3.107 \(\int \frac{d+e x+f x^2+g x^3}{\left (a+b x^2+c x^4\right )^{5/2}} \, dx\)

Optimal. Leaf size=680 \[ -\frac{\sqrt [4]{c} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (6 a^{3/2} \sqrt{c} f-3 \sqrt{a} b \sqrt{c} d+a b f-10 a c d+2 b^2 d\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{6 a^{7/4} \left (b-2 \sqrt{a} \sqrt{c}\right ) \left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}}-\frac{\sqrt{c} x \sqrt{a+b x^2+c x^4} \left (12 a^2 c f+a b^2 f-16 a b c d+2 b^3 d\right )}{3 a^2 \left (b^2-4 a c\right )^2 \left (\sqrt{a}+\sqrt{c} x^2\right )}+\frac{x \left (c x^2 \left (12 a^2 c f+a b^2 f-16 a b c d+2 b^3 d\right )+4 a^2 b c f+20 a^2 c^2 d+a b^3 f-17 a b^2 c d+2 b^4 d\right )}{3 a^2 \left (b^2-4 a c\right )^2 \sqrt{a+b x^2+c x^4}}+\frac{\sqrt [4]{c} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (12 a^2 c f+a b^2 f-16 a b c d+2 b^3 d\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{3 a^{7/4} \left (b^2-4 a c\right )^2 \sqrt{a+b x^2+c x^4}}+\frac{x \left (c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right )}{3 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^{3/2}}+\frac{4 \left (b+2 c x^2\right ) (2 c e-b g)}{3 \left (b^2-4 a c\right )^2 \sqrt{a+b x^2+c x^4}}-\frac{-2 a g+x^2 (2 c e-b g)+b e}{3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^{3/2}} \]

[Out]

(x*(b^2*d - 2*a*c*d - a*b*f + c*(b*d - 2*a*f)*x^2))/(3*a*(b^2 - 4*a*c)*(a + b*x^
2 + c*x^4)^(3/2)) - (b*e - 2*a*g + (2*c*e - b*g)*x^2)/(3*(b^2 - 4*a*c)*(a + b*x^
2 + c*x^4)^(3/2)) + (4*(2*c*e - b*g)*(b + 2*c*x^2))/(3*(b^2 - 4*a*c)^2*Sqrt[a +
b*x^2 + c*x^4]) + (x*(2*b^4*d - 17*a*b^2*c*d + 20*a^2*c^2*d + a*b^3*f + 4*a^2*b*
c*f + c*(2*b^3*d - 16*a*b*c*d + a*b^2*f + 12*a^2*c*f)*x^2))/(3*a^2*(b^2 - 4*a*c)
^2*Sqrt[a + b*x^2 + c*x^4]) - (Sqrt[c]*(2*b^3*d - 16*a*b*c*d + a*b^2*f + 12*a^2*
c*f)*x*Sqrt[a + b*x^2 + c*x^4])/(3*a^2*(b^2 - 4*a*c)^2*(Sqrt[a] + Sqrt[c]*x^2))
+ (c^(1/4)*(2*b^3*d - 16*a*b*c*d + a*b^2*f + 12*a^2*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])/(3*a^(7/4)*(b^2 - 4*a*c)^2*Sqrt[a +
b*x^2 + c*x^4]) - (c^(1/4)*(2*b^2*d - 3*Sqrt[a]*b*Sqrt[c]*d - 10*a*c*d + a*b*f +
6*a^(3/2)*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])/(6*a^(7/4)*(b - 2*Sqrt[a]*Sqrt[c])*(b^2 - 4*a*c)*Sqrt[a + b*x^2 + c*x^4
])

_______________________________________________________________________________________

Rubi [A]  time = 1.23534, antiderivative size = 680, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{\sqrt [4]{c} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (6 a^{3/2} \sqrt{c} f-3 \sqrt{a} b \sqrt{c} d+a b f-10 a c d+2 b^2 d\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{6 a^{7/4} \left (b-2 \sqrt{a} \sqrt{c}\right ) \left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}}-\frac{\sqrt{c} x \sqrt{a+b x^2+c x^4} \left (12 a^2 c f+a b^2 f-16 a b c d+2 b^3 d\right )}{3 a^2 \left (b^2-4 a c\right )^2 \left (\sqrt{a}+\sqrt{c} x^2\right )}+\frac{x \left (c x^2 \left (12 a^2 c f+a b^2 f-16 a b c d+2 b^3 d\right )+4 a^2 b c f+20 a^2 c^2 d+a b^3 f-17 a b^2 c d+2 b^4 d\right )}{3 a^2 \left (b^2-4 a c\right )^2 \sqrt{a+b x^2+c x^4}}+\frac{\sqrt [4]{c} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (12 a^2 c f+a b^2 f-16 a b c d+2 b^3 d\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{3 a^{7/4} \left (b^2-4 a c\right )^2 \sqrt{a+b x^2+c x^4}}+\frac{x \left (c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right )}{3 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^{3/2}}+\frac{4 \left (b+2 c x^2\right ) (2 c e-b g)}{3 \left (b^2-4 a c\right )^2 \sqrt{a+b x^2+c x^4}}-\frac{-2 a g+x^2 (2 c e-b g)+b e}{3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]  Int[(d + e*x + f*x^2 + g*x^3)/(a + b*x^2 + c*x^4)^(5/2),x]

