3.30.31 \(\int \frac {a b c-b^2 x+a^2 x^2}{\sqrt {c+b x+a x^2} (c+b x^2)} \, dx\)

Optimal. Leaf size=343 \[ \frac {\text {RootSum}\left [\text {$\#$1}^4 b+4 \text {$\#$1}^2 a c-2 \text {$\#$1}^2 b c-4 \text {$\#$1} \sqrt {a} b c+b^2 c+b c^2\& ,\frac {\text {$\#$1}^2 b^3 \left (-\log \left (-\text {$\#$1}+\sqrt {a x^2+b x+c}-\sqrt {a} x\right )\right )-2 \text {$\#$1} a^{3/2} b^2 c \log \left (-\text {$\#$1}+\sqrt {a x^2+b x+c}-\sqrt {a} x\right )+2 \text {$\#$1} a^{5/2} c \log \left (-\text {$\#$1}+\sqrt {a x^2+b x+c}-\sqrt {a} x\right )-a^2 b c \log \left (-\text {$\#$1}+\sqrt {a x^2+b x+c}-\sqrt {a} x\right )+a b^3 c \log \left (-\text {$\#$1}+\sqrt {a x^2+b x+c}-\sqrt {a} x\right )+b^3 c \log \left (-\text {$\#$1}+\sqrt {a x^2+b x+c}-\sqrt {a} x\right )}{\text {$\#$1}^3 b+2 \text {$\#$1} a c-\text {$\#$1} b c-\sqrt {a} b c}\& \right ]}{2 b}-\frac {a^{3/2} \log \left (-2 \sqrt {a} \sqrt {a x^2+b x+c}+2 a x+b\right )}{b} \]

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Rubi [B]  time = 5.11, antiderivative size = 1002, normalized size of antiderivative = 2.92, number of steps used = 8, number of rules used = 6, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1078, 621, 206, 1036, 1030, 205} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {b+2 a x}{2 \sqrt {a} \sqrt {a x^2+b x+c}}\right ) a^{3/2}}{b}+\frac {\sqrt {\sqrt {c} a^2+(1-b) b \sqrt {c} a-b \left (b \sqrt {c}-\sqrt {b^3+c b^2-2 a c b+a^2 c}\right )} \sqrt {b^4-a c b^3+\left (a^2 c-a \sqrt {c} \sqrt {b^3+c b^2-2 a c b+a^2 c}\right ) b^2+a^2 c b-a^3 c+a^2 \sqrt {c} \sqrt {b^3+c b^2-2 a c b+a^2 c}} \tan ^{-1}\left (\frac {b \sqrt {c} \left (\sqrt {c} a^2+(1-b) b \sqrt {c} a-b^2 \sqrt {c}+b \sqrt {b^3+c b^2-2 a c b+a^2 c}\right )-\left (b^4-a \left (a-b^2\right ) \left (a c-b c-\sqrt {b^3+c b^2-2 a c b+a^2 c} \sqrt {c}\right )\right ) x}{\sqrt {2} \sqrt [4]{c} \sqrt {b^4-a c b^3+a^2 c b^2+a^2 c b-a^3 c+a \left (a-b^2\right ) \sqrt {c} \sqrt {b^3+c b^2-2 a c b+a^2 c}} \sqrt {\sqrt {c} a^2+(1-b) b \sqrt {c} a-b \left (b \sqrt {c}-\sqrt {b^3+c b^2-2 a c b+a^2 c}\right )} \sqrt {a x^2+b x+c}}\right )}{\sqrt {2} b \sqrt [4]{c} \sqrt {b^3+c b^2-2 a c b+a^2 c}}-\frac {\sqrt {\sqrt {c} a^2+(1-b) b \sqrt {c} a-b \left (\sqrt {c} b+\sqrt {b^3+c b^2-2 a c b+a^2 c}\right )} \sqrt {b^4-a c b^3+\left (c a^2+\sqrt {c} \sqrt {b^3+c b^2-2 a c b+a^2 c} a\right ) b^2+a^2 c b-a^2 \sqrt {c} \left (\sqrt {c} a+\sqrt {b^3+c b^2-2 a c b+a^2 c}\right )} \tan ^{-1}\left (\frac {b \sqrt {c} \left (\sqrt {c} a^2+(1-b) b \sqrt {c} a-b \left (\sqrt {c} b+\sqrt {b^3+c b^2-2 a c b+a^2 c}\right )\right )-\left (b^4-a \left (a-b^2\right ) \left (a c-b c+\sqrt {b^3+c b^2-2 a c b+a^2 c} \sqrt {c}\right )\right ) x}{\sqrt {2} \sqrt [4]{c} \sqrt {b^4-a c b^3+a^2 c b^2+a^2 c b-a^3 c-a \left (a-b^2\right ) \sqrt {c} \sqrt {b^3+c b^2-2 a c b+a^2 c}} \sqrt {\sqrt {c} a^2+(1-b) b \sqrt {c} a-b \left (\sqrt {c} b+\sqrt {b^3+c b^2-2 a c b+a^2 c}\right )} \sqrt {a x^2+b x+c}}\right )}{\sqrt {2} b \sqrt [4]{c} \sqrt {b^3+c b^2-2 a c b+a^2 c}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a^2*Sqrt[c] + a*(1 - b)*b*Sqrt[c] - b*(b*Sqrt[c] - Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c])]*Sqrt[b^4 - a^3*
c + a^2*b*c - a*b^3*c + a^2*Sqrt[c]*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c] + b^2*(a^2*c - a*Sqrt[c]*Sqrt[b^3 + a^
2*c - 2*a*b*c + b^2*c])]*ArcTan[(b*Sqrt[c]*(a^2*Sqrt[c] + a*(1 - b)*b*Sqrt[c] - b^2*Sqrt[c] + b*Sqrt[b^3 + a^2
*c - 2*a*b*c + b^2*c]) - (b^4 - a*(a - b^2)*(a*c - b*c - Sqrt[c]*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c]))*x)/(Sqr
t[2]*c^(1/4)*Sqrt[b^4 - a^3*c + a^2*b*c + a^2*b^2*c - a*b^3*c + a*(a - b^2)*Sqrt[c]*Sqrt[b^3 + a^2*c - 2*a*b*c
+ b^2*c]]*Sqrt[a^2*Sqrt[c] + a*(1 - b)*b*Sqrt[c] - b*(b*Sqrt[c] - Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c])]*Sqrt[
c + b*x + a*x^2])])/(Sqrt[2]*b*c^(1/4)*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c]) - (Sqrt[a^2*Sqrt[c] + a*(1 - b)*b*
Sqrt[c] - b*(b*Sqrt[c] + Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c])]*Sqrt[b^4 + a^2*b*c - a*b^3*c - a^2*Sqrt[c]*(a*S
qrt[c] + Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c]) + b^2*(a^2*c + a*Sqrt[c]*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c])]*A
rcTan[(b*Sqrt[c]*(a^2*Sqrt[c] + a*(1 - b)*b*Sqrt[c] - b*(b*Sqrt[c] + Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c])) - (
b^4 - a*(a - b^2)*(a*c - b*c + Sqrt[c]*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c]))*x)/(Sqrt[2]*c^(1/4)*Sqrt[b^4 - a^
3*c + a^2*b*c + a^2*b^2*c - a*b^3*c - a*(a - b^2)*Sqrt[c]*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c]]*Sqrt[a^2*Sqrt[c
] + a*(1 - b)*b*Sqrt[c] - b*(b*Sqrt[c] + Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c])]*Sqrt[c + b*x + a*x^2])])/(Sqrt[
2]*b*c^(1/4)*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c]) + (a^(3/2)*ArcTanh[(b + 2*a*x)/(2*Sqrt[a]*Sqrt[c + b*x + a*x
^2])])/b

