3.104 \(\int \left (d+e x+f x^2+g x^3\right ) \sqrt{a+b x^2+c x^4} \, dx\)

Optimal. Leaf size=505 \[ \frac{x \sqrt{a+b x^2+c x^4} \left (6 a c f-2 b^2 f+5 b c d\right )}{15 c^{3/2} \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{\sqrt [4]{a} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (6 a c f-2 b^2 f+5 b c d\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{15 c^{7/4} \sqrt{a+b x^2+c x^4}}-\frac{\left (b^2-4 a c\right ) (2 c e-b g) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{32 c^{5/2}}+\frac{\sqrt [4]{a} \left (2 \sqrt{a} \sqrt{c}+b\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (3 \sqrt{a} \sqrt{c} f-2 b f+5 c d\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{30 c^{7/4} \sqrt{a+b x^2+c x^4}}+\frac{\left (b+2 c x^2\right ) \sqrt{a+b x^2+c x^4} (2 c e-b g)}{16 c^2}+\frac{x \sqrt{a+b x^2+c x^4} \left (b f+5 c d+3 c f x^2\right )}{15 c}+\frac{g \left (a+b x^2+c x^4\right )^{3/2}}{6 c} \]

[Out]

((5*b*c*d - 2*b^2*f + 6*a*c*f)*x*Sqrt[a + b*x^2 + c*x^4])/(15*c^(3/2)*(Sqrt[a] +
Sqrt[c]*x^2)) + ((2*c*e - b*g)*(b + 2*c*x^2)*Sqrt[a + b*x^2 + c*x^4])/(16*c^2)
+ (x*(5*c*d + b*f + 3*c*f*x^2)*Sqrt[a + b*x^2 + c*x^4])/(15*c) + (g*(a + b*x^2 +
c*x^4)^(3/2))/(6*c) - ((b^2 - 4*a*c)*(2*c*e - b*g)*ArcTanh[(b + 2*c*x^2)/(2*Sqr
t[c]*Sqrt[a + b*x^2 + c*x^4])])/(32*c^(5/2)) - (a^(1/4)*(5*b*c*d - 2*b^2*f + 6*a
*c*f)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2
]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(15*c^(
7/4)*Sqrt[a + b*x^2 + c*x^4]) + (a^(1/4)*(b + 2*Sqrt[a]*Sqrt[c])*(5*c*d - 2*b*f
+ 3*Sqrt[a]*Sqrt[c]*f)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a]
+ Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt
[c]))/4])/(30*c^(7/4)*Sqrt[a + b*x^2 + c*x^4])

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Rubi [A]  time = 0.771819, antiderivative size = 505, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312 \[ \frac{x \sqrt{a+b x^2+c x^4} \left (6 a c f-2 b^2 f+5 b c d\right )}{15 c^{3/2} \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{\sqrt [4]{a} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (6 a c f-2 b^2 f+5 b c d\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{15 c^{7/4} \sqrt{a+b x^2+c x^4}}-\frac{\left (b^2-4 a c\right ) (2 c e-b g) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{32 c^{5/2}}+\frac{\sqrt [4]{a} \left (2 \sqrt{a} \sqrt{c}+b\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (3 \sqrt{a} \sqrt{c} f-2 b f+5 c d\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{30 c^{7/4} \sqrt{a+b x^2+c x^4}}+\frac{\left (b+2 c x^2\right ) \sqrt{a+b x^2+c x^4} (2 c e-b g)}{16 c^2}+\frac{x \sqrt{a+b x^2+c x^4} \left (b f+5 c d+3 c f x^2\right )}{15 c}+\frac{g \left (a+b x^2+c x^4\right )^{3/2}}{6 c} \]

Warning: Unable to verify antiderivative.

[In]  Int[(d + e*x + f*x^2 + g*x^3)*Sqrt[a + b*x^2 + c*x^4],x]

[Out]

