3.111 \(\int \frac{A+B x^2}{a-\sqrt{a c} x^2+c x^4} \, dx\)

Optimal. Leaf size=414 \[ -\frac{\left (A-\frac{\sqrt{a} B}{\sqrt{c}}\right ) \log \left (-x \sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}}+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{a} \sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}}}+\frac{\left (A-\frac{\sqrt{a} B}{\sqrt{c}}\right ) \log \left (x \sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}}+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{a} \sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}}}-\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}}-2 \sqrt{c} x}{\sqrt{2 \sqrt{a} \sqrt{c}-\sqrt{a c}}}\right )}{2 \sqrt{a} \sqrt{c} \sqrt{2 \sqrt{a} \sqrt{c}-\sqrt{a c}}}+\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}}+2 \sqrt{c} x}{\sqrt{2 \sqrt{a} \sqrt{c}-\sqrt{a c}}}\right )}{2 \sqrt{a} \sqrt{c} \sqrt{2 \sqrt{a} \sqrt{c}-\sqrt{a c}}} \]

[Out]

-((Sqrt[a]*B + A*Sqrt[c])*ArcTan[(Sqrt[2*Sqrt[a]*Sqrt[c] + Sqrt[a*c]] - 2*Sqrt[c]*x)/Sqrt[2*Sqrt[a]*Sqrt[c] -
Sqrt[a*c]]])/(2*Sqrt[a]*Sqrt[c]*Sqrt[2*Sqrt[a]*Sqrt[c] - Sqrt[a*c]]) + ((Sqrt[a]*B + A*Sqrt[c])*ArcTan[(Sqrt[2
*Sqrt[a]*Sqrt[c] + Sqrt[a*c]] + 2*Sqrt[c]*x)/Sqrt[2*Sqrt[a]*Sqrt[c] - Sqrt[a*c]]])/(2*Sqrt[a]*Sqrt[c]*Sqrt[2*S
qrt[a]*Sqrt[c] - Sqrt[a*c]]) - ((A - (Sqrt[a]*B)/Sqrt[c])*Log[Sqrt[a] - Sqrt[2*Sqrt[a]*Sqrt[c] + Sqrt[a*c]]*x
+ Sqrt[c]*x^2])/(4*Sqrt[a]*Sqrt[2*Sqrt[a]*Sqrt[c] + Sqrt[a*c]]) + ((A - (Sqrt[a]*B)/Sqrt[c])*Log[Sqrt[a] + Sqr
t[2*Sqrt[a]*Sqrt[c] + Sqrt[a*c]]*x + Sqrt[c]*x^2])/(4*Sqrt[a]*Sqrt[2*Sqrt[a]*Sqrt[c] + Sqrt[a*c]])

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Rubi [A]  time = 0.453327, antiderivative size = 414, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {1169, 634, 618, 204, 628} \[ -\frac{\left (A-\frac{\sqrt{a} B}{\sqrt{c}}\right ) \log \left (-x \sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}}+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{a} \sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}}}+\frac{\left (A-\frac{\sqrt{a} B}{\sqrt{c}}\right ) \log \left (x \sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}}+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{a} \sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}}}-\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}}-2 \sqrt{c} x}{\sqrt{2 \sqrt{a} \sqrt{c}-\sqrt{a c}}}\right )}{2 \sqrt{a} \sqrt{c} \sqrt{2 \sqrt{a} \sqrt{c}-\sqrt{a c}}}+\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}}+2 \sqrt{c} x}{\sqrt{2 \sqrt{a} \sqrt{c}-\sqrt{a c}}}\right )}{2 \sqrt{a} \sqrt{c} \sqrt{2 \sqrt{a} \sqrt{c}-\sqrt{a c}}} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x^2)/(a - Sqrt[a*c]*x^2 + c*x^4),x]

[Out]

