\(\int \frac {1}{\sqrt {1-d x} \sqrt {1+d x} (a+b x+c x^2)^2} \, dx\) [796]

Optimal result

Integrand size = 32, antiderivative size = 571 \[ \int \frac {1}{\sqrt {1-d x} \sqrt {1+d x} \left (a+b x+c x^2\right )^2} \, dx=-\frac {\left (b \left (b^2 d^2-c \left (c+3 a d^2\right )\right )-c \left (2 c^2-b^2 d^2+2 a c d^2\right ) x\right ) \sqrt {1-d^2 x^2}}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (c+a d^2\right )^2\right ) \left (a+b x+c x^2\right )}-\frac {c \left (4 c^3+12 a c^2 d^2-a b \left (b+\sqrt {b^2-4 a c}\right ) d^4-c d^2 \left (5 b^2-b \sqrt {b^2-4 a c}-8 a^2 d^2\right )\right ) \text {arctanh}\left (\frac {2 c+\left (b-\sqrt {b^2-4 a c}\right ) d^2 x}{\sqrt {2} \sqrt {2 c^2+2 a c d^2-b \left (b-\sqrt {b^2-4 a c}\right ) d^2} \sqrt {1-d^2 x^2}}\right )}{\sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c^2+2 a c d^2-b \left (b-\sqrt {b^2-4 a c}\right ) d^2} \left (b^2 d^2-\left (c+a d^2\right )^2\right )}+\frac {c \left (4 c^3+12 a c^2 d^2-2 a b^2 d^4-b \left (b+\sqrt {b^2-4 a c}\right ) d^2 \left (c-a d^2\right )-4 c d^2 \left (b^2-2 a^2 d^2\right )\right ) \text {arctanh}\left (\frac {2 c+\left (b+\sqrt {b^2-4 a c}\right ) d^2 x}{\sqrt {2} \sqrt {2 c^2+2 a c d^2-b \left (b+\sqrt {b^2-4 a c}\right ) d^2} \sqrt {1-d^2 x^2}}\right )}{\sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c^2+2 a c d^2-b \left (b+\sqrt {b^2-4 a c}\right ) d^2} \left (b^2 d^2-\left (c+a d^2\right )^2\right )} \]

[Out]

-(b*(b^2*d^2-c*(3*a*d^2+c))-c*(2*a*c*d^2-b^2*d^2+2*c^2)*x)*(-d^2*x^2+1)^(1/2)/(-4*a*c+b^2)/(b^2*d^2-(a*d^2+c)^
2)/(c*x^2+b*x+a)-1/2*c*arctanh(1/2*(2*c+d^2*x*(b-(-4*a*c+b^2)^(1/2)))*2^(1/2)/(-d^2*x^2+1)^(1/2)/(2*c^2+2*a*c*
d^2-b*d^2*(b-(-4*a*c+b^2)^(1/2)))^(1/2))*(4*c^3+12*a*c^2*d^2-a*b*d^4*(b+(-4*a*c+b^2)^(1/2))-c*d^2*(5*b^2-8*a^2
*d^2-b*(-4*a*c+b^2)^(1/2)))/(-4*a*c+b^2)^(3/2)/(b^2*d^2-(a*d^2+c)^2)*2^(1/2)/(2*c^2+2*a*c*d^2-b*d^2*(b-(-4*a*c
+b^2)^(1/2)))^(1/2)+1/2*c*arctanh(1/2*(2*c+d^2*x*(b+(-4*a*c+b^2)^(1/2)))*2^(1/2)/(-d^2*x^2+1)^(1/2)/(2*c^2+2*a
*c*d^2-b*d^2*(b+(-4*a*c+b^2)^(1/2)))^(1/2))*(4*c^3+12*a*c^2*d^2-2*a*b^2*d^4-4*c*d^2*(-2*a^2*d^2+b^2)-b*d^2*(-a
*d^2+c)*(b+(-4*a*c+b^2)^(1/2)))/(-4*a*c+b^2)^(3/2)/(b^2*d^2-(a*d^2+c)^2)*2^(1/2)/(2*c^2+2*a*c*d^2-b*d^2*(b+(-4
*a*c+b^2)^(1/2)))^(1/2)

