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

Optimal. Leaf size=717 \[ -\frac{x \sqrt{a+b x^2+c x^4} \left (-84 a^2 c^2 f+57 a b^2 c f-144 a b c^2 d-8 b^4 f+18 b^3 c d\right )}{315 c^{5/2} \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{\sqrt [4]{a} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (-84 a^2 c^2 f+57 a b^2 c f+\sqrt{a} \sqrt{c} \left (24 a b c f-180 a c^2 d-4 b^3 f+9 b^2 c d\right )-144 a b c^2 d-8 b^4 f+18 b^3 c d\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{630 c^{11/4} \sqrt{a+b x^2+c x^4}}+\frac{\sqrt [4]{a} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (-84 a^2 c^2 f+57 a b^2 c f-144 a b c^2 d-8 b^4 f+18 b^3 c d\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{315 c^{11/4} \sqrt{a+b x^2+c x^4}}+\frac{3 \left (b^2-4 a c\right )^2 (2 c e-b g) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{512 c^{7/2}}-\frac{3 \left (b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt{a+b x^2+c x^4} (2 c e-b g)}{256 c^3}+\frac{x \sqrt{a+b x^2+c x^4} \left (3 c x^2 \left (14 a c f-4 b^2 f+9 b c d\right )+9 a b c f+90 a c^2 d-4 b^3 f+9 b^2 c d\right )}{315 c^2}+\frac{\left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2} (2 c e-b g)}{32 c^2}+\frac{x \left (a+b x^2+c x^4\right )^{3/2} \left (3 (b f+3 c d)+7 c f x^2\right )}{63 c}+\frac{g \left (a+b x^2+c x^4\right )^{5/2}}{10 c} \]

[Out]

-((18*b^3*c*d - 144*a*b*c^2*d - 8*b^4*f + 57*a*b^2*c*f - 84*a^2*c^2*f)*x*Sqrt[a
+ b*x^2 + c*x^4])/(315*c^(5/2)*(Sqrt[a] + Sqrt[c]*x^2)) - (3*(b^2 - 4*a*c)*(2*c*
e - b*g)*(b + 2*c*x^2)*Sqrt[a + b*x^2 + c*x^4])/(256*c^3) + (x*(9*b^2*c*d + 90*a
*c^2*d - 4*b^3*f + 9*a*b*c*f + 3*c*(9*b*c*d - 4*b^2*f + 14*a*c*f)*x^2)*Sqrt[a +
b*x^2 + c*x^4])/(315*c^2) + ((2*c*e - b*g)*(b + 2*c*x^2)*(a + b*x^2 + c*x^4)^(3/
2))/(32*c^2) + (x*(3*(3*c*d + b*f) + 7*c*f*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(63*c
) + (g*(a + b*x^2 + c*x^4)^(5/2))/(10*c) + (3*(b^2 - 4*a*c)^2*(2*c*e - b*g)*ArcT
anh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/(512*c^(7/2)) + (a^(1/4)
*(18*b^3*c*d - 144*a*b*c^2*d - 8*b^4*f + 57*a*b^2*c*f - 84*a^2*c^2*f)*(Sqrt[a] +
Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*Ar
cTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(315*c^(11/4)*Sqrt[a +
b*x^2 + c*x^4]) - (a^(1/4)*(18*b^3*c*d - 144*a*b*c^2*d - 8*b^4*f + 57*a*b^2*c*f
- 84*a^2*c^2*f + Sqrt[a]*Sqrt[c]*(9*b^2*c*d - 180*a*c^2*d - 4*b^3*f + 24*a*b*c*f
))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*E
llipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(630*c^(11
/4)*Sqrt[a + b*x^2 + c*x^4])

