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

Optimal. Leaf size=571 \[ -\frac {\left (b \left (b^2 d^2-c \left (c+3 a d^2\right )\right )-c \left (2 c^2-b^2 d^2+2 a c d^2\right ) x\right ) \sqrt {1-d^2 x^2}}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (c+a d^2\right )^2\right ) \left (a+b x+c x^2\right )}-\frac {c \left (4 c^3+12 a c^2 d^2-a b \left (b+\sqrt {b^2-4 a c}\right ) d^4-c d^2 \left (5 b^2-b \sqrt {b^2-4 a c}-8 a^2 d^2\right )\right ) \tanh ^{-1}\left (\frac {2 c+\left (b-\sqrt {b^2-4 a c}\right ) d^2 x}{\sqrt {2} \sqrt {2 c^2+2 a c d^2-b \left (b-\sqrt {b^2-4 a c}\right ) d^2} \sqrt {1-d^2 x^2}}\right )}{\sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c^2+2 a c d^2-b \left (b-\sqrt {b^2-4 a c}\right ) d^2} \left (b^2 d^2-\left (c+a d^2\right )^2\right )}+\frac {c \left (4 c^3+12 a c^2 d^2-2 a b^2 d^4-b \left (b+\sqrt {b^2-4 a c}\right ) d^2 \left (c-a d^2\right )-4 c d^2 \left (b^2-2 a^2 d^2\right )\right ) \tanh ^{-1}\left (\frac {2 c+\left (b+\sqrt {b^2-4 a c}\right ) d^2 x}{\sqrt {2} \sqrt {2 c^2+2 a c d^2-b \left (b+\sqrt {b^2-4 a c}\right ) d^2} \sqrt {1-d^2 x^2}}\right )}{\sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c^2+2 a c d^2-b \left (b+\sqrt {b^2-4 a c}\right ) d^2} \left (b^2 d^2-\left (c+a d^2\right )^2\right )} \]

[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)

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Rubi [A]
time = 4.04, antiderivative size = 571, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {913, 989, 1048, 739, 212} \begin {gather*} -\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 ) \tanh ^{-1}\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 ) \tanh ^{-1}\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 )} \end {gather*}

Antiderivative was successfully verified.

[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*} \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 (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*}

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Mathematica [A]
time = 11.06, size = 800, normalized size = 1.40 \begin {gather*} -\frac {-\frac {2 \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 ) (a+x (b+c x))}+\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 ) \log \left (-b+\sqrt {b^2-4 a c}-2 c x\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {c^2+a c d^2+\frac {1}{2} b \left (-b+\sqrt {b^2-4 a c}\right ) d^2}}+\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 ) \log \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {c^2+a c d^2-\frac {1}{2} b \left (b+\sqrt {b^2-4 a c}\right ) d^2}}-\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 ) \log \left (-2 c-b d^2 x+\sqrt {b^2-4 a c} d^2 x-\sqrt {4 c^2+4 a c d^2+2 b \left (-b+\sqrt {b^2-4 a c}\right ) d^2} \sqrt {1-d^2 x^2}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {c^2+a c d^2+\frac {1}{2} b \left (-b+\sqrt {b^2-4 a c}\right ) d^2}}+\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 ) \log \left (2 c+b d^2 x+\sqrt {b^2-4 a c} d^2 x+\sqrt {4 c^2+4 a c d^2-2 b \left (b+\sqrt {b^2-4 a c}\right ) d^2} \sqrt {1-d^2 x^2}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {c^2+a c d^2-\frac {1}{2} b \left (b+\sqrt {b^2-4 a c}\right ) d^2}}}{2 \left (c^2-b^2 d^2+2 a c d^2+a^2 d^4\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*((-2*(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)*
(a + x*(b + c*x))) + (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))*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x])/((b^2 - 4*a*c)^(3/2)*Sqrt[c^2 + a*c*d^2 + (b*(-b +
Sqrt[b^2 - 4*a*c])*d^2)/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*S
qrt[b^2 - 4*a*c] - 8*a^2*d^2))*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x])/((b^2 - 4*a*c)^(3/2)*Sqrt[c^2 + a*c*d^2 - (
b*(b + Sqrt[b^2 - 4*a*c])*d^2)/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))*Log[-2*c - b*d^2*x + Sqrt[b^2 - 4*a*c]*d^2*x - Sqrt[4*c^2 + 4*a*c*d^2 +
2*b*(-b + Sqrt[b^2 - 4*a*c])*d^2]*Sqrt[1 - d^2*x^2]])/((b^2 - 4*a*c)^(3/2)*Sqrt[c^2 + a*c*d^2 + (b*(-b + Sqrt[
b^2 - 4*a*c])*d^2)/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))*Log[2*c + b*d^2*x + Sqrt[b^2 - 4*a*c]*d^2*x + Sqrt[4*c^2 + 4*a*c*d^2 - 2*b*(b + Sqr
t[b^2 - 4*a*c])*d^2]*Sqrt[1 - d^2*x^2]])/((b^2 - 4*a*c)^(3/2)*Sqrt[c^2 + a*c*d^2 - (b*(b + Sqrt[b^2 - 4*a*c])*
d^2)/2]))/(c^2 - b^2*d^2 + 2*a*c*d^2 + a^2*d^4)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.60, size = 41834, normalized size = 73.26