Rubi [A] (verified)

Time = 3.91 (sec) , antiderivative size = 571, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {913, 989, 1048, 739, 212} \[ \int \frac {1}{\sqrt {1-d x} \sqrt {1+d x} \left (a+b x+c x^2\right )^2} \, dx=-\frac {c \left (-c d^2 \left (-8 a^2 d^2-b \sqrt {b^2-4 a c}+5 b^2\right )-a b d^4 \left (\sqrt {b^2-4 a c}+b\right )+12 a c^2 d^2+4 c^3\right ) \text {arctanh}\left (\frac {d^2 x \left (b-\sqrt {b^2-4 a c}\right )+2 c}{\sqrt {2} \sqrt {1-d^2 x^2} \sqrt {-b d^2 \left (b-\sqrt {b^2-4 a c}\right )+2 a c d^2+2 c^2}}\right )}{\sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {-b d^2 \left (b-\sqrt {b^2-4 a c}\right )+2 a c d^2+2 c^2} \left (b^2 d^2-\left (a d^2+c\right )^2\right )}+\frac {c \left (-4 c d^2 \left (b^2-2 a^2 d^2\right )-b d^2 \left (\sqrt {b^2-4 a c}+b\right ) \left (c-a d^2\right )-2 a b^2 d^4+12 a c^2 d^2+4 c^3\right ) \text {arctanh}\left (\frac {d^2 x \left (\sqrt {b^2-4 a c}+b\right )+2 c}{\sqrt {2} \sqrt {1-d^2 x^2} \sqrt {-b d^2 \left (\sqrt {b^2-4 a c}+b\right )+2 a c d^2+2 c^2}}\right )}{\sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {-b d^2 \left (\sqrt {b^2-4 a c}+b\right )+2 a c d^2+2 c^2} \left (b^2 d^2-\left (a d^2+c\right )^2\right )}-\frac {\sqrt {1-d^2 x^2} \left (b \left (b^2 d^2-c \left (3 a d^2+c\right )\right )-c x \left (2 a c d^2-b^2 d^2+2 c^2\right )\right )}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (a d^2+c\right )^2\right ) \left (a+b x+c x^2\right )} \]

[In]

Int[1/(Sqrt[1 - d*x]*Sqrt[1 + d*x]*(a + b*x + c*x^2)^2),x]

[Out]

-(((b*(b^2*d^2 - c*(c + 3*a*d^2)) - c*(2*c^2 - b^2*d^2 + 2*a*c*d^2)*x)*Sqrt[1 - d^2*x^2])/((b^2 - 4*a*c)*(b^2*
d^2 - (c + a*d^2)^2)*(a + b*x + c*x^2))) - (c*(4*c^3 + 12*a*c^2*d^2 - a*b*(b + Sqrt[b^2 - 4*a*c])*d^4 - c*d^2*
(5*b^2 - b*Sqrt[b^2 - 4*a*c] - 8*a^2*d^2))*ArcTanh[(2*c + (b - Sqrt[b^2 - 4*a*c])*d^2*x)/(Sqrt[2]*Sqrt[2*c^2 +
2*a*c*d^2 - b*(b - Sqrt[b^2 - 4*a*c])*d^2]*Sqrt[1 - d^2*x^2])])/(Sqrt[2]*(b^2 - 4*a*c)^(3/2)*Sqrt[2*c^2 + 2*a
*c*d^2 - b*(b - Sqrt[b^2 - 4*a*c])*d^2]*(b^2*d^2 - (c + a*d^2)^2)) + (c*(4*c^3 + 12*a*c^2*d^2 - 2*a*b^2*d^4 -
b*(b + Sqrt[b^2 - 4*a*c])*d^2*(c - a*d^2) - 4*c*d^2*(b^2 - 2*a^2*d^2))*ArcTanh[(2*c + (b + Sqrt[b^2 - 4*a*c])*
d^2*x)/(Sqrt[2]*Sqrt[2*c^2 + 2*a*c*d^2 - b*(b + Sqrt[b^2 - 4*a*c])*d^2]*Sqrt[1 - d^2*x^2])])/(Sqrt[2]*(b^2 - 4
*a*c)^(3/2)*Sqrt[2*c^2 + 2*a*c*d^2 - b*(b + Sqrt[b^2 - 4*a*c])*d^2]*(b^2*d^2 - (c + a*d^2)^2))