[Out]

(x*(b^2*d - 2*a*c*d - a*b*f + c*(b*d - 2*a*f)*x^2))/(3*a*(b^2 - 4*a*c)*(a + b*x^
2 + c*x^4)^(3/2)) - (b*e - 2*a*g + (2*c*e - b*g)*x^2)/(3*(b^2 - 4*a*c)*(a + b*x^
2 + c*x^4)^(3/2)) + (4*(2*c*e - b*g)*(b + 2*c*x^2))/(3*(b^2 - 4*a*c)^2*Sqrt[a +
b*x^2 + c*x^4]) + (x*(2*b^4*d - 17*a*b^2*c*d + 20*a^2*c^2*d + a*b^3*f + 4*a^2*b*
c*f + c*(2*b^3*d - 16*a*b*c*d + a*b^2*f + 12*a^2*c*f)*x^2))/(3*a^2*(b^2 - 4*a*c)
^2*Sqrt[a + b*x^2 + c*x^4]) - (Sqrt[c]*(2*b^3*d - 16*a*b*c*d + a*b^2*f + 12*a^2*
c*f)*x*Sqrt[a + b*x^2 + c*x^4])/(3*a^2*(b^2 - 4*a*c)^2*(Sqrt[a] + Sqrt[c]*x^2))
+ (c^(1/4)*(2*b^3*d - 16*a*b*c*d + a*b^2*f + 12*a^2*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])/(3*a^(7/4)*(b^2 - 4*a*c)^2*Sqrt[a +
b*x^2 + c*x^4]) - (c^(1/4)*(2*b^2*d - 3*Sqrt[a]*b*Sqrt[c]*d - 10*a*c*d + a*b*f +
6*a^(3/2)*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])/(6*a^(7/4)*(b - 2*Sqrt[a]*Sqrt[c])*(b^2 - 4*a*c)*Sqrt[a + b*x^2 + c*x^4
])

_______________________________________________________________________________________

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**3+f*x**2+e*x+d)/(c*x**4+b*x**2+a)**(5/2),x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [C]  time = 4.79479, size = 598, normalized size = 0.88 \[ \frac{-4 a \left (b^2-4 a c\right ) \left (-2 a^2 g+a b (e+x (f-g x))+2 a c x (d+x (e+f x))-b d x \left (b+c x^2\right )\right )+4 \left (a+b x^2+c x^4\right ) \left (4 a^2 \left (b^2 (-g)+b c (2 e+x (f-2 g x))+c^2 x (5 d+x (4 e+3 f x))\right )+a b x \left (b^2 f-17 b c d+b c f x^2-16 c^2 d x^2\right )+2 b^3 d x \left (b+c x^2\right )\right )+\frac{i \sqrt{2} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \left (a+b x^2+c x^4\right ) \left (\left (4 a^2 c \left (3 f \sqrt{b^2-4 a c}-10 c d\right )+a b^2 \left (f \sqrt{b^2-4 a c}+18 c d\right )+4 a b c \left (a f-4 d \sqrt{b^2-4 a c}\right )+b^3 \left (2 d \sqrt{b^2-4 a c}-a f\right )-2 b^4 d\right ) F\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )-\left (\sqrt{b^2-4 a c}-b\right ) \left (12 a^2 c f+a b^2 f-16 a b c d+2 b^3 d\right ) E\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )\right )}{\sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}}}}{12 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x + f*x^2 + g*x^3)/(a + b*x^2 + c*x^4)^(5/2),x]

[Out]

