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

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] - Sqrt[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))

_______________________________________________________________________________________

Rubi [A]  time = 0.39161, 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 \[ -\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} c^{3/4}}+\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (\frac{2 \sqrt [4]{c} x}{\sqrt [4]{a}}+\sqrt{3}\right )}{2 a^{3/4} c^{3/4}}-\frac{\left (A-\frac{\sqrt{a} B}{\sqrt{c}}\right ) \log \left (-\sqrt{3} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{3} a^{3/4} \sqrt [4]{c}}+\frac{\left (A-\frac{\sqrt{a} B}{\sqrt{c}}\right ) \log \left (\sqrt{3} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{3} a^{3/4} \sqrt [4]{c}} \]

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] - Sqrt[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))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 88.3607, size = 240, normalized size = 1.03 \[ - \frac{\sqrt{3} \left (A \sqrt{c} - B \sqrt{a}\right ) \log{\left (- \frac{\sqrt{3} \sqrt [4]{a} x}{\sqrt [4]{c}} + \frac{\sqrt{a}}{\sqrt{c}} + x^{2} \right )}}{12 a^{\frac{3}{4}} c^{\frac{3}{4}}} + \frac{\sqrt{3} \left (A \sqrt{c} - B \sqrt{a}\right ) \log{\left (\frac{\sqrt{3} \sqrt [4]{a} x}{\sqrt [4]{c}} + \frac{\sqrt{a}}{\sqrt{c}} + x^{2} \right )}}{12 a^{\frac{3}{4}} c^{\frac{3}{4}}} - \frac{\left (A \sqrt{c} + B \sqrt{a}\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\sqrt [4]{a} - \frac{2 \sqrt{3} \sqrt [4]{c} x}{3}\right )}{\sqrt [4]{a}} \right )}}{2 a^{\frac{3}{4}} c^{\frac{3}{4}}} + \frac{\left (A \sqrt{c} + B \sqrt{a}\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\sqrt [4]{a} + \frac{2 \sqrt{3} \sqrt [4]{c} x}{3}\right )}{\sqrt [4]{a}} \right )}}{2 a^{\frac{3}{4}} c^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)/(a+c*x**4-x**2*a**(1/2)*c**(1/2)),x)

[Out]

-sqrt(3)*(A*sqrt(c) - B*sqrt(a))*log(-sqrt(3)*a**(1/4)*x/c**(1/4) + sqrt(a)/sqrt
(c) + x**2)/(12*a**(3/4)*c**(3/4)) + sqrt(3)*(A*sqrt(c) - B*sqrt(a))*log(sqrt(3)
*a**(1/4)*x/c**(1/4) + sqrt(a)/sqrt(c) + x**2)/(12*a**(3/4)*c**(3/4)) - (A*sqrt(
c) + B*sqrt(a))*atan(sqrt(3)*(a**(1/4) - 2*sqrt(3)*c**(1/4)*x/3)/a**(1/4))/(2*a*
*(3/4)*c**(3/4)) + (A*sqrt(c) + B*sqrt(a))*atan(sqrt(3)*(a**(1/4) + 2*sqrt(3)*c*
*(1/4)*x/3)/a**(1/4))/(2*a**(3/4)*c**(3/4))

_______________________________________________________________________________________

Mathematica [C]  time = 0.309243, 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))

_______________________________________________________________________________________

Maple [A]  time = 0.07, size = 320, normalized size = 1.4 \[ -{\frac{A\sqrt{3}}{12}\ln \left ( \sqrt [4]{a}\sqrt [4]{c}x\sqrt{3}-{x}^{2}\sqrt{c}-\sqrt{a} \right ){\frac{1}{\sqrt [4]{c}}}{a}^{-{\frac{3}{4}}}}+{\frac{B\sqrt{3}}{12}\ln \left ( \sqrt [4]{a}\sqrt [4]{c}x\sqrt{3}-{x}^{2}\sqrt{c}-\sqrt{a} \right ){c}^{-{\frac{3}{4}}}{\frac{1}{\sqrt [4]{a}}}}-{\frac{A}{2}\arctan \left ({1 \left ( \sqrt{3}\sqrt [4]{c}\sqrt [4]{a}-2\,x\sqrt{c} \right ){\frac{1}{\sqrt{\sqrt{a}\sqrt{c}}}}} \right ){\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{\sqrt{a}\sqrt{c}}}}}-{\frac{B}{2}\arctan \left ({1 \left ( \sqrt{3}\sqrt [4]{c}\sqrt [4]{a}-2\,x\sqrt{c} \right ){\frac{1}{\sqrt{\sqrt{a}\sqrt{c}}}}} \right ){\frac{1}{\sqrt{c}}}{\frac{1}{\sqrt{\sqrt{a}\sqrt{c}}}}}+{\frac{A\sqrt{3}}{12}\ln \left ( \sqrt [4]{a}\sqrt [4]{c}x\sqrt{3}+\sqrt{a}+{x}^{2}\sqrt{c} \right ){\frac{1}{\sqrt [4]{c}}}{a}^{-{\frac{3}{4}}}}-{\frac{B\sqrt{3}}{12}\ln \left ( \sqrt [4]{a}\sqrt [4]{c}x\sqrt{3}+\sqrt{a}+{x}^{2}\sqrt{c} \right ){c}^{-{\frac{3}{4}}}{\frac{1}{\sqrt [4]{a}}}}+{\frac{A}{2}\arctan \left ({1 \left ( 2\,x\sqrt{c}+\sqrt{3}\sqrt [4]{c}\sqrt [4]{a} \right ){\frac{1}{\sqrt{\sqrt{a}\sqrt{c}}}}} \right ){\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{\sqrt{a}\sqrt{c}}}}}+{\frac{B}{2}\arctan \left ({1 \left ( 2\,x\sqrt{c}+\sqrt{3}\sqrt [4]{c}\sqrt [4]{a} \right ){\frac{1}{\sqrt{\sqrt{a}\sqrt{c}}}}} \right ){\frac{1}{\sqrt{c}}}{\frac{1}{\sqrt{\sqrt{a}\sqrt{c}}}}} \]

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)-x^2*c^(1/2)-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)-x^2*c^(1/2)-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*x*c^(1/
2))/(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*x*c^(1/2))/(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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

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

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

_______________________________________________________________________________________

GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError