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

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

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Rubi [A]  time = 0.891831, 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 \[ -\frac{\left (A-\frac{\sqrt{a} B}{\sqrt{c}}\right ) \log \left (-x \sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}}+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{a} \sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}}}+\frac{\left (A-\frac{\sqrt{a} B}{\sqrt{c}}\right ) \log \left (x \sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}}+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{a} \sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}}}-\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}}-2 \sqrt{c} x}{\sqrt{2 \sqrt{a} \sqrt{c}-\sqrt{a c}}}\right )}{2 \sqrt{a} \sqrt{c} \sqrt{2 \sqrt{a} \sqrt{c}-\sqrt{a c}}}+\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}}+2 \sqrt{c} x}{\sqrt{2 \sqrt{a} \sqrt{c}-\sqrt{a c}}}\right )}{2 \sqrt{a} \sqrt{c} \sqrt{2 \sqrt{a} \sqrt{c}-\sqrt{a c}}} \]

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

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Rubi in Sympy [A]  time = 106.045, size = 386, normalized size = 0.93 \[ - \frac{\left (A \sqrt{c} - B \sqrt{a}\right ) \log{\left (\frac{\sqrt{a}}{\sqrt{c}} + x^{2} - \frac{x \sqrt{2 \sqrt{a} \sqrt{c} + \sqrt{a c}}}{\sqrt{c}} \right )}}{4 \sqrt{a} \sqrt{c} \sqrt{2 \sqrt{a} \sqrt{c} + \sqrt{a c}}} + \frac{\left (A \sqrt{c} - B \sqrt{a}\right ) \log{\left (\frac{\sqrt{a}}{\sqrt{c}} + x^{2} + \frac{x \sqrt{2 \sqrt{a} \sqrt{c} + \sqrt{a c}}}{\sqrt{c}} \right )}}{4 \sqrt{a} \sqrt{c} \sqrt{2 \sqrt{a} \sqrt{c} + \sqrt{a c}}} + \frac{\left (A \sqrt{c} + B \sqrt{a}\right ) \operatorname{atan}{\left (\frac{2 \sqrt{c} x - \sqrt{2 \sqrt{a} \sqrt{c} + \sqrt{a c}}}{\sqrt{2 \sqrt{a} \sqrt{c} - \sqrt{a c}}} \right )}}{2 \sqrt{a} \sqrt{c} \sqrt{2 \sqrt{a} \sqrt{c} - \sqrt{a c}}} + \frac{\left (A \sqrt{c} + B \sqrt{a}\right ) \operatorname{atan}{\left (\frac{2 \sqrt{c} x + \sqrt{2 \sqrt{a} \sqrt{c} + \sqrt{a c}}}{\sqrt{2 \sqrt{a} \sqrt{c} - \sqrt{a c}}} \right )}}{2 \sqrt{a} \sqrt{c} \sqrt{2 \sqrt{a} \sqrt{c} - \sqrt{a c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(A*sqrt(c) - B*sqrt(a))*log(sqrt(a)/sqrt(c) + x**2 - x*sqrt(2*sqrt(a)*sqrt(c) +
sqrt(a*c))/sqrt(c))/(4*sqrt(a)*sqrt(c)*sqrt(2*sqrt(a)*sqrt(c) + sqrt(a*c))) + (
A*sqrt(c) - B*sqrt(a))*log(sqrt(a)/sqrt(c) + x**2 + x*sqrt(2*sqrt(a)*sqrt(c) + s
qrt(a*c))/sqrt(c))/(4*sqrt(a)*sqrt(c)*sqrt(2*sqrt(a)*sqrt(c) + sqrt(a*c))) + (A*
sqrt(c) + B*sqrt(a))*atan((2*sqrt(c)*x - sqrt(2*sqrt(a)*sqrt(c) + sqrt(a*c)))/sq
rt(2*sqrt(a)*sqrt(c) - sqrt(a*c)))/(2*sqrt(a)*sqrt(c)*sqrt(2*sqrt(a)*sqrt(c) - s
qrt(a*c))) + (A*sqrt(c) + B*sqrt(a))*atan((2*sqrt(c)*x + sqrt(2*sqrt(a)*sqrt(c)
+ sqrt(a*c)))/sqrt(2*sqrt(a)*sqrt(c) - sqrt(a*c)))/(2*sqrt(a)*sqrt(c)*sqrt(2*sqr
t(a)*sqrt(c) - sqrt(a*c)))