-((Sqrt[a]*B + A*Sqrt[c])*ArcTan[(Sqrt[2*Sqrt[a]*Sqrt[c] + Sqrt[a*c]] - 2*Sqrt[c]*x)/Sqrt[2*Sqrt[a]*Sqrt[c] -
Sqrt[a*c]]])/(2*Sqrt[a]*Sqrt[c]*Sqrt[2*Sqrt[a]*Sqrt[c] - Sqrt[a*c]]) + ((Sqrt[a]*B + A*Sqrt[c])*ArcTan[(Sqrt[2
*Sqrt[a]*Sqrt[c] + Sqrt[a*c]] + 2*Sqrt[c]*x)/Sqrt[2*Sqrt[a]*Sqrt[c] - Sqrt[a*c]]])/(2*Sqrt[a]*Sqrt[c]*Sqrt[2*S
qrt[a]*Sqrt[c] - Sqrt[a*c]]) - ((A - (Sqrt[a]*B)/Sqrt[c])*Log[Sqrt[a] - Sqrt[2*Sqrt[a]*Sqrt[c] + Sqrt[a*c]]*x
+ Sqrt[c]*x^2])/(4*Sqrt[a]*Sqrt[2*Sqrt[a]*Sqrt[c] + Sqrt[a*c]]) + ((A - (Sqrt[a]*B)/Sqrt[c])*Log[Sqrt[a] + Sqr
t[2*Sqrt[a]*Sqrt[c] + Sqrt[a*c]]*x + Sqrt[c]*x^2])/(4*Sqrt[a]*Sqrt[2*Sqrt[a]*Sqrt[c] + Sqrt[a*c]])

Rule 1169

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r =
Rt[2*q - b/c, 2]}, Dist[1/(2*c*q*r), Int[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(
d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]

Rule 634

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

Rule 618

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

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 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{A+B x^2}{a-\sqrt{a c} x^2+c x^4} \, dx &=\frac{\int \frac{\frac{A \sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}}}{\sqrt{c}}-\left (A-\frac{\sqrt{a} B}{\sqrt{c}}\right ) x}{\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}} x}{\sqrt{c}}+x^2} \, dx}{2 \sqrt{a} \sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}}}+\frac{\int \frac{\frac{A \sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}}}{\sqrt{c}}+\left (A-\frac{\sqrt{a} B}{\sqrt{c}}\right ) x}{\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}} x}{\sqrt{c}}+x^2} \, dx}{2 \sqrt{a} \sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}}}\\ &=\frac{\left (B+\frac{A \sqrt{c}}{\sqrt{a}}\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}} x}{\sqrt{c}}+x^2} \, dx}{4 c}+\frac{\left (B+\frac{A \sqrt{c}}{\sqrt{a}}\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}} x}{\sqrt{c}}+x^2} \, dx}{4 c}-\frac{\left (A-\frac{\sqrt{a} B}{\sqrt{c}}\right ) \int \frac{-\frac{\sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}}}{\sqrt{c}}+2 x}{\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}} x}{\sqrt{c}}+x^2} \, dx}{4 \sqrt{a} \sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}}}+\frac{\left (A-\frac{\sqrt{a} B}{\sqrt{c}}\right ) \int \frac{\frac{\sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}}}{\sqrt{c}}+2 x}{\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}} x}{\sqrt{c}}+x^2} \, dx}{4 \sqrt{a} \sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}}}\\ &=-\frac{\left (A-\frac{\sqrt{a} B}{\sqrt{c}}\right ) \log \left (\sqrt{a}-\sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}} x+\sqrt{c} x^2\right )}{4 \sqrt{a} \sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}}}+\frac{\left (A-\frac{\sqrt{a} B}{\sqrt{c}}\right ) \log \left (\sqrt{a}+\sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}} x+\sqrt{c} x^2\right )}{4 \sqrt{a} \sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}}}-\frac{\left (B+\frac{A \sqrt{c}}{\sqrt{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{2 \sqrt{a} \sqrt{c}-\sqrt{a c}}{c}-x^2} \, dx,x,-\frac{\sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}}}{\sqrt{c}}+2 x\right )}{2 c}-\frac{\left (B+\frac{A \sqrt{c}}{\sqrt{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{2 \sqrt{a} \sqrt{c}-\sqrt{a c}}{c}-x^2} \, dx,x,\frac{\sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}}}{\sqrt{c}}+2 x\right )}{2 c}\\ &=-\frac{\left (B+\frac{A \sqrt{c}}{\sqrt{a}}\right ) \tan ^{-1}\left (\frac{\sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}}-2 \sqrt{c} x}{\sqrt{2 \sqrt{a} \sqrt{c}-\sqrt{a c}}}\right )}{2 \sqrt{c} \sqrt{2 \sqrt{a} \sqrt{c}-\sqrt{a c}}}+\frac{\left (B+\frac{A \sqrt{c}}{\sqrt{a}}\right ) \tan ^{-1}\left (\frac{\sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}}+2 \sqrt{c} x}{\sqrt{2 \sqrt{a} \sqrt{c}-\sqrt{a c}}}\right )}{2 \sqrt{c} \sqrt{2 \sqrt{a} \sqrt{c}-\sqrt{a c}}}-\frac{\left (A-\frac{\sqrt{a} B}{\sqrt{c}}\right ) \log \left (\sqrt{a}-\sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}} x+\sqrt{c} x^2\right )}{4 \sqrt{a} \sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}}}+\frac{\left (A-\frac{\sqrt{a} B}{\sqrt{c}}\right ) \log \left (\sqrt{a}+\sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}} x+\sqrt{c} x^2\right )}{4 \sqrt{a} \sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}}}\\ \end{align*}

