3.31.52 \(\int \frac {1}{\sqrt {c+b x+a x^2} (c^3+a^3 b^3 x^3)} \, dx\)

Optimal. Leaf size=463 \[ \frac {2 \tanh ^{-1}\left (-\frac {\sqrt {a} b \sqrt {a x^2+b x+c}}{\sqrt {c} \sqrt {a b^2-b^2+c}}+\frac {a b x}{\sqrt {c} \sqrt {a b^2-b^2+c}}+\frac {\sqrt {c}}{\sqrt {a b^2-b^2+c}}\right )}{3 \sqrt {a} c^{5/2} \sqrt {a b^2-b^2+c}}-\frac {\text {RootSum}\left [\text {$\#$1}^4 a^2 b^2+2 \text {$\#$1}^3 a^{3/2} b c-2 \text {$\#$1}^2 a^2 b^2 c-\text {$\#$1}^2 a b^2 c+4 \text {$\#$1}^2 a c^2-2 \text {$\#$1} a^{3/2} b c^2-4 \text {$\#$1} \sqrt {a} b c^2+a^2 b^2 c^2+a b^2 c^2+b^2 c^2\& ,\frac {\text {$\#$1}^2 (-a) b \log \left (-\text {$\#$1}+\sqrt {a x^2+b x+c}-\sqrt {a} x\right )-4 \text {$\#$1} \sqrt {a} c \log \left (-\text {$\#$1}+\sqrt {a x^2+b x+c}-\sqrt {a} x\right )+a b c \log \left (-\text {$\#$1}+\sqrt {a x^2+b x+c}-\sqrt {a} x\right )+2 b c \log \left (-\text {$\#$1}+\sqrt {a x^2+b x+c}-\sqrt {a} x\right )}{-2 \text {$\#$1}^3 a^{3/2} b^2-3 \text {$\#$1}^2 a b c+2 \text {$\#$1} a^{3/2} b^2 c+\text {$\#$1} \sqrt {a} b^2 c-4 \text {$\#$1} \sqrt {a} c^2+a b c^2+2 b c^2}\& \right ]}{3 \sqrt {a} c^2} \]

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Rubi [A]  time = 2.47, antiderivative size = 518, normalized size of antiderivative = 1.12, number of steps used = 9, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {2074, 724, 204, 1035, 1029, 208} \begin {gather*} \frac {\sqrt {2 \left (\sqrt {\left (a^2+a+1\right ) b^4+b^2 (c-a c)+c^2}+c\right )+(1-a) b^2} \tanh ^{-1}\left (\frac {(a+1) b c-a x \left (-\sqrt {\left (a^2+a+1\right ) b^4+b^2 (c-a c)+c^2}+a b^2-c\right )}{\sqrt {a} \sqrt {c} \sqrt {2 \left (\sqrt {\left (a^2+a+1\right ) b^4+b^2 (c-a c)+c^2}+c\right )+(1-a) b^2} \sqrt {a x^2+b x+c}}\right )}{3 \sqrt {a} c^{5/2} \sqrt {\left (a^2+a+1\right ) b^4+b^2 (c-a c)+c^2}}-\frac {\sqrt {-2 \sqrt {\left (a^2+a+1\right ) b^4+b^2 (c-a c)+c^2}+(1-a) b^2+2 c} \tanh ^{-1}\left (\frac {(a+1) b c-a x \left (\sqrt {\left (a^2+a+1\right ) b^4+b^2 (c-a c)+c^2}+a b^2-c\right )}{\sqrt {a} \sqrt {c} \sqrt {-2 \sqrt {\left (a^2+a+1\right ) b^4+b^2 (c-a c)+c^2}+(1-a) b^2+2 c} \sqrt {a x^2+b x+c}}\right )}{3 \sqrt {a} c^{5/2} \sqrt {\left (a^2+a+1\right ) b^4+b^2 (c-a c)+c^2}}-\frac {\tan ^{-1}\left (\frac {(1-2 a) b c-a x \left (b^2-2 c\right )}{2 \sqrt {a} \sqrt {c} \sqrt {(1-a) b^2-c} \sqrt {a x^2+b x+c}}\right )}{3 \sqrt {a} c^{5/2} \sqrt {(1-a) b^2-c}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

