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

Optimal. Leaf size=234 \[ -\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} c^{3/4}}+\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (\frac{2 \sqrt [4]{c} x}{\sqrt [4]{a}}+\sqrt{3}\right )}{2 a^{3/4} c^{3/4}}-\frac{\left (A-\frac{\sqrt{a} B}{\sqrt{c}}\right ) \log \left (-\sqrt{3} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{3} a^{3/4} \sqrt [4]{c}}+\frac{\left (A-\frac{\sqrt{a} B}{\sqrt{c}}\right ) \log \left (\sqrt{3} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{3} a^{3/4} \sqrt [4]{c}} \]

[Out]

-((Sqrt[a]*B + A*Sqrt[c])*ArcTan[Sqrt[3] - (2*c^(1/4)*x)/a^(1/4)])/(2*a^(3/4)*c^(3/4)) + ((Sqrt[a]*B + A*Sqrt[
c])*ArcTan[Sqrt[3] + (2*c^(1/4)*x)/a^(1/4)])/(2*a^(3/4)*c^(3/4)) - ((A - (Sqrt[a]*B)/Sqrt[c])*Log[Sqrt[a] - Sq
rt[3]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[3]*a^(3/4)*c^(1/4)) + ((A - (Sqrt[a]*B)/Sqrt[c])*Log[Sqrt[a] +
Sqrt[3]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[3]*a^(3/4)*c^(1/4))

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Rubi [A]  time = 0.171647, antiderivative size = 234, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156, Rules used = {1169, 634, 617, 204, 628} \[ -\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} c^{3/4}}+\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (\frac{2 \sqrt [4]{c} x}{\sqrt [4]{a}}+\sqrt{3}\right )}{2 a^{3/4} c^{3/4}}-\frac{\left (A-\frac{\sqrt{a} B}{\sqrt{c}}\right ) \log \left (-\sqrt{3} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{3} a^{3/4} \sqrt [4]{c}}+\frac{\left (A-\frac{\sqrt{a} B}{\sqrt{c}}\right ) \log \left (\sqrt{3} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{3} a^{3/4} \sqrt [4]{c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[a]*B + A*Sqrt[c])*ArcTan[Sqrt[3] - (2*c^(1/4)*x)/a^(1/4)])/(2*a^(3/4)*c^(3/4)) + ((Sqrt[a]*B + A*Sqrt[
c])*ArcTan[Sqrt[3] + (2*c^(1/4)*x)/a^(1/4)])/(2*a^(3/4)*c^(3/4)) - ((A - (Sqrt[a]*B)/Sqrt[c])*Log[Sqrt[a] - Sq
rt[3]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[3]*a^(3/4)*c^(1/4)) + ((A - (Sqrt[a]*B)/Sqrt[c])*Log[Sqrt[a] +
Sqrt[3]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[3]*a^(3/4)*c^(1/4))