Mathematica [C]  time = 0.197111, size = 247, normalized size = 0.6 \[ \frac{\frac{\left (\sqrt{3} \sqrt{a} B \sqrt{c}-i \left (B \sqrt{a c}+2 A c\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{-\sqrt{a c}-i \sqrt{3} \sqrt{a} \sqrt{c}}}\right )}{\sqrt{-\sqrt{a c}-i \sqrt{3} \sqrt{a} \sqrt{c}}}+\frac{\left (\sqrt{3} \sqrt{a} B \sqrt{c}+i \left (B \sqrt{a c}+2 A c\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{-\sqrt{a c}+i \sqrt{3} \sqrt{a} \sqrt{c}}}\right )}{\sqrt{-\sqrt{a c}+i \sqrt{3} \sqrt{a} \sqrt{c}}}}{\sqrt{6} \sqrt{a} c} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x^2)/(a - Sqrt[a*c]*x^2 + c*x^4),x]

[Out]

(((Sqrt[3]*Sqrt[a]*B*Sqrt[c] - I*(2*A*c + B*Sqrt[a*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[(-I)*Sqrt[3]*Sqrt[a]*S
qrt[c] - Sqrt[a*c]]])/Sqrt[(-I)*Sqrt[3]*Sqrt[a]*Sqrt[c] - Sqrt[a*c]] + ((Sqrt[3]*Sqrt[a]*B*Sqrt[c] + I*(2*A*c
+ B*Sqrt[a*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[I*Sqrt[3]*Sqrt[a]*Sqrt[c] - Sqrt[a*c]]])/Sqrt[I*Sqrt[3]*Sqrt[a
]*Sqrt[c] - Sqrt[a*c]])/(Sqrt[6]*Sqrt[a]*c)