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Rubi [A]  time = 1.39402, antiderivative size = 717, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 10, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312 \[ -\frac{x \sqrt{a+b x^2+c x^4} \left (-84 a^2 c^2 f+57 a b^2 c f-144 a b c^2 d-8 b^4 f+18 b^3 c d\right )}{315 c^{5/2} \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{\sqrt [4]{a} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (-84 a^2 c^2 f+57 a b^2 c f+\sqrt{a} \sqrt{c} \left (24 a b c f-180 a c^2 d-4 b^3 f+9 b^2 c d\right )-144 a b c^2 d-8 b^4 f+18 b^3 c d\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{630 c^{11/4} \sqrt{a+b x^2+c x^4}}+\frac{\sqrt [4]{a} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (-84 a^2 c^2 f+57 a b^2 c f-144 a b c^2 d-8 b^4 f+18 b^3 c d\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{315 c^{11/4} \sqrt{a+b x^2+c x^4}}+\frac{3 \left (b^2-4 a c\right )^2 (2 c e-b g) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{512 c^{7/2}}-\frac{3 \left (b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt{a+b x^2+c x^4} (2 c e-b g)}{256 c^3}+\frac{x \sqrt{a+b x^2+c x^4} \left (3 c x^2 \left (14 a c f-4 b^2 f+9 b c d\right )+9 a b c f+90 a c^2 d-4 b^3 f+9 b^2 c d\right )}{315 c^2}+\frac{\left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2} (2 c e-b g)}{32 c^2}+\frac{x \left (a+b x^2+c x^4\right )^{3/2} \left (3 (b f+3 c d)+7 c f x^2\right )}{63 c}+\frac{g \left (a+b x^2+c x^4\right )^{5/2}}{10 c} \]

Warning: Unable to verify antiderivative.

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

[Out]

-((18*b^3*c*d - 144*a*b*c^2*d - 8*b^4*f + 57*a*b^2*c*f - 84*a^2*c^2*f)*x*Sqrt[a
+ b*x^2 + c*x^4])/(315*c^(5/2)*(Sqrt[a] + Sqrt[c]*x^2)) - (3*(b^2 - 4*a*c)*(2*c*
e - b*g)*(b + 2*c*x^2)*Sqrt[a + b*x^2 + c*x^4])/(256*c^3) + (x*(9*b^2*c*d + 90*a
*c^2*d - 4*b^3*f + 9*a*b*c*f + 3*c*(9*b*c*d - 4*b^2*f + 14*a*c*f)*x^2)*Sqrt[a +
b*x^2 + c*x^4])/(315*c^2) + ((2*c*e - b*g)*(b + 2*c*x^2)*(a + b*x^2 + c*x^4)^(3/
2))/(32*c^2) + (x*(3*(3*c*d + b*f) + 7*c*f*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(63*c
) + (g*(a + b*x^2 + c*x^4)^(5/2))/(10*c) + (3*(b^2 - 4*a*c)^2*(2*c*e - b*g)*ArcT
anh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/(512*c^(7/2)) + (a^(1/4)
*(18*b^3*c*d - 144*a*b*c^2*d - 8*b^4*f + 57*a*b^2*c*f - 84*a^2*c^2*f)*(Sqrt[a] +
Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*Ar
cTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(315*c^(11/4)*Sqrt[a +
b*x^2 + c*x^4]) - (a^(1/4)*(18*b^3*c*d - 144*a*b*c^2*d - 8*b^4*f + 57*a*b^2*c*f
- 84*a^2*c^2*f + Sqrt[a]*Sqrt[c]*(9*b^2*c*d - 180*a*c^2*d - 4*b^3*f + 24*a*b*c*f
))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*E
llipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(630*c^(11
/4)*Sqrt[a + b*x^2 + c*x^4])