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 739

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 913

Int[((d_) + (e_.)*(x_))^(m_)*((f_) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :>
Int[(d*f + e*g*x^2)^m*(a + b*x + c*x^2)^p, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[m - n, 0] &&
EqQ[e*f + d*g, 0] && (IntegerQ[m] || (GtQ[d, 0] && GtQ[f, 0]))

Rule 989

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_.) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp[(b^3*f + b*c*(c*d
- 3*a*f) + c*(2*c^2*d + b^2*f - c*(2*a*f))*x)*(a + b*x + c*x^2)^(p + 1)*((d + f*x^2)^(q + 1)/((b^2 - 4*a*c)*(b
^2*d*f + (c*d - a*f)^2)*(p + 1))), x] - Dist[1/((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f)^2)*(p + 1)), Int[(a + b*x
+ c*x^2)^(p + 1)*(d + f*x^2)^q*Simp[2*c*(b^2*d*f + (c*d - a*f)^2)*(p + 1) - (2*c^2*d + b^2*f - c*(2*a*f))*(a*
f*(p + 1) - c*d*(p + 2)) + (2*f*(b^3*f + b*c*(c*d - 3*a*f))*(p + q + 2) - (2*c^2*d + b^2*f - c*(2*a*f))*(b*f*(
p + 1)))*x + c*f*(2*c^2*d + b^2*f - c*(2*a*f))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, f, q}, x]
&& NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[b^2*d*f + (c*d - a*f)^2, 0] &&  !( !IntegerQ[p] && ILtQ[q, -1]) &
&  !IGtQ[q, 0]

