3.2.25 \(\int \frac {1}{x (a+b x+c x^2)^{3/2} (d+e x+f x^2)} \, dx\)

Optimal. Leaf size=816 \[ \frac {2 \left (b^2+c x b-2 a c\right )}{a \left (b^2-4 a c\right ) d \sqrt {c x^2+b x+a}}-\frac {\tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {c x^2+b x+a}}\right )}{a^{3/2} d}+\frac {f \left (\left (e-\sqrt {e^2-4 d f}\right ) \left (f (b e-a f)-c \left (e^2-d f\right )\right )-2 \left (f \left (b e^2-a f e-b d f\right )-c \left (e^3-2 d e f\right )\right )\right ) \tanh ^{-1}\left (\frac {4 a f-b \left (e-\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-b f e+2 a f^2-2 c d f-(c e-b f) \sqrt {e^2-4 d f}} \sqrt {c x^2+b x+a}}\right )}{\sqrt {2} d \sqrt {e^2-4 d f} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \sqrt {c e^2-b f e+2 a f^2-2 c d f-(c e-b f) \sqrt {e^2-4 d f}}}-\frac {f \left (\left (e+\sqrt {e^2-4 d f}\right ) \left (f (b e-a f)-c \left (e^2-d f\right )\right )-2 \left (f \left (b e^2-a f e-b d f\right )-c \left (e^3-2 d e f\right )\right )\right ) \tanh ^{-1}\left (\frac {4 a f-b \left (e+\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e+\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-b f e+2 a f^2-2 c d f+(c e-b f) \sqrt {e^2-4 d f}} \sqrt {c x^2+b x+a}}\right )}{\sqrt {2} d \sqrt {e^2-4 d f} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \sqrt {c e^2-b f e+2 a f^2-2 c d f+(c e-b f) \sqrt {e^2-4 d f}}}+\frac {2 \left (c e (2 a c e-b (c d+a f))+(b e-a f) \left (f b^2+2 c^2 d-c (b e+2 a f)\right )+c \left (2 d e c^2-b \left (e^2+d f\right ) c+b f (b e-a f)\right ) x\right )}{\left (b^2-4 a c\right ) d \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \sqrt {c x^2+b x+a}} \]

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Rubi [A]  time = 15.92, antiderivative size = 814, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {6728, 740, 12, 724, 206, 1016, 1032} \begin {gather*} \frac {2 \left (b^2+c x b-2 a c\right )}{a \left (b^2-4 a c\right ) d \sqrt {c x^2+b x+a}}-\frac {\tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {c x^2+b x+a}}\right )}{a^{3/2} d}-\frac {f \left (2 f \left (b e^2-a f e-b d f\right )-2 c \left (e^3-2 d e f\right )-\left (e-\sqrt {e^2-4 d f}\right ) \left (f (b e-a f)-c \left (e^2-d f\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} d \sqrt {e^2-4 d f} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \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 {f \left (2 f \left (b e^2-a f e-b d f\right )-2 c \left (e^3-2 d e f\right )-\left (e+\sqrt {e^2-4 d f}\right ) \left (f (b e-a f)-c \left (e^2-d f\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} d \sqrt {e^2-4 d f} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \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 {2 \left (c e (2 a c e-b (c d+a f))+(b e-a f) \left (f b^2+2 c^2 d-c (b e+2 a f)\right )+c \left (2 d e c^2-b \left (e^2+d f\right ) c+b f (b e-a f)\right ) x\right )}{\left (b^2-4 a c\right ) d \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \sqrt {c x^2+b x+a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*(a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2)),x]

[Out]

