3.2.8 \(\int \frac {(a+b x+c x^2)^{3/2}}{(d+e x+f x^2)^3} \, dx\) [108]

Optimal. Leaf size=671 \[ -\frac {(e+2 f x) \left (a+b x+c x^2\right )^{3/2}}{2 \left (e^2-4 d f\right ) \left (d+e x+f x^2\right )^2}+\frac {3 \left (4 c d e+4 a e f-b \left (e^2+4 d f\right )+2 \left (c e^2-2 b e f+4 a f^2\right ) x\right ) \sqrt {a+b x+c x^2}}{4 \left (e^2-4 d f\right )^2 \left (d+e x+f x^2\right )}-\frac {3 \left (2 (2 c d-b e+2 a f) (c e-b f) \left (e-\sqrt {e^2-4 d f}\right )-f \left (4 b e (c d+3 a f)-b^2 \left (e^2+4 d f\right )-4 a \left (c e^2+4 a f^2\right )\right )\right ) \tanh ^{-1}\left (\frac {4 a f-b \left (e-\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-2 c d f-b e f+2 a f^2-(c e-b f) \sqrt {e^2-4 d f}} \sqrt {a+b x+c x^2}}\right )}{4 \sqrt {2} \left (e^2-4 d f\right )^{5/2} \sqrt {c e^2-2 c d f-b e f+2 a f^2-(c e-b f) \sqrt {e^2-4 d f}}}+\frac {3 \left (2 (2 c d-b e+2 a f) (c e-b f) \left (e+\sqrt {e^2-4 d f}\right )-f \left (4 b e (c d+3 a f)-b^2 \left (e^2+4 d f\right )-4 a \left (c e^2+4 a f^2\right )\right )\right ) \tanh ^{-1}\left (\frac {4 a f-b \left (e+\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e+\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-2 c d f-b e f+2 a f^2+(c e-b f) \sqrt {e^2-4 d f}} \sqrt {a+b x+c x^2}}\right )}{4 \sqrt {2} \left (e^2-4 d f\right )^{5/2} \sqrt {c e^2-2 c d f-b e f+2 a f^2+(c e-b f) \sqrt {e^2-4 d f}}} \]

[Out]

-1/2*(2*f*x+e)*(c*x^2+b*x+a)^(3/2)/(-4*d*f+e^2)/(f*x^2+e*x+d)^2+3/4*(4*c*d*e+4*a*e*f-b*(4*d*f+e^2)+2*(4*a*f^2-
2*b*e*f+c*e^2)*x)*(c*x^2+b*x+a)^(1/2)/(-4*d*f+e^2)^2/(f*x^2+e*x+d)-3/8*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))*(-f*(4*b*e*(3*a*f+c*d)-b^2*(4*d*f+e^2)-4*a*(4*a*f^2+c*e^2))+2*(2*a*f-b*e+2*c*d)*(-b*
f+c*e)*(e-(-4*d*f+e^2)^(1/2)))/(-4*d*f+e^2)^(5/2)*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)+3/8*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))*(-f*(4*b*e*(3*a*f+c*d)-b^2*
(4*d*f+e^2)-4*a*(4*a*f^2+c*e^2))+2*(2*a*f-b*e+2*c*d)*(-b*f+c*e)*(e+(-4*d*f+e^2)^(1/2)))/(-4*d*f+e^2)^(5/2)*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)