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 1029

Int[((g_.) + (h_.)*(x_))/(((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symb
ol] :> Dist[-2*g*(g*b - 2*a*h), Subst[Int[1/Simp[g*(g*b - 2*a*h)*(b^2 - 4*a*c) - (b*d - a*e)*x^2, x], x], x, S
imp[g*b - 2*a*h - (b*h - 2*g*c)*x, x]/Sqrt[d + e*x + f*x^2]], 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] && NeQ[b*d - a*e, 0] && EqQ[h^2*(b*d - a*e) - 2*g*h*(c*d - a*f) + g^2*(
c*e - b*f), 0]

Rule 1035

Int[((g_.) + (h_.)*(x_))/(((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symb
ol] :> With[{q = Rt[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 2]}, Dist[1/(2*q), Int[Simp[h*(b*d - a*e) - g*(c*
d - a*f - q) - (g*(c*e - b*f) - h*(c*d - a*f + q))*x, x]/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] - D
ist[1/(2*q), Int[Simp[h*(b*d - a*e) - g*(c*d - a*f + q) - (g*(c*e - b*f) - h*(c*d - a*f - q))*x, x]/((a + b*x
+ c*x^2)*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] && NeQ[b*d - a*e, 0] && NegQ[b^2 - 4*a*c]

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {c+b x+a x^2} \left (c^3+a^3 b^3 x^3\right )} \, dx &=\int \left (\frac {1}{3 c^2 (c+a b x) \sqrt {c+b x+a x^2}}+\frac {2 c-a b x}{3 c^2 \sqrt {c+b x+a x^2} \left (c^2-a b c x+a^2 b^2 x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {1}{(c+a b x) \sqrt {c+b x+a x^2}} \, dx}{3 c^2}+\frac {\int \frac {2 c-a b x}{\sqrt {c+b x+a x^2} \left (c^2-a b c x+a^2 b^2 x^2\right )} \, dx}{3 c^2}\\ &=-\frac {2 \operatorname {Subst}\left (\int \frac {1}{-4 a b^2 c+4 a^2 b^2 c+4 a c^2-x^2} \, dx,x,\frac {-b c+2 a b c-\left (-a b^2+2 a c\right ) x}{\sqrt {c+b x+a x^2}}\right )}{3 c^2}-\frac {\int \frac {a c^2 \left ((1-a) b^2+2 c-2 \sqrt {\left (1+a+a^2\right ) b^4+c^2+b^2 (c-a c)}\right )-a^2 b c \left ((2+a) b^2+c-\sqrt {\left (1+a+a^2\right ) b^4+c^2+b^2 (c-a c)}\right ) x}{\sqrt {c+b x+a x^2} \left (c^2-a b c x+a^2 b^2 x^2\right )} \, dx}{6 a c^3 \sqrt {\left (1+a+a^2\right ) b^4+c^2+b^2 (c-a c)}}+\frac {\int \frac {a c^2 \left ((1-a) b^2+2 \left (c+\sqrt {\left (1+a+a^2\right ) b^4+c^2+b^2 (c-a c)}\right )\right )-a^2 b c \left ((2+a) b^2+c+\sqrt {\left (1+a+a^2\right ) b^4+c^2+b^2 (c-a c)}\right ) x}{\sqrt {c+b x+a x^2} \left (c^2-a b c x+a^2 b^2 x^2\right )} \, dx}{6 a c^3 \sqrt {\left (1+a+a^2\right ) b^4+c^2+b^2 (c-a c)}}\\ &=-\frac {\tan ^{-1}\left (\frac {(1-2 