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

Optimal. Leaf size=367 \[ \frac{\left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (\sqrt{a} f+3 \sqrt{b} d\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 a^{7/4} \sqrt [4]{b} \sqrt{a+b x^4}}-\frac{3 \sqrt [4]{b} d \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 a^{7/4} \sqrt{a+b x^4}}-\frac{e \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{2 a^{3/2}}+\frac{x \left (a f-b c x-b d x^2-b e x^3\right )}{2 a^2 \sqrt{a+b x^4}}-\frac{c \sqrt{a+b x^4}}{2 a^2 x^2}-\frac{d \sqrt{a+b x^4}}{a^2 x}+\frac{3 \sqrt{b} d x \sqrt{a+b x^4}}{2 a^2 \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{e \sqrt{a+b x^4}}{2 a^2} \]

[Out]

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

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Rubi [A]  time = 0.837959, antiderivative size = 367, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 12, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ \frac{\left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (\sqrt{a} f+3 \sqrt{b} d\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 a^{7/4} \sqrt [4]{b} \sqrt{a+b x^4}}-\frac{3 \sqrt [4]{b} d \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 a^{7/4} \sqrt{a+b x^4}}-\frac{e \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{2 a^{3/2}}+\frac{x \left (a f-b c x-b d x^2-b e x^3\right )}{2 a^2 \sqrt{a+b x^4}}-\frac{c \sqrt{a+b x^4}}{2 a^2 x^2}-\frac{d \sqrt{a+b x^4}}{a^2 x}+\frac{3 \sqrt{b} d x \sqrt{a+b x^4}}{2 a^2 \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{e \sqrt{a+b x^4}}{2 a^2} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x + e*x^2 + f*x^3)/(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^2*Sqrt[a + b*x^4]) + (e*Sqrt[a + b*x^
4])/(2*a^2) - (c*Sqrt[a + b*x^4])/(2*a^2*x^2) - (d*Sqrt[a + b*x^4])/(a^2*x) + (3
*Sqrt[b]*d*x*Sqrt[a + b*x^4])/(2*a^2*(Sqrt[a] + Sqrt[b]*x^2)) - (e*ArcTanh[Sqrt[
a + b*x^4]/Sqrt[a]])/(2*a^(3/2)) - (3*b^(1/4)*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^(7/4)*Sqrt[a + b*x^4]) + ((3*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^(7/4)*b^(1/4)*Sqrt[a + b*x^4])

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Rubi in Sympy [A]  time = 23.9306, size = 110, normalized size = 0.3 \[ \frac{x \left (\frac{c}{x^{3}} + \frac{d}{x^{2}} + \frac{e}{x} + f\right )}{2 a \sqrt{a + b x^{4}}} + \frac{f \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{4 a^{\frac{5}{4}} \sqrt [4]{b} \sqrt{a + b x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(c/x**3 + d/x**2 + e/x + f)/(2*a*sqrt(a + b*x**4)) + f*sqrt((a + b*x**4)/(sqrt
(a) + sqrt(b)*x**2)**2)*(sqrt(a) + sqrt(b)*x**2)*elliptic_f(2*atan(b**(1/4)*x/a*
*(1/4)), 1/2)/(4*a**(5/4)*b**(1/4)*sqrt(a + b*x**4))

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Mathematica [C]  time = 0.710298, size = 259, normalized size = 0.71 \[ \frac{-\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \left (\sqrt{a} e x^2 \sqrt{a+b x^4} \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )+a \left (c+2 d x+x^2 (-(e+f x))\right )+b x^4 (2 c+3 d x)\right )-i \sqrt{a} x^2 \sqrt{\frac{b x^4}{a}+1} \left (\sqrt{a} f-3 i \sqrt{b} d\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )+3 \sqrt{a} \sqrt{b} d x^2 \sqrt{\frac{b x^4}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )}{2 a^2 x^2 \sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(-(Sqrt[(I*Sqrt[b])/Sqrt[a]]*(b*x^4*(2*c + 3*d*x) + a*(c + 2*d*x - x^2*(e + f*x)
) + Sqrt[a]*e*x^2*Sqrt[a + b*x^4]*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]])) + 3*Sqrt[a]
*Sqrt[b]*d*x^2*Sqrt[1 + (b*x^4)/a]*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]
*x], -1] - I*Sqrt[a]*((-3*I)*Sqrt[b]*d + Sqrt[a]*f)*x^2*Sqrt[1 + (b*x^4)/a]*Elli
pticF[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -1])/(2*a^2*Sqrt[(I*Sqrt[b])/Sqrt[
a]]*x^2*Sqrt[a + b*x^4])

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Maple [C]  time = 0.024, size = 363, normalized size = 1. \[{\frac{fx}{2\,a}{\frac{1}{\sqrt{ \left ({x}^{4}+{\frac{a}{b}} \right ) b}}}}+{\frac{f}{2\,a}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{\frac{c \left ( 2\,b{x}^{4}+a \right ) }{2\,{x}^{2}{a}^{2}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{\frac{{x}^{3}bd}{2\,{a}^{2}}{\frac{1}{\sqrt{ \left ({x}^{4}+{\frac{a}{b}} \right ) b}}}}-{\frac{d}{{a}^{2}x}\sqrt{b{x}^{4}+a}}+{{\frac{3\,i}{2}}d\sqrt{b}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){a}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{{\frac{3\,i}{2}}d\sqrt{b}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticE} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){a}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}+{\frac{e}{2\,a}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{\frac{e}{2}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{4}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 84.4303, size = 316, normalized size = 0.86 \[ c \left (- \frac{1}{2 a \sqrt{b} x^{4} \sqrt{\frac{a}{b x^{4}} + 1}} - \frac{\sqrt{b}}{a^{2} \sqrt{\frac{a}{b x^{4}} + 1}}\right ) + e \left (\frac{2 a^{3} \sqrt{1 + \frac{b x^{4}}{a}}}{4 a^{\frac{9}{2}} + 4 a^{\frac{7}{2}} b x^{4}} + \frac{a^{3} \log{\left (\frac{b x^{4}}{a} \right )}}{4 a^{\frac{9}{2}} + 4 a^{\frac{7}{2}} b x^{4}} - \frac{2 a^{3} \log{\left (\sqrt{1 + \frac{b x^{4}}{a}} + 1 \right )}}{4 a^{\frac{9}{2}} + 4 a^{\frac{7}{2}} b x^{4}} + \frac{a^{2} b x^{4} \log{\left (\frac{b x^{4}}{a} \right )}}{4 a^{\frac{9}{2}} + 4 a^{\frac{7}{2}} b x^{4}} - \frac{2 a^{2} b x^{4} \log{\left (\sqrt{1 + \frac{b x^{4}}{a}} + 1 \right )}}{4 a^{\frac{9}{2}} + 4 a^{\frac{7}{2}} b x^{4}}\right ) + \frac{d \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{3}{2} \\ \frac{3}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{3}{2}} x \Gamma \left (\frac{3}{4}\right )} + \frac{f x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{3}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{3}{2}} \Gamma \left (\frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*(-1/(2*a*sqrt(b)*x**4*sqrt(a/(b*x**4) + 1)) - sqrt(b)/(a**2*sqrt(a/(b*x**4) +
1))) + e*(2*a**3*sqrt(1 + b*x**4/a)/(4*a**(9/2) + 4*a**(7/2)*b*x**4) + a**3*log(
b*x**4/a)/(4*a**(9/2) + 4*a**(7/2)*b*x**4) - 2*a**3*log(sqrt(1 + b*x**4/a) + 1)/
(4*a**(9/2) + 4*a**(7/2)*b*x**4) + a**2*b*x**4*log(b*x**4/a)/(4*a**(9/2) + 4*a**
(7/2)*b*x**4) - 2*a**2*b*x**4*log(sqrt(1 + b*x**4/a) + 1)/(4*a**(9/2) + 4*a**(7/
2)*b*x**4)) + d*gamma(-1/4)*hyper((-1/4, 3/2), (3/4,), b*x**4*exp_polar(I*pi)/a)
/(4*a**(3/2)*x*gamma(3/4)) + f*x*gamma(1/4)*hyper((1/4, 3/2), (5/4,), b*x**4*exp
_polar(I*pi)/a)/(4*a**(3/2)*gamma(5/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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