(2*(b^2 - 2*a*c + b*c*x))/(a*(b^2 - 4*a*c)*d*Sqrt[a + b*x + c*x^2]) + (2*(c*e*(2*a*c*e - b*(c*d + a*f)) + (b*e
- a*f)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) + c*(2*c^2*d*e + b*f*(b*e - a*f) - b*c*(e^2 + d*f))*x))/((b^2 - 4*
a*c)*d*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*Sqrt[a + b*x + c*x^2]) - ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[
a + b*x + c*x^2])]/(a^(3/2)*d) - (f*(2*f*(b*e^2 - b*d*f - a*e*f) - 2*c*(e^3 - 2*d*e*f) - (e - Sqrt[e^2 - 4*d*f
])*(f*(b*e - a*f) - c*(e^2 - 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])])/(Sqrt[2]*d*Sqrt[e^2 - 4*d*f]*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*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]]) + (f*(2*f*(b*e^2 - b*d*f - a*e*f) - 2*c*(e^3 - 2*d*e*f) - (e + Sqrt[
e^2 - 4*d*f])*(f*(b*e - a*f) - c*(e^2 - d*f)))*ArcTanh[(4*a*f - b*(e + Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e + Sq
rt[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]*d*Sqrt[e^2 - 4*d*f]*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 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 740

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e
+ a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m
+ 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x
, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, 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] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 1016

Int[((g_.) + (h_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Sy
mbol] :> Simp[((a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q + 1)*(g*c*(2*a*c*e - b*(c*d + a*f)) + (g*b - a*h
)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) + c*(g*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) - h*(b*c*d - 2*a*c*e + a*b*f)
)*x))/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)), x] + Dist[1/((b^2 - 4*a*c)*((c*d - a*
f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)), Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q*Simp[(b*h - 2*g*c)
*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1) + (b^2*(g*f) - b*(h*c*d + g*c*e + a*h*f) + 2*(g*c*(c*d - a*
f) - a*(-(h*c*e))))*(a*f*(p + 1) - c*d*(p + 2)) - e*((g*c)*(2*a*c*e - b*(c*d + a*f)) + (g*b - a*h)*(2*c^2*d +
b^2*f - c*(b*e + 2*a*f)))*(p + q + 2) - (2*f*((g*c)*(2*a*c*e - b*(c*d + a*f)) + (g*b - a*h)*(2*c^2*d + b^2*f -
c*(b*e + 2*a*f)))*(p + q + 2) - (b^2*g*f - b*(h*c*d + g*c*e + a*h*f) + 2*(g*c*(c*d - a*f) - a*(-(h*c*e))))*(b
*f*(p + 1) - c*e*(2*p + q + 4)))*x - c*f*(b^2*(g*f) - b*(h*c*d + g*c*e + a*h*f) + 2*(g*c*(c*d - a*f) + a*h*c*e
))*(2*p + 2*q + 5)*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] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 0] &&  !( !IntegerQ[p] && ILtQ[q, -1
])

