∫ 1___________ (1−dx)3∕2(1+dx)3∕2(a+bx+cx2) dx [800]

3.8.100 $\int \frac {1}{(1-d x)^{3/2} (1+d x)^{3/2} (a+b x+c x^2)} \, dx$ [800]

Optimal. Leaf size=443 \[ \frac {d^2 \left (b-\left (c+a d^2\right ) x\right )}{\left (b^2 d^2-\left (c+a d^2\right )^2\right ) \sqrt {1-d^2 x^2}}+\frac {c \left (2 c^2+2 a c d^2-b \left (b+\sqrt {b^2-4 a c}\right ) d^2\right ) \tanh ^{-1}\left (\frac {2 c+\left (b-\sqrt {b^2-4 a c}\right ) d^2 x}{\sqrt {2} \sqrt {2 c^2+2 a c d^2-b \left (b-\sqrt {b^2-4 a c}\right ) d^2} \sqrt {1-d^2 x^2}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {2 c^2+2 a c d^2-b \left (b-\sqrt {b^2-4 a c}\right ) d^2} \left (b^2 d^2-\left (c+a d^2\right )^2\right )}-\frac {c \left (2 c^2+2 a c d^2-b \left (b-\sqrt {b^2-4 a c}\right ) d^2\right ) \tanh ^{-1}\left (\frac {2 c+\left (b+\sqrt {b^2-4 a c}\right ) d^2 x}{\sqrt {2} \sqrt {2 c^2+2 a c d^2-b \left (b+\sqrt {b^2-4 a c}\right ) d^2} \sqrt {1-d^2 x^2}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {2 c^2+2 a c d^2-b \left (b+\sqrt {b^2-4 a c}\right ) d^2} \left (b^2 d^2-\left (c+a d^2\right )^2\right )} \]

[Out]

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

+(-4*a*c+b^2)^(1/2)))/(b^2*d^2-(a*d^2+c)^2)*2^(1/2)/(-4*a*c+b^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))*(2*c^2+2*a*c*d^2-b*d^2*(b-(-4*a*c+b^2)^(1/2)))/(b^2*d^2-(a*d^2+c)^2)*2^(

1/2)/(-4*a*c+b^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 = 0.97, antiderivative size = 443, 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, 990, 1048, 739, 212} \begin {gather*} \frac {c \left (-b d^2 \left (\sqrt {b^2-4 a c}+b\right )+2 a c d^2+2 c^2\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} \sqrt {b^2-4 a c} \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 (-b d^2 \left (b-\sqrt {b^2-4 a c}\right )+2 a c d^2+2 c^2\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} \sqrt {b^2-4 a c} \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 {d^2 \left (b-x \left (a d^2+c\right )\right )}{\sqrt {1-d^2 x^2} \left (b^2 d^2-\left (a d^2+c\right )^2\right )} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

b^2 - 4*a*c])*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]*Sqrt[b^2 - 4*a*c]*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*(2*c^2 + 2*a*c*d^2 - b*(b - Sqrt[b^2 - 4*a*c])*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]*Sqrt[b^2 - 4*a*c]*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 990

Int[((a_.) + (c_.)*(x_)^2)^(p_)*((d_.) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp[(2*a*c^2*e + c*(2*

c^2*d - c*(2*a*f))*x)*(a + c*x^2)^(p + 1)*((d + e*x + f*x^2)^(q + 1)/((-4*a*c)*(a*c*e^2 + (c*d - a*f)^2)*(p +

1))), x] - Dist[1/((-4*a*c)*(a*c*e^2 + (c*d - a*f)^2)*(p + 1)), Int[(a + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q*Si

mp[2*c*((c*d - a*f)^2 - ((-a)*e)*(c*e))*(p + 1) - (2*c^2*d - c*(2*a*f))*(a*f*(p + 1) - c*d*(p + 2)) - e*(-2*a*

c^2*e)*(p + q + 2) + (2*f*(2*a*c^2*e)*(p + q + 2) - (2*c^2*d - c*(2*a*f))*((-c)*e*(2*p + q + 4)))*x + c*f*(2*c

^2*d - c*(2*a*f))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, c, d, e, f, q}, x] && NeQ[e^2 - 4*d*f, 0] && Lt

