3.11 \(\int \frac{1+\sqrt{\frac{c}{a}} x^2}{\left (d+e x^2\right ) \sqrt{a+b x^2+c x^4}} \, dx\)

Optimal. Leaf size=257 \[ \frac{\left (x^2 \sqrt{\frac{c}{a}}+1\right ) \sqrt{\frac{a+b x^2+c x^4}{a \left (x^2 \sqrt{\frac{c}{a}}+1\right )^2}} \left (d \sqrt{\frac{c}{a}}+e\right ) \Pi \left (-\frac{\left (\sqrt{\frac{c}{a}} d-e\right )^2}{4 \sqrt{\frac{c}{a}} d e};2 \tan ^{-1}\left (\sqrt [4]{\frac{c}{a}} x\right )|\frac{1}{4} \left (2-\frac{b \sqrt{\frac{c}{a}}}{c}\right )\right )}{4 d e \sqrt [4]{\frac{c}{a}} \sqrt{a+b x^2+c x^4}}-\frac{\left (d \sqrt{\frac{c}{a}}-e\right ) \tan ^{-1}\left (\frac{x \sqrt{\frac{a e}{d}-b+\frac{c d}{e}}}{\sqrt{a+b x^2+c x^4}}\right )}{2 d e \sqrt{\frac{a e}{d}-b+\frac{c d}{e}}} \]

[Out]

-((Sqrt[c/a]*d - e)*ArcTan[(Sqrt[-b + (c*d)/e + (a*e)/d]*x)/Sqrt[a + b*x^2 + c*x
^4]])/(2*d*e*Sqrt[-b + (c*d)/e + (a*e)/d]) + ((Sqrt[c/a]*d + e)*(1 + Sqrt[c/a]*x
^2)*Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + Sqrt[c/a]*x^2)^2)]*EllipticPi[-(Sqrt[c/a]*d
- e)^2/(4*Sqrt[c/a]*d*e), 2*ArcTan[(c/a)^(1/4)*x], (2 - (b*Sqrt[c/a])/c)/4])/(4
*(c/a)^(1/4)*d*e*Sqrt[a + b*x^2 + c*x^4])

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Rubi [A]  time = 0.457724, antiderivative size = 257, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.024 \[ \frac{\left (x^2 \sqrt{\frac{c}{a}}+1\right ) \sqrt{\frac{a+b x^2+c x^4}{a \left (x^2 \sqrt{\frac{c}{a}}+1\right )^2}} \left (d \sqrt{\frac{c}{a}}+e\right ) \Pi \left (-\frac{\left (\sqrt{\frac{c}{a}} d-e\right )^2}{4 \sqrt{\frac{c}{a}} d e};2 \tan ^{-1}\left (\sqrt [4]{\frac{c}{a}} x\right )|\frac{1}{4} \left (2-\frac{b \sqrt{\frac{c}{a}}}{c}\right )\right )}{4 d e \sqrt [4]{\frac{c}{a}} \sqrt{a+b x^2+c x^4}}-\frac{\left (d \sqrt{\frac{c}{a}}-e\right ) \tan ^{-1}\left (\frac{x \sqrt{\frac{a e}{d}-b+\frac{c d}{e}}}{\sqrt{a+b x^2+c x^4}}\right )}{2 d e \sqrt{\frac{a e}{d}-b+\frac{c d}{e}}} \]

Warning: Unable to verify antiderivative.