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 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {1}{x \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx &=\int \left (\frac {1}{d x \left (a+b x+c x^2\right )^{3/2}}+\frac {-e-f x}{d \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {1}{x \left (a+b x+c x^2\right )^{3/2}} \, dx}{d}+\frac {\int \frac {-e-f x}{\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx}{d}\\ &=\frac {2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) d \sqrt {a+b x+c x^2}}+\frac {2 \left (c e (2 a c e-b (c d+a f))+(b e-a f) \left (2 c^2 d+b^2 f-c (b e+2 a f)\right )+c \left (2 c^2 d e+b f (b e-a f)-b c \left (e^2+d f\right )\right ) x\right )}{\left (b^2-4 a c\right ) d \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \sqrt {a+b x+c x^2}}-\frac {2 \int \frac {-\frac {b^2}{2}+2 a c}{x \sqrt {a+b x+c x^2}} \, dx}{a \left (b^2-4 a c\right ) d}-\frac {2 \int \frac {-\frac {1}{2} \left (b^2-4 a c\right ) \left (f \left (b e^2-b d f-a e f\right )-c \left (e^3-2 d e f\right )\right )+\frac {1}{2} \left (b^2-4 a c\right ) f \left (c e^2-c d f-b e f+a f^2\right ) x}{\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )} \, dx}{\left (b^2-4 a c\right ) d \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}\\ &=\frac {2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) d \sqrt {a+b x+c x^2}}+\frac {2 \left (c e (2 a c e-b (c d+a f))+(b e-a f) \left (2 c^2 d+b^2 f-c (b e+2 a f)\right )+c \left (2 c^2 d e+b f (b e-a f)-b c \left (e^2+d f\right )\right ) x\right )}{\left (b^2-4 a c\right ) d \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \sqrt {a+b x+c x^2}}+\frac {\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx}{a d}+\frac {\left (f \left (2 f \left (b e^2-b d f-a e f\right )-2 c \left (e^3-2 d e f\right )-\left (e-\sqrt {e^2-4 d f}\right ) \left (f (b e-a f)-c \left (e^2-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}{d \sqrt {e^2-4 d f} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac {\left (f \left (2 f \left (b e^2-b d f-a e f\right )-2 c \left (e^3-2 d e f\right )-\left (e+\sqrt {e^2-4 d f}\right ) \left (f (b e-a f)-c \left (e^2-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}{d \sqrt {e^2-4 d f} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}\\ &=\frac {2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) d \sqrt {a+b x+c x^2}}+\frac {2 \left (c e (2 a c e-b (c d+a f))+(b e-a f) \left (2 c^2 d+b^2 f-c (b e+2 a f)\right )+c \left (2 c^2 d e+b f (b e-a f)-b c \left (e^2+d f\right )\right ) x\right )}{\left (b^2-4 a c\right ) d \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \sqrt {a+b x+c x^2}}-\frac {2 \operatorname {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b x}{\sqrt {a+b x+c x^2}}\right )}{a d}-\frac {\left (2 f \left (2 f \left (b e^2-b d f-a e f\right )-2 c \left (e^3-2 d e f\right )-\left (e-\sqrt {e^2-4 d f}\right ) \left (f (b e-a f)-c \left (e^2-d f\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 )}{d \sqrt {e^2-4 d f} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}+\frac {\left (2 f \left (2 f \left (b e^2-b d f-a e f\right )-2 c \left (e^3-2 d e f\right )-\left (e+\sqrt {e^2-4 d f}\right ) \left (f (b e-a f)-c \left (e^2-d f\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 )}{d \sqrt {e^2-4 d f} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}\\ &=\frac {2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) d \sqrt {a+b x+c x^2}}+\frac {2 \left (c e (2 a c e-b (c d+a f))+(b e-a f) \left (2 c^2 d+b^2 f-c (b e+2 a f)\right )+c \left (2 c^2 d e+b f (b e-a f)-b c \left (e^2+d f\right )\right ) x\right )}{\left (b^2-4 a c\right ) d \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \sqrt {a+b x+c x^2}}-\frac {\tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{a^{3/2} d}-\frac {f \left (2 f \left (b e^2-b d f-a e f\right )-2 c \left (e^3-2 d e f\right )-\left (e-\sqrt {e^2-4 d f}\right ) \left (f (b e-a f)-c \left (e^2-d f\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} d \sqrt {e^2-4 d f} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \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 {f \left (2 f \left (b e^2-b d f-a e f\right )-2 c \left (e^3-2 d e f\right )-\left (e+\sqrt {e^2-4 d f}\right ) \left (f (b e-a f)-c \left (e^2-d f\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} d \sqrt {e^2-4 d f} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \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 = 1121, normalized size = 1.37 \begin {gather*} \frac {16 \sqrt {2} \left (\frac {e f}{\sqrt {e^2-4 d f}}+f\right ) \sqrt {c e^2-b f e-c \sqrt {e^2-4 d f} e+2 a f^2-2 c d f+b f \sqrt {e^2-4 d f}} \left (c x^2+b x+a\right )^{3/2} \tanh ^{-1}\left (\frac {4 a f-b \left (e-\sqrt {e^2-4 d f}\right )-\left (2 c \left (e-\sqrt {e^2-4 d f}\right )-2 b f\right ) x}{2 \sqrt {2} \sqrt {c e^2-b f e-c \sqrt {e^2-4 d f} e+2 a f^2-2 c d f+b f \sqrt {e^2-4 d f}} \sqrt {c x^2+b x+a}}\right ) f^2}{d \left (4 a f^2-2 b \left (e-\sqrt {e^2-4 d f}\right ) f+c \left (e-\sqrt {e^2-4 d f}\right )^2\right ) \left (16 a f^2-8 b \left (e-\sqrt {e^2-4 d f}\right ) f+4 c \left (e-\sqrt {e^2-4 d f}\right )^2\right ) (a+x (b+c x))^{3/2}}-\frac {16 \sqrt {2} \left (\frac {e f}{\sqrt {e^2-4 d f}}-f\right ) \sqrt {c e^2-b f e+c \sqrt {e^2-4 d f} e+2 a f^2-2 c d f-b f \sqrt {e^2-4 d f}} \left (c x^2+b x+a\right )^{3/2} \tanh ^{-1}\left (\frac {4 a f-b \left (e+\sqrt {e^2-4 d f}\right )-\left (2 c \left (e+\sqrt {e^2-4 d f}\right )-2 b f\right ) x}{2 \sqrt {2} \sqrt {c e^2-b f e+c \sqrt {e^2-4 d f} e+2 a f^2-2 c d f-b f \sqrt {e^2-4 d f}} \sqrt {c x^2+b x+a}}\right ) f^2}{d \left (4 a f^2-2 b \left (e+\sqrt {e^2-4 d f}\right ) f+c \left (e+\sqrt {e^2-4 d f}\right )^2\right ) \left (16 a f^2-8 b \left (e+\sqrt {e^2-4 d f}\right ) f+4 c \left (e+\sqrt {e^2-4 d f}\right )^2\right ) (a+x (b+c x))^{3/2}}-\frac {2 \left (\frac {e}{\sqrt {e^2-4 d f}}+1\right ) \left (2 f b^2-c \left (e-\sqrt {e^2-4 d f}\right ) b-4 a c f+2 c \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right ) x\right ) \left (c x^2+b x+a\right ) f}{\left (b^2-4 a c\right ) d \left (4 a f^2-2 b \left (e-\sqrt {e^2-4 d f}\right ) f+c \left (e-\sqrt {e^2-4 d f}\right )^2\right ) (a+x (b+c x))^{3/2}}-\frac {2 \left (1-\frac {e}{\sqrt {e^2-4 d f}}\right ) \left (2 f b^2-c \left (e+\sqrt {e^2-4 d f}\right ) b-4 a c f+2 c \left (b f-c \left (e+\sqrt {e^2-4 d f}\right )\right ) x\right ) \left (c x^2+b x+a\right ) f}{\left (b^2-4 a c\right ) d \left (4 a f^2-2 b \left (e+\sqrt {e^2-4 d f}\right ) f+c \left (e+\sqrt {e^2-4 d f}\right )^2\right ) (a+x (b+c x))^{3/2}}-\frac {\left (c x^2+b x+a\right )^{3/2} \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {c x^2+b x+a}}\right )}{a^{3/2} d (a+x (b+c x))^{3/2}}+\frac {2 \left (b^2+c x b-2 a c\right ) \left (c x^2+b x+a\right )}{a \left (b^2-4 a c\right ) d (a+x (b+c x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2)),x]