Q[p, -1] && NeQ[a*c*e^2 + (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}{(1-d x)^{3/2} (1+d x)^{3/2} \left (a+b x+c x^2\right )} \, dx &=\int \frac {1}{\left (a+b x+c x^2\right ) \left (1-d^2 x^2\right )^{3/2}} \, dx\\ &=\frac {d^2 \left (b-\left (c+a d^2\right ) x\right )}{\left (b^2 d^2-\left (c+a d^2\right )^2\right ) \sqrt {1-d^2 x^2}}-\frac {\int \frac {2 d^2 \left (c^2-b^2 d^2+a c d^2\right )-2 b c d^4 x}{\left (a+b x+c x^2\right ) \sqrt {1-d^2 x^2}} \, dx}{2 d^2 \left (b^2 d^2-\left (c+a d^2\right )^2\right )}\\ &=\frac {d^2 \left (b-\left (c+a d^2\right ) x\right )}{\left (b^2 d^2-\left (c+a d^2\right )^2\right ) \sqrt {1-d^2 x^2}}+\frac {\left (c \left (2 c^2+2 a c d^2-b \left (b-\sqrt {b^2-4 a c}\right ) d^2\right )\right ) \int \frac {1}{\left (b+\sqrt {b^2-4 a c}+2 c x\right ) \sqrt {1-d^2 x^2}} \, dx}{\sqrt {b^2-4 a c} \left (b^2 d^2-\left (c+a d^2\right )^2\right )}-\frac {\left (c \left (2 c^2+2 a c d^2-b \left (b+\sqrt {b^2-4 a c}\right ) d^2\right )\right ) \int \frac {1}{\left (b-\sqrt {b^2-4 a c}+2 c x\right ) \sqrt {1-d^2 x^2}} \, dx}{\sqrt {b^2-4 a c} \left (b^2 d^2-\left (c+a d^2\right )^2\right )}\\ &=\frac {d^2 \left (b-\left (c+a d^2\right ) x\right )}{\left (b^2 d^2-\left (c+a d^2\right )^2\right ) \sqrt {1-d^2 x^2}}-\frac {\left (c \left (2 c^2+2 a c d^2-b \left (b-\sqrt {b^2-4 a c}\right ) d^2\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 )}{\sqrt {b^2-4 a c} \left (b^2 d^2-\left (c+a d^2\right )^2\right )}+\frac {\left (c \left (2 c^2+2 a c d^2-b \left (b+\sqrt {b^2-4 a c}\right ) d^2\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 )}{\sqrt {b^2-4 a c} \left (b^2 d^2-\left (c+a d^2\right )^2\right )}\\ &=\frac {d^2 \left (b-\left (c+a d^2\right ) x\right )}{\left (b^2 d^2-\left (c+a d^2\right )^2\right ) \sqrt {1-d^2 x^2}}+\frac {c \left (2 c^2+2 a c d^2-b \left (b+\sqrt {b^2-4 a c}\right ) d^2\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} \sqrt {b^2-4 a c} \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 (2 c^2+2 a c d^2-b \left (b-\sqrt {b^2-4 a c}\right ) d^2\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} \sqrt {b^2-4 a c} \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 [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 2.68, size = 1546, normalized size = 3.49 \begin {gather*} \frac {4 d^2 \left (-b+\left (c+a d^2\right ) x\right ) \sqrt {1-d^2 x^2}+2 c^2 \sqrt {-d^2} \left (c+a d^2\right ) (-1+d x) (1+d x) \text {RootSum}\left [c^2-4 c^2 \text {$\#$1}+4 b^2 d^2 \text {$\#$1}-8 a c d^2 \text {$\#$1}+6 c^2 \text {$\#$1}^2-8 b^2 d^2 \text {$\#$1}^2+16 a c d^2 \text {$\#$1}^2+16 a^2 d^4 \text {$\#$1}^2-4 c^2 \text {$\#$1}^3+4 b^2 d^2 \text {$\#$1}^3-8 a c d^2 \text {$\#$1}^3+c^2 \text {$\#$1}^4\&,\frac {\log \left (-1+2 d^2 x^2+2 \sqrt {-d^2} x \sqrt {1-d^2 x^2}+\text {$\#$1}\right )}{-c^2+b^2 d^2-2 a c d^2+3 c^2 \text {$\#$1}-4 b^2 d^2 \text {$\#$1}+8 a c d^2 \text {$\#$1}+8 a^2 d^4 \text {$\#$1}-3 c^2 \text {$\#$1}^2+3 b^2 d^2 \text {$\#$1}^2-6 a c d^2 \text {$\#$1}^2+c^2 \text {$\#$1}^3}\&\right ]-4 \sqrt {-d^2} \left (c^3+3 a c^2 d^2-2 a b^2 d^4+2 a^2 c d^4\right ) (-1+d x) (1+d x) \text {RootSum}\left [c^2-4 c^2 \text {$\#$1}+4 b^2 d^2 \text {$\#$1}-8 a c d^2 \text {$\#$1}+6 c^2 \text {$\#$1}^2-8 b^2 d^2 \text {$\#$1}^2+16 a c d^2 \text {$\#$1}^2+16 a^2 d^4 \text {$\#$1}^2-4 c^2 \text {$\#$1}^3+4 b^2 d^2 \text {$\#$1}^3-8 a c d^2 \text {$\#$1}^3+c^2 \text {$\#$1}^4\&,\frac {\log \left (-1+2 d^2 x^2+2 \sqrt {-d^2} x \sqrt {1-d^2 x^2}+\text {$\#$1}\right ) \text {$\#$1}}{-c^2+b^2 d^2-2 a c d^2+3 c^2 \text {$\#$1}-4 b^2 d^2 \text {$\#$1}+8 a c d^2 \text {$\#$1}+8 a^2 d^4 \text {$\#$1}-3 c^2 \text {$\#$1}^2+3 b^2 d^2 \text {$\#$1}^2-6 a c d^2 \text {$\#$1}^2+c^2 \text {$\#$1}^3}\&\right ]+2 c^2 \sqrt {-d^2} \left (c+a d^2\right ) (-1+d x) (1+d x) \text {RootSum}\left [c^2-4 c^2 \text {$\#$1}+4 b^2 d^2 \text {$\#$1}-8 a c d^2 \text {$\#$1}+6 c^2 \text {$\#$1}^2-8 b^2 d^2 \text {$\#$1}^2+16 a c d^2 \text {$\#$1}^2+16 a^2 d^4 \text {$\#$1}^2-4 c^2 \text {$\#$1}^3+4 b^2 d^2 \text {$\#$1}^3-8 a c d^2 \text {$\#$1}^3+c^2 \text {$\#$1}^4\&,\frac {\log \left (-1+2 d^2 x^2+2 \sqrt {-d^2} x \sqrt {1-d^2 x^2}+\text {$\#$1}\right ) \text {$\#$1}^2}{-c^2+b^2 d^2-2 a c d^2+3 c^2 \text {$\#$1}-4 b^2 d^2 \text {$\#$1}+8 a c d^2 \text {$\#$1}+8 a^2 d^4 \text {$\#$1}-3 c^2 \text {$\#$1}^2+3 b^2 d^2 \text {$\#$1}^2-6 a c d^2 \text {$\#$1}^2+c^2 \text {$\#$1}^3}\&\right ]+b d^2 (-1+d x) (1+d x) \text {RootSum}\left [c^2-4 c^2 \text {$\#$1}^2+4 b^2 d^2 \text {$\#$1}^2-8 a c d^2 \text {$\#$1}^2+6 c^2 \text {$\#$1}^4-8 b^2 d^2 \text {$\#$1}^4+16 a c d^2 \text {$\#$1}^4+16 a^2 d^4 \text {$\#$1}^4-4 c^2 \text {$\#$1}^6+4 b^2 d^2 \text {$\#$1}^6-8 a c d^2 \text {$\#$1}^6+c^2 \text {$\#$1}^8\&,\frac {-c^2 \log \left (-\sqrt {-d^2} x+\sqrt {1-d^2 x^2}-\text {$\#$1}\right )+7 c^2 \log \left (-\sqrt {-d^2} x+\sqrt {1-d^2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2-4 b^2 d^2 \log \left (-\sqrt {-d^2} x+\sqrt {1-d^2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+8 a c d^2 \log \left (-\sqrt {-d^2} x+\sqrt {1-d^2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2-7 c^2 \log \left (-\sqrt {-d^2} x+\sqrt {1-d^2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^4+4 b^2 d^2 \log \left (-\sqrt {-d^2} x+\sqrt {1-d^2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^4-8 a c d^2 \log \left (-\sqrt {-d^2} x+\sqrt {1-d^2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^4+c^2 \log \left (-\sqrt {-d^2} x+\sqrt {1-d^2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^6}{-c^2 \text {$\#$1}+b^2 d^2 \text {$\#$1}-2 a c d^2 \text {$\#$1}+3 c^2 \text {$\#$1}^3-4 b^2 d^2 \text {$\#$1}^3+8 a c d^2 \text {$\#$1}^3+8 a^2 d^4 \text {$\#$1}^3-3 c^2 \text {$\#$1}^5+3 b^2 d^2 \text {$\#$1}^5-6 a c d^2 \text {$\#$1}^5+c^2 \text {$\#$1}^7}\&\right ]}{4 (c+d (-b+a d)) (c+d (b+a d)) (1-d x) (1+d x)} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(4*d^2*(-b + (c + a*d^2)*x)*Sqrt[1 - d^2*x^2] + 2*c^2*Sqrt[-d^2]*(c + a*d^2)*(-1 + d*x)*(1 + d*x)*RootSum[c^2

