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]

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

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Rubi [A]  time = 0.17, antiderivative size = 234, normalized size of antiderivative = 1.00, 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 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 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 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]

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 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]

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*}

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Mathematica [C]  time = 0.19, size = 163, normalized size = 0.70 \[ \frac {\sqrt [4]{-1} \left (\frac {\left (\left (\sqrt {3}-i\right ) \sqrt {a} B-2 i A \sqrt {c}\right ) \tan ^{-1}\left (\frac {(1+i) \sqrt [4]{c} x}{\sqrt {\sqrt {3}-i} \sqrt [4]{a}}\right )}{\sqrt {\sqrt {3}-i}}-\frac {\left (\left (\sqrt {3}+i\right ) \sqrt {a} B+2 i A \sqrt {c}\right ) \tanh ^{-1}\left (\frac {(1+i) \sqrt [4]{c} x}{\sqrt {\sqrt {3}+i} \sqrt [4]{a}}\right )}{\sqrt {\sqrt {3}+i}}\right )}{\sqrt {6} a^{3/4} c^{3/4}} \]

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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fricas [B]  time = 1.17, size = 1469, normalized size = 6.28 \[ \text {result too large to display} \]

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

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 >> 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 [A]  time = 0.07, size = 320, normalized size = 1.37 \[ \frac {A \arctan \left (\frac {2 \sqrt {c}\, x +\sqrt {3}\, a^{\frac {1}{4}} c^{\frac {1}{4}}}{\sqrt {\sqrt {a}\, \sqrt {c}}}\right )}{2 \sqrt {\sqrt {a}\, \sqrt {c}}\, \sqrt {a}}-\frac {A \arctan \left (\frac {-2 \sqrt {c}\, x +\sqrt {3}\, a^{\frac {1}{4}} c^{\frac {1}{4}}}{\sqrt {\sqrt {a}\, \sqrt {c}}}\right )}{2 \sqrt {\sqrt {a}\, \sqrt {c}}\, \sqrt {a}}+\frac {B \arctan \left (\frac {2 \sqrt {c}\, x +\sqrt {3}\, a^{\frac {1}{4}} c^{\frac {1}{4}}}{\sqrt {\sqrt {a}\, \sqrt {c}}}\right )}{2 \sqrt {\sqrt {a}\, \sqrt {c}}\, \sqrt {c}}-\frac {B \arctan \left (\frac {-2 \sqrt {c}\, x +\sqrt {3}\, a^{\frac {1}{4}} c^{\frac {1}{4}}}{\sqrt {\sqrt {a}\, \sqrt {c}}}\right )}{2 \sqrt {\sqrt {a}\, \sqrt {c}}\, \sqrt {c}}+\frac {\sqrt {3}\, A \ln \left (\sqrt {c}\, x^{2}+\sqrt {3}\, a^{\frac {1}{4}} c^{\frac {1}{4}} x +\sqrt {a}\right )}{12 a^{\frac {3}{4}} c^{\frac {1}{4}}}-\frac {\sqrt {3}\, A \ln \left (-\sqrt {c}\, x^{2}+\sqrt {3}\, a^{\frac {1}{4}} c^{\frac {1}{4}} x -\sqrt {a}\right )}{12 a^{\frac {3}{4}} c^{\frac {1}{4}}}-\frac {\sqrt {3}\, B \ln \left (\sqrt {c}\, x^{2}+\sqrt {3}\, a^{\frac {1}{4}} c^{\frac {1}{4}} x +\sqrt {a}\right )}{12 a^{\frac {1}{4}} c^{\frac {3}{4}}}+\frac {\sqrt {3}\, B \ln \left (-\sqrt {c}\, x^{2}+\sqrt {3}\, a^{\frac {1}{4}} c^{\frac {1}{4}} x -\sqrt {a}\right )}{12 a^{\frac {1}{4}} c^{\frac {3}{4}}} \]