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Rubi in Sympy [A]  time = 132.275, size = 709, normalized size = 0.99 \[ - \frac{\sqrt [4]{a} \sqrt{\frac{a + b x^{2} + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) \left (84 a^{2} c^{2} f - 57 a b^{2} c f + 144 a b c^{2} d + 8 b^{4} f - 18 b^{3} c d\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2} - \frac{b}{4 \sqrt{a} \sqrt{c}}\right )}{315 c^{\frac{11}{4}} \sqrt{a + b x^{2} + c x^{4}}} + \frac{\sqrt [4]{a} \sqrt{\frac{a + b x^{2} + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) \left (\sqrt{a} \sqrt{c} \left (- 24 a b c f + 180 a c^{2} d + 4 b^{3} f - 9 b^{2} c d\right ) + 84 a^{2} c^{2} f - 57 a b^{2} c f + 144 a b c^{2} d + 8 b^{4} f - 18 b^{3} c d\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2} - \frac{b}{4 \sqrt{a} \sqrt{c}}\right )}{630 c^{\frac{11}{4}} \sqrt{a + b x^{2} + c x^{4}}} + \frac{g \left (a + b x^{2} + c x^{4}\right )^{\frac{5}{2}}}{10 c} + \frac{x \left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}} \left (3 b f + 9 c d + 7 c f x^{2}\right )}{63 c} - \frac{x \sqrt{a + b x^{2} + c x^{4}} \left (- 9 a b c f - 90 a c^{2} d + 4 b^{3} f - 9 b^{2} c d + 3 c x^{2} \left (- 14 a c f + 4 b^{2} f - 9 b c d\right )\right )}{315 c^{2}} - \frac{\left (b + 2 c x^{2}\right ) \left (b g - 2 c e\right ) \left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{32 c^{2}} + \frac{3 \left (b + 2 c x^{2}\right ) \left (- 4 a c + b^{2}\right ) \left (b g - 2 c e\right ) \sqrt{a + b x^{2} + c x^{4}}}{256 c^{3}} + \frac{x \sqrt{a + b x^{2} + c x^{4}} \left (84 a^{2} c^{2} f - 57 a b^{2} c f + 144 a b c^{2} d + 8 b^{4} f - 18 b^{3} c d\right )}{315 c^{\frac{5}{2}} \left (\sqrt{a} + \sqrt{c} x^{2}\right )} - \frac{3 \left (- 4 a c + b^{2}\right )^{2} \left (b g - 2 c e\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{2 \sqrt{c} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{512 c^{\frac{7}{2}}} \]

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)**(3/2),x)

[Out]

-a**(1/4)*sqrt((a + b*x**2 + c*x**4)/(sqrt(a) + sqrt(c)*x**2)**2)*(sqrt(a) + sqr
t(c)*x**2)*(84*a**2*c**2*f - 57*a*b**2*c*f + 144*a*b*c**2*d + 8*b**4*f - 18*b**3
*c*d)*elliptic_e(2*atan(c**(1/4)*x/a**(1/4)), 1/2 - b/(4*sqrt(a)*sqrt(c)))/(315*
c**(11/4)*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)*(-24*a*b*c*f +
180*a*c**2*d + 4*b**3*f - 9*b**2*c*d) + 84*a**2*c**2*f - 57*a*b**2*c*f + 144*a*
b*c**2*d + 8*b**4*f - 18*b**3*c*d)*elliptic_f(2*atan(c**(1/4)*x/a**(1/4)), 1/2 -
b/(4*sqrt(a)*sqrt(c)))/(630*c**(11/4)*sqrt(a + b*x**2 + c*x**4)) + g*(a + b*x**
2 + c*x**4)**(5/2)/(10*c) + x*(a + b*x**2 + c*x**4)**(3/2)*(3*b*f + 9*c*d + 7*c*
f*x**2)/(63*c) - x*sqrt(a + b*x**2 + c*x**4)*(-9*a*b*c*f - 90*a*c**2*d + 4*b**3*
f - 9*b**2*c*d + 3*c*x**2*(-14*a*c*f + 4*b**2*f - 9*b*c*d))/(315*c**2) - (b + 2*
c*x**2)*(b*g - 2*c*e)*(a + b*x**2 + c*x**4)**(3/2)/(32*c**2) + 3*(b + 2*c*x**2)*
(-4*a*c + b**2)*(b*g - 2*c*e)*sqrt(a + b*x**2 + c*x**4)/(256*c**3) + x*sqrt(a +
b*x**2 + c*x**4)*(84*a**2*c**2*f - 57*a*b**2*c*f + 144*a*b*c**2*d + 8*b**4*f - 1
8*b**3*c*d)/(315*c**(5/2)*(sqrt(a) + sqrt(c)*x**2)) - 3*(-4*a*c + b**2)**2*(b*g
- 2*c*e)*atanh((b + 2*c*x**2)/(2*sqrt(c)*sqrt(a + b*x**2 + c*x**4)))/(512*c**(7/
2))