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Maple [A]  time = 0.194, size = 404, normalized size = 1. \begin{align*}{\frac{B\sqrt{3}}{12\,a}\ln \left ( x\sqrt{3}\sqrt [4]{ac}-{x}^{2}\sqrt{c}-\sqrt{a} \right ) \left ( ac \right ) ^{{\frac{3}{4}}}{c}^{-{\frac{3}{2}}}}-{\frac{A\sqrt{3}}{12\,c}\ln \left ( x\sqrt{3}\sqrt [4]{ac}-{x}^{2}\sqrt{c}-\sqrt{a} \right ) \left ( ac \right ) ^{{\frac{3}{4}}}{a}^{-{\frac{3}{2}}}}-{\frac{A}{2}\arctan \left ({ \left ( \sqrt{3}\sqrt [4]{ac}-2\,x\sqrt{c} \right ){\frac{1}{\sqrt{4\,\sqrt{a}\sqrt{c}-3\,\sqrt{ac}}}}} \right ){\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{4\,\sqrt{a}\sqrt{c}-3\,\sqrt{ac}}}}}-{\frac{B}{2}\arctan \left ({ \left ( \sqrt{3}\sqrt [4]{ac}-2\,x\sqrt{c} \right ){\frac{1}{\sqrt{4\,\sqrt{a}\sqrt{c}-3\,\sqrt{ac}}}}} \right ){\frac{1}{\sqrt{c}}}{\frac{1}{\sqrt{4\,\sqrt{a}\sqrt{c}-3\,\sqrt{ac}}}}}-{\frac{B\sqrt{3}}{12\,a}\ln \left ({x}^{2}\sqrt{c}+x\sqrt{3}\sqrt [4]{ac}+\sqrt{a} \right ) \left ( ac \right ) ^{{\frac{3}{4}}}{c}^{-{\frac{3}{2}}}}+{\frac{A\sqrt{3}}{12\,c}\ln \left ({x}^{2}\sqrt{c}+x\sqrt{3}\sqrt [4]{ac}+\sqrt{a} \right ) \left ( ac \right ) ^{{\frac{3}{4}}}{a}^{-{\frac{3}{2}}}}+{\frac{A}{2}\arctan \left ({ \left ( 2\,x\sqrt{c}+\sqrt{3}\sqrt [4]{ac} \right ){\frac{1}{\sqrt{4\,\sqrt{a}\sqrt{c}-3\,\sqrt{ac}}}}} \right ){\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{4\,\sqrt{a}\sqrt{c}-3\,\sqrt{ac}}}}}+{\frac{B}{2}\arctan \left ({ \left ( 2\,x\sqrt{c}+\sqrt{3}\sqrt [4]{ac} \right ){\frac{1}{\sqrt{4\,\sqrt{a}\sqrt{c}-3\,\sqrt{ac}}}}} \right ){\frac{1}{\sqrt{c}}}{\frac{1}{\sqrt{4\,\sqrt{a}\sqrt{c}-3\,\sqrt{ac}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x^2+A)/(a+c*x^4-x^2*(a*c)^(1/2)),x)

[Out]

1/12/a/c^(3/2)*ln(x*3^(1/2)*(a*c)^(1/4)-x^2*c^(1/2)-a^(1/2))*B*3^(1/2)*(a*c)^(3/4)-1/12/a^(3/2)/c*ln(x*3^(1/2)
*(a*c)^(1/4)-x^2*c^(1/2)-a^(1/2))*A*3^(1/2)*(a*c)^(3/4)-1/2/a^(1/2)/(4*a^(1/2)*c^(1/2)-3*(a*c)^(1/2))^(1/2)*ar
ctan((3^(1/2)*(a*c)^(1/4)-2*x*c^(1/2))/(4*a^(1/2)*c^(1/2)-3*(a*c)^(1/2))^(1/2))*A-1/2/c^(1/2)/(4*a^(1/2)*c^(1/
2)-3*(a*c)^(1/2))^(1/2)*arctan((3^(1/2)*(a*c)^(1/4)-2*x*c^(1/2))/(4*a^(1/2)*c^(1/2)-3*(a*c)^(1/2))^(1/2))*B-1/
12/a/c^(3/2)*ln(x^2*c^(1/2)+x*3^(1/2)*(a*c)^(1/4)+a^(1/2))*B*3^(1/2)*(a*c)^(3/4)+1/12/a^(3/2)/c*ln(x^2*c^(1/2)
+x*3^(1/2)*(a*c)^(1/4)+a^(1/2))*A*3^(1/2)*(a*c)^(3/4)+1/2/a^(1/2)/(4*a^(1/2)*c^(1/2)-3*(a*c)^(1/2))^(1/2)*arct
an((2*x*c^(1/2)+3^(1/2)*(a*c)^(1/4))/(4*a^(1/2)*c^(1/2)-3*(a*c)^(1/2))^(1/2))*A+1/2/c^(1/2)/(4*a^(1/2)*c^(1/2)
-3*(a*c)^(1/2))^(1/2)*arctan((2*x*c^(1/2)+3^(1/2)*(a*c)^(1/4))/(4*a^(1/2)*c^(1/2)-3*(a*c)^(1/2))^(1/2))*B

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B x^{2} + A}{c x^{4} - \sqrt{a c} x^{2} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/(a+c*x^4-x^2*(a*c)^(1/2)),x, algorithm="maxima")

[Out]

integrate((B*x^2 + A)/(c*x^4 - sqrt(a*c)*x^2 + a), x)

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Fricas [B]  time = 3.61015, size = 3148, normalized size = 7.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/(a+c*x^4-x^2*(a*c)^(1/2)),x, algorithm="fricas")

[Out]