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)-c^(1/2)*x^2-a^(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)-c^(1/2)*x^2-a^(1/2))*B*3^(1/2)-1/2/a^(1/2)/(a^(1/2)*c^(1/2))^(1/2)*arctan((3^(1/2)*c^(1/
4)*a^(1/4)-2*c^(1/2)*x)/(a^(1/2)*c^(1/2))^(1/2))*A-1/2/c^(1/2)/(a^(1/2)*c^(1/2))^(1/2)*arctan((3^(1/2)*c^(1/4)
*a^(1/4)-2*c^(1/2)*x)/(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)+c^(
1/2)*x^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)+c^(1/2)*x^2)*B*3^(1/2)+1/2/a^(1/
2)/(a^(1/2)*c^(1/2))^(1/2)*arctan((2*c^(1/2)*x+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*c^(1/2)*x+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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {B x^{2} + A}{c x^{4} - \sqrt {a} \sqrt {c} x^{2} + a}\,{d x} \]

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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mupad [B]  time = 5.29, size = 1575, normalized size = 6.73 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- 2*atanh((6*A^2*x*((B^2*(-27*a^3*c^3)^(1/2))/(72*a^2*c^3) - B^2/(24*a^(1/2)*c^(3/2)) - (A*B)/(6*a*c) - (A^2*(
-27*a^3*c^3)^(1/2))/(72*a^3*c^2) - A^2/(24*a^(3/2)*c^(1/2)))^(1/2))/((2*A^2*B)/c - (2*B^3*a)/c^2 + A^3/(a^(1/2
)*c^(1/2)) + (A^3*(-27*a^3*c^3)^(1/2))/(3*a^2*c^2) - (A*B^2*a^(1/2))/c^(3/2) - (A*B^2*(-27*a^3*c^3)^(1/2))/(3*
a*c^3)) - (6*B^2*a*x*((B^2*(-27*a^3*c^3)^(1/2))/(72*a^2*c^3) - B^2/(24*a^(1/2)*c^(3/2)) - (A*B)/(6*a*c) - (A^2
*(-27*a^3*c^3)^(1/2))/(72*a^3*c^2) - A^2/(24*a^(3/2)*c^(1/2)))^(1/2))/(2*A^2*B - (2*B^3*a)/c + (A^3*c^(1/2))/a
^(1/2) + (A^3*(-27*a^3*c^3)^(1/2))/(3*a^2*c) - (A*B^2*a^(1/2))/c^(1/2) - (A*B^2*(-27*a^3*c^3)^(1/2))/(3*a*c^2)
) - (2*A^2*x*(-27*a^3*c^3)^(1/2)*((B^2*(-27*a^3*c^3)^(1/2))/(72*a^2*c^3) - B^2/(24*a^(1/2)*c^(3/2)) - (A*B)/(6
*a*c) - (A^2*(-27*a^3*c^3)^(1/2))/(72*a^3*c^2) - A^2/(24*a^(3/2)*c^(1/2)))^(1/2))/(3*a^(3/2)*c^(7/2)*((2*A^2*B
)/c^3 - (2*B^3*a)/c^4 + A^3/(a^(1/2)*c^(5/2)) + (A^3*(-27*a^3*c^3)^(1/2))/(3*a^2*c^4) - (A*B^2*a^(1/2))/c^(7/2
) - (A*B^2*(-27*a^3*c^3)^(1/2))/(3*a*c^5))) + (2*B^2*x*(-27*a^3*c^3)^(1/2)*((B^2*(-27*a^3*c^3)^(1/2))/(72*a^2*
c^3) - B^2/(24*a^(1/2)*c^(3/2)) - (A*B)/(6*a*c) - (A^2*(-27*a^3*c^3)^(1/2))/(72*a^3*c^2) - A^2/(24*a^(3/2)*c^(
