3.8 \(\int \frac{f+g x}{(d+e x) \sqrt{-a+b x^2+c x^4}} \, dx\)

Optimal. Leaf size=527 \[ \frac{\sqrt{\sqrt{4 a c+b^2}-b} \sqrt{\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}+1} \sqrt{\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}+1} (e f-d g) \Pi \left (-\frac{\left (b-\sqrt{b^2+4 a c}\right ) e^2}{2 c d^2};\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2+4 a c}-b}}\right )|\frac{b-\sqrt{b^2+4 a c}}{b+\sqrt{b^2+4 a c}}\right )}{\sqrt{2} \sqrt{c} d e \sqrt{-a+b x^2+c x^4}}+\frac{g \sqrt{\sqrt{4 a c+b^2}+b} \left (\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}+1\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right )|-\frac{2 \sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )}{\sqrt{2} \sqrt{c} e \sqrt{\frac{\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}+1}{\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}+1}} \sqrt{-a+b x^2+c x^4}}-\frac{(e f-d g) \tanh ^{-1}\left (\frac{-2 a e^2+x^2 \left (b e^2+2 c d^2\right )+b d^2}{2 \sqrt{-a+b x^2+c x^4} \sqrt{-a e^4+b d^2 e^2+c d^4}}\right )}{2 \sqrt{-a e^4+b d^2 e^2+c d^4}} \]

[Out]

-((e*f - d*g)*ArcTanh[(b*d^2 - 2*a*e^2 + (2*c*d^2 + b*e^2)*x^2)/(2*Sqrt[c*d^4 +
b*d^2*e^2 - a*e^4]*Sqrt[-a + b*x^2 + c*x^4])])/(2*Sqrt[c*d^4 + b*d^2*e^2 - a*e^4
]) + (Sqrt[b + Sqrt[b^2 + 4*a*c]]*g*(1 + (2*c*x^2)/(b - Sqrt[b^2 + 4*a*c]))*Elli
pticF[ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 + 4*a*c]]], (-2*Sqrt[b^2 + 4*
a*c])/(b - Sqrt[b^2 + 4*a*c])])/(Sqrt[2]*Sqrt[c]*e*Sqrt[(1 + (2*c*x^2)/(b - Sqrt
[b^2 + 4*a*c]))/(1 + (2*c*x^2)/(b + Sqrt[b^2 + 4*a*c]))]*Sqrt[-a + b*x^2 + c*x^4
]) + (Sqrt[-b + Sqrt[b^2 + 4*a*c]]*(e*f - d*g)*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2
+ 4*a*c])]*Sqrt[1 + (2*c*x^2)/(b + Sqrt[b^2 + 4*a*c])]*EllipticPi[-((b - Sqrt[b^
2 + 4*a*c])*e^2)/(2*c*d^2), ArcSin[(Sqrt[2]*Sqrt[c]*x)/Sqrt[-b + Sqrt[b^2 + 4*a*
c]]], (b - Sqrt[b^2 + 4*a*c])/(b + Sqrt[b^2 + 4*a*c])])/(Sqrt[2]*Sqrt[c]*d*e*Sqr
t[-a + b*x^2 + c*x^4])

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Rubi [A]  time = 1.85558, antiderivative size = 527, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.29 \[ \frac{\sqrt{\sqrt{4 a c+b^2}-b} \sqrt{\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}+1} \sqrt{\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}+1} (e f-d g) \Pi \left (-\frac{\left (b-\sqrt{b^2+4 a c}\right ) e^2}{2 c d^2};\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2+4 a c}-b}}\right )|\frac{b-\sqrt{b^2+4 a c}}{b+\sqrt{b^2+4 a c}}\right )}{\sqrt{2} \sqrt{c} d e \sqrt{-a+b x^2+c x^4}}+\frac{g \sqrt{\sqrt{4 a c+b^2}+b} \left (\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}+1\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right )|-\frac{2 \sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )}{\sqrt{2} \sqrt{c} e \sqrt{\frac{\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}+1}{\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}+1}} \sqrt{-a+b x^2+c x^4}}-\frac{(e f-d g) \tanh ^{-1}\left (\frac{-2 a e^2+x^2 \left (b e^2+2 c d^2\right )+b d^2}{2 \sqrt{-a+b x^2+c x^4} \sqrt{-a e^4+b d^2 e^2+c d^4}}\right )}{2 \sqrt{-a e^4+b d^2 e^2+c d^4}} \]

Antiderivative was successfully verified.