-1/2*sqrt(1/6)*sqrt(-(3*sqrt(1/3)*a^2*c^2*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^3*c^3)) + 4*A*B*a*c + (
B^2*a + A^2*c)*sqrt(a*c))/(a^2*c^2))*log(-2*(B^6*a^3 - A^6*c^3)*x + 3*sqrt(1/6)*(A*B^4*a^3*c - A^5*a*c^3 - sqr
t(1/3)*(2*B^3*a^4*c^2 + A^2*B*a^3*c^3)*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^3*c^3)) - (A^2*B^3*a^2*c -
A^4*B*a*c^2 - sqrt(1/3)*(A*B^2*a^3*c^2 - A^3*a^2*c^3)*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^3*c^3)))*s
qrt(a*c))*sqrt(-(3*sqrt(1/3)*a^2*c^2*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^3*c^3)) + 4*A*B*a*c + (B^2*a
+ A^2*c)*sqrt(a*c))/(a^2*c^2))) + 1/2*sqrt(1/6)*sqrt(-(3*sqrt(1/3)*a^2*c^2*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A
^4*c^2)/(a^3*c^3)) + 4*A*B*a*c + (B^2*a + A^2*c)*sqrt(a*c))/(a^2*c^2))*log(-2*(B^6*a^3 - A^6*c^3)*x - 3*sqrt(1
/6)*(A*B^4*a^3*c - A^5*a*c^3 - sqrt(1/3)*(2*B^3*a^4*c^2 + A^2*B*a^3*c^3)*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*
c^2)/(a^3*c^3)) - (A^2*B^3*a^2*c - A^4*B*a*c^2 - sqrt(1/3)*(A*B^2*a^3*c^2 - A^3*a^2*c^3)*sqrt(-(B^4*a^2 - 2*A^
2*B^2*a*c + A^4*c^2)/(a^3*c^3)))*sqrt(a*c))*sqrt(-(3*sqrt(1/3)*a^2*c^2*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^
2)/(a^3*c^3)) + 4*A*B*a*c + (B^2*a + A^2*c)*sqrt(a*c))/(a^2*c^2))) - 1/2*sqrt(1/6)*sqrt((3*sqrt(1/3)*a^2*c^2*s
qrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^3*c^3)) - 4*A*B*a*c - (B^2*a + A^2*c)*sqrt(a*c))/(a^2*c^2))*log(-2
*(B^6*a^3 - A^6*c^3)*x + 3*sqrt(1/6)*(A*B^4*a^3*c - A^5*a*c^3 + sqrt(1/3)*(2*B^3*a^4*c^2 + A^2*B*a^3*c^3)*sqrt
(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^3*c^3)) - (A^2*B^3*a^2*c - A^4*B*a*c^2 + sqrt(1/3)*(A*B^2*a^3*c^2 - A
^3*a^2*c^3)*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^3*c^3)))*sqrt(a*c))*sqrt((3*sqrt(1/3)*a^2*c^2*sqrt(-(
B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^3*c^3)) - 4*A*B*a*c - (B^2*a + A^2*c)*sqrt(a*c))/(a^2*c^2))) + 1/2*sqrt(
1/6)*sqrt((3*sqrt(1/3)*a^2*c^2*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^3*c^3)) - 4*A*B*a*c - (B^2*a + A^2
*c)*sqrt(a*c))/(a^2*c^2))*log(-2*(B^6*a^3 - A^6*c^3)*x - 3*sqrt(1/6)*(A*B^4*a^3*c - A^5*a*c^3 + sqrt(1/3)*(2*B
^3*a^4*c^2 + A^2*B*a^3*c^3)*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^3*c^3)) - (A^2*B^3*a^2*c - A^4*B*a*c^
2 + sqrt(1/3)*(A*B^2*a^3*c^2 - A^3*a^2*c^3)*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^3*c^3)))*sqrt(a*c))*s
qrt((3*sqrt(1/3)*a^2*c^2*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^3*c^3)) - 4*A*B*a*c - (B^2*a + A^2*c)*sq
rt(a*c))/(a^2*c^2)))

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: PolynomialError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x**2+A)/(a+c*x**4-x**2*(a*c)**(1/2)),x)

[Out]

Exception raised: PolynomialError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/(a+c*x^4-x^2*(a*c)^(1/2)),x, algorithm="giac")

[Out]

Exception raised: TypeError