- 4*c^2*#1 + 4*b^2*d^2*#1 - 8*a*c*d^2*#1 + 6*c^2*#1^2 - 8*b^2*d^2*#1^2 + 16*a*c*d^2*#1^2 + 16*a^2*d^4*#1^2 - 4

*c^2*#1^3 + 4*b^2*d^2*#1^3 - 8*a*c*d^2*#1^3 + c^2*#1^4 & , Log[-1 + 2*d^2*x^2 + 2*Sqrt[-d^2]*x*Sqrt[1 - d^2*x^

2] + #1]/(-c^2 + b^2*d^2 - 2*a*c*d^2 + 3*c^2*#1 - 4*b^2*d^2*#1 + 8*a*c*d^2*#1 + 8*a^2*d^4*#1 - 3*c^2*#1^2 + 3*

b^2*d^2*#1^2 - 6*a*c*d^2*#1^2 + c^2*#1^3) & ] - 4*Sqrt[-d^2]*(c^3 + 3*a*c^2*d^2 - 2*a*b^2*d^4 + 2*a^2*c*d^4)*(

-1 + d*x)*(1 + d*x)*RootSum[c^2 - 4*c^2*#1 + 4*b^2*d^2*#1 - 8*a*c*d^2*#1 + 6*c^2*#1^2 - 8*b^2*d^2*#1^2 + 16*a*

c*d^2*#1^2 + 16*a^2*d^4*#1^2 - 4*c^2*#1^3 + 4*b^2*d^2*#1^3 - 8*a*c*d^2*#1^3 + c^2*#1^4 & , (Log[-1 + 2*d^2*x^2

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

2*#1 + 8*a^2*d^4*#1 - 3*c^2*#1^2 + 3*b^2*d^2*#1^2 - 6*a*c*d^2*#1^2 + c^2*#1^3) & ] + 2*c^2*Sqrt[-d^2]*(c + a*d

^2)*(-1 + d*x)*(1 + d*x)*RootSum[c^2 - 4*c^2*#1 + 4*b^2*d^2*#1 - 8*a*c*d^2*#1 + 6*c^2*#1^2 - 8*b^2*d^2*#1^2 +

16*a*c*d^2*#1^2 + 16*a^2*d^4*#1^2 - 4*c^2*#1^3 + 4*b^2*d^2*#1^3 - 8*a*c*d^2*#1^3 + c^2*#1^4 & , (Log[-1 + 2*d^

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

*a*c*d^2*#1 + 8*a^2*d^4*#1 - 3*c^2*#1^2 + 3*b^2*d^2*#1^2 - 6*a*c*d^2*#1^2 + c^2*#1^3) & ] + b*d^2*(-1 + d*x)*(

1 + d*x)*RootSum[c^2 - 4*c^2*#1^2 + 4*b^2*d^2*#1^2 - 8*a*c*d^2*#1^2 + 6*c^2*#1^4 - 8*b^2*d^2*#1^4 + 16*a*c*d^2

