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

Optimal. Leaf size=303 \[ -\frac{\left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (\sqrt{b} d-\sqrt{a} f\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 a^{3/4} b^{5/4} \sqrt{a+b x^4}}+\frac{d \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 a^{3/4} b^{3/4} \sqrt{a+b x^4}}-\frac{x \left (a f-b c x-b d x^2-b e x^3\right )}{2 a b \sqrt{a+b x^4}}-\frac{d x \sqrt{a+b x^4}}{2 a \sqrt{b} \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{e \sqrt{a+b x^4}}{2 a b} \]

[Out]

-(x*(a*f - b*c*x - b*d*x^2 - b*e*x^3))/(2*a*b*Sqrt[a + b*x^4]) - (e*Sqrt[a + b*x
^4])/(2*a*b) - (d*x*Sqrt[a + b*x^4])/(2*a*Sqrt[b]*(Sqrt[a] + Sqrt[b]*x^2)) + (d*
(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticE[2*
ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(3/4)*b^(3/4)*Sqrt[a + b*x^4]) - ((Sqrt[
b]*d - Sqrt[a]*f)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^
2)^2]*EllipticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(3/4)*b^(5/4)*Sqrt[a +
b*x^4])

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Rubi [A]  time = 0.403118, antiderivative size = 303, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ -\frac{\left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (\sqrt{b} d-\sqrt{a} f\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 a^{3/4} b^{5/4} \sqrt{a+b x^4}}+\frac{d \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 a^{3/4} b^{3/4} \sqrt{a+b x^4}}-\frac{x \left (a f-b c x-b d x^2-b e x^3\right )}{2 a b \sqrt{a+b x^4}}-\frac{d x \sqrt{a+b x^4}}{2 a \sqrt{b} \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{e \sqrt{a+b x^4}}{2 a b} \]

Antiderivative was successfully verified.

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

[Out]

-(x*(a*f - b*c*x - b*d*x^2 - b*e*x^3))/(2*a*b*Sqrt[a + b*x^4]) - (e*Sqrt[a + b*x
^4])/(2*a*b) - (d*x*Sqrt[a + b*x^4])/(2*a*Sqrt[b]*(Sqrt[a] + Sqrt[b]*x^2)) + (d*
(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticE[2*
ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(3/4)*b^(3/4)*Sqrt[a + b*x^4]) - ((Sqrt[
b]*d - Sqrt[a]*f)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^
2)^2]*EllipticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(3/4)*b^(5/4)*Sqrt[a +
b*x^4])

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Rubi in Sympy [A]  time = 56.3881, size = 265, normalized size = 0.87 \[ - \frac{e \sqrt{a + b x^{4}}}{2 a b} - \frac{x \left (a f - b c x - b d x^{2} - b e x^{3}\right )}{2 a b \sqrt{a + b x^{4}}} - \frac{d x \sqrt{a + b x^{4}}}{2 a \sqrt{b} \left (\sqrt{a} + \sqrt{b} x^{2}\right )} + \frac{d \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{2 a^{\frac{3}{4}} b^{\frac{3}{4}} \sqrt{a + b x^{4}}} + \frac{\sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) \left (\sqrt{a} f - \sqrt{b} d\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{4 a^{\frac{3}{4}} b^{\frac{5}{4}} \sqrt{a + b x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-e*sqrt(a + b*x**4)/(2*a*b) - x*(a*f - b*c*x - b*d*x**2 - b*e*x**3)/(2*a*b*sqrt(
a + b*x**4)) - d*x*sqrt(a + b*x**4)/(2*a*sqrt(b)*(sqrt(a) + sqrt(b)*x**2)) + d*s
qrt((a + b*x**4)/(sqrt(a) + sqrt(b)*x**2)**2)*(sqrt(a) + sqrt(b)*x**2)*elliptic_
e(2*atan(b**(1/4)*x/a**(1/4)), 1/2)/(2*a**(3/4)*b**(3/4)*sqrt(a + b*x**4)) + sqr
t((a + b*x**4)/(sqrt(a) + sqrt(b)*x**2)**2)*(sqrt(a) + sqrt(b)*x**2)*(sqrt(a)*f
- sqrt(b)*d)*elliptic_f(2*atan(b**(1/4)*x/a**(1/4)), 1/2)/(4*a**(3/4)*b**(5/4)*s
qrt(a + b*x**4))