(-4*a*(b^2 - 4*a*c)*(-2*a^2*g - b*d*x*(b + c*x^2) + 2*a*c*x*(d + x*(e + f*x)) +
a*b*(e + x*(f - g*x))) + 4*(a + b*x^2 + c*x^4)*(2*b^3*d*x*(b + c*x^2) + a*b*x*(-
17*b*c*d + b^2*f - 16*c^2*d*x^2 + b*c*f*x^2) + 4*a^2*(-(b^2*g) + c^2*x*(5*d + x*
(4*e + 3*f*x)) + b*c*(2*e + x*(f - 2*g*x)))) + (I*Sqrt[2]*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])]*(a + b*x^2 + c*x^4)*(-((-b + Sqrt[b^2 - 4*a*c])*(2*b^3*d - 16*a*b*c*d + a*
b^2*f + 12*a^2*c*f)*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])]) + (-2*b^4*d + b^3*(2*Sqrt[
b^2 - 4*a*c]*d - a*f) + 4*a*b*c*(-4*Sqrt[b^2 - 4*a*c]*d + a*f) + a*b^2*(18*c*d +
Sqrt[b^2 - 4*a*c]*f) + 4*a^2*c*(-10*c*d + 3*Sqrt[b^2 - 4*a*c]*f))*EllipticF[I*A
rcSinh[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])])/(12*a^2*(b^2 - 4*a*c)^2*(
a + b*x^2 + c*x^4)^(3/2))

_______________________________________________________________________________________

Maple [B]  time = 0.082, size = 1395, normalized size = 2.1 \[ \text{result too large to display} \]

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)^(5/2),x)

[Out]

d*((-1/3*b/a/(4*a*c-b^2)/c*x^3+1/3*(2*a*c-b^2)/a/(4*a*c-b^2)/c^2*x)*(c*x^4+b*x^2
+a)^(1/2)/(x^4+b/c*x^2+1/c*a)^2-2*c*(1/3*b*(8*a*c-b^2)/(4*a*c-b^2)^2/a^2*x^3-1/6
*(20*a^2*c^2-17*a*b^2*c+2*b^4)/a^2/(4*a*c-b^2)^2/c*x)/((x^4+b/c*x^2+1/c*a)*c)^(1
/2)+1/4*(2/3*(5*a*c-b^2)/(4*a*c-b^2)/a^2-1/3*(20*a^2*c^2-17*a*b^2*c+2*b^4)/a^2/(
4*a*c-b^2)^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)
*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))-1/3*b*c*(8*a*c-b^2)/(4*a*c-b^2)^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)/(b+(-4*a*c+b^2)^(1/2))*(Ellipti
cF(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))-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))))+1/3*e*(16*c^3*x^6+24*b*c^2*x^4+24*a
*c^2*x^2+6*b^2*c*x^2+12*a*b*c-b^3)/(c*x^4+b*x^2+a)^(3/2)/(16*a^2*c^2-8*a*b^2*c+b
^4)+f*((2/3/c/(4*a*c-b^2)*x^3+1/3*b/(4*a*c-b^2)/c^2*x)*(c*x^4+b*x^2+a)^(1/2)/(x^
4+b/c*x^2+1/c*a)^2-2*c*(-1/6*(12*a*c+b^2)/(4*a*c-b^2)^2/a*x^3-1/6*b*(4*a*c+b^2)/
c/(4*a*c-b^2)^2/a*x)/((x^4+b/c*x^2+1/c*a)*c)^(1/2)+1/4*(-1/3*b/(4*a*c-b^2)/a-1/3
*b*(4*a*c+b^2)/(4*a*c-b^2)^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))+1/6*c*(12*a*c+b^2)/(4*a*c-b^2)^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))-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))))-1/3*g*(8*b*c^2*x^6+1
2*b^2*c*x^4+12*a*b*c*x^2+3*b^3*x^2+8*a^2*c+2*a*b^2)/(c*x^4+b*x^2+a)^(3/2)/(16*a^
2*c^2-8*a*b^2*c+b^4)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{g x^{3} + f x^{2} + e x + d}{{\left (c x^{4} + b x^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \]

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)^(5/2),x, algorithm="maxima")

[Out]

integrate((g*x^3 + f*x^2 + e*x + d)/(c*x^4 + b*x^2 + a)^(5/2), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{g x^{3} + f x^{2} + e x + d}{{\left (c^{2} x^{8} + 2 \, b c x^{6} +{\left (b^{2} + 2 \, a c\right )} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt{c x^{4} + b x^{2} + a}}, x\right ) \]

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)^(5/2),x, algorithm="fricas")

[Out]

integral((g*x^3 + f*x^2 + e*x + d)/((c^2*x^8 + 2*b*c*x^6 + (b^2 + 2*a*c)*x^4 + 2
*a*b*x^2 + a^2)*sqrt(c*x^4 + b*x^2 + a)), x)

_______________________________________________________________________________________

Sympy [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**3+f*x**2+e*x+d)/(c*x**4+b*x**2+a)**(5/2),x)

[Out]

Timed out

_______________________________________________________________________________________

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((g*x^3 + f*x^2 + e*x + d)/(c*x^4 + b*x^2 + a)^(5/2),x, algorithm="giac")

[Out]

Exception raised: TypeError