[Out]

(2*(b^2 - 2*a*c + b*c*x)*(a + b*x + c*x^2))/(a*(b^2 - 4*a*c)*d*(a + x*(b + c*x))^(3/2)) - (2*f*(1 + e/Sqrt[e^2
- 4*d*f])*(2*b^2*f - 4*a*c*f - b*c*(e - Sqrt[e^2 - 4*d*f]) + 2*c*(b*f - c*(e - Sqrt[e^2 - 4*d*f]))*x)*(a + b*
x + c*x^2))/((b^2 - 4*a*c)*d*(4*a*f^2 - 2*b*f*(e - Sqrt[e^2 - 4*d*f]) + c*(e - Sqrt[e^2 - 4*d*f])^2)*(a + x*(b
+ c*x))^(3/2)) - (2*f*(1 - e/Sqrt[e^2 - 4*d*f])*(2*b^2*f - 4*a*c*f - b*c*(e + Sqrt[e^2 - 4*d*f]) + 2*c*(b*f -
c*(e + Sqrt[e^2 - 4*d*f]))*x)*(a + b*x + c*x^2))/((b^2 - 4*a*c)*d*(4*a*f^2 - 2*b*f*(e + Sqrt[e^2 - 4*d*f]) +
c*(e + Sqrt[e^2 - 4*d*f])^2)*(a + x*(b + c*x))^(3/2)) - ((a + b*x + c*x^2)^(3/2)*ArcTanh[(2*a + b*x)/(2*Sqrt[a
]*Sqrt[a + b*x + c*x^2])])/(a^(3/2)*d*(a + x*(b + c*x))^(3/2)) + (16*Sqrt[2]*f^2*(f + (e*f)/Sqrt[e^2 - 4*d*f])
*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)*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*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(d*(4*
a*f^2 - 2*b*f*(e - Sqrt[e^2 - 4*d*f]) + c*(e - Sqrt[e^2 - 4*d*f])^2)*(16*a*f^2 - 8*b*f*(e - Sqrt[e^2 - 4*d*f])
+ 4*c*(e - Sqrt[e^2 - 4*d*f])^2)*(a + x*(b + c*x))^(3/2)) - (16*Sqrt[2]*f^2*(-f + (e*f)/Sqrt[e^2 - 4*d*f])*Sq
rt[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)*
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*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(d*(4*a*f
^2 - 2*b*f*(e + Sqrt[e^2 - 4*d*f]) + c*(e + Sqrt[e^2 - 4*d*f])^2)*(16*a*f^2 - 8*b*f*(e + Sqrt[e^2 - 4*d*f]) +
4*c*(e + Sqrt[e^2 - 4*d*f])^2)*(a + x*(b + c*x))^(3/2))

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IntegrateAlgebraic [C]  time = 6.98, size = 1054, normalized size = 1.29 \begin {gather*} \frac {2 \left (f b^4-c e b^3+c f x b^3+c^2 d b^2-4 a c f b^2-c^2 e x b^2+3 a c^2 e b+c^3 d x b-3 a c^2 f x b-2 a c^3 d+2 a^2 c^2 f+2 a c^3 e x\right )}{a \left (b^2-4 a c\right ) \left (d f b^2-c d e b-a e f b+c^2 d^2+a c e^2+a^2 f^2-2 a c d f\right ) \sqrt {c x^2+b x+a}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}-\frac {\sqrt {c x^2+b x+a}}{\sqrt {a}}\right )}{a^{3/2} d}-\frac {\text {RootSum}\left [f \text {$\#$1}^4-2 \sqrt {c} e \text {$\#$1}^3+4 c d \text {$\#$1}^2+b e \text {$\#$1}^2-2 a f \text {$\#$1}^2-4 b \sqrt {c} d \text {$\#$1}+2 a \sqrt {c} e \text {$\#$1}+b^2 d-a b e+a^2 f\&,\frac {-b c \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right ) e^3+2 c^{3/2} \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right ) \text {$\#$1} e^3-c f \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right ) \text {$\#$1}^2 e^2+b^2 f \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right ) e^2+a c f \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right ) e^2-2 b \sqrt {c} f \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right ) \text {$\#$1} e^2+b f^2 \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right ) \text {$\#$1}^2 e-2 a b f^2 \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right ) e+2 b c d f \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right ) e+2 a \sqrt {c} f^2 \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right ) \text {$\#$1} e-4 c^{3/2} d f \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right ) \text {$\#$1} e-a f^3 \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right ) \text {$\#$1}^2+c d f^2 \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right ) \text {$\#$1}^2+a^2 f^3 \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right )-b^2 d f^2 \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right )-a c d f^2 \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right )+2 b \sqrt {c} d f^2 \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right ) \text {$\#$1}}{-2 f \text {$\#$1}^3+3 \sqrt {c} e \text {$\#$1}^2-4 c d \text {$\#$1}-b e \text {$\#$1}+2 a f \text {$\#$1}+2 b \sqrt {c} d-a \sqrt {c} e}\&\right ]}{d \left (d f b^2-c d e b-a e f b+c^2 d^2+a c e^2+a^2 f^2-2 a c d f\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x*(a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2)),x]

