((B*x+A)*(c*x**2+b*x+a)**(3/2)/(f*x**2+e*x+d),x)

[Out]

Timed out

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