((5*b*c*d - 2*b^2*f + 6*a*c*f)*x*Sqrt[a + b*x^2 + c*x^4])/(15*c^(3/2)*(Sqrt[a] +
Sqrt[c]*x^2)) + ((2*c*e - b*g)*(b + 2*c*x^2)*Sqrt[a + b*x^2 + c*x^4])/(16*c^2)
+ (x*(5*c*d + b*f + 3*c*f*x^2)*Sqrt[a + b*x^2 + c*x^4])/(15*c) + (g*(a + b*x^2 +
c*x^4)^(3/2))/(6*c) - ((b^2 - 4*a*c)*(2*c*e - b*g)*ArcTanh[(b + 2*c*x^2)/(2*Sqr
t[c]*Sqrt[a + b*x^2 + c*x^4])])/(32*c^(5/2)) - (a^(1/4)*(5*b*c*d - 2*b^2*f + 6*a
*c*f)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2
]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(15*c^(
7/4)*Sqrt[a + b*x^2 + c*x^4]) + (a^(1/4)*(b + 2*Sqrt[a]*Sqrt[c])*(5*c*d - 2*b*f
+ 3*Sqrt[a]*Sqrt[c]*f)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a]
+ Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt
[c]))/4])/(30*c^(7/4)*Sqrt[a + b*x^2 + c*x^4])

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Rubi in Sympy [A]  time = 85.1694, size = 474, normalized size = 0.94 \[ \frac{\sqrt [4]{a} \sqrt{\frac{a + b x^{2} + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) \left (- 6 a c f + 2 b^{2} f - 5 b c d\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2} - \frac{b}{4 \sqrt{a} \sqrt{c}}\right )}{15 c^{\frac{7}{4}} \sqrt{a + b x^{2} + c x^{4}}} - \frac{\sqrt [4]{a} \sqrt{\frac{a + b x^{2} + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) \left (\sqrt{a} \sqrt{c} \left (b f - 10 c d\right ) - 6 a c f + 2 b^{2} f - 5 b c d\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2} - \frac{b}{4 \sqrt{a} \sqrt{c}}\right )}{30 c^{\frac{7}{4}} \sqrt{a + b x^{2} + c x^{4}}} + \frac{g \left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{6 c} + \frac{x \sqrt{a + b x^{2} + c x^{4}} \left (b f + 5 c d + 3 c f x^{2}\right )}{15 c} - \frac{\left (b + 2 c x^{2}\right ) \left (b g - 2 c e\right ) \sqrt{a + b x^{2} + c x^{4}}}{16 c^{2}} - \frac{x \sqrt{a + b x^{2} + c x^{4}} \left (- 6 a c f + 2 b^{2} f - 5 b c d\right )}{15 c^{\frac{3}{2}} \left (\sqrt{a} + \sqrt{c} x^{2}\right )} + \frac{\left (- 4 a c + b^{2}\right ) \left (b g - 2 c e\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{2 \sqrt{c} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{32 c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((g*x**3+f*x**2+e*x+d)*(c*x**4+b*x**2+a)**(1/2),x)

[Out]

a**(1/4)*sqrt((a + b*x**2 + c*x**4)/(sqrt(a) + sqrt(c)*x**2)**2)*(sqrt(a) + sqrt
(c)*x**2)*(-6*a*c*f + 2*b**2*f - 5*b*c*d)*elliptic_e(2*atan(c**(1/4)*x/a**(1/4))
, 1/2 - b/(4*sqrt(a)*sqrt(c)))/(15*c**(7/4)*sqrt(a + b*x**2 + c*x**4)) - a**(1/4
)*sqrt((a + b*x**2 + c*x**4)/(sqrt(a) + sqrt(c)*x**2)**2)*(sqrt(a) + sqrt(c)*x**
2)*(sqrt(a)*sqrt(c)*(b*f - 10*c*d) - 6*a*c*f + 2*b**2*f - 5*b*c*d)*elliptic_f(2*
atan(c**(1/4)*x/a**(1/4)), 1/2 - b/(4*sqrt(a)*sqrt(c)))/(30*c**(7/4)*sqrt(a + b*
x**2 + c*x**4)) + g*(a + b*x**2 + c*x**4)**(3/2)/(6*c) + x*sqrt(a + b*x**2 + c*x
**4)*(b*f + 5*c*d + 3*c*f*x**2)/(15*c) - (b + 2*c*x**2)*(b*g - 2*c*e)*sqrt(a + b
*x**2 + c*x**4)/(16*c**2) - x*sqrt(a + b*x**2 + c*x**4)*(-6*a*c*f + 2*b**2*f - 5
*b*c*d)/(15*c**(3/2)*(sqrt(a) + sqrt(c)*x**2)) + (-4*a*c + b**2)*(b*g - 2*c*e)*a
tanh((b + 2*c*x**2)/(2*sqrt(c)*sqrt(a + b*x**2 + c*x**4)))/(32*c**(5/2))

