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

Optimal. Leaf size=314 \[ \frac{\left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (3 \sqrt{a} e+\sqrt{b} c\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{a} b^{7/4} \sqrt{a+b x^4}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{2 b^{3/2}}+\frac{3 e x \sqrt{a+b x^4}}{2 b^{3/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{3 \sqrt [4]{a} e \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 b^{7/4} \sqrt{a+b x^4}}+\frac{f \sqrt{a+b x^4}}{b^2}-\frac{x \left (c+d x+e x^2+f x^3\right )}{2 b \sqrt{a+b x^4}} \]

[Out]

-(x*(c + d*x + e*x^2 + f*x^3))/(2*b*Sqrt[a + b*x^4]) + (f*Sqrt[a + b*x^4])/b^2 +
(3*e*x*Sqrt[a + b*x^4])/(2*b^(3/2)*(Sqrt[a] + Sqrt[b]*x^2)) + (d*ArcTanh[(Sqrt[
b]*x^2)/Sqrt[a + b*x^4]])/(2*b^(3/2)) - (3*a^(1/4)*e*(Sqrt[a] + Sqrt[b]*x^2)*Sqr
t[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticE[2*ArcTan[(b^(1/4)*x)/a^(1/4)]
, 1/2])/(2*b^(7/4)*Sqrt[a + b*x^4]) + ((Sqrt[b]*c + 3*Sqrt[a]*e)*(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^(1/4)*b^(7/4)*Sqrt[a + b*x^4])

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Rubi [A]  time = 0.57713, antiderivative size = 314, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{\left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (3 \sqrt{a} e+\sqrt{b} c\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{a} b^{7/4} \sqrt{a+b x^4}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{2 b^{3/2}}+\frac{3 e x \sqrt{a+b x^4}}{2 b^{3/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{3 \sqrt [4]{a} e \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 b^{7/4} \sqrt{a+b x^4}}+\frac{f \sqrt{a+b x^4}}{b^2}-\frac{x \left (c+d x+e x^2+f x^3\right )}{2 b \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

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

[Out]

-(x*(c + d*x + e*x^2 + f*x^3))/(2*b*Sqrt[a + b*x^4]) + (f*Sqrt[a + b*x^4])/b^2 +
(3*e*x*Sqrt[a + b*x^4])/(2*b^(3/2)*(Sqrt[a] + Sqrt[b]*x^2)) + (d*ArcTanh[(Sqrt[
b]*x^2)/Sqrt[a + b*x^4]])/(2*b^(3/2)) - (3*a^(1/4)*e*(Sqrt[a] + Sqrt[b]*x^2)*Sqr
t[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticE[2*ArcTan[(b^(1/4)*x)/a^(1/4)]
, 1/2])/(2*b^(7/4)*Sqrt[a + b*x^4]) + ((Sqrt[b]*c + 3*Sqrt[a]*e)*(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^(1/4)*b^(7/4)*Sqrt[a + b*x^4])

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Rubi in Sympy [A]  time = 92.6243, size = 287, normalized size = 0.91 \[ - \frac{3 \sqrt [4]{a} e \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{2 b^{\frac{7}{4}} \sqrt{a + b x^{4}}} - \frac{x \left (c + d x + e x^{2} + f x^{3}\right )}{2 b \sqrt{a + b x^{4}}} + \frac{f \sqrt{a + b x^{4}}}{b^{2}} + \frac{d \operatorname{atanh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a + b x^{4}}} \right )}}{2 b^{\frac{3}{2}}} + \frac{3 e x \sqrt{a + b x^{4}}}{2 b^{\frac{3}{2}} \left (\sqrt{a} + \sqrt{b} x^{2}\right )} + \frac{\sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) \left (3 \sqrt{a} e + \sqrt{b} c\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{4 \sqrt [4]{a} b^{\frac{7}{4}} \sqrt{a + b x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*a**(1/4)*e*sqrt((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*b**(7/4)*sqrt(a + b*x**4))
- x*(c + d*x + e*x**2 + f*x**3)/(2*b*sqrt(a + b*x**4)) + f*sqrt(a + b*x**4)/b**
2 + d*atanh(sqrt(b)*x**2/sqrt(a + b*x**4))/(2*b**(3/2)) + 3*e*x*sqrt(a + b*x**4)
/(2*b**(3/2)*(sqrt(a) + sqrt(b)*x**2)) + sqrt((a + b*x**4)/(sqrt(a) + sqrt(b)*x*
*2)**2)*(sqrt(a) + sqrt(b)*x**2)*(3*sqrt(a)*e + sqrt(b)*c)*elliptic_f(2*atan(b**
(1/4)*x/a**(1/4)), 1/2)/(4*a**(1/4)*b**(7/4)*sqrt(a + b*x**4))

