3.524 \(\int \frac{c+d x+e x^2+f x^3}{x^3 \sqrt{a+b x^4}} \, dx\)

Optimal. Leaf size=300 \[ \frac{\left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (\sqrt{a} f+\sqrt{b} d\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 a^{3/4} \sqrt [4]{b} \sqrt{a+b x^4}}-\frac{\sqrt [4]{b} d \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{a^{3/4} \sqrt{a+b x^4}}-\frac{c \sqrt{a+b x^4}}{2 a x^2}-\frac{d \sqrt{a+b x^4}}{a x}+\frac{\sqrt{b} d x \sqrt{a+b x^4}}{a \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{e \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{2 \sqrt{a}} \]

[Out]

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

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Rubi [A]  time = 0.547339, antiderivative size = 300, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{\left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (\sqrt{a} f+\sqrt{b} d\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 a^{3/4} \sqrt [4]{b} \sqrt{a+b x^4}}-\frac{\sqrt [4]{b} d \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{a^{3/4} \sqrt{a+b x^4}}-\frac{c \sqrt{a+b x^4}}{2 a x^2}-\frac{d \sqrt{a+b x^4}}{a x}+\frac{\sqrt{b} d x \sqrt{a+b x^4}}{a \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{e \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{2 \sqrt{a}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 60.4383, size = 265, normalized size = 0.88 \[ \frac{\sqrt{b} d x \sqrt{a + b x^{4}}}{a \left (\sqrt{a} + \sqrt{b} x^{2}\right )} - \frac{c \sqrt{a + b x^{4}}}{2 a x^{2}} - \frac{d \sqrt{a + b x^{4}}}{a x} - \frac{e \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{4}}}{\sqrt{a}} \right )}}{2 \sqrt{a}} - \frac{\sqrt [4]{b} d \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{a^{\frac{3}{4}} \sqrt{a + b x^{4}}} + \frac{\sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) \left (\sqrt{a} f + \sqrt{b} d\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{2 a^{\frac{3}{4}} \sqrt [4]{b} \sqrt{a + b x^{4}}} \]

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)**(1/2),x)

[Out]

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

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Mathematica [C]  time = 0.760966, size = 242, normalized size = 0.81 \[ \frac{-\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \left (\left (a+b x^4\right ) (c+2 d x)+\sqrt{a} e x^2 \sqrt{a+b x^4} \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )\right )-2 i \sqrt{a} x^2 \sqrt{\frac{b x^4}{a}+1} \left (\sqrt{a} f-i \sqrt{b} d\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )+2 \sqrt{a} \sqrt{b} d x^2 \sqrt{\frac{b x^4}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )}{2 a x^2 \sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.021, size = 293, normalized size = 1. \[{f\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{\frac{c}{2\,a{x}^{2}}\sqrt{b{x}^{4}+a}}-{\frac{d}{ax}\sqrt{b{x}^{4}+a}}+{id\sqrt{b}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{id\sqrt{b}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticE} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{\frac{e}{2}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{4}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}} \]

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)^(1/2),x)

[Out]

f/(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*(b*x^
4+a)^(1/2)/a/x^2-d*(b*x^4+a)^(1/2)/a/x+I*d*b^(1/2)/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)-I*d*b^(1/2)/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*e/a^(1/2)*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. \[ \int \frac{f x^{3} + e x^{2} + d x + c}{\sqrt{b x^{4} + a} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.40882, size = 126, normalized size = 0.42 \[ - \frac{\sqrt{b} c \sqrt{\frac{a}{b x^{4}} + 1}}{2 a} + \frac{d \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{2} \\ \frac{3}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt{a} x \Gamma \left (\frac{3}{4}\right )} - \frac{e \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2 \sqrt{a}} + \frac{f x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{5}{4}\right )} \]

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)**(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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