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Rubi [A]
time = 11.09, antiderivative size = 669, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {985, 1027, 1046, 738, 212} \begin {gather*} \frac {3 \left (-2 \left (e-\sqrt {e^2-4 d f}\right ) (c e-b f) (2 a f-b e+2 c d)+4 b e f (3 a f+c d)-4 a f \left (4 a f^2+c e^2\right )+b^2 (-f) \left (4 d f+e^2\right )\right ) \tanh ^{-1}\left (\frac {4 a f+2 x \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right )-b \left (e-\sqrt {e^2-4 d f}\right )}{2 \sqrt {2} \sqrt {a+b x+c x^2} \sqrt {2 a f^2-\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{4 \sqrt {2} \left (e^2-4 d f\right )^{5/2} \sqrt {2 a f^2-\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}-\frac {3 \left (-2 \left (\sqrt {e^2-4 d f}+e\right ) (c e-b f) (2 a f-b e+2 c d)+4 b e f (3 a f+c d)-4 a f \left (4 a f^2+c e^2\right )+b^2 (-f) \left (4 d f+e^2\right )\right ) \tanh ^{-1}\left (\frac {4 a f+2 x \left (b f-c \left (\sqrt {e^2-4 d f}+e\right )\right )-b \left (\sqrt {e^2-4 d f}+e\right )}{2 \sqrt {2} \sqrt {a+b x+c x^2} \sqrt {2 a f^2+\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{4 \sqrt {2} \left (e^2-4 d f\right )^{5/2} \sqrt {2 a f^2+\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}+\frac {3 \sqrt {a+b x+c x^2} \left (2 x \left (4 a f^2-2 b e f+c e^2\right )+4 a e f-b \left (4 d f+e^2\right )+4 c d e\right )}{4 \left (e^2-4 d f\right )^2 \left (d+e x+f x^2\right )}-\frac {(e+2 f x) \left (a+b x+c x^2\right )^{3/2}}{2 \left (e^2-4 d f\right ) \left (d+e x+f x^2\right )^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/2*((e + 2*f*x)*(a + b*x + c*x^2)^(3/2))/((e^2 - 4*d*f)*(d + e*x + f*x^2)^2) + (3*(4*c*d*e + 4*a*e*f - b*(e^
2 + 4*d*f) + 2*(c*e^2 - 2*b*e*f + 4*a*f^2)*x)*Sqrt[a + b*x + c*x^2])/(4*(e^2 - 4*d*f)^2*(d + e*x + f*x^2)) + (
3*(4*b*e*f*(c*d + 3*a*f) - b^2*f*(e^2 + 4*d*f) - 4*a*f*(c*e^2 + 4*a*f^2) - 2*(2*c*d - b*e + 2*a*f)*(c*e - b*f)
*(e - Sqrt[e^2 - 4*d*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])])/(4
*Sqrt[2]*(e^2 - 4*d*f)^(5/2)*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]]) - (3*(4*
b*e*f*(c*d + 3*a*f) - b^2*f*(e^2 + 4*d*f) - 4*a*f*(c*e^2 + 4*a*f^2) - 2*(2*c*d - b*e + 2*a*f)*(c*e - b*f)*(e +
Sqrt[e^2 - 4*d*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*Sq
rt[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])])/(4*Sqrt
[2]*(e^2 - 4*d*f)^(5/2)*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]])

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 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 985

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

Rule 1027

Int[((g_.) + (h_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Sy
mbol] :> Simp[(g*b - 2*a*h - (b*h - 2*g*c)*x)*(a + b*x + c*x^2)^(p + 1)*((d + e*x + f*x^2)^q/((b^2 - 4*a*c)*(p
+ 1))), x] - Dist[1/((b^2 - 4*a*c)*(p + 1)), Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q - 1)*Simp[e*q
*(g*b - 2*a*h) - d*(b*h - 2*g*c)*(2*p + 3) + (2*f*q*(g*b - 2*a*h) - e*(b*h - 2*g*c)*(2*p + q + 3))*x - f*(b*h
- 2*g*c)*(2*p + 2*q + 3)*x^2, x], 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] && LtQ[p, -1] && GtQ[q, 0]

Rule 1046

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]

Rubi steps

\begin {align*} \int \frac {\left (a+b x+c x^2\right )^{3/2}}{\left (d+e x+f x^2\right )^3} \, dx &=-\frac {(e+2 f x) \left (a+b x+c x^2\right )^{3/2}}{2 \left (e^2-4 d f\right ) \left (d+e x+f x^2\right )^2}+\frac {\int \frac {\left (\frac {3}{2} (b e-4 a f)+3 (c e-b f) x\right ) \sqrt {a+b x+c x^2}}{\left (d+e x+f x^2\right )^2} \, dx}{2 \left (e^2-4 d f\right )}\\ &=-\frac {(e+2 f x) \left (a+b x+c x^2\right )^{3/2}}{2 \left (e^2-4 d f\right ) \left (d+e x+f x^2\right )^2}+\frac {3 \left (4 c d e+4 a e f-b \left (e^2+4 d f\right )+2 \left (c e^2-2 b e f+4 a f^2\right ) x\right ) \sqrt {a+b x+c x^2}}{4 \left (e^2-4 d f\right )^2 \left (d+e x+f x^2\right )}+\frac {\int \frac {-\frac {3}{4} \left (4 b e (c d+3 a f)-b^2 \left (e^2+4 d f\right )-4 a \left (c e^2+4 a f^2\right )\right )-3 (2 c d-b e+2 a f) (c e-b f) x}{\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )} \, dx}{2 \left (e^2-4 d f\right )^2}\\ &=-\frac {(e+2 f x) \left (a+b x+c x^2\right )^{3/2}}{2 \left (e^2-4 d f\right ) \left (d+e x+f x^2\right )^2}+\frac {3 \left (4 c d e+4 a e f-b \left (e^2+4 d f\right )+2 \left (c e^2-2 b e f+4 a f^2\right ) x\right ) \sqrt {a+b x+c x^2}}{4 \left (e^2-4 d f\right )^2 \left (d+e x+f x^2\right )}-\frac {\left (3 \left (4 b e f (c d+3 a f)-b^2 f \left (e^2+4 d f\right )-4 a f \left (c e^2+4 a f^2\right )-2 (2 c d-b e+2 a f) (c e-b f) \left (e-\sqrt {e^2-4 d f}\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}{4 \left (e^2-4 d f\right )^{5/2}}+\frac {\left (3 \left (4 b e f (c d+3 a f)-b^2 f \left (e^2+4 d f\right )-4 a f \left (c e^2+4 a f^2\right )-2 (2 c d-b e+2 a f) (c e-b f) \left (e+\sqrt {e^2-4 d f}\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}{4 \left (e^2-4 d f\right )^{5/2}}\\ &=-\frac {(e+2 f x) \left (a+b x+c x^2\right )^{3/2}}{2 \left (e^2-4 d f\right ) \left (d+e x+f x^2\right )^2}+\frac {3 \left (4 c d e+4 a e f-b \left (e^2+4 d f\right )+2 \left (c e^2-2 b e f+4 a f^2\right ) x\right ) \sqrt {a+b x+c x^2}}{4 \left (e^2-4 d f\right )^2 \left (d+e x+f x^2\right )}+\frac {\left (3 \left (4 b e f (c d+3 a f)-b^2 f \left (e^2+4 d f\right )-4 a f \left (c e^2+4 a f^2\right )-2 (2 c d-b e+2 a f) (c e-b f) \left (e-\sqrt {e^2-4 d f}\right )\right )\right ) \text {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 )}{2 \left (e^2-4 d f\right )^{5/2}}-\frac {\left (3 \left (4 b e f (c d+3 a f)-b^2 f \left (e^2+4 d f\right )-4 a f \left (c e^2+4 a f^2\right )-2 (2 c d-b e+2 a f) (c e-b f) \left (e+\sqrt {e^2-4 d f}\right )\right )\right ) \text {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 )}{2 \left (e^2-4 d f\right )^{5/2}}\\ &=-\frac {(e+2 f x) \left (a+b x+c x^2\right )^{3/2}}{2 \left (e^2-4 d f\right ) \left (d+e x+f x^2\right )^2}+\frac {3 \left (4 c d e+4 a e f-b \left (e^2+4 d f\right )+2 \left (c e^2-2 b e f+4 a f^2\right ) x\right ) \sqrt {a+b x+c x^2}}{4 \left (e^2-4 d f\right )^2 \left (d+e x+f x^2\right )}+\frac {3 \left (4 b e f (c d+3 a f)-b^2 f \left (e^2+4 d f\right )-4 a f \left (c e^2+4 a f^2\right )-2 (2 c d-b e+2 a f) (c e-b f) \left (e-\sqrt {e^2-4 d f}\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 )}{4 \sqrt {2} \left (e^2-4 d f\right )^{5/2} \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}}}-\frac {3 \left (4 b e f (c d+3 a f)-b^2 f \left (e^2+4 d f\right )-4 a f \left (c e^2+4 a f^2\right )-2 (2 c d-b e+2 a f) (c e-b f) \left (e+\sqrt {e^2-4 d f}\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 )}{4 \sqrt {2} \left (e^2-4 d f\right )^{5/2} \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}}}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1621\) vs. \(2(671)=1342\).