1/2)))^(1/2))/(3*a^(1/2)*c^(9/2)*((2*A^2*B)/c^3 - (2*B^3*a)/c^4 + A^3/(a^(1/2)*c^(5/2)) + (A^3*(-27*a^3*c^3)^(
1/2))/(3*a^2*c^4) - (A*B^2*a^(1/2))/c^(7/2) - (A*B^2*(-27*a^3*c^3)^(1/2))/(3*a*c^5))))*((B^2*(-27*a^3*c^3)^(1/
2))/(72*a^2*c^3) - B^2/(24*a^(1/2)*c^(3/2)) - (A*B)/(6*a*c) - (A^2*(-27*a^3*c^3)^(1/2))/(72*a^3*c^2) - A^2/(24
*a^(3/2)*c^(1/2)))^(1/2) - 2*atanh((6*A^2*x*((A^2*(-27*a^3*c^3)^(1/2))/(72*a^3*c^2) - B^2/(24*a^(1/2)*c^(3/2))
- (A*B)/(6*a*c) - A^2/(24*a^(3/2)*c^(1/2)) - (B^2*(-27*a^3*c^3)^(1/2))/(72*a^2*c^3))^(1/2))/((2*A^2*B)/c - (2
*B^3*a)/c^2 + A^3/(a^(1/2)*c^(1/2)) - (A^3*(-27*a^3*c^3)^(1/2))/(3*a^2*c^2) - (A*B^2*a^(1/2))/c^(3/2) + (A*B^2
*(-27*a^3*c^3)^(1/2))/(3*a*c^3)) - (6*B^2*a*x*((A^2*(-27*a^3*c^3)^(1/2))/(72*a^3*c^2) - B^2/(24*a^(1/2)*c^(3/2
)) - (A*B)/(6*a*c) - A^2/(24*a^(3/2)*c^(1/2)) - (B^2*(-27*a^3*c^3)^(1/2))/(72*a^2*c^3))^(1/2))/(2*A^2*B - (2*B
^3*a)/c + (A^3*c^(1/2))/a^(1/2) - (A^3*(-27*a^3*c^3)^(1/2))/(3*a^2*c) - (A*B^2*a^(1/2))/c^(1/2) + (A*B^2*(-27*
a^3*c^3)^(1/2))/(3*a*c^2)) + (2*A^2*x*(-27*a^3*c^3)^(1/2)*((A^2*(-27*a^3*c^3)^(1/2))/(72*a^3*c^2) - B^2/(24*a^
(1/2)*c^(3/2)) - (A*B)/(6*a*c) - A^2/(24*a^(3/2)*c^(1/2)) - (B^2*(-27*a^3*c^3)^(1/2))/(72*a^2*c^3))^(1/2))/(3*
a^(3/2)*c^(7/2)*((2*A^2*B)/c^3 - (2*B^3*a)/c^4 + A^3/(a^(1/2)*c^(5/2)) - (A^3*(-27*a^3*c^3)^(1/2))/(3*a^2*c^4)
- (A*B^2*a^(1/2))/c^(7/2) + (A*B^2*(-27*a^3*c^3)^(1/2))/(3*a*c^5))) - (2*B^2*x*(-27*a^3*c^3)^(1/2)*((A^2*(-27
*a^3*c^3)^(1/2))/(72*a^3*c^2) - B^2/(24*a^(1/2)*c^(3/2)) - (A*B)/(6*a*c) - A^2/(24*a^(3/2)*c^(1/2)) - (B^2*(-2
7*a^3*c^3)^(1/2))/(72*a^2*c^3))^(1/2))/(3*a^(1/2)*c^(9/2)*((2*A^2*B)/c^3 - (2*B^3*a)/c^4 + A^3/(a^(1/2)*c^(5/2
)) - (A^3*(-27*a^3*c^3)^(1/2))/(3*a^2*c^4) - (A*B^2*a^(1/2))/c^(7/2) + (A*B^2*(-27*a^3*c^3)^(1/2))/(3*a*c^5)))
)*((A^2*(-27*a^3*c^3)^(1/2))/(72*a^3*c^2) - B^2/(24*a^(1/2)*c^(3/2)) - (A*B)/(6*a*c) - A^2/(24*a^(3/2)*c^(1/2)
) - (B^2*(-27*a^3*c^3)^(1/2))/(72*a^2*c^3))^(1/2)

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: PolynomialError} \]

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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