[Out]

(2*(b^2*c^2*d - 2*a*c^3*d - b^3*c*e + 3*a*b*c^2*e + b^4*f - 4*a*b^2*c*f + 2*a^2*c^2*f + b*c^3*d*x - b^2*c^2*e*
x + 2*a*c^3*e*x + b^3*c*f*x - 3*a*b*c^2*f*x))/(a*(b^2 - 4*a*c)*(c^2*d^2 - b*c*d*e + a*c*e^2 + b^2*d*f - 2*a*c*
d*f - a*b*e*f + a^2*f^2)*Sqrt[a + b*x + c*x^2]) + (2*ArcTanh[(Sqrt[c]*x)/Sqrt[a] - Sqrt[a + b*x + c*x^2]/Sqrt[
a]])/(a^(3/2)*d) - RootSum[b^2*d - a*b*e + a^2*f - 4*b*Sqrt[c]*d*#1 + 2*a*Sqrt[c]*e*#1 + 4*c*d*#1^2 + b*e*#1^2
- 2*a*f*#1^2 - 2*Sqrt[c]*e*#1^3 + f*#1^4 & , (-(b*c*e^3*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]) + 2*b
*c*d*e*f*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1] + b^2*e^2*f*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] -
#1] + a*c*e^2*f*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1] - b^2*d*f^2*Log[-(Sqrt[c]*x) + Sqrt[a + b*x +
c*x^2] - #1] - a*c*d*f^2*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1] - 2*a*b*e*f^2*Log[-(Sqrt[c]*x) + Sqrt[
a + b*x + c*x^2] - #1] + a^2*f^3*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1] + 2*c^(3/2)*e^3*Log[-(Sqrt[c]*
x) + Sqrt[a + b*x + c*x^2] - #1]*#1 - 4*c^(3/2)*d*e*f*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1 - 2*b*
Sqrt[c]*e^2*f*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1 + 2*b*Sqrt[c]*d*f^2*Log[-(Sqrt[c]*x) + Sqrt[a
+ b*x + c*x^2] - #1]*#1 + 2*a*Sqrt[c]*e*f^2*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1 - c*e^2*f*Log[-(
Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1^2 + c*d*f^2*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1^2 +
b*e*f^2*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1^2 - a*f^3*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] -
#1]*#1^2)/(2*b*Sqrt[c]*d - a*Sqrt[c]*e - 4*c*d*#1 - b*e*#1 + 2*a*f*#1 + 3*Sqrt[c]*e*#1^2 - 2*f*#1^3) & ]/(d*(
c^2*d^2 - b*c*d*e + a*c*e^2 + b^2*d*f - 2*a*c*d*f - a*b*e*f + a^2*f^2))

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fricas [F(-1)]  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(1/x/(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]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(c*x^2+b*x+a)^(3/2)/(f*x^2+e*x+d),x, algorithm="giac")

[Out]

sage0*x

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maple [B]  time = 0.04, size = 4594, normalized size = 5.63 \begin {gather*} \text {output too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(c*x^2+b*x+a)^(3/2)/(f*x^2+e*x+d),x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}} {\left (f x^{2} + e x + d\right )} x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(c*x^2+b*x+a)^(3/2)/(f*x^2+e*x+d),x, algorithm="maxima")

[Out]

integrate(1/((c*x^2 + b*x + a)^(3/2)*(f*x^2 + e*x + d)*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2)),x)

[Out]

int(1/(x*(a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \left (a + b x + c x^{2}\right )^{\frac {3}{2}} \left (d + e x + f x^{2}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(c*x**2+b*x+a)**(3/2)/(f*x**2+e*x+d),x)

[Out]

Integral(1/(x*(a + b*x + c*x**2)**(3/2)*(d + e*x + f*x**2)), x)

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