*#1^4 + 16*a^2*d^4*#1^4 - 4*c^2*#1^6 + 4*b^2*d^2*#1^6 - 8*a*c*d^2*#1^6 + c^2*#1^8 & , (-(c^2*Log[-(Sqrt[-d^2]*

x) + Sqrt[1 - d^2*x^2] - #1]) + 7*c^2*Log[-(Sqrt[-d^2]*x) + Sqrt[1 - d^2*x^2] - #1]*#1^2 - 4*b^2*d^2*Log[-(Sqr

t[-d^2]*x) + Sqrt[1 - d^2*x^2] - #1]*#1^2 + 8*a*c*d^2*Log[-(Sqrt[-d^2]*x) + Sqrt[1 - d^2*x^2] - #1]*#1^2 - 7*c

^2*Log[-(Sqrt[-d^2]*x) + Sqrt[1 - d^2*x^2] - #1]*#1^4 + 4*b^2*d^2*Log[-(Sqrt[-d^2]*x) + Sqrt[1 - d^2*x^2] - #1

]*#1^4 - 8*a*c*d^2*Log[-(Sqrt[-d^2]*x) + Sqrt[1 - d^2*x^2] - #1]*#1^4 + c^2*Log[-(Sqrt[-d^2]*x) + Sqrt[1 - d^2

*x^2] - #1]*#1^6)/(-(c^2*#1) + b^2*d^2*#1 - 2*a*c*d^2*#1 + 3*c^2*#1^3 - 4*b^2*d^2*#1^3 + 8*a*c*d^2*#1^3 + 8*a^

2*d^4*#1^3 - 3*c^2*#1^5 + 3*b^2*d^2*#1^5 - 6*a*c*d^2*#1^5 + c^2*#1^7) & ])/(4*(c + d*(-b + a*d))*(c + d*(b + a

*d))*(1 - d*x)*(1 + d*x))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.15, size = 11142, normalized size = 25.15


method	result	size

default	$\text {Expression too large to display}$	$11142$

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-d*x+1)^(3/2)/(d*x+1)^(3/2)/(c*x^2+b*x+a),x,method=_RETURNVERBOSE)

[Out]

result too large to display

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 21628 vs. $2 (404) = 808$.
time = 28.51, size = 21628, normalized size = 48.82 \begin {gather*} \text {Too large to display} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-d*x+1)^(3/2)/(d*x+1)^(3/2)/(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

1/2*(2*b*d^4*x^2 - 2*b*d^2 - sqrt(2)*(a^2*d^4 - (b^2 - 2*a*c)*d^2 - (a^2*d^6 - (b^2 - 2*a*c)*d^4 + c^2*d^2)*x^

2 + c^2)*sqrt(-((b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 - 2*a^3*c^3)*d^6 - 2*c^6 - 3*(b^4*c^2 - 4*a*b^2*c^3 + 2*a^2*c

^4)*d^4 + 3*(b^2*c^4 - 2*a*c^5)*d^2 + ((a^6*b^2 - 4*a^7*c)*d^12 - 3*(a^4*b^4 - 6*a^5*b^2*c + 8*a^6*c^2)*d^10 +

3*(a^2*b^6 - 8*a^3*b^4*c + 21*a^4*b^2*c^2 - 20*a^5*c^3)*d^8 + b^2*c^6 - 4*a*c^7 - (b^8 - 10*a*b^6*c + 42*a^2*

b^4*c^2 - 92*a^3*b^2*c^3 + 80*a^4*c^4)*d^6 + 3*(b^6*c^2 - 8*a*b^4*c^3 + 21*a^2*b^2*c^4 - 20*a^3*c^5)*d^4 - 3*(

b^4*c^4 - 6*a*b^2*c^5 + 8*a^2*c^6)*d^2)*sqrt((9*b^2*c^8*d^4 + (b^10 - 8*a*b^8*c + 22*a^2*b^6*c^2 - 24*a^3*b^4*

c^3 + 9*a^4*b^2*c^4)*d^12 - 6*(b^8*c^2 - 6*a*b^6*c^3 + 11*a^2*b^4*c^4 - 6*a^3*b^2*c^5)*d^10 + 3*(5*b^6*c^4 - 2

0*a*b^4*c^5 + 18*a^2*b^2*c^6)*d^8 - 18*(b^4*c^6 - 2*a*b^2*c^7)*d^6)/((a^12*b^2 - 4*a^13*c)*d^24 - 6*(a^10*b^4