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Mathematica [C]  time = 0.433688, size = 197, normalized size = 0.65 \[ \frac{\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \left (b x^2 (c+d x)-a (e+f x)\right )+\sqrt{a} \sqrt{\frac{b x^4}{a}+1} \left (\sqrt{b} d-i \sqrt{a} f\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )-\sqrt{a} \sqrt{b} d \sqrt{\frac{b x^4}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )}{2 a b \sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(I*Sqrt[b])/Sqrt[a]]*(b*x^2*(c + d*x) - a*(e + f*x)) - Sqrt[a]*Sqrt[b]*d*S
qrt[1 + (b*x^4)/a]*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -1] + Sqrt[
a]*(Sqrt[b]*d - I*Sqrt[a]*f)*Sqrt[1 + (b*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqr
t[b])/Sqrt[a]]*x], -1])/(2*a*Sqrt[(I*Sqrt[b])/Sqrt[a]]*b*Sqrt[a + b*x^4])

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Maple [C]  time = 0.014, size = 250, normalized size = 0.8 \[ d \left ({\frac{{x}^{3}}{2\,a}{\frac{1}{\sqrt{ \left ({x}^{4}+{\frac{a}{b}} \right ) b}}}}-{{\frac{i}{2}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}} \left ({\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ) \right ){\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}{\frac{1}{\sqrt{b}}}} \right ) +{\frac{c{x}^{2}}{2\,a}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{\frac{e}{2\,b}{\frac{1}{\sqrt{b{x}^{4}+a}}}}+f \left ( -{\frac{x}{2\,b}{\frac{1}{\sqrt{ \left ({x}^{4}+{\frac{a}{b}} \right ) b}}}}+{\frac{1}{2\,b}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*(1/2/a*x^3/((x^4+a/b)*b)^(1/2)-1/2*I/a^(1/2)/(I/a^(1/2)*b^(1/2))^(1/2)*(1-I/a^
(1/2)*b^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*b^(1/2)*x^2)^(1/2)/(b*x^4+a)^(1/2)/b^(1/2)
*(EllipticF(x*(I/a^(1/2)*b^(1/2))^(1/2),I)-EllipticE(x*(I/a^(1/2)*b^(1/2))^(1/2)
,I)))+1/2*c*x^2/a/(b*x^4+a)^(1/2)-1/2*e/b/(b*x^4+a)^(1/2)+f*(-1/2/b*x/((x^4+a/b)
*b)^(1/2)+1/2/b/(I/a^(1/2)*b^(1/2))^(1/2)*(1-I/a^(1/2)*b^(1/2)*x^2)^(1/2)*(1+I/a
^(1/2)*b^(1/2)*x^2)^(1/2)/(b*x^4+a)^(1/2)*EllipticF(x*(I/a^(1/2)*b^(1/2))^(1/2),
I))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*c*x^2/(sqrt(b*x^4 + a)*a) + integrate((f*x^4 + e*x^3 + d*x^2)/(b*x^4 + a)^(3
/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 26.5585, size = 133, normalized size = 0.44 \[ e \left (\begin{cases} - \frac{1}{2 b \sqrt{a + b x^{4}}} & \text{for}\: b \neq 0 \\\frac{x^{4}}{4 a^{\frac{3}{2}}} & \text{otherwise} \end{cases}\right ) + \frac{c x^{2}}{2 a^{\frac{3}{2}} \sqrt{1 + \frac{b x^{4}}{a}}} + \frac{d x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{3}{2} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{3}{2}} \Gamma \left (\frac{7}{4}\right )} + \frac{f x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{5}{4}, \frac{3}{2} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{3}{2}} \Gamma \left (\frac{9}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e*Piecewise((-1/(2*b*sqrt(a + b*x**4)), Ne(b, 0)), (x**4/(4*a**(3/2)), True)) +
c*x**2/(2*a**(3/2)*sqrt(1 + b*x**4/a)) + d*x**3*gamma(3/4)*hyper((3/4, 3/2), (7/
4,), b*x**4*exp_polar(I*pi)/a)/(4*a**(3/2)*gamma(7/4)) + f*x**5*gamma(5/4)*hyper
((5/4, 3/2), (9/4,), b*x**4*exp_polar(I*pi)/a)/(4*a**(3/2)*gamma(9/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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