a) b c-a \left (b^2-2 c\right ) x}{2 \sqrt {a} \sqrt {(1-a) b^2-c} \sqrt {c} \sqrt {c+b x+a x^2}}\right )}{3 \sqrt {a} \sqrt {(1-a) b^2-c} c^{5/2}}+\frac {\left (\left ((1-a) b^2+2 c-2 \sqrt {\left (1+a+a^2\right ) b^4+c^2+b^2 (c-a c)}\right ) \left (-a^2 b c^3 \left ((1-a) b^2+2 c-2 \sqrt {\left (1+a+a^2\right ) b^4+c^2+b^2 (c-a c)}\right )+2 a^2 b c^3 \left ((2+a) b^2+c-\sqrt {\left (1+a+a^2\right ) b^4+c^2+b^2 (c-a c)}\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-9 a^5 (1+a) b^5 c^7 \left ((1-a) b^2+2 \left (c-\sqrt {\left (1+a+a^2\right ) b^4+c^2+b^2 (c-a c)}\right )\right )+(1+a) b c^2 x^2} \, dx,x,\frac {3 a^2 (1+a) b^3 c^3-3 a^3 b^2 c^2 \left (a b^2-c+\sqrt {\left (1+a+a^2\right ) b^4+c^2+b^2 (c-a c)}\right ) x}{\sqrt {c+b x+a x^2}}\right )}{3 c \sqrt {\left (1+a+a^2\right ) b^4+c^2+b^2 (c-a c)}}-\frac {\left (a^2 (1+a) b^3 c^2 \left ((1-a) b^2+2 \left (c+\sqrt {\left (1+a+a^2\right ) b^4+c^2+b^2 (c-a c)}\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-9 a^5 (1+a) b^5 c^7 \left ((1-a) b^2+2 \left (c+\sqrt {\left (1+a+a^2\right ) b^4+c^2+b^2 (c-a c)}\right )\right )+(1+a) b c^2 x^2} \, dx,x,\frac {3 a^2 (1+a) b^3 c^3-3 a^3 b^2 c^2 \left (a b^2-c-\sqrt {\left (1+a+a^2\right ) b^4+c^2+b^2 (c-a c)}\right ) x}{\sqrt {c+b x+a x^2}}\right )}{\sqrt {\left (1+a+a^2\right ) b^4+c^2+b^2 (c-a c)}}\\ &=-\frac {\tan ^{-1}\left (\frac {(1-2 a) b c-a \left (b^2-2 c\right ) x}{2 \sqrt {a} \sqrt {(1-a) b^2-c} \sqrt {c} \sqrt {c+b x+a x^2}}\right )}{3 \sqrt {a} \sqrt {(1-a) b^2-c} c^{5/2}}+\frac {\sqrt {(1-a) b^2+2 \left (c+\sqrt {\left (1+a+a^2\right ) b^4+c^2+b^2 (c-a c)}\right )} \tanh ^{-1}\left (\frac {(1+a) b c-a \left (a b^2-c-\sqrt {\left (1+a+a^2\right ) b^4+c^2+b^2 (c-a c)}\right ) x}{\sqrt {a} \sqrt {c} \sqrt {(1-a) b^2+2 \left (c+\sqrt {\left (1+a+a^2\right ) b^4+c^2+b^2 (c-a c)}\right )} \sqrt {c+b x+a x^2}}\right )}{3 \sqrt {a} c^{5/2} \sqrt {\left (1+a+a^2\right ) b^4+c^2+b^2 (c-a c)}}-\frac {\sqrt {(1-a) b^2+2 c-2 \sqrt {\left (1+a+a^2\right ) b^4+c^2+b^2 (c-a c)}} \tanh ^{-1}\left (\frac {(1+a) b c-a \left (a b^2-c+\sqrt {\left (1+a+a^2\right ) b^4+c^2+b^2 (c-a c)}\right ) x}{\sqrt {a} \sqrt {c} \sqrt {(1-a) b^2+2 c-2 \sqrt {\left (1+a+a^2\right ) b^4+c^2+b^2 (c-a c)}} \sqrt {c+b x+a x^2}}\right )}{3 \sqrt {a} c^{5/2} \sqrt {\left (1+a+a^2\right ) b^4+c^2+b^2 (c-a c)}}\\ \end {align*}