Rule 1048

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\left (a+b x+c x^2\right )^2 \sqrt {1-d^2 x^2}} \, dx \\ & = -\frac {\left (b \left (b^2 d^2-c \left (c+3 a d^2\right )\right )-c \left (2 c^2-b^2 d^2+2 a c d^2\right ) x\right ) \sqrt {1-d^2 x^2}}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (c+a d^2\right )^2\right ) \left (a+b x+c x^2\right )}-\frac {\int \frac {-2 c^3-6 a c^2 d^2+a b^2 d^4+2 c d^2 \left (b^2-2 a^2 d^2\right )-b c d^2 \left (c-a d^2\right ) x}{\left (a+b x+c x^2\right ) \sqrt {1-d^2 x^2}} \, dx}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (c+a d^2\right )^2\right )} \\ & = -\frac {\left (b \left (b^2 d^2-c \left (c+3 a d^2\right )\right )-c \left (2 c^2-b^2 d^2+2 a c d^2\right ) x\right ) \sqrt {1-d^2 x^2}}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (c+a d^2\right )^2\right ) \left (a+b x+c x^2\right )}+\frac {\left (c \left (4 c^3+12 a c^2 d^2-a b \left (b+\sqrt {b^2-4 a c}\right ) d^4-c d^2 \left (5 b^2-b \sqrt {b^2-4 a c}-8 a^2 d^2\right )\right )\right ) \int \frac {1}{\left (b-\sqrt {b^2-4 a c}+2 c x\right ) \sqrt {1-d^2 x^2}} \, dx}{\left (b^2-4 a c\right )^{3/2} \left (b^2 d^2-\left (c+a d^2\right )^2\right )}+\frac {\left (b c \left (b+\sqrt {b^2-4 a c}\right ) d^2 \left (c-a d^2\right )+2 c \left (-2 c^3-6 a c^2 d^2+a b^2 d^4+2 c d^2 \left (b^2-2 a^2 d^2\right )\right )\right ) \int \frac {1}{\left (b+\sqrt {b^2-4 a c}+2 c x\right ) \sqrt {1-d^2 x^2}} \, dx}{\left (b^2-4 a c\right )^{3/2} \left (b^2 d^2-\left (c+a d^2\right )^2\right )} \\ & = -\frac {\left (b \left (b^2 d^2-c \left (c+3 a d^2\right )\right )-c \left (2 c^2-b^2 d^2+2 a c d^2\right ) x\right ) \sqrt {1-d^2 x^2}}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (c+a d^2\right )^2\right ) \left (a+b x+c x^2\right )}-\frac {\left (c \left (4 c^3+12 a c^2 d^2-a b \left (b+\sqrt {b^2-4 a c}\right ) d^4-c d^2 \left (5 b^2-b \sqrt {b^2-4 a c}-8 a^2 d^2\right )\right )\right ) \text {Subst}\left (\int \frac {1}{4 c^2-\left (b-\sqrt {b^2-4 a c}\right )^2 d^2-x^2} \, dx,x,\frac {2 c+\left (b-\sqrt {b^2-4 a c}\right ) d^2 x}{\sqrt {1-d^2 x^2}}\right )}{\left (b^2-4 a c\right )^{3/2} \left (b^2 d^2-\left (c+a d^2\right )^2\right )}-\frac {\left (b c \left (b+\sqrt {b^2-4 a c}\right ) d^2 \left (c-a d^2\right )+2 c \left (-2 c^3-6 a c^2 d^2+a b^2 d^4+2 c d^2 \left (b^2-2 a^2 d^2\right )\right )\right ) \text {Subst}\left (\int \frac {1}{4 c^2-\left (b+\sqrt {b^2-4 a c}\right )^2 d^2-x^2} \, dx,x,\frac {2 c+\left (b+\sqrt {b^2-4 a c}\right ) d^2 x}{\sqrt {1-d^2 x^2}}\right )}{\left (b^2-4 a c\right )^{3/2} \left (b^2 d^2-\left (c+a d^2\right )^2\right )} \\ & = -\frac {\left (b \left (b^2 d^2-c \left (c+3 a d^2\right )\right )-c \left (2 c^2-b^2 d^2+2 a c d^2\right ) x\right ) \sqrt {1-d^2 x^2}}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (c+a d^2\right )^2\right ) \left (a+b x+c x^2\right )}-\frac {c \left (4 c^3+12 a c^2 d^2-a b \left (b+\sqrt {b^2-4 a c}\right ) d^4-c d^2 \left (5 b^2-b \sqrt {b^2-4 a c}-8 a^2 d^2\right )\right ) \tanh ^{-1}\left (\frac {2 c+\left (b-\sqrt {b^2-4 a c}\right ) d^2 x}{\sqrt {2} \sqrt {2 c^2+2 a c d^2-b \left (b-\sqrt {b^2-4 a c}\right ) d^2} \sqrt {1-d^2 x^2}}\right )}{\sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c^2+2 a c d^2-b \left (b-\sqrt {b^2-4 a c}\right ) d^2} \left (b^2 d^2-\left (c+a d^2\right )^2\right )}-\frac {c \left (b \left (b+\sqrt {b^2-4 a c}\right ) d^2 \left (c-a d^2\right )-2 \left (2 c^3+6 a c^2 d^2-a b^2 d^4-2 c d^2 \left (b^2-2 a^2 d^2\right )\right )\right ) \tanh ^{-1}\left (\frac {2 c+\left (b+\sqrt {b^2-4 a c}\right ) d^2 x}{\sqrt {2} \sqrt {2 c^2+2 a c d^2-b \left (b+\sqrt {b^2-4 a c}\right ) d^2} \sqrt {1-d^2 x^2}}\right )}{\sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c^2+2 a c d^2-b \left (b+\sqrt {b^2-4 a c}\right ) d^2} \left (b^2 d^2-\left (c+a d^2\right )^2\right )} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 2.67 (sec) , antiderivative size = 1548, normalized size of antiderivative = 2.71 \[ \int \frac {1}{\sqrt {1-d x} \sqrt {1+d x} \left (a+b x+c x^2\right )^2} \, dx=\frac {\left (-b^3 d^2+b c \left (c+3 a d^2\right )-b^2 c d^2 x+2 c^2 \left (c+a d^2\right ) x\right ) \sqrt {1-d^2 x^2}}{\left (b^2-4 a c\right ) (-c+d (b-a d)) (c+d (b+a d)) (a+x (b+c x))}+\frac {\text {RootSum}\left [a d^4-2 b d^2 \text {$\#$1}+4 c \text {$\#$1}^2+2 a d^2 \text {$\#$1}^2-2 b \text {$\#$1}^3+a \text {$\#$1}^4\&,\frac {-4 b^2 \log (x)+4 a c \log (x)-a^2 d^2 \log (x)+4 b^2 \log \left (-1+\sqrt {1-d^2 x^2}-x \text {$\#$1}\right )-4 a c \log \left (-1+\sqrt {1-d^2 x^2}-x \text {$\#$1}\right )+a^2 d^2 \log \left (-1+\sqrt {1-d^2 x^2}-x \text {$\#$1}\right )-2 a b \log (x) \text {$\#$1}+2 a b \log \left (-1+\sqrt {1-d^2 x^2}-x \text {$\#$1}\right ) \text {$\#$1}-a^2 \log (x) \text {$\#$1}^2+a^2 \log \left (-1+\sqrt {1-d^2 x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2}{b d^2-4 c \text {$\#$1}-2 a d^2 \text {$\#$1}+3 b \text {$\#$1}^2-2 a \text {$\#$1}^3}\&\right ]}{a^3}-\frac {\text {RootSum}\left [a d^4-2 b d^2 \text {$\#$1}+4 c \text {$\#$1}^2+2 a d^2 \text {$\#$1}^2-2 b \text {$\#$1}^3+a \text {$\#$1}^4\&,\frac {4 b^4 c^2 \log (x)-20 a b^2 c^3 \log (x)+16 a^2 c^4 \log (x)-4 b^6 d^2 \log (x)+28 a b^4 c d^2 \log (x)-55 a^2 b^2 c^2 d^2 \log (x)+30 a^3 c^3 d^2 \log (x)+3 a^2 b^4 d^4 \log (x)-16 a^3 b^2 c d^4 \log (x)+14 a^4 c^2 d^4 \log (x)-4 b^4 c^2 \log \left (-1+\sqrt {1-d^2 x^2}-x \text {$\#$1}\right )+20 a b^2 c^3 \log \left (-1+\sqrt {1-d^2 x^2}-x \text {$\#$1}\right )-16 a^2 c^4 \log \left (-1+\sqrt {1-d^2 x^2}-x \text {$\#$1}\right )+4 b^6 d^2 \log \left (-1+\sqrt {1-d^2 x^2}-x \text {$\#$1}\right )-28 a b^4 c d^2 \log \left (-1+\sqrt {1-d^2 x^2}-x \text {$\#$1}\right )+55 a^2 b^2 c^2 d^2 \log \left (-1+\sqrt {1-d^2 x^2}-x \text {$\#$1}\right )-30 a^3 c^3 d^2 \log \left (-1+\sqrt {1-d^2 x^2}-x \text {$\#$1}\right )-3 a^2 b^4 d^4 \log \left (-1+\sqrt {1-d^2 x^2}-x \text {$\#$1}\right )+16 a^3 b^2 c d^4 \log \left (-1+\sqrt {1-d^2 x^2}-x \text {$\#$1}\right )-14 a^4 c^2 d^4 \log \left (-1+\sqrt {1-d^2 x^2}-x \text {$\#$1}\right )+2 a b^3 c^2 \log (x) \text {$\#$1}-8 a^2 b c^3 \log (x) \text {$\#$1}-2 a b^5 d^2 \log (x) \text {$\#$1}+12 a^2 b^3 c d^2 \log (x) \text {$\#$1}-18 a^3 b c^2 d^2 \log (x) \text {$\#$1}+2 a^3 b^3 d^4 \log (x) \text {$\#$1}-6 a^4 b c d^4 \log (x) \text {$\#$1}-2 a b^3 c^2 \log \left (-1+\sqrt {1-d^2 x^2}-x \text {$\#$1}\right ) \text {$\#$1}+8 a^2 b c^3 \log \left (-1+\sqrt {1-d^2 x^2}-x \text {$\#$1}\right ) \text {$\#$1}+2 a b^5 d^2 \log \left (-1+\sqrt {1-d^2 x^2}-x \text {$\#$1}\right ) \text {$\#$1}-12 a^2 b^3 c d^2 \log \left (-1+\sqrt {1-d^2 x^2}-x \text {$\#$1}\right ) \text {$\#$1}+18 a^3 b c^2 d^2 \log \left (-1+\sqrt {1-d^2 x^2}-x \text {$\#$1}\right ) \text {$\#$1}-2 a^3 b^3 d^4 \log \left (-1+\sqrt {1-d^2 x^2}-x \text {$\#$1}\right ) \text {$\#$1}+6 a^4 b c d^4 \log \left (-1+\sqrt {1-d^2 x^2}-x \text {$\#$1}\right ) \text {$\#$1}+a^2 b^2 c^2 \log (x) \text {$\#$1}^2-2 a^3 c^3 \log (x) \text {$\#$1}^2-a^2 b^4 d^2 \log (x) \text {$\#$1}^2+4 a^3 b^2 c d^2 \log (x) \text {$\#$1}^2-2 a^4 c^2 d^2 \log (x) \text {$\#$1}^2-a^2 b^2 c^2 \log \left (-1+\sqrt {1-d^2 x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2+2 a^3 c^3 \log \left (-1+\sqrt {1-d^2 x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2+a^2 b^4 d^2 \log \left (-1+\sqrt {1-d^2 x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2-4 a^3 b^2 c d^2 \log \left (-1+\sqrt {1-d^2 x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2+2 a^4 c^2 d^2 \log \left (-1+\sqrt {1-d^2 x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2}{b d^2-4 c \text {$\#$1}-2 a d^2 \text {$\#$1}+3 b \text {$\#$1}^2-2 a \text {$\#$1}^3}\&\right ]}{a^3 \left (-b^2+4 a c\right ) (c+d (-b+a d)) (c+d (b+a d))} \]