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Mathematica [C]  time = 0.317388, 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]*Sqrt[c] - Sqrt[a*c]]])/Sqrt[(-I)*Sqrt[3]*Sqrt[a]*Sq
rt[c] - Sqrt[a*c]] + ((Sqrt[3]*Sqrt[a]*B*Sqrt[c] + I*(2*A*c + B*Sqrt[a*c]))*ArcT
an[(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 = 0.074, size = 404, normalized size = 1. \[ -{\frac{B\sqrt{3}}{12\,a}\ln \left ({x}^{2}\sqrt{c}+\sqrt{3}\sqrt [4]{ac}x+\sqrt{a} \right ) \left ( ac \right ) ^{{\frac{3}{4}}}{c}^{-{\frac{3}{2}}}}+{\frac{A\sqrt{3}}{12\,c}\ln \left ({x}^{2}\sqrt{c}+\sqrt{3}\sqrt [4]{ac}x+\sqrt{a} \right ) \left ( ac \right ) ^{{\frac{3}{4}}}{a}^{-{\frac{3}{2}}}}+{\frac{A}{2}\arctan \left ({1 \left ( 2\,x\sqrt{c}+\sqrt{3}\sqrt [4]{ac} \right ){\frac{1}{\sqrt{4\,\sqrt{a}\sqrt{c}-3\,\sqrt{ac}}}}} \right ){\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{4\,\sqrt{a}\sqrt{c}-3\,\sqrt{ac}}}}}+{\frac{B}{2}\arctan \left ({1 \left ( 2\,x\sqrt{c}+\sqrt{3}\sqrt [4]{ac} \right ){\frac{1}{\sqrt{4\,\sqrt{a}\sqrt{c}-3\,\sqrt{ac}}}}} \right ){\frac{1}{\sqrt{c}}}{\frac{1}{\sqrt{4\,\sqrt{a}\sqrt{c}-3\,\sqrt{ac}}}}}+{\frac{B\sqrt{3}}{12\,a}\ln \left ( \sqrt{3}\sqrt [4]{ac}x-{x}^{2}\sqrt{c}-\sqrt{a} \right ) \left ( ac \right ) ^{{\frac{3}{4}}}{c}^{-{\frac{3}{2}}}}-{\frac{A\sqrt{3}}{12\,c}\ln \left ( \sqrt{3}\sqrt [4]{ac}x-{x}^{2}\sqrt{c}-\sqrt{a} \right ) \left ( ac \right ) ^{{\frac{3}{4}}}{a}^{-{\frac{3}{2}}}}-{\frac{A}{2}\arctan \left ({1 \left ( \sqrt{3}\sqrt [4]{ac}-2\,x\sqrt{c} \right ){\frac{1}{\sqrt{4\,\sqrt{a}\sqrt{c}-3\,\sqrt{ac}}}}} \right ){\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{4\,\sqrt{a}\sqrt{c}-3\,\sqrt{ac}}}}}-{\frac{B}{2}\arctan \left ({1 \left ( \sqrt{3}\sqrt [4]{ac}-2\,x\sqrt{c} \right ){\frac{1}{\sqrt{4\,\sqrt{a}\sqrt{c}-3\,\sqrt{ac}}}}} \right ){\frac{1}{\sqrt{c}}}{\frac{1}{\sqrt{4\,\sqrt{a}\sqrt{c}-3\,\sqrt{ac}}}}} \]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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*c)*x^2 + a),x, algorithm="fricas")

[Out]

Timed out

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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*c)**(1/2)),x)

[Out]

Exception raised: PolynomialError

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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*c)*x^2 + a),x, algorithm="giac")

[Out]

Exception raised: TypeError