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Mathematica [A]  time = 1.25, size = 309, normalized size = 0.67 \begin {gather*} \frac {-\frac {\tan ^{-1}\left (\frac {-a b^2 x+b (c-2 a c)+2 a c x}{2 \sqrt {a} \sqrt {c} \sqrt {-\left ((a-1) b^2\right )-c} \sqrt {x (a x+b)+c}}\right )}{\sqrt {-\left ((a-1) b^2\right )-c}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [3]{-1} a b^2 x+b \left (2 \sqrt [3]{-1} a c+c\right )+2 a c x}{2 \sqrt {a} \sqrt {c} \sqrt {\left ((-1)^{2/3} a+\sqrt [3]{-1}\right ) b^2+c} \sqrt {x (a x+b)+c}}\right )}{\sqrt {\left ((-1)^{2/3} a+\sqrt [3]{-1}\right ) b^2+c}}+\frac {\tanh ^{-1}\left (\frac {-(-1)^{2/3} a b^2 x+b \left (c-2 (-1)^{2/3} a c\right )+2 a c x}{2 \sqrt {a} \sqrt {c} \sqrt {c-\sqrt [3]{-1} \left (a+\sqrt [3]{-1}\right ) b^2} \sqrt {x (a x+b)+c}}\right )}{\sqrt {c-\sqrt [3]{-1} \left (a+\sqrt [3]{-1}\right ) b^2}}}{3 \sqrt {a} c^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-(ArcTan[(b*(c - 2*a*c) - a*b^2*x + 2*a*c*x)/(2*Sqrt[a]*Sqrt[-((-1 + a)*b^2) - c]*Sqrt[c]*Sqrt[c + x*(b + a*x
)])]/Sqrt[-((-1 + a)*b^2) - c]) + ArcTanh[(b*(c + 2*(-1)^(1/3)*a*c) + (-1)^(1/3)*a*b^2*x + 2*a*c*x)/(2*Sqrt[a]
*Sqrt[c]*Sqrt[((-1)^(1/3) + (-1)^(2/3)*a)*b^2 + c]*Sqrt[c + x*(b + a*x)])]/Sqrt[((-1)^(1/3) + (-1)^(2/3)*a)*b^
2 + c] + ArcTanh[(b*(c - 2*(-1)^(2/3)*a*c) - (-1)^(2/3)*a*b^2*x + 2*a*c*x)/(2*Sqrt[a]*Sqrt[c]*Sqrt[-((-1)^(1/3
)*((-1)^(1/3) + a)*b^2) + c]*Sqrt[c + x*(b + a*x)])]/Sqrt[-((-1)^(1/3)*((-1)^(1/3) + a)*b^2) + c])/(3*Sqrt[a]*
c^(5/2))

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IntegrateAlgebraic [A]  time = 0.72, size = 463, normalized size = 1.00 \begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {\sqrt {c}}{\sqrt {-b^2+a b^2+c}}+\frac {a b x}{\sqrt {c} \sqrt {-b^2+a b^2+c}}-\frac {\sqrt {a} b \sqrt {c+b x+a x^2}}{\sqrt {c} \sqrt {-b^2+a b^2+c}}\right )}{3 \sqrt {a} c^{5/2} \sqrt {-b^2+a b^2+c}}-\frac {\text {RootSum}\left [b^2 c^2+a b^2 c^2+a^2 b^2 c^2-4 \sqrt {a} b c^2 \text {$\#$1}-2 a^{3/2} b c^2 \text {$\#$1}-a b^2 c \text {$\#$1}^2-2 a^2 b^2 c \text {$\#$1}^2+4 a c^2 \text {$\#$1}^2+2 a^{3/2} b c \text {$\#$1}^3+a^2 b^2 \text {$\#$1}^4\&,\frac {2 b c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )+a b c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )-4 \sqrt {a} c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}-a b \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{2 b c^2+a b c^2+\sqrt {a} b^2 c \text {$\#$1}+2 a^{3/2} b^2 c \text {$\#$1}-4 \sqrt {a} c^2 \text {$\#$1}-3 a b c \text {$\#$1}^2-2 a^{3/2} b^2 \text {$\#$1}^3}\&\right ]}{3 \sqrt {a} c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*ArcTanh[Sqrt[c]/Sqrt[-b^2 + a*b^2 + c] + (a*b*x)/(Sqrt[c]*Sqrt[-b^2 + a*b^2 + c]) - (Sqrt[a]*b*Sqrt[c + b*x
+ a*x^2])/(Sqrt[c]*Sqrt[-b^2 + a*b^2 + c])])/(3*Sqrt[a]*c^(5/2)*Sqrt[-b^2 + a*b^2 + c]) - RootSum[b^2*c^2 + a
*b^2*c^2 + a^2*b^2*c^2 - 4*Sqrt[a]*b*c^2*#1 - 2*a^(3/2)*b*c^2*#1 - a*b^2*c*#1^2 - 2*a^2*b^2*c*#1^2 + 4*a*c^2*#
1^2 + 2*a^(3/2)*b*c*#1^3 + a^2*b^2*#1^4 & , (2*b*c*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1] + a*b*c*Log[
-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1] - 4*Sqrt[a]*c*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1 - a
*b*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1^2)/(2*b*c^2 + a*b*c^2 + Sqrt[a]*b^2*c*#1 + 2*a^(3/2)*b^2*
c*#1 - 4*Sqrt[a]*c^2*#1 - 3*a*b*c*#1^2 - 2*a^(3/2)*b^2*#1^3) & ]/(3*Sqrt[a]*c^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/(a*x^2+b*x+c)^(1/2)/(a^3*b^3*x^3+c^3),x, algorithm="fricas")

[Out]

Timed out

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [B]  time = 1.02, size = 1195, normalized size = 2.58