method	result	size
default	\(\text {Expression too large to display}\)	\(41834\)

Verification of antiderivative is not currently implemented for this CAS.

[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

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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(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)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 35403 vs. \(2 (529) = 1058\).
time = 81.34, size = 35403, normalized size = 62.00 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*a*b*c^2 - 2*(a*b^3 - 3*a^2*b*c)*d^2 + 2*(b*c^3 - (b^3*c - 3*a*b*c^2)*d^2)*x^2 + sqrt(2)*(a^2*b^2*c^2 -
4*a^3*c^3 + (a^4*b^2 - 4*a^5*c)*d^4 - (a^2*b^4 - 6*a^3*b^2*c + 8*a^4*c^2)*d^2 + (a*b^2*c^3 - 4*a^2*c^4 + (a^3
*b^2*c - 4*a^4*c^2)*d^4 - (a*b^4*c - 6*a^2*b^2*c^2 + 8*a^3*c^3)*d^2)*x^2 + (a*b^3*c^2 - 4*a^2*b*c^3 + (a^3*b^3
- 4*a^4*b*c)*d^4 - (a*b^5 - 6*a^2*b^3*c + 8*a^3*b*c^2)*d^2)*x)*sqrt(-((a^2*b^6 - 12*a^3*b^4*c + 42*a^4*b^2*c^
2 - 32*a^5*c^3)*d^10 + (4*a*b^6*c - 39*a^2*b^4*c^2 + 114*a^3*b^2*c^3 - 128*a^4*c^4)*d^8 - 8*c^8 + 2*(2*b^6*c^2
- 18*a*b^4*c^3 + 75*a^2*b^2*c^4 - 100*a^3*c^5)*d^6 - (21*b^4*c^4 - 102*a*b^2*c^5 + 152*a^2*c^6)*d^4 + 8*(3*b^
2*c^6 - 7*a*c^7)*d^2 + ((a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*d^12 + b^6*c^6 - 12*a*b^4*c^7 +
48*a^2*b^2*c^8 - 64*a^3*c^9 - 3*(a^4*b^8 - 14*a^5*b^6*c + 72*a^6*b^4*c^2 - 160*a^7*b^2*c^3 + 128*a^8*c^4)*d^1
0 + 3*(a^2*b^10 - 16*a^3*b^8*c + 101*a^4*b^6*c^2 - 316*a^5*b^4*c^3 + 496*a^6*b^2*c^4 - 320*a^7*c^5)*d^8 - (b^1
2 - 18*a*b^10*c + 138*a^2*b^8*c^2 - 588*a^3*b^6*c^3 + 1488*a^4*b^4*c^4 - 2112*a^5*b^2*c^5 + 1280*a^6*c^6)*d^6
+ 3*(b^10*c^2 - 16*a*b^8*c^3 + 101*a^2*b^6*c^4 - 316*a^3*b^4*c^5 + 496*a^4*b^2*c^6 - 320*a^5*c^7)*d^4 - 3*(b^8
*c^4 - 14*a*b^6*c^5 + 72*a^2*b^4*c^6 - 160*a^3*b^2*c^7 + 128*a^4*c^8)*d^2)*sqrt(((a^4*b^6 - 12*a^5*b^4*c + 36*
a^6*b^2*c^2)*d^20 + 2*(4*a^3*b^6*c - 39*a^4*b^4*c^2 + 90*a^5*b^2*c^3)*d^18 + 9*b^2*c^8*d^8 + 3*(8*a^2*b^6*c^2