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Mathematica [C]  time = 6.34107, size = 1534, normalized size = 3.04 \[ \sqrt{c x^4+b x^2+a} \left (\frac{g x^4}{6}+\frac{f x^3}{5}+\frac{(6 c e+b g) x^2}{24 c}+\frac{(5 c d+b f) x}{15 c}+\frac{-3 g b^2+6 c e b+8 a c g}{48 c^2}\right )+\frac{\frac{15 g \log \left (2 c x^2+b+2 \sqrt{c} \sqrt{c x^4+b x^2+a}\right ) b^3}{2 \sqrt{c}}-\frac{8 i \sqrt{2} \left (\sqrt{b^2-4 a c}-b\right ) f \sqrt{1-\frac{2 c x^2}{-b-\sqrt{b^2-4 a c}}} \sqrt{1-\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}} \left (E\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}}{\sqrt{b^2-4 a c}-b}\right )-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}}{\sqrt{b^2-4 a c}-b}\right )\right ) b^2}{\sqrt{-\frac{c}{-b-\sqrt{b^2-4 a c}}} \sqrt{c x^4+b x^2+a}}-15 \sqrt{c} e \log \left (2 c x^2+b+2 \sqrt{c} \sqrt{c x^4+b x^2+a}\right ) b^2+\frac{20 i \sqrt{2} c \left (\sqrt{b^2-4 a c}-b\right ) d \sqrt{1-\frac{2 c x^2}{-b-\sqrt{b^2-4 a c}}} \sqrt{1-\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}} \left (E\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}}{\sqrt{b^2-4 a c}-b}\right )-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}}{\sqrt{b^2-4 a c}-b}\right )\right ) b}{\sqrt{-\frac{c}{-b-\sqrt{b^2-4 a c}}} \sqrt{c x^4+b x^2+a}}+\frac{8 i \sqrt{2} a c f \sqrt{1-\frac{2 c x^2}{-b-\sqrt{b^2-4 a c}}} \sqrt{1-\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}} 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}}{\sqrt{b^2-4 a c}-b}\right ) b}{\sqrt{-\frac{c}{-b-\sqrt{b^2-4 a c}}} \sqrt{c x^4+b x^2+a}}-30 a \sqrt{c} g \log \left (2 c x^2+b+2 \sqrt{c} \sqrt{c x^4+b x^2+a}\right ) b+\frac{24 i \sqrt{2} a c \left (\sqrt{b^2-4 a c}-b\right ) f \sqrt{1-\frac{2 c x^2}{-b-\sqrt{b^2-4 a c}}} \sqrt{1-\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}} \left (E\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}}{\sqrt{b^2-4 a c}-b}\right )-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}}{\sqrt{b^2-4 a c}-b}\right )\right )}{\sqrt{-\frac{c}{-b-\sqrt{b^2-4 a c}}} \sqrt{c x^4+b x^2+a}}-\frac{80 i \sqrt{2} a c^2 d \sqrt{1-\frac{2 c x^2}{-b-\sqrt{b^2-4 a c}}} \sqrt{1-\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}} 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}}{\sqrt{b^2-4 a c}-b}\right )}{\sqrt{-\frac{c}{-b-\sqrt{b^2-4 a c}}} \sqrt{c x^4+b x^2+a}}+60 a c^{3/2} e \log \left (2 c x^2+b+2 \sqrt{c} \sqrt{c x^4+b x^2+a}\right )}{240 c^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x + f*x^2 + g*x^3)*Sqrt[a + b*x^2 + c*x^4],x]

[Out]