[In]

Integrate[1/(Sqrt[1 - d*x]*Sqrt[1 + d*x]*(a + b*x + c*x^2)^2),x]

[Out]

((-(b^3*d^2) + b*c*(c + 3*a*d^2) - b^2*c*d^2*x + 2*c^2*(c + a*d^2)*x)*Sqrt[1 - d^2*x^2])/((b^2 - 4*a*c)*(-c +
d*(b - a*d))*(c + d*(b + a*d))*(a + x*(b + c*x))) + RootSum[a*d^4 - 2*b*d^2*#1 + 4*c*#1^2 + 2*a*d^2*#1^2 - 2*b
*#1^3 + a*#1^4 & , (-4*b^2*Log[x] + 4*a*c*Log[x] - a^2*d^2*Log[x] + 4*b^2*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1] -
4*a*c*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1] + a^2*d^2*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1] - 2*a*b*Log[x]*#1 + 2*a
*b*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1]*#1 - a^2*Log[x]*#1^2 + a^2*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1]*#1^2)/(b*d
^2 - 4*c*#1 - 2*a*d^2*#1 + 3*b*#1^2 - 2*a*#1^3) & ]/a^3 - RootSum[a*d^4 - 2*b*d^2*#1 + 4*c*#1^2 + 2*a*d^2*#1^2
- 2*b*#1^3 + a*#1^4 & , (4*b^4*c^2*Log[x] - 20*a*b^2*c^3*Log[x] + 16*a^2*c^4*Log[x] - 4*b^6*d^2*Log[x] + 28*a
*b^4*c*d^2*Log[x] - 55*a^2*b^2*c^2*d^2*Log[x] + 30*a^3*c^3*d^2*Log[x] + 3*a^2*b^4*d^4*Log[x] - 16*a^3*b^2*c*d^
4*Log[x] + 14*a^4*c^2*d^4*Log[x] - 4*b^4*c^2*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1] + 20*a*b^2*c^3*Log[-1 + Sqrt[1
- d^2*x^2] - x*#1] - 16*a^2*c^4*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1] + 4*b^6*d^2*Log[-1 + Sqrt[1 - d^2*x^2] - x
*#1] - 28*a*b^4*c*d^2*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1] + 55*a^2*b^2*c^2*d^2*Log[-1 + Sqrt[1 - d^2*x^2] - x*#
1] - 30*a^3*c^3*d^2*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1] - 3*a^2*b^4*d^4*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1] + 16
*a^3*b^2*c*d^4*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1] - 14*a^4*c^2*d^4*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1] + 2*a*b^
3*c^2*Log[x]*#1 - 8*a^2*b*c^3*Log[x]*#1 - 2*a*b^5*d^2*Log[x]*#1 + 12*a^2*b^3*c*d^2*Log[x]*#1 - 18*a^3*b*c^2*d^
2*Log[x]*#1 + 2*a^3*b^3*d^4*Log[x]*#1 - 6*a^4*b*c*d^4*Log[x]*#1 - 2*a*b^3*c^2*Log[-1 + Sqrt[1 - d^2*x^2] - x*#
1]*#1 + 8*a^2*b*c^3*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1]*#1 + 2*a*b^5*d^2*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1]*#1
- 12*a^2*b^3*c*d^2*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1]*#1 + 18*a^3*b*c^2*d^2*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1]
*#1 - 2*a^3*b^3*d^4*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1]*#1 + 6*a^4*b*c*d^4*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1]*#
1 + a^2*b^2*c^2*Log[x]*#1^2 - 2*a^3*c^3*Log[x]*#1^2 - a^2*b^4*d^2*Log[x]*#1^2 + 4*a^3*b^2*c*d^2*Log[x]*#1^2 -
2*a^4*c^2*d^2*Log[x]*#1^2 - a^2*b^2*c^2*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1]*#1^2 + 2*a^3*c^3*Log[-1 + Sqrt[1 -
d^2*x^2] - x*#1]*#1^2 + a^2*b^4*d^2*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1]*#1^2 - 4*a^3*b^2*c*d^2*Log[-1 + Sqrt[1
- d^2*x^2] - x*#1]*#1^2 + 2*a^4*c^2*d^2*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1]*#1^2)/(b*d^2 - 4*c*#1 - 2*a*d^2*#1
+ 3*b*#1^2 - 2*a*#1^3) & ]/(a^3*(-b^2 + 4*a*c)*(c + d*(-b + a*d))*(c + d*(b + a*d)))