time = 16.30, size = 1621, normalized size = 2.42 \begin {gather*} \frac {(a+x (b+c x))^{3/2} \left (\frac {c d e-2 b d f+a e f+c e^2 x-2 c d f x-b e f x+2 a f^2 x}{2 f \left (-e^2+4 d f\right ) \left (d+e x+f x^2\right )^2}+\frac {2 c e^3+4 c d e f-7 b e^2 f+4 b d f^2+12 a e f^2+2 c e^2 f x+16 c d f^2 x-12 b e f^2 x+24 a f^3 x}{4 f \left (-e^2+4 d f\right )^2 \left (d+e x+f x^2\right )}\right )}{a+b x+c x^2}+\frac {3 \left (4 c^2 d e^2-2 b c e^3-8 b c d e f+3 b^2 e^2 f+8 a c e^2 f+4 b^2 d f^2-16 a b e f^2+16 a^2 f^3-4 c^2 d e \sqrt {e^2-4 d f}+2 b c e^2 \sqrt {e^2-4 d f}+4 b c d f \sqrt {e^2-4 d f}-2 b^2 e f \sqrt {e^2-4 d f}-4 a c e f \sqrt {e^2-4 d f}+4 a b f^2 \sqrt {e^2-4 d f}\right ) (a+x (b+c x))^{3/2} \log \left (-e+\sqrt {e^2-4 d f}-2 f x\right )}{4 \sqrt {2} \left (e^2-4 d f\right )^{5/2} \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}} \left (a+b x+c x^2\right )^{3/2}}+\frac {3 \left (-4 c^2 d e^2+2 b c e^3+8 b c d e f-3 b^2 e^2 f-8 a c e^2 f-4 b^2 d f^2+16 a b e f^2-16 a^2 f^3-4 c^2 d e \sqrt {e^2-4 d f}+2 b c e^2 \sqrt {e^2-4 d f}+4 b c d f \sqrt {e^2-4 d f}-2 b^2 e f \sqrt {e^2-4 d f}-4 a c e f \sqrt {e^2-4 d f}+4 a b f^2 \sqrt {e^2-4 d f}\right ) (a+x (b+c x))^{3/2} \log \left (e+\sqrt {e^2-4 d f}+2 f x\right )}{4 \sqrt {2} \left (e^2-4 d f\right )^{5/2} \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}} \left (a+b x+c x^2\right )^{3/2}}-\frac {3 \left (-4 c^2 d e^2+2 b c e^3+8 b c d e f-3 b^2 e^2 f-8 a c e^2 f-4 b^2 d f^2+16 a b e f^2-16 a^2 f^3-4 c^2 d e \sqrt {e^2-4 d f}+2 b c e^2 \sqrt {e^2-4 d f}+4 b c d f \sqrt {e^2-4 d f}-2 b^2 e f \sqrt {e^2-4 d f}-4 a c e f \sqrt {e^2-4 d f}+4 a b f^2 \sqrt {e^2-4 d f}\right ) (a+x (b+c x))^{3/2} \log \left (b e-4 a f+b \sqrt {e^2-4 d f}+2 c e x-2 b f x+2 c \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 \sqrt {e^2-4 d f}} \sqrt {a+b x+c x^2}\right )}{4 \sqrt {2} \left (e^2-4 d f\right )^{5/2} \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}} \left (a+b x+c x^2\right )^{3/2}}-\frac {3 \left (4 c^2 d e^2-2 b c e^3-8 b c d e f+3 b^2 e^2 f+8 a c e^2 f+4 b^2 d f^2-16 a b e f^2+16 a^2 f^3-4 c^2 d e \sqrt {e^2-4 d f}+2 b c e^2 \sqrt {e^2-4 d f}+4 b c d f \sqrt {e^2-4 d f}-2 b^2 e f \sqrt {e^2-4 d f}-4 a c e f \sqrt {e^2-4 d f}+4 a b f^2 \sqrt {e^2-4 d f}\right ) (a+x (b+c x))^{3/2} \log \left (-b e+4 a f+b \sqrt {e^2-4 d f}-2 c e x+2 b f x+2 c \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 \sqrt {e^2-4 d f}} \sqrt {a+b x+c x^2}\right )}{4 \sqrt {2} \left (e^2-4 d f\right )^{5/2} \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}} \left (a+b x+c x^2\right )^{3/2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((a + x*(b + c*x))^(3/2)*((c*d*e - 2*b*d*f + a*e*f + c*e^2*x - 2*c*d*f*x - b*e*f*x + 2*a*f^2*x)/(2*f*(-e^2 + 4
*d*f)*(d + e*x + f*x^2)^2) + (2*c*e^3 + 4*c*d*e*f - 7*b*e^2*f + 4*b*d*f^2 + 12*a*e*f^2 + 2*c*e^2*f*x + 16*c*d*
f^2*x - 12*b*e*f^2*x + 24*a*f^3*x)/(4*f*(-e^2 + 4*d*f)^2*(d + e*x + f*x^2))))/(a + b*x + c*x^2) + (3*(4*c^2*d*