[In]  Int[(1 + Sqrt[c/a]*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]),x]

[Out]

-((Sqrt[c/a]*d - e)*ArcTan[(Sqrt[-b + (c*d)/e + (a*e)/d]*x)/Sqrt[a + b*x^2 + c*x
^4]])/(2*d*e*Sqrt[-b + (c*d)/e + (a*e)/d]) + ((Sqrt[c/a]*d + e)*(1 + Sqrt[c/a]*x
^2)*Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + Sqrt[c/a]*x^2)^2)]*EllipticPi[-(Sqrt[c/a]*d
- e)^2/(4*Sqrt[c/a]*d*e), 2*ArcTan[(c/a)^(1/4)*x], (2 - (b*Sqrt[c/a])/c)/4])/(4
*(c/a)^(1/4)*d*e*Sqrt[a + b*x^2 + c*x^4])

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Rubi in Sympy [A]  time = 61.0579, size = 479, normalized size = 1.86 \[ \frac{\left (- d \sqrt{\frac{c}{a}} + e\right ) \operatorname{atan}{\left (\frac{x \sqrt{\frac{a e}{d} - b + \frac{c d}{e}}}{\sqrt{a + b x^{2} + c x^{4}}} \right )}}{2 d e \sqrt{\frac{a e}{d} - b + \frac{c d}{e}}} - \frac{\sqrt [4]{c} \sqrt{\frac{a + b x^{2} + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) \left (- d \sqrt{\frac{c}{a}} + e\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2} - \frac{b}{4 \sqrt{a} \sqrt{c}}\right )}{2 \sqrt [4]{a} e \left (\sqrt{a} e - \sqrt{c} d\right ) \sqrt{a + b x^{2} + c x^{4}}} + \frac{\sqrt{\frac{c}{a}} \sqrt{\frac{a + b x^{2} + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2} - \frac{b}{4 \sqrt{a} \sqrt{c}}\right )}{2 \sqrt [4]{a} \sqrt [4]{c} e \sqrt{a + b x^{2} + c x^{4}}} + \frac{\sqrt{\frac{a + b x^{2} + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) \left (\sqrt{a} e + \sqrt{c} d\right ) \left (- d \sqrt{\frac{c}{a}} + e\right ) \Pi \left (- \frac{\sqrt{a} \left (e - \frac{\sqrt{c} d}{\sqrt{a}}\right )^{2}}{4 \sqrt{c} d e}; 2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2} - \frac{b}{4 \sqrt{a} \sqrt{c}}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} d e \left (\sqrt{a} e - \sqrt{c} d\right ) \sqrt{a + b x^{2} + c x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-d*sqrt(c/a) + e)*atan(x*sqrt(a*e/d - b + c*d/e)/sqrt(a + b*x**2 + c*x**4))/(2*
d*e*sqrt(a*e/d - b + c*d/e)) - c**(1/4)*sqrt((a + b*x**2 + c*x**4)/(sqrt(a) + sq
rt(c)*x**2)**2)*(sqrt(a) + sqrt(c)*x**2)*(-d*sqrt(c/a) + e)*elliptic_f(2*atan(c*
*(1/4)*x/a**(1/4)), 1/2 - b/(4*sqrt(a)*sqrt(c)))/(2*a**(1/4)*e*(sqrt(a)*e - sqrt
(c)*d)*sqrt(a + b*x**2 + c*x**4)) + sqrt(c/a)*sqrt((a + b*x**2 + c*x**4)/(sqrt(a
) + sqrt(c)*x**2)**2)*(sqrt(a) + sqrt(c)*x**2)*elliptic_f(2*atan(c**(1/4)*x/a**(
1/4)), 1/2 - b/(4*sqrt(a)*sqrt(c)))/(2*a**(1/4)*c**(1/4)*e*sqrt(a + b*x**2 + c*x
**4)) + sqrt((a + b*x**2 + c*x**4)/(sqrt(a) + sqrt(c)*x**2)**2)*(sqrt(a) + sqrt(
c)*x**2)*(sqrt(a)*e + sqrt(c)*d)*(-d*sqrt(c/a) + e)*elliptic_pi(-sqrt(a)*(e - sq
rt(c)*d/sqrt(a))**2/(4*sqrt(c)*d*e), 2*atan(c**(1/4)*x/a**(1/4)), 1/2 - b/(4*sqr
t(a)*sqrt(c)))/(4*a**(1/4)*c**(1/4)*d*e*(sqrt(a)*e - sqrt(c)*d)*sqrt(a + b*x**2
+ c*x**4))