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 1.02 (sec) , antiderivative size = 41837, normalized size of antiderivative = 73.27

method	result	size
default	\(\text {Expression too large to display}\)	\(41837\)

[In]

int(1/(c*x^2+b*x+a)^2/(-d*x+1)^(1/2)/(d*x+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

result too large to display

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 35403 vs. \(2 (529) = 1058\).

Time = 33.26 (sec) , antiderivative size = 35403, normalized size of antiderivative = 62.00 \[ \int \frac {1}{\sqrt {1-d x} \sqrt {1+d x} \left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

integrate(1/(c*x^2+b*x+a)^2/(-d*x+1)^(1/2)/(d*x+1)^(1/2),x, algorithm="fricas")

[Out]

Too large to include

Sympy [F]

\[ \int \frac {1}{\sqrt {1-d x} \sqrt {1+d x} \left (a+b x+c x^2\right )^2} \, dx=\int \frac {1}{\sqrt {- d x + 1} \sqrt {d x + 1} \left (a + b x + c x^{2}\right )^{2}}\, dx \]

[In]

integrate(1/(c*x**2+b*x+a)**2/(-d*x+1)**(1/2)/(d*x+1)**(1/2),x)

[Out]

Integral(1/(sqrt(-d*x + 1)*sqrt(d*x + 1)*(a + b*x + c*x**2)**2), x)

Maxima [F]

\[ \int \frac {1}{\sqrt {1-d x} \sqrt {1+d x} \left (a+b x+c x^2\right )^2} \, dx=\int { \frac {1}{{\left (c x^{2} + b x + a\right )}^{2} \sqrt {d x + 1} \sqrt {-d x + 1}} \,d x } \]

[In]

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

[Out]

integrate(1/((c*x^2 + b*x + a)^2*sqrt(d*x + 1)*sqrt(-d*x + 1)), x)

Giac [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {1-d x} \sqrt {1+d x} \left (a+b x+c x^2\right )^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {1-d x} \sqrt {1+d x} \left (a+b x+c x^2\right )^2} \, dx=\text {Hanged} \]

[In]

int(1/((1 - d*x)^(1/2)*(d*x + 1)^(1/2)*(a + b*x + c*x^2)^2),x)

[Out]

\text{Hanged}