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 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
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} \sqrt{c} x^2+c x^4} \, dx &=\frac{\int \frac{\frac{\sqrt{3} \sqrt [4]{a} A}{\sqrt [4]{c}}-\left (A-\frac{\sqrt{a} B}{\sqrt{c}}\right ) x}{\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{3} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{2 \sqrt{3} a^{3/4} \sqrt [4]{c}}+\frac{\int \frac{\frac{\sqrt{3} \sqrt [4]{a} A}{\sqrt [4]{c}}+\left (A-\frac{\sqrt{a} B}{\sqrt{c}}\right ) x}{\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{3} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{2 \sqrt{3} a^{3/4} \sqrt [4]{c}}\\ &=\frac{\left (B+\frac{A \sqrt{c}}{\sqrt{a}}\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{3} \sqrt [4]{a} x}{\sqrt [4]{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{3} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 c}+\frac{\left (\sqrt{a} B-A \sqrt{c}\right ) \int \frac{-\frac{\sqrt{3} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{3} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 \sqrt{3} a^{3/4} c^{3/4}}+\frac{\left (A-\frac{\sqrt{a} B}{\sqrt{c}}\right ) \int \frac{\frac{\sqrt{3} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{3} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 \sqrt{3} a^{3/4} \sqrt [4]{c}}\\ &=\frac{\left (\sqrt{a} B-A \sqrt{c}\right ) \log \left (\sqrt{a}-\sqrt{3} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{4 \sqrt{3} a^{3/4} c^{3/4}}+\frac{\left (A-\frac{\sqrt{a} B}{\sqrt{c}}\right ) \log \left (\sqrt{a}+\sqrt{3} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{4 \sqrt{3} a^{3/4} \sqrt [4]{c}}+\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{3}-x^2} \, dx,x,1-\frac{2 \sqrt [4]{c} x}{\sqrt{3} \sqrt [4]{a}}\right )}{2 \sqrt{3} a^{3/4} c^{3/4}}-\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{3}-x^2} \, dx,x,1+\frac{2 \sqrt [4]{c} x}{\sqrt{3} \sqrt [4]{a}}\right )}{2 \sqrt{3} a^{3/4} c^{3/4}}\\ &=-\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} c^{3/4}}+\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (\sqrt{3}+\frac{2 \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} c^{3/4}}+\frac{\left (\sqrt{a} B-A \sqrt{c}\right ) \log \left (\sqrt{a}-\sqrt{3} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{4 \sqrt{3} a^{3/4} c^{3/4}}+\frac{\left (A-\frac{\sqrt{a} B}{\sqrt{c}}\right ) \log \left (\sqrt{a}+\sqrt{3} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{4 \sqrt{3} a^{3/4} \sqrt [4]{c}}\\ \end{align*}

Mathematica [C]  time = 0.184857, size = 163, normalized size = 0.7 \[ \frac{\sqrt [4]{-1} \left (\frac{\left (\left (\sqrt{3}-i\right ) \sqrt{a} B-2 i A \sqrt{c}\right ) \tan ^{-1}\left (\frac{(1+i) \sqrt [4]{c} x}{\sqrt{\sqrt{3}-i} \sqrt [4]{a}}\right )}{\sqrt{\sqrt{3}-i}}-\frac{\left (\left (\sqrt{3}+i\right ) \sqrt{a} B+2 i A \sqrt{c}\right ) \tanh ^{-1}\left (\frac{(1+i) \sqrt [4]{c} x}{\sqrt{\sqrt{3}+i} \sqrt [4]{a}}\right )}{\sqrt{\sqrt{3}+i}}\right )}{\sqrt{6} a^{3/4} c^{3/4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-1)^(1/4)*((((-I + Sqrt[3])*Sqrt[a]*B - (2*I)*A*Sqrt[c])*ArcTan[((1 + I)*c^(1/4)*x)/(Sqrt[-I + Sqrt[3]]*a^(1
/4))])/Sqrt[-I + Sqrt[3]] - (((I + Sqrt[3])*Sqrt[a]*B + (2*I)*A*Sqrt[c])*ArcTanh[((1 + I)*c^(1/4)*x)/(Sqrt[I +
Sqrt[3]]*a^(1/4))])/Sqrt[I + Sqrt[3]]))/(Sqrt[6]*a^(3/4)*c^(3/4))