Rule 205

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

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 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 1030

Int[((g_) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[-2*
a*g*h, Subst[Int[1/Simp[2*a^2*g*h*c + a*e*x^2, x], x], x, Simp[a*h - g*c*x, x]/Sqrt[d + e*x + f*x^2]], x] /; F
reeQ[{a, c, d, e, f, g, h}, x] && EqQ[a*h^2*e + 2*g*h*(c*d - a*f) - g^2*c*e, 0]

Rule 1036

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

Rule 1078

Int[((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Sym
bol] :> Dist[C/c, Int[1/Sqrt[d + e*x + f*x^2], x], x] + Dist[1/c, Int[(A*c - a*C + B*c*x)/((a + c*x^2)*Sqrt[d
+ e*x + f*x^2]), x], x] /; FreeQ[{a, c, d, e, f, A, B, C}, x] && NeQ[e^2 - 4*d*f, 0]

Rubi steps

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

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Mathematica [A]  time = 0.82, size = 375, normalized size = 1.09 \begin {gather*} \frac {1}{2} \left (\frac {2 a^{3/2} \tanh ^{-1}\left (\frac {2 a x+b}{2 \sqrt {a} \sqrt {x (a x+b)+c}}\right )}{b}+\frac {\left (-a^2 \sqrt {c}+a b^2 \sqrt {c}+(-b)^{5/2}\right ) \tan ^{-1}\left (\frac {2 a \sqrt {-b} \sqrt {c} x+b^2 (-x)-2 b c+\sqrt {-b} b \sqrt {c}}{2 \sqrt [4]{c} \sqrt {b \left (a \sqrt {c}+b \left (\sqrt {-b}-\sqrt {c}\right )\right )} \sqrt {x (a x+b)+c}}\right )}{\sqrt {-b} \sqrt [4]{c} \sqrt {b \left (a \sqrt {c}+b \left (\sqrt {-b}-\sqrt {c}\right )\right )}}+\frac {\left (a^2 \sqrt {c}-a b^2 \sqrt {c}+(-b)^{5/2}\right ) \tanh ^{-1}\left (\frac {2 a \sqrt {-b} \sqrt {c} x+b^2 x+2 b c+\sqrt {-b} b \sqrt {c}}{2 \sqrt [4]{c} \sqrt {b \left (b \left (\sqrt {-b}+\sqrt {c}\right )-a \sqrt {c}\right )} \sqrt {x (a x+b)+c}}\right )}{\sqrt {-b} \sqrt [4]{c} \sqrt {b \left (b \left (\sqrt {-b}+\sqrt {c}\right )-a \sqrt {c}\right )}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.53, size = 347, normalized size = 1.01 \begin {gather*} -\frac {a^{3/2} \log \left (b^2+2 a b x-2 \sqrt {a} b \sqrt {c+b x+a x^2}\right )}{b}+\frac {\text {RootSum}\left [b^2 c+b c^2-4 \sqrt {a} b c \text {$\#$1}+4 a c \text {$\#$1}^2-2 b c \text {$\#$1}^2+b \text {$\#$1}^4\&,\frac {-a^2 b c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )+b^3 c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )+a b^3 c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )+2 a^{5/2} c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}-2 a^{3/2} b^2 c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}-b^3 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-\sqrt {a} b c+2 a c \text {$\#$1}-b c \text {$\#$1}+b \text {$\#$1}^3}\&\right ]}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((a^(3/2)*Log[b^2 + 2*a*b*x - 2*Sqrt[a]*b*Sqrt[c + b*x + a*x^2]])/b) + RootSum[b^2*c + b*c^2 - 4*Sqrt[a]*b*c*
#1 + 4*a*c*#1^2 - 2*b*c*#1^2 + b*#1^4 & , (-(a^2*b*c*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]) + b^3*c*L
og[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1] + a*b^3*c*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1] + 2*a^(
5/2)*c*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1 - 2*a^(3/2)*b^2*c*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a
*x^2] - #1]*#1 - b^3*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1^2)/(-(Sqrt[a]*b*c) + 2*a*c*#1 - b*c*#1
+ b*#1^3) & ]/(2*b)

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Erro
r: Bad Argument Type

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maple [B]  time = 0.64, size = 520, normalized size = 1.52

method	result	size
default	\(\frac {a^{\frac {3}{2}} \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x +c}\right )}{b}-\frac {\left (a \,b^{2} c -\sqrt {-b c}\, b^{2}-a^{2} c \right ) \ln \left (\frac {-\frac {2 \left (-\sqrt {-b c}\, b +a c -b c \right )}{b}+\frac {\sqrt {-b c}\, \left (-\sqrt {-b c}\, b +2 a c \right ) \left (x -\frac {\sqrt {-b c}}{b}\right )}{b c}+2 \sqrt {-\frac {-\sqrt {-b c}\, b +a c -b c}{b}}\, \sqrt {\left (x -\frac {\sqrt {-b c}}{b}\right )^{2} a +\frac {\sqrt {-b c}\, \left (-\sqrt {-b c}\, b +2 a c \right ) \left (x -\frac {\sqrt {-b c}}{b}\right )}{b c}-\frac {-\sqrt {-b c}\, b +a c -b c}{b}}}{x -\frac {\sqrt {-b c}}{b}}\right )}{2 \sqrt {-b c}\, b \sqrt {-\frac {-\sqrt {-b c}\, b +a c -b c}{b}}}-\frac {\left (-a \,b^{2} c -\sqrt {-b c}\, b^{2}+a^{2} c \right ) \ln \left (\frac {-\frac {2 \left (\sqrt {-b c}\, b +a c -b c \right )}{b}-\frac {\sqrt {-b c}\, \left (\sqrt {-b c}\, b +2 a c \right ) \left (x +\frac {\sqrt {-b c}}{b}\right )}{b c}+2 \sqrt {-\frac {\sqrt {-b c}\, b +a c -b c}{b}}\, \sqrt {\left (x +\frac {\sqrt {-b c}}{b}\right )^{2} a -\frac {\sqrt {-b c}\, \left (\sqrt {-b c}\, b +2 a c \right ) \left (x +\frac {\sqrt {-b c}}{b}\right )}{b c}-\frac {\sqrt {-b c}\, b +a c -b c}{b}}}{x +\frac {\sqrt {-b c}}{b}}\right )}{2 \sqrt {-b c}\, b \sqrt {-\frac {\sqrt {-b c}\, b +a c -b c}{b}}}\)	\(520\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a^2*x^2 - b^2*x + a*b*c)/((c + b*x^2)*(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 {a^{2} x^{2} + a b c - b^{2} x}{\left (b x^{2} + c\right ) \sqrt {a x^{2} + b x + c}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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