e^2 - 2*b*c*e^3 - 8*b*c*d*e*f + 3*b^2*e^2*f + 8*a*c*e^2*f + 4*b^2*d*f^2 - 16*a*b*e*f^2 + 16*a^2*f^3 - 4*c^2*d*
e*Sqrt[e^2 - 4*d*f] + 2*b*c*e^2*Sqrt[e^2 - 4*d*f] + 4*b*c*d*f*Sqrt[e^2 - 4*d*f] - 2*b^2*e*f*Sqrt[e^2 - 4*d*f]
- 4*a*c*e*f*Sqrt[e^2 - 4*d*f] + 4*a*b*f^2*Sqrt[e^2 - 4*d*f])*(a + x*(b + c*x))^(3/2)*Log[-e + Sqrt[e^2 - 4*d*f
] - 2*f*x])/(4*Sqrt[2]*(e^2 - 4*d*f)^(5/2)*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]]*(a + b*x + c*x^2)^(3/2)) + (3*(-4*c^2*d*e^2 + 2*b*c*e^3 + 8*b*c*d*e*f - 3*b^2*e^2*f - 8*a
*c*e^2*f - 4*b^2*d*f^2 + 16*a*b*e*f^2 - 16*a^2*f^3 - 4*c^2*d*e*Sqrt[e^2 - 4*d*f] + 2*b*c*e^2*Sqrt[e^2 - 4*d*f]
+ 4*b*c*d*f*Sqrt[e^2 - 4*d*f] - 2*b^2*e*f*Sqrt[e^2 - 4*d*f] - 4*a*c*e*f*Sqrt[e^2 - 4*d*f] + 4*a*b*f^2*Sqrt[e^
2 - 4*d*f])*(a + x*(b + c*x))^(3/2)*Log[e + Sqrt[e^2 - 4*d*f] + 2*f*x])/(4*Sqrt[2]*(e^2 - 4*d*f)^(5/2)*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]]*(a + b*x + c*x^2)^(3/2)) - (3
*(-4*c^2*d*e^2 + 2*b*c*e^3 + 8*b*c*d*e*f - 3*b^2*e^2*f - 8*a*c*e^2*f - 4*b^2*d*f^2 + 16*a*b*e*f^2 - 16*a^2*f^3
- 4*c^2*d*e*Sqrt[e^2 - 4*d*f] + 2*b*c*e^2*Sqrt[e^2 - 4*d*f] + 4*b*c*d*f*Sqrt[e^2 - 4*d*f] - 2*b^2*e*f*Sqrt[e^
2 - 4*d*f] - 4*a*c*e*f*Sqrt[e^2 - 4*d*f] + 4*a*b*f^2*Sqrt[e^2 - 4*d*f])*(a + x*(b + c*x))^(3/2)*Log[b*e - 4*a*
f + b*Sqrt[e^2 - 4*d*f] + 2*c*e*x - 2*b*f*x + 2*c*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*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2]])/(4*Sqrt[2]*(e^2 - 4*d*f)^(
5/2)*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]]*(a + b*x + c*x^2)
^(3/2)) - (3*(4*c^2*d*e^2 - 2*b*c*e^3 - 8*b*c*d*e*f + 3*b^2*e^2*f + 8*a*c*e^2*f + 4*b^2*d*f^2 - 16*a*b*e*f^2 +
16*a^2*f^3 - 4*c^2*d*e*Sqrt[e^2 - 4*d*f] + 2*b*c*e^2*Sqrt[e^2 - 4*d*f] + 4*b*c*d*f*Sqrt[e^2 - 4*d*f] - 2*b^2*
e*f*Sqrt[e^2 - 4*d*f] - 4*a*c*e*f*Sqrt[e^2 - 4*d*f] + 4*a*b*f^2*Sqrt[e^2 - 4*d*f])*(a + x*(b + c*x))^(3/2)*Log
[-(b*e) + 4*a*f + b*Sqrt[e^2 - 4*d*f] - 2*c*e*x + 2*b*f*x + 2*c*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*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2]])/(4*Sqrt[2]*(
e^2 - 4*d*f)^(5/2)*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]]*(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. \(16308\) vs. \(2(611)=1222\).
time = 0.16, size = 16309, normalized size = 24.31

method	result	size
default	\(\text {Expression too large to display}\)	\(16309\)

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)^3,x,method=_RETURNVERBOSE)

[Out]

result too large to display

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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)^3,x, algorithm="maxima")

[Out]

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

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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)^3,x, algorithm="fricas")

[Out]

Timed out

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Sympy [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)**3,x)

[Out]

Timed out

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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)^3,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 )}^3} \,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)^3,x)

[Out]

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

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