- 64*a^3*b^4*c^3 + 123*a^4*b^2*c^4)*d^16 + 2*(16*a*b^6*c^3 - 111*a^2*b^4*c^4 + 198*a^3*b^2*c^5)*d^14 + 2*(8*b^
6*c^4 - 60*a*b^4*c^5 + 117*a^2*b^2*c^6)*d^12 - 24*(b^4*c^6 - 3*a*b^2*c^7)*d^10)/((a^12*b^6 - 12*a^13*b^4*c + 4
8*a^14*b^2*c^2 - 64*a^15*c^3)*d^24 - 6*(a^10*b^8 - 14*a^11*b^6*c + 72*a^12*b^4*c^2 - 160*a^13*b^2*c^3 + 128*a^
14*c^4)*d^22 + 3*(5*a^8*b^10 - 80*a^9*b^8*c + 502*a^10*b^6*c^2 - 1544*a^11*b^4*c^3 + 2336*a^12*b^2*c^4 - 1408*
a^13*c^5)*d^20 - 10*(2*a^6*b^12 - 36*a^7*b^10*c + 267*a^8*b^8*c^2 - 1050*a^9*b^6*c^3 + 2328*a^10*b^4*c^4 - 278
4*a^11*b^2*c^5 + 1408*a^12*c^6)*d^18 + b^6*c^12 - 12*a*b^4*c^13 + 48*a^2*b^2*c^14 - 64*a^3*c^15 + 15*(a^4*b^14
- 20*a^5*b^12*c + 172*a^6*b^10*c^2 - 832*a^7*b^8*c^3 + 2465*a^8*b^6*c^4 - 4492*a^9*b^4*c^5 + 4656*a^10*b^2*c^
6 - 2112*a^11*c^7)*d^16 - 6*(a^2*b^16 - 22*a^3*b^14*c + 218*a^4*b^12*c^2 - 1284*a^5*b^10*c^3 + 4930*a^6*b^8*c^
4 - 12572*a^7*b^6*c^5 + 20624*a^8*b^4*c^6 - 19776*a^9*b^2*c^7 + 8448*a^10*c^8)*d^14 + (b^18 - 24*a*b^16*c + 28
2*a^2*b^14*c^2 - 2120*a^3*b^12*c^3 + 10938*a^4*b^10*c^4 - 39072*a^5*b^8*c^5 + 95068*a^6*b^6*c^6 - 150864*a^7*b
^4*c^7 + 141120*a^8*b^2*c^8 - 59136*a^9*c^9)*d^12 - 6*(b^16*c^2 - 22*a*b^14*c^3 + 218*a^2*b^12*c^4 - 1284*a^3*
b^10*c^5 + 4930*a^4*b^8*c^6 - 12572*a^5*b^6*c^7 + 20624*a^6*b^4*c^8 - 19776*a^7*b^2*c^9 + 8448*a^8*c^10)*d^10
+ 15*(b^14*c^4 - 20*a*b^12*c^5 + 172*a^2*b^10*c^6 - 832*a^3*b^8*c^7 + 2465*a^4*b^6*c^8 - 4492*a^5*b^4*c^9 + 46
56*a^6*b^2*c^10 - 2112*a^7*c^11)*d^8 - 10*(2*b^12*c^6 - 36*a*b^10*c^7 + 267*a^2*b^8*c^8 - 1050*a^3*b^6*c^9 + 2
328*a^4*b^4*c^10 - 2784*a^5*b^2*c^11 + 1408*a^6*c^12)*d^6 + 3*(5*b^10*c^8 - 80*a*b^8*c^9 + 502*a^2*b^6*c^10 -
1544*a^3*b^4*c^11 + 2336*a^4*b^2*c^12 - 1408*a^5*c^13)*d^4 - 6*(b^8*c^10 - 14*a*b^6*c^11 + 72*a^2*b^4*c^12 - 1
60*a^3*b^2*c^13 + 128*a^4*c^14)*d^2)))/((a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*d^12 + b^6*c^6