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Mathematica [C]  time = 5.37154, size = 2588, normalized size = 3.61 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[c]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*(a + b*x^2 + c*x^4)*(-945*b^4*g + 2*
b^3*c*(945*e + x*(512*f + 315*g*x)) - 12*b^2*c*(-525*a*g + c*x*(192*d + 105*e*x
+ 64*f*x^2 + 42*g*x^3)) - 8*b*c^2*(3*a*(525*e + 256*f*x + 147*g*x^2) + 2*c*x^3*(
1152*d + 945*e*x + 800*f*x^2 + 693*g*x^3)) - 16*c^2*(504*a^2*g + 2*c^2*x^5*(360*
d + 7*x*(45*e + 40*f*x + 36*g*x^2)) + a*c*x*(2160*d + 7*x*(225*e + 16*x*(11*f +
9*g*x))))) + (2304*I)*Sqrt[2]*b^3*c^(3/2)*(b - Sqrt[b^2 - 4*a*c])*d*Sqrt[(b + Sq
rt[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])]*(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])]) + (18432*I)*Sqrt[2]*a*b*c^(5/2)*(-b + Sqrt[b^2 - 4*a*c])*d*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])]*(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]*Sqr
t[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])
]) + (7296*I)*Sqrt[2]*a*b^2*c^(3/2)*(b - Sqrt[b^2 - 4*a*c])*f*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])]*(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])])
+ (1024*I)*Sqrt[2]*b^4*Sqrt[c]*(-b + Sqrt[b^2 - 4*a*c])*f*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])]*(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])]) + (1
0752*I)*Sqrt[2]*a^2*c^(5/2)*(-b + Sqrt[b^2 - 4*a*c])*f*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]
)]*(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 + S
qrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])]) + (2304
*I)*Sqrt[2]*a*b^2*c^(5/2)*d*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])]*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])] - (46080*I)*Sqrt[2]*a^2*c^(7/2)*d*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])]*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])] - (1024*I)*Sqrt[2]*a*b^3*c^(3/2)*f*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])]*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])] + (6144*I)*Sqrt[2]*a^2*b*c^(5/2)*f*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])]*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])] + 1890*b^4*c*Sqrt[c/(b +
Sqrt[b^2 - 4*a*c])]*e*Sqrt[a + b*x^2 + c*x^4]*Log[b + 2*c*x^2 + 2*Sqrt[c]*Sqrt[
a + b*x^2 + c*x^4]] - 15120*a*b^2*c^2*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*e*Sqrt[a +
b*x^2 + c*x^4]*Log[b + 2*c*x^2 + 2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4]] + 30240*a^2
*c^3*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*e*Sqrt[a + b*x^2 + c*x^4]*Log[b + 2*c*x^2 +
2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4]] - 945*b^5*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*g*
Sqrt[a + b*x^2 + c*x^4]*Log[b + 2*c*x^2 + 2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4]] + 7
560*a*b^3*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*g*Sqrt[a + b*x^2 + c*x^4]*Log[b + 2*
c*x^2 + 2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4]] - 15120*a^2*b*c^2*Sqrt[c/(b + Sqrt[b^
2 - 4*a*c])]*g*Sqrt[a + b*x^2 + c*x^4]*Log[b + 2*c*x^2 + 2*Sqrt[c]*Sqrt[a + b*x^
2 + c*x^4]])/(161280*c^(7/2)*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[a + b*x^2 + c*
x^4])