Sqrt[a + b*x^2 + c*x^4]*((6*b*c*e - 3*b^2*g + 8*a*c*g)/(48*c^2) + ((5*c*d + b*f)
*x)/(15*c) + ((6*c*e + b*g)*x^2)/(24*c) + (f*x^3)/5 + (g*x^4)/6) + (((20*I)*Sqrt
[2]*b*c*(-b + Sqrt[b^2 - 4*a*c])*d*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])]*(EllipticE[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])] - 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/(-b - Sqrt[b^2 - 4*a*c
]))]*Sqrt[a + b*x^2 + c*x^4]) - ((8*I)*Sqrt[2]*b^2*(-b + Sqrt[b^2 - 4*a*c])*f*Sq
rt[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])]*(EllipticE[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])] - EllipticF[I*ArcSinh[Sqrt[2]*Sq
rt[-(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/(-b - Sqrt[b^2 - 4*a*c]))]*Sqrt[a + b*x^2 + c*x^4]) + ((24*
I)*Sqrt[2]*a*c*(-b + Sqrt[b^2 - 4*a*c])*f*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])]*(EllipticE[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])] - 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/(-b - Sqrt[b^2
- 4*a*c]))]*Sqrt[a + b*x^2 + c*x^4]) - ((80*I)*Sqrt[2]*a*c^2*d*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])]*Ellipti
cF[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/(-b - Sqrt[b^2 - 4*a*c]))]*Sqrt[a + b
*x^2 + c*x^4]) + ((8*I)*Sqrt[2]*a*b*c*f*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])]*EllipticF[I*ArcSinh[Sqrt[2]*Sq
rt[-(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/(-b - Sqrt[b^2 - 4*a*c]))]*Sqrt[a + b*x^2 + c*x^4]) - 15*b^2
*Sqrt[c]*e*Log[b + 2*c*x^2 + 2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4]] + 60*a*c^(3/2)*e
*Log[b + 2*c*x^2 + 2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4]] + (15*b^3*g*Log[b + 2*c*x^
2 + 2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4]])/(2*Sqrt[c]) - 30*a*b*Sqrt[c]*g*Log[b + 2
*c*x^2 + 2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4]])/(240*c^2)

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Maple [B]  time = 0.01, size = 1585, normalized size = 3.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((g*x^3+f*x^2+e*x+d)*(c*x^4+b*x^2+a)^(1/2),x)

[Out]

1/3*x*d*(c*x^4+b*x^2+a)^(1/2)+1/6*d*a*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))-1/6*d*b*a*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)/(b+(-4*a*c+b^2)^(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))+1/6*d*b*a*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)/(b+(-4*a*c+b^2)^(1/2))*EllipticE(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))+1/4*e*(c*x^4+b*x^2+a
)^(1/2)*x^2+1/8*e/c*(c*x^4+b*x^2+a)^(1/2)*b+1/4*e/c^(1/2)*ln((1/2*b+c*x^2)/c^(1/
2)+(c*x^4+b*x^2+a)^(1/2))*a-1/16*e/c^(3/2)*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2
+a)^(1/2))*b^2+1/5*f*x^3*(c*x^4+b*x^2+a)^(1/2)+1/15*f*b/c*x*(c*x^4+b*x^2+a)^(1/2
)-1/60*f*b/c*a*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))-1/5*f*a^2*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)/(b+(-4*a*c+b^2)^(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))+1/5*f*a^
2*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)/(b+(-4*a*c+
b^2)^(1/2))*EllipticE(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))+1/15*f*a*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)/(b+(-4*a*c+b^2)^(1/2))*b^2/c*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))-1/15*f*a*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)/(b+(-4*a*c+b^2)^(1/2))*b^2/c*EllipticE(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))+1/6*g*(c*x^4+b*x^2+a)^(
3/2)/c-1/8*g*b/c*(c*x^4+b*x^2+a)^(1/2)*x^2-1/16*g*b^2/c^2*(c*x^4+b*x^2+a)^(1/2)-
1/8*g*b/c^(3/2)*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))*a+1/32*g*b^3/c^(
5/2)*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{c x^{4} + b x^{2} + a}{\left (g x^{3} + f x^{2} + e x + d\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + b*x^2 + a)*(g*x^3 + f*x^2 + e*x + d),x, algorithm="maxima")

[Out]

integrate(sqrt(c*x^4 + b*x^2 + a)*(g*x^3 + f*x^2 + e*x + d), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{c x^{4} + b x^{2} + a}{\left (g x^{3} + f x^{2} + e x + d\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + b*x^2 + a)*(g*x^3 + f*x^2 + e*x + d),x, algorithm="fricas")

[Out]

integral(sqrt(c*x^4 + b*x^2 + a)*(g*x^3 + f*x^2 + e*x + d), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{a + b x^{2} + c x^{4}} \left (d + e x + f x^{2} + g x^{3}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x**3+f*x**2+e*x+d)*(c*x**4+b*x**2+a)**(1/2),x)

[Out]

Integral(sqrt(a + b*x**2 + c*x**4)*(d + e*x + f*x**2 + g*x**3), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{c x^{4} + b x^{2} + a}{\left (g x^{3} + f x^{2} + e x + d\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + b*x^2 + a)*(g*x^3 + f*x^2 + e*x + d),x, algorithm="giac")

[Out]

integrate(sqrt(c*x^4 + b*x^2 + a)*(g*x^3 + f*x^2 + e*x + d), x)