- 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9 - 3*(a^4*b^8 - 14*a^5*b^6*c + 72*a^6*b^4*c^2 - 160*a^7*b^2*c^3 +
128*a^8*c^4)*d^10 + 3*(a^2*b^10 - 16*a^3*b^8*c + 101*a^4*b^6*c^2 - 316*a^5*b^4*c^3 + 496*a^6*b^2*c^4 - 320*a^7
*c^5)*d^8 - (b^12 - 18*a*b^10*c + 138*a^2*b^8*c^2 - 588*a^3*b^6*c^3 + 1488*a^4*b^4*c^4 - 2112*a^5*b^2*c^5 + 12
80*a^6*c^6)*d^6 + 3*(b^10*c^2 - 16*a*b^8*c^3 + 101*a^2*b^6*c^4 - 316*a^3*b^4*c^5 + 496*a^4*b^2*c^6 - 320*a^5*c
^7)*d^4 - 3*(b^8*c^4 - 14*a*b^6*c^5 + 72*a^2*b^4*c^6 - 160*a^3*b^2*c^7 + 128*a^4*c^8)*d^2))*log(-(48*a*b*c^9*d
^4 + 4*(3*a^4*b^5*c^2 - 34*a^5*b^3*c^3 + 96*a^6*b*c^4)*d^14 + 4*(18*a^3*b^5*c^3 - 161*a^4*b^3*c^4 + 336*a^5*b*
c^5)*d^12 + 24*(6*a^2*b^5*c^4 - 43*a^3*b^3*c^5 + 76*a^4*b*c^6)*d^10 + 4*(24*a*b^5*c^5 - 161*a^2*b^3*c^6 + 300*
a^3*b*c^7)*d^8 - 8*(17*a*b^3*c^7 - 48*a^2*b*c^8)*d^6 + 2*((3*a^7*b^6*c^2 - 40*a^8*b^4*c^3 + 176*a^9*b^2*c^4 -
256*a^10*c^5)*d^16 - 4*b^4*c^11 + 32*a*b^2*c^12 - 64*a^2*c^13 - (9*a^5*b^8*c^2 - 144*a^6*b^6*c^3 + 832*a^7*b^4
*c^4 - 2048*a^8*b^2*c^5 + 1792*a^9*c^6)*d^14 + (9*a^3*b^10*c^2 - 174*a^4*b^8*c^3 + 1281*a^5*b^6*c^4 - 4540*a^6
*b^4*c^5 + 7856*a^7*b^2*c^6 - 5440*a^8*c^7)*d^12 - (3*a*b^12*c^2 - 76*a^2*b^10*c^3 + 734*a^3*b^8*c^4 - 3634*a^
4*b^6*c^5 + 10040*a^5*b^4*c^6 - 14944*a^6*b^2*c^7 + 9344*a^7*c^8)*d^10 - (6*b^12*c^3 - 109*a*b^10*c^4 + 884*a^
2*b^8*c^5 - 4069*a^3*b^6*c^6 + 10964*a^4*b^4*c^7 - 16048*a^5*b^2*c^8 + 9920*a^6*c^9)*d^8 + (22*b^10*c^5 - 329*
a*b^8*c^6 + 1996*a^2*b^6*c^7 - 6224*a^3*b^4*c^8 + 10048*a^4*b^2*c^9 - 6656*a^5*c^10)*d^6 - (30*b^8*c^7 - 375*a
*b^6*c^8 + 1732*a^2*b^4*c^9 - 3536*a^3*b^2*c^10 + 2752*a^4*c^11)*d^4 + 2*(9*b^6*c^9 - 92*a*b^4*c^10 + 304*a^2*
b^2*c^11 - 320*a^3*c^12)*d^2)*x*sqrt(((a^4*b^6 ...

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

[Out]

Timed out

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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

\text{Hanged}

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