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Maple [B]  time = 0.013, size = 3038, normalized size = 4.2 \[ \text{output too large to display} \]

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)^(3/2),x)

[Out]

1/8*e*c*x^6*(c*x^4+b*x^2+a)^(1/2)+3/16*e*b*x^4*(c*x^4+b*x^2+a)^(1/2)+3/16*e*a^2*
ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))/c^(1/2)+3/256*e/c^(5/2)*b^4*ln((
1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))-3/128*e/c^2*b^3*(c*x^4+b*x^2+a)^(1/2
)+1/10*g*a^2/c*(c*x^4+b*x^2+a)^(1/2)+5/16*e*a*x^2*(c*x^4+b*x^2+a)^(1/2)+1/10*g*c
*x^8*(c*x^4+b*x^2+a)^(1/2)+11/80*g*b*x^6*(c*x^4+b*x^2+a)^(1/2)-2/15*f*a^3*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))+2/15*f*a^3*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))+1/7*d
*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))*a^2+10/63*f*b*x^5*(c*x^4+b*x^2+a)^(1/2)+1/9*f*c*x^7*(c*x^4+b*x^2+a)
^(1/2)+11/45*f*x^3*(c*x^4+b*x^2+a)^(1/2)*a+7/160*g/c*b*a*x^2*(c*x^4+b*x^2+a)^(1/
2)+8/105*f/c*x*(c*x^4+b*x^2+a)^(1/2)*a*b+5/32*e/c*b*a*(c*x^4+b*x^2+a)^(1/2)+3/7*
d*x*(c*x^4+b*x^2+a)^(1/2)*a+8/35*d*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))*b*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))-2/10
5*f/c*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)*Ell
ipticF(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))*b+1/315*f/c^2*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))*b^3-1/140*d*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))/c*a*b^
2-8/35*d*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))*b*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*g*a*x^4*(c*x^4+b*x^2+a)^(1
/2)+4/315*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^4/c^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))+19/210*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))/
c*b^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))-1/35*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)/(b+(-4*a*c+b^2)^(1/2))*b^3/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))
-19/210*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))/c*b^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))-4/315*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^4/c
^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))+1/35*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)/(b+(-4*a*c+b^2)^(1/2))*b^3/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/
64*e/c*b^2*x^2*(c*x^4+b*x^2+a)^(1/2)+1/7*d*c*x^5*(c*x^4+b*x^2+a)^(1/2)+8/35*d*b*
x^3*(c*x^4+b*x^2+a)^(1/2)+3/256*g/c^3*b^4*(c*x^4+b*x^2+a)^(1/2)-3/512*g/c^(7/2)*
b^5*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))+1/105*f/c*x^3*(c*x^4+b*x^2+a
)^(1/2)*b^2+3/64*g/c^(5/2)*b^3*a*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))
-3/32*g*a^2*b/c^(3/2)*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))+1/160*g/c*
b^2*x^4*(c*x^4+b*x^2+a)^(1/2)-1/128*g/c^2*b^3*x^2*(c*x^4+b*x^2+a)^(1/2)+1/35*d/c
*x*(c*x^4+b*x^2+a)^(1/2)*b^2-5/64*g/c^2*b^2*a*(c*x^4+b*x^2+a)^(1/2)-3/32*e/c^(3/
2)*b^2*a*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))-4/315*f/c^2*x*(c*x^4+b*
x^2+a)^(1/2)*b^3

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (c g x^{7} + c f x^{6} +{\left (c e + b g\right )} x^{5} +{\left (c d + b f\right )} x^{4} +{\left (b e + a g\right )} x^{3} + a e x +{\left (b d + a f\right )} x^{2} + a d\right )} \sqrt{c x^{4} + b x^{2} + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((c*g*x^7 + c*f*x^6 + (c*e + b*g)*x^5 + (c*d + b*f)*x^4 + (b*e + a*g)*x^
3 + a*e*x + (b*d + a*f)*x^2 + a*d)*sqrt(c*x^4 + b*x^2 + a), x)

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

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)**(3/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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