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Maple [A]  time = 0.092, size = 318, normalized size = 1.4 \begin{align*} -{\frac{A\sqrt{3}}{12}\ln \left ( -\sqrt [4]{a}\sqrt [4]{c}x\sqrt{3}+\sqrt{a}+{x}^{2}\sqrt{c} \right ){\frac{1}{\sqrt [4]{c}}}{a}^{-{\frac{3}{4}}}}+{\frac{B\sqrt{3}}{12}\ln \left ( -\sqrt [4]{a}\sqrt [4]{c}x\sqrt{3}+\sqrt{a}+{x}^{2}\sqrt{c} \right ){c}^{-{\frac{3}{4}}}{\frac{1}{\sqrt [4]{a}}}}+{\frac{A}{2}\arctan \left ({ \left ( 2\,x\sqrt{c}-\sqrt{3}\sqrt [4]{c}\sqrt [4]{a} \right ){\frac{1}{\sqrt{\sqrt{a}\sqrt{c}}}}} \right ){\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{\sqrt{a}\sqrt{c}}}}}+{\frac{B}{2}\arctan \left ({ \left ( 2\,x\sqrt{c}-\sqrt{3}\sqrt [4]{c}\sqrt [4]{a} \right ){\frac{1}{\sqrt{\sqrt{a}\sqrt{c}}}}} \right ){\frac{1}{\sqrt{c}}}{\frac{1}{\sqrt{\sqrt{a}\sqrt{c}}}}}+{\frac{A\sqrt{3}}{12}\ln \left ( \sqrt [4]{a}\sqrt [4]{c}x\sqrt{3}+\sqrt{a}+{x}^{2}\sqrt{c} \right ){\frac{1}{\sqrt [4]{c}}}{a}^{-{\frac{3}{4}}}}-{\frac{B\sqrt{3}}{12}\ln \left ( \sqrt [4]{a}\sqrt [4]{c}x\sqrt{3}+\sqrt{a}+{x}^{2}\sqrt{c} \right ){c}^{-{\frac{3}{4}}}{\frac{1}{\sqrt [4]{a}}}}+{\frac{A}{2}\arctan \left ({ \left ( 2\,x\sqrt{c}+\sqrt{3}\sqrt [4]{c}\sqrt [4]{a} \right ){\frac{1}{\sqrt{\sqrt{a}\sqrt{c}}}}} \right ){\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{\sqrt{a}\sqrt{c}}}}}+{\frac{B}{2}\arctan \left ({ \left ( 2\,x\sqrt{c}+\sqrt{3}\sqrt [4]{c}\sqrt [4]{a} \right ){\frac{1}{\sqrt{\sqrt{a}\sqrt{c}}}}} \right ){\frac{1}{\sqrt{c}}}{\frac{1}{\sqrt{\sqrt{a}\sqrt{c}}}}} \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^(1/2)*c^(1/2)),x)

[Out]

-1/12/c^(1/4)/a^(3/4)*ln(-a^(1/4)*c^(1/4)*x*3^(1/2)+a^(1/2)+x^2*c^(1/2))*A*3^(1/2)+1/12/c^(3/4)/a^(1/4)*ln(-a^
(1/4)*c^(1/4)*x*3^(1/2)+a^(1/2)+x^2*c^(1/2))*B*3^(1/2)+1/2/a^(1/2)/(a^(1/2)*c^(1/2))^(1/2)*arctan((2*x*c^(1/2)
-3^(1/2)*c^(1/4)*a^(1/4))/(a^(1/2)*c^(1/2))^(1/2))*A+1/2/c^(1/2)/(a^(1/2)*c^(1/2))^(1/2)*arctan((2*x*c^(1/2)-3
^(1/2)*c^(1/4)*a^(1/4))/(a^(1/2)*c^(1/2))^(1/2))*B+1/12/c^(1/4)/a^(3/4)*ln(a^(1/4)*c^(1/4)*x*3^(1/2)+a^(1/2)+x
^2*c^(1/2))*A*3^(1/2)-1/12/c^(3/4)/a^(1/4)*ln(a^(1/4)*c^(1/4)*x*3^(1/2)+a^(1/2)+x^2*c^(1/2))*B*3^(1/2)+1/2/a^(
1/2)/(a^(1/2)*c^(1/2))^(1/2)*arctan((2*x*c^(1/2)+3^(1/2)*c^(1/4)*a^(1/4))/(a^(1/2)*c^(1/2))^(1/2))*A+1/2/c^(1/
2)/(a^(1/2)*c^(1/2))^(1/2)*arctan((2*x*c^(1/2)+3^(1/2)*c^(1/4)*a^(1/4))/(a^(1/2)*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} \sqrt{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^(1/2)*c^(1/2)),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 7.48679, size = 3245, normalized size = 13.87 \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^(1/2)*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)*sqrt(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
- (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)*sqrt(c) - 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)))*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)*sqrt(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)*sqrt(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 - (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)*sqrt(c) - 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)))*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)*sqrt(c))/(a^2*c^2))) - 1/2*s
qrt(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)*sqrt(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 - (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)*sqrt(c) + 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)))*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)*sqrt(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)*sqrt(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 - (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)*sqrt(c) + 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)))*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)*sqrt(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**(1/2)*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^(1/2)*c^(1/2)),x, algorithm="giac")

[Out]

Exception raised: TypeError