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Mathematica [C]  time = 0.592966, size = 312, normalized size = 1.21 \[ -\frac{i \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \left (\left (e-d \sqrt{\frac{c}{a}}\right ) \Pi \left (\frac{\left (b+\sqrt{b^2-4 a c}\right ) e}{2 c d};i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )+d \sqrt{\frac{c}{a}} F\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )\right )}{\sqrt{2} d e \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + Sqrt[c/a]*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]),x]

[Out]

((-I)*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (
2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*(Sqrt[c/a]*d*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[
c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])]
+ (-(Sqrt[c/a]*d) + e)*EllipticPi[((b + Sqrt[b^2 - 4*a*c])*e)/(2*c*d), I*ArcSinh
[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b
^2 - 4*a*c])]))/(Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d*e*Sqrt[a + b*x^2 + c*
x^4])

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Maple [A]  time = 0.063, size = 369, normalized size = 1.4 \[{\frac{\sqrt{2}}{4\,e}\sqrt{{\frac{c}{a}}}\sqrt{4-2\,{\frac{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ){x}^{2}}{a}}}\sqrt{4+2\,{\frac{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ){x}^{2}}{a}}}{\it EllipticF} \left ({\frac{\sqrt{2}x}{2}\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}},{\frac{1}{2}\sqrt{-4+2\,{\frac{b \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) }{ac}}}} \right ){\frac{1}{\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}+{\frac{\sqrt{2}}{de} \left ( -d\sqrt{{\frac{c}{a}}}+e \right ) \sqrt{1+{\frac{b{x}^{2}}{2\,a}}-{\frac{{x}^{2}}{2\,a}\sqrt{-4\,ac+{b}^{2}}}}\sqrt{1+{\frac{b{x}^{2}}{2\,a}}+{\frac{{x}^{2}}{2\,a}\sqrt{-4\,ac+{b}^{2}}}}{\it EllipticPi} \left ({\frac{\sqrt{2}x}{2}\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}},-2\,{\frac{ae}{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) d}},{\sqrt{2}\sqrt{-{\frac{1}{2\,a} \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) }}{\frac{1}{\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}}}} \right ){\frac{1}{\sqrt{-{\frac{b}{a}}+{\frac{1}{a}\sqrt{-4\,ac+{b}^{2}}}}}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(c/a)^(1/2)/e*2^(1/2)/((-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^(1/2)*((-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))+1/e*(-d*(c/a)^(1/2)+e)/d*2^(1/2)/(-b/a+1/a*(-4*a*c
+b^2)^(1/2))^(1/2)*(1+1/2*b*x^2/a-1/2/a*x^2*(-4*a*c+b^2)^(1/2))^(1/2)*(1+1/2*b*x
^2/a+1/2/a*x^2*(-4*a*c+b^2)^(1/2))^(1/2)/(c*x^4+b*x^2+a)^(1/2)*EllipticPi(1/2*x*
2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),-2/(-b+(-4*a*c+b^2)^(1/2))*a*e/d,(-1/2
*(b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*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. \[ \text{Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2*sqrt(c/a) + 1)/(sqrt(c*x^4 + b*x^2 + a)*(e*x^2 + d)),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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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((x^2*sqrt(c/a) + 1)/(sqrt(c*x^4 + b*x^2 + a)*(e*x^2 + d)),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2} \sqrt{\frac{c}{a}} + 1}{\left (d + e x^{2}\right ) \sqrt{a + b x^{2} + c x^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((x**2*sqrt(c/a) + 1)/((d + e*x**2)*sqrt(a + b*x**2 + c*x**4)), x)

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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((x^2*sqrt(c/a) + 1)/(sqrt(c*x^4 + b*x^2 + a)*(e*x^2 + d)),x, algorithm="giac")

[Out]

Exception raised: TypeError