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Mathematica [C]  time = 0.584146, size = 243, normalized size = 0.77 \[ \frac{\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \left (\sqrt{b} d \sqrt{a+b x^4} \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )+2 a f+b x \left (-c-d x-e x^2+f x^3\right )\right )-\sqrt{b} \sqrt{\frac{b x^4}{a}+1} \left (3 \sqrt{a} e+i \sqrt{b} c\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )+3 \sqrt{a} \sqrt{b} e \sqrt{\frac{b x^4}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )}{2 b^2 \sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(I*Sqrt[b])/Sqrt[a]]*(2*a*f + b*x*(-c - d*x - e*x^2 + f*x^3) + Sqrt[b]*d*S
qrt[a + b*x^4]*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a + b*x^4]]) + 3*Sqrt[a]*Sqrt[b]*e*Sqr
t[1 + (b*x^4)/a]*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -1] - Sqrt[b]
*(I*Sqrt[b]*c + 3*Sqrt[a]*e)*Sqrt[1 + (b*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqr
t[b])/Sqrt[a]]*x], -1])/(2*Sqrt[(I*Sqrt[b])/Sqrt[a]]*b^2*Sqrt[a + b*x^4])

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Maple [C]  time = 0.011, size = 340, normalized size = 1.1 \[ -{\frac{cx}{2\,b}{\frac{1}{\sqrt{ \left ({x}^{4}+{\frac{a}{b}} \right ) b}}}}+{\frac{c}{2\,b}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{\frac{d{x}^{2}}{2\,b}{\frac{1}{\sqrt{b{x}^{4}+a}}}}+{\frac{d}{2}\ln \left ( \sqrt{b}{x}^{2}+\sqrt{b{x}^{4}+a} \right ){b}^{-{\frac{3}{2}}}}-{\frac{e{x}^{3}}{2\,b}{\frac{1}{\sqrt{ \left ({x}^{4}+{\frac{a}{b}} \right ) b}}}}+{{\frac{3\,i}{2}}e\sqrt{a}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{{\frac{3\,i}{2}}e\sqrt{a}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticE} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}+{\frac{f \left ( b{x}^{4}+2\,a \right ) }{2\,{b}^{2}}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 38.2042, size = 172, normalized size = 0.55 \[ d \left (\frac{\operatorname{asinh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2 b^{\frac{3}{2}}} - \frac{x^{2}}{2 \sqrt{a} b \sqrt{1 + \frac{b x^{4}}{a}}}\right ) + f \left (\begin{cases} \frac{a}{b^{2} \sqrt{a + b x^{4}}} + \frac{x^{4}}{2 b \sqrt{a + b x^{4}}} & \text{for}\: b \neq 0 \\\frac{x^{8}}{8 a^{\frac{3}{2}}} & \text{otherwise} \end{cases}\right ) + \frac{c x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{5}{4}, \frac{3}{2} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{3}{2}} \Gamma \left (\frac{9}{4}\right )} + \frac{e x^{7} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{3}{2}} \Gamma \left (\frac{11}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*(asinh(sqrt(b)*x**2/sqrt(a))/(2*b**(3/2)) - x**2/(2*sqrt(a)*b*sqrt(1 + b*x**4/
a))) + f*Piecewise((a/(b**2*sqrt(a + b*x**4)) + x**4/(2*b*sqrt(a + b*x**4)), Ne(
b, 0)), (x**8/(8*a**(3/2)), True)) + c*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)) + e*x**7*gamma(7/4)*hyper((3/
2, 7/4), (11/4,), b*x**4*exp_polar(I*pi)/a)/(4*a**(3/2)*gamma(11/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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