method	result	size
default	\(\frac {4 a b \ln \left (\frac {\frac {2 c \left (a \,b^{2}-b^{2}+c \right )}{a \,b^{2}}+\frac {\left (b^{2}-2 c \right ) \left (x +\frac {c}{a b}\right )}{b}+2 \sqrt {\frac {c \left (a \,b^{2}-b^{2}+c \right )}{a \,b^{2}}}\, \sqrt {\left (x +\frac {c}{a b}\right )^{2} a +\frac {\left (b^{2}-2 c \right ) \left (x +\frac {c}{a b}\right )}{b}+\frac {c \left (a \,b^{2}-b^{2}+c \right )}{a \,b^{2}}}}{x +\frac {c}{a b}}\right )}{\left (3 a b c +\sqrt {-3 a^{2} b^{2} c^{2}}\right ) \left (-3 a b c +\sqrt {-3 a^{2} b^{2} c^{2}}\right ) \sqrt {\frac {c \left (a \,b^{2}-b^{2}+c \right )}{a \,b^{2}}}}-\frac {2 a b \sqrt {2}\, \ln \left (\frac {\frac {2 a^{2} b^{3} c +a \,b^{3} c -a b \,c^{2}-\sqrt {-3 a^{2} b^{2} c^{2}}\, b^{2}-\sqrt {-3 a^{2} b^{2} c^{2}}\, c}{a^{2} b^{3}}+\frac {\left (a \,b^{3}+a b c -\sqrt {-3 a^{2} b^{2} c^{2}}\right ) \left (x +\frac {-a b c +\sqrt {-3 a^{2} b^{2} c^{2}}}{2 a^{2} b^{2}}\right )}{a \,b^{2}}+\frac {\sqrt {2}\, \sqrt {\frac {2 a^{2} b^{3} c +a \,b^{3} c -a b \,c^{2}-\sqrt {-3 a^{2} b^{2} c^{2}}\, b^{2}-\sqrt {-3 a^{2} b^{2} c^{2}}\, c}{a^{2} b^{3}}}\, \sqrt {4 \left (x +\frac {-a b c +\sqrt {-3 a^{2} b^{2} c^{2}}}{2 a^{2} b^{2}}\right )^{2} a +\frac {4 \left (a \,b^{3}+a b c -\sqrt {-3 a^{2} b^{2} c^{2}}\right ) \left (x +\frac {-a b c +\sqrt {-3 a^{2} b^{2} c^{2}}}{2 a^{2} b^{2}}\right )}{a \,b^{2}}+\frac {4 a^{2} b^{3} c +2 a \,b^{3} c -2 a b \,c^{2}-2 \sqrt {-3 a^{2} b^{2} c^{2}}\, b^{2}-2 \sqrt {-3 a^{2} b^{2} c^{2}}\, c}{a^{2} b^{3}}}}{2}}{x +\frac {-a b c +\sqrt {-3 a^{2} b^{2} c^{2}}}{2 a^{2} b^{2}}}\right )}{\left (-3 a b c +\sqrt {-3 a^{2} b^{2} c^{2}}\right ) \sqrt {-3 a^{2} b^{2} c^{2}}\, \sqrt {\frac {2 a^{2} b^{3} c +a \,b^{3} c -a b \,c^{2}-\sqrt {-3 a^{2} b^{2} c^{2}}\, b^{2}-\sqrt {-3 a^{2} b^{2} c^{2}}\, c}{a^{2} b^{3}}}}-\frac {2 a b \sqrt {2}\, \ln \left (\frac {\frac {2 a^{2} b^{3} c +a \,b^{3} c -a b \,c^{2}+\sqrt {-3 a^{2} b^{2} c^{2}}\, b^{2}+\sqrt {-3 a^{2} b^{2} c^{2}}\, c}{a^{2} b^{3}}+\frac {\left (a \,b^{3}+a b c +\sqrt {-3 a^{2} b^{2} c^{2}}\right ) \left (x -\frac {a b c +\sqrt {-3 a^{2} b^{2} c^{2}}}{2 a^{2} b^{2}}\right )}{a \,b^{2}}+\frac {\sqrt {2}\, \sqrt {\frac {2 a^{2} b^{3} c +a \,b^{3} c -a b \,c^{2}+\sqrt {-3 a^{2} b^{2} c^{2}}\, b^{2}+\sqrt {-3 a^{2} b^{2} c^{2}}\, c}{a^{2} b^{3}}}\, \sqrt {4 \left (x -\frac {a b c +\sqrt {-3 a^{2} b^{2} c^{2}}}{2 a^{2} b^{2}}\right )^{2} a +\frac {4 \left (a \,b^{3}+a b c +\sqrt {-3 a^{2} b^{2} c^{2}}\right ) \left (x -\frac {a b c +\sqrt {-3 a^{2} b^{2} c^{2}}}{2 a^{2} b^{2}}\right )}{a \,b^{2}}+\frac {4 a^{2} b^{3} c +2 a \,b^{3} c -2 a b \,c^{2}+2 \sqrt {-3 a^{2} b^{2} c^{2}}\, b^{2}+2 \sqrt {-3 a^{2} b^{2} c^{2}}\, c}{a^{2} b^{3}}}}{2}}{x -\frac {a b c +\sqrt {-3 a^{2} b^{2} c^{2}}}{2 a^{2} b^{2}}}\right )}{\left (3 a b c +\sqrt {-3 a^{2} b^{2} c^{2}}\right ) \sqrt {-3 a^{2} b^{2} c^{2}}\, \sqrt {\frac {2 a^{2} b^{3} c +a \,b^{3} c -a b \,c^{2}+\sqrt {-3 a^{2} b^{2} c^{2}}\, b^{2}+\sqrt {-3 a^{2} b^{2} c^{2}}\, c}{a^{2} b^{3}}}}\)	\(1195\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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