- 6*a^11*b^2*c + 8*a^12*c^2)*d^22 + 3*(5*a^8*b^6 - 40*a^9*b^4*c + 102*a^10*b^2*c^2 - 88*a^11*c^3)*d^20 - 10*(2

*a^6*b^8 - 20*a^7*b^6*c + 75*a^8*b^4*c^2 - 130*a^9*b^2*c^3 + 88*a^10*c^4)*d^18 + 15*(a^4*b^10 - 12*a^5*b^8*c +

60*a^6*b^6*c^2 - 160*a^7*b^4*c^3 + 225*a^8*b^2*c^4 - 132*a^9*c^5)*d^16 - 6*(a^2*b^12 - 14*a^3*b^10*c + 90*a^4

*b^8*c^2 - 340*a^5*b^6*c^3 + 770*a^6*b^4*c^4 - 972*a^7*b^2*c^5 + 528*a^8*c^6)*d^14 + b^2*c^12 - 4*a*c^13 + (b^

14 - 16*a*b^12*c + 138*a^2*b^10*c^2 - 760*a^3*b^8*c^3 + 2650*a^4*b^6*c^4 - 5712*a^5*b^4*c^5 + 6972*a^6*b^2*c^6

- 3696*a^7*c^7)*d^12 - 6*(b^12*c^2 - 14*a*b^10*c^3 + 90*a^2*b^8*c^4 - 340*a^3*b^6*c^5 + 770*a^4*b^4*c^6 - 972

*a^5*b^2*c^7 + 528*a^6*c^8)*d^10 + 15*(b^10*c^4 - 12*a*b^8*c^5 + 60*a^2*b^6*c^6 - 160*a^3*b^4*c^7 + 225*a^4*b^

2*c^8 - 132*a^5*c^9)*d^8 - 10*(2*b^8*c^6 - 20*a*b^6*c^7 + 75*a^2*b^4*c^8 - 130*a^3*b^2*c^9 + 88*a^4*c^10)*d^6

+ 3*(5*b^6*c^8 - 40*a*b^4*c^9 + 102*a^2*b^2*c^10 - 88*a^3*c^11)*d^4 - 6*(b^4*c^10 - 6*a*b^2*c^11 + 8*a^2*c^12)

*d^2)))/((a^6*b^2 - 4*a^7*c)*d^12 - 3*(a^4*b^4 - 6*a^5*b^2*c + 8*a^6*c^2)*d^10 + 3*(a^2*b^6 - 8*a^3*b^4*c + 21

*a^4*b^2*c^2 - 20*a^5*c^3)*d^8 + b^2*c^6 - 4*a*c^7 - (b^8 - 10*a*b^6*c + 42*a^2*b^4*c^2 - 92*a^3*b^2*c^3 + 80*

a^4*c^4)*d^6 + 3*(b^6*c^2 - 8*a*b^4*c^3 + 21*a^2*b^2*c^4 - 20*a^3*c^5)*d^4 - 3*(b^4*c^4 - 6*a*b^2*c^5 + 8*a^2*

c^6)*d^2))*log(-(12*a*b*c^7*d^2 + 4*(a*b^5*c^3 - 4*a^2*b^3*c^4 + 3*a^3*b*c^5)*d^6 - 12*(a*b^3*c^5 - 2*a^2*b*c^

6)*d^4 + 2*((a^6*b^2*c^3 - 4*a^7*c^4)*d^12 + b^2*c^9 - 4*a*c^10 - 3*(a^4*b^4*c^3 - 6*a^5*b^2*c^4 + 8*a^6*c^5)*

d^10 + 3*(a^2*b^6*c^3 - 8*a^3*b^4*c^4 + 21*a^4*b^2*c^5 - 20*a^5*c^6)*d^8 - (b^8*c^3 - 10*a*b^6*c^4 + 42*a^2*b^

4*c^5 - 92*a^3*b^2*c^6 + 80*a^4*c^7)*d^6 + 3*(b^6*c^5 - 8*a*b^4*c^6 + 21*a^2*b^2*c^7 - 20*a^3*c^8)*d^4 - 3*(b^

4*c^7 - 6*a*b^2*c^8 + 8*a^2*c^9)*d^2)*x*sqrt((9*b^2*c^8*d^4 + (b^10 - 8*a*b^8*c + 22*a^2*b^6*c^2 - 24*a^3*b^4*