[In]  Int[(f + g*x)/((d + e*x)*Sqrt[-a + b*x^2 + c*x^4]),x]

[Out]

-((e*f - d*g)*ArcTanh[(b*d^2 - 2*a*e^2 + (2*c*d^2 + b*e^2)*x^2)/(2*Sqrt[c*d^4 +
b*d^2*e^2 - a*e^4]*Sqrt[-a + b*x^2 + c*x^4])])/(2*Sqrt[c*d^4 + b*d^2*e^2 - a*e^4
]) + (Sqrt[b + Sqrt[b^2 + 4*a*c]]*g*(1 + (2*c*x^2)/(b - Sqrt[b^2 + 4*a*c]))*Elli
pticF[ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 + 4*a*c]]], (-2*Sqrt[b^2 + 4*
a*c])/(b - Sqrt[b^2 + 4*a*c])])/(Sqrt[2]*Sqrt[c]*e*Sqrt[(1 + (2*c*x^2)/(b - Sqrt
[b^2 + 4*a*c]))/(1 + (2*c*x^2)/(b + Sqrt[b^2 + 4*a*c]))]*Sqrt[-a + b*x^2 + c*x^4
]) + (Sqrt[-b + Sqrt[b^2 + 4*a*c]]*(e*f - d*g)*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2
+ 4*a*c])]*Sqrt[1 + (2*c*x^2)/(b + Sqrt[b^2 + 4*a*c])]*EllipticPi[-((b - Sqrt[b^
2 + 4*a*c])*e^2)/(2*c*d^2), ArcSin[(Sqrt[2]*Sqrt[c]*x)/Sqrt[-b + Sqrt[b^2 + 4*a*
c]]], (b - Sqrt[b^2 + 4*a*c])/(b + Sqrt[b^2 + 4*a*c])])/(Sqrt[2]*Sqrt[c]*d*e*Sqr
t[-a + b*x^2 + c*x^4])

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((g*x+f)/(e*x+d)/(c*x**4+b*x**2-a)**(1/2),x)

[Out]

Timed out

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Mathematica [A]  time = 0.865718, size = 0, normalized size = 0. \[ \int \frac{f+g x}{(d+e x) \sqrt{-a+b x^2+c x^4}} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[(f + g*x)/((d + e*x)*Sqrt[-a + b*x^2 + c*x^4]),x]

[Out]

Integrate[(f + g*x)/((d + e*x)*Sqrt[-a + b*x^2 + c*x^4]), x]

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Maple [A]  time = 0.057, size = 439, normalized size = 0.8 \[{\frac{g}{2\,e}\sqrt{4+2\,{\frac{ \left ( -b+\sqrt{4\,ac+{b}^{2}} \right ){x}^{2}}{a}}}\sqrt{4-2\,{\frac{ \left ( b+\sqrt{4\,ac+{b}^{2}} \right ){x}^{2}}{a}}}{\it EllipticF} \left ({\frac{x}{2}\sqrt{-2\,{\frac{-b+\sqrt{4\,ac+{b}^{2}}}{a}}}},{\frac{1}{2}\sqrt{-4-2\,{\frac{b \left ( b+\sqrt{4\,ac+{b}^{2}} \right ) }{ac}}}} \right ){\frac{1}{\sqrt{-2\,{\frac{-b+\sqrt{4\,ac+{b}^{2}}}{a}}}}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}-a}}}}+{\frac{-dg+ef}{{e}^{2}} \left ( -{\frac{1}{2}{\it Artanh} \left ({\frac{1}{2} \left ( 2\,{\frac{c{x}^{2}{d}^{2}}{{e}^{2}}}+b{x}^{2}+{\frac{b{d}^{2}}{{e}^{2}}}-2\,a \right ){\frac{1}{\sqrt{{\frac{c{d}^{4}}{{e}^{4}}}+{\frac{b{d}^{2}}{{e}^{2}}}-a}}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}-a}}}} \right ){\frac{1}{\sqrt{{\frac{c{d}^{4}}{{e}^{4}}}+{\frac{b{d}^{2}}{{e}^{2}}}-a}}}}+{\frac{e}{d}\sqrt{1+{\frac{{x}^{2}}{2\,a} \left ( -b+\sqrt{4\,ac+{b}^{2}} \right ) }}\sqrt{1-{\frac{{x}^{2}}{2\,a} \left ( b+\sqrt{4\,ac+{b}^{2}} \right ) }}{\it EllipticPi} \left ( \sqrt{-{\frac{1}{2\,a} \left ( -b+\sqrt{4\,ac+{b}^{2}} \right ) }}x,-2\,{\frac{a{e}^{2}}{ \left ( -b+\sqrt{4\,ac+{b}^{2}} \right ){d}^{2}}},{\frac{\sqrt{2}}{2}\sqrt{{\frac{1}{a} \left ( b+\sqrt{4\,ac+{b}^{2}} \right ) }}{\frac{1}{\sqrt{-{\frac{1}{2\,a} \left ( -b+\sqrt{4\,ac+{b}^{2}} \right ) }}}}} \right ){\frac{1}{\sqrt{-{\frac{1}{2\,a} \left ( -b+\sqrt{4\,ac+{b}^{2}} \right ) }}}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}-a}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((g*x+f)/(e*x+d)/(c*x^4+b*x^2-a)^(1/2),x)

[Out]

1/2/e*g/(-2*(-b+(4*a*c+b^2)^(1/2))/a)^(1/2)*(4+2*(-b+(4*a*c+b^2)^(1/2))/a*x^2)^(
1/2)*(4-2*(b+(4*a*c+b^2)^(1/2))/a*x^2)^(1/2)/(c*x^4+b*x^2-a)^(1/2)*EllipticF(1/2
*x*(-2*(-b+(4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4-2*b*(b+(4*a*c+b^2)^(1/2))/a/c)^(1
/2))+(-d*g+e*f)/e^2*(-1/2/(c*d^4/e^4+b*d^2/e^2-a)^(1/2)*arctanh(1/2*(2*c*x^2*d^2
/e^2+b*x^2+b*d^2/e^2-2*a)/(c*d^4/e^4+b*d^2/e^2-a)^(1/2)/(c*x^4+b*x^2-a)^(1/2))+1
/(-1/2*(-b+(4*a*c+b^2)^(1/2))/a)^(1/2)/d*e*(1+1/2*(-b+(4*a*c+b^2)^(1/2))/a*x^2)^
(1/2)*(1-1/2*(b+(4*a*c+b^2)^(1/2))/a*x^2)^(1/2)/(c*x^4+b*x^2-a)^(1/2)*EllipticPi
((-1/2*(-b+(4*a*c+b^2)^(1/2))/a)^(1/2)*x,-2/(-b+(4*a*c+b^2)^(1/2))*a/d^2*e^2,1/2
*2^(1/2)*((b+(4*a*c+b^2)^(1/2))/a)^(1/2)/(-1/2*(-b+(4*a*c+b^2)^(1/2))/a)^(1/2)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{g x + f}{\sqrt{c x^{4} + b x^{2} - a}{\left (e x + d\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x + f)/(sqrt(c*x^4 + b*x^2 - a)*(e*x + d)),x, algorithm="maxima")

[Out]

integrate((g*x + f)/(sqrt(c*x^4 + b*x^2 - a)*(e*x + d)), 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((g*x + f)/(sqrt(c*x^4 + b*x^2 - a)*(e*x + d)),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{f + g x}{\left (d + e x\right ) \sqrt{- a + b x^{2} + c x^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x+f)/(e*x+d)/(c*x**4+b*x**2-a)**(1/2),x)

[Out]

Integral((f + g*x)/((d + e*x)*sqrt(-a + b*x**2 + c*x**4)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x + f)/(sqrt(c*x^4 + b*x^2 - a)*(e*x + d)),x, algorithm="giac")

[Out]

Timed out