3.526 \(\int \frac{c+d x+e x^2+f x^3}{x^5 \sqrt{a+b x^4}} \, dx\)

Optimal. Leaf size=346 \[ \frac{b c \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{4 a^{3/2}}-\frac{\sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (\sqrt{b} d-3 \sqrt{a} f\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{6 a^{5/4} \sqrt{a+b x^4}}-\frac{\sqrt [4]{b} f \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{a^{3/4} \sqrt{a+b x^4}}-\frac{c \sqrt{a+b x^4}}{4 a x^4}-\frac{d \sqrt{a+b x^4}}{3 a x^3}-\frac{e \sqrt{a+b x^4}}{2 a x^2}-\frac{f \sqrt{a+b x^4}}{a x}+\frac{\sqrt{b} f x \sqrt{a+b x^4}}{a \left (\sqrt{a}+\sqrt{b} x^2\right )} \]

[Out]

-(c*Sqrt[a + b*x^4])/(4*a*x^4) - (d*Sqrt[a + b*x^4])/(3*a*x^3) - (e*Sqrt[a + b*x
^4])/(2*a*x^2) - (f*Sqrt[a + b*x^4])/(a*x) + (Sqrt[b]*f*x*Sqrt[a + b*x^4])/(a*(S
qrt[a] + Sqrt[b]*x^2)) + (b*c*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]])/(4*a^(3/2)) - (b
^(1/4)*f*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*Ell
ipticE[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(a^(3/4)*Sqrt[a + b*x^4]) - (b^(1/4)
*(Sqrt[b]*d - 3*Sqrt[a]*f)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + S
qrt[b]*x^2)^2]*EllipticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(6*a^(5/4)*Sqrt[a
+ b*x^4])

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Rubi [A]  time = 0.749794, antiderivative size = 346, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 11, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.367 \[ \frac{b c \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{4 a^{3/2}}-\frac{\sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (\sqrt{b} d-3 \sqrt{a} f\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{6 a^{5/4} \sqrt{a+b x^4}}-\frac{\sqrt [4]{b} f \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{a^{3/4} \sqrt{a+b x^4}}-\frac{c \sqrt{a+b x^4}}{4 a x^4}-\frac{d \sqrt{a+b x^4}}{3 a x^3}-\frac{e \sqrt{a+b x^4}}{2 a x^2}-\frac{f \sqrt{a+b x^4}}{a x}+\frac{\sqrt{b} f x \sqrt{a+b x^4}}{a \left (\sqrt{a}+\sqrt{b} x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

-(c*Sqrt[a + b*x^4])/(4*a*x^4) - (d*Sqrt[a + b*x^4])/(3*a*x^3) - (e*Sqrt[a + b*x
^4])/(2*a*x^2) - (f*Sqrt[a + b*x^4])/(a*x) + (Sqrt[b]*f*x*Sqrt[a + b*x^4])/(a*(S
qrt[a] + Sqrt[b]*x^2)) + (b*c*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]])/(4*a^(3/2)) - (b
^(1/4)*f*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*Ell
ipticE[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(a^(3/4)*Sqrt[a + b*x^4]) - (b^(1/4)
*(Sqrt[b]*d - 3*Sqrt[a]*f)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + S
qrt[b]*x^2)^2]*EllipticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(6*a^(5/4)*Sqrt[a
+ b*x^4])

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Rubi in Sympy [A]  time = 81.2337, size = 306, normalized size = 0.88 \[ \frac{\sqrt{b} f x \sqrt{a + b x^{4}}}{a \left (\sqrt{a} + \sqrt{b} x^{2}\right )} - \frac{c \sqrt{a + b x^{4}}}{4 a x^{4}} - \frac{d \sqrt{a + b x^{4}}}{3 a x^{3}} - \frac{e \sqrt{a + b x^{4}}}{2 a x^{2}} - \frac{f \sqrt{a + b x^{4}}}{a x} + \frac{b c \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{4}}}{\sqrt{a}} \right )}}{4 a^{\frac{3}{2}}} - \frac{\sqrt [4]{b} f \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{a^{\frac{3}{4}} \sqrt{a + b x^{4}}} + \frac{\sqrt [4]{b} \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) \left (3 \sqrt{a} f - \sqrt{b} d\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{6 a^{\frac{5}{4}} \sqrt{a + b x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((f*x**3+e*x**2+d*x+c)/x**5/(b*x**4+a)**(1/2),x)

[Out]

sqrt(b)*f*x*sqrt(a + b*x**4)/(a*(sqrt(a) + sqrt(b)*x**2)) - c*sqrt(a + b*x**4)/(
4*a*x**4) - d*sqrt(a + b*x**4)/(3*a*x**3) - e*sqrt(a + b*x**4)/(2*a*x**2) - f*sq
rt(a + b*x**4)/(a*x) + b*c*atanh(sqrt(a + b*x**4)/sqrt(a))/(4*a**(3/2)) - b**(1/
4)*f*sqrt((a + b*x**4)/(sqrt(a) + sqrt(b)*x**2)**2)*(sqrt(a) + sqrt(b)*x**2)*ell
iptic_e(2*atan(b**(1/4)*x/a**(1/4)), 1/2)/(a**(3/4)*sqrt(a + b*x**4)) + b**(1/4)
*sqrt((a + b*x**4)/(sqrt(a) + sqrt(b)*x**2)**2)*(sqrt(a) + sqrt(b)*x**2)*(3*sqrt
(a)*f - sqrt(b)*d)*elliptic_f(2*atan(b**(1/4)*x/a**(1/4)), 1/2)/(6*a**(5/4)*sqrt
(a + b*x**4))

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Mathematica [C]  time = 0.893223, size = 259, normalized size = 0.75 \[ \frac{\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \left (3 b c x^4 \sqrt{a+b x^4} \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )-\sqrt{a} \left (a+b x^4\right ) \left (3 c+4 d x+6 x^2 (e+2 f x)\right )\right )-4 \sqrt{a} \sqrt{b} x^4 \sqrt{\frac{b x^4}{a}+1} \left (3 \sqrt{a} f-i \sqrt{b} d\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )+12 a \sqrt{b} f x^4 \sqrt{\frac{b x^4}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )}{12 a^{3/2} x^4 \sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(I*Sqrt[b])/Sqrt[a]]*(-(Sqrt[a]*(a + b*x^4)*(3*c + 4*d*x + 6*x^2*(e + 2*f*
x))) + 3*b*c*x^4*Sqrt[a + b*x^4]*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]]) + 12*a*Sqrt[b
]*f*x^4*Sqrt[1 + (b*x^4)/a]*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -1
] - 4*Sqrt[a]*Sqrt[b]*((-I)*Sqrt[b]*d + 3*Sqrt[a]*f)*x^4*Sqrt[1 + (b*x^4)/a]*Ell
ipticF[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -1])/(12*a^(3/2)*Sqrt[(I*Sqrt[b])
/Sqrt[a]]*x^4*Sqrt[a + b*x^4])

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Maple [C]  time = 0.015, size = 335, normalized size = 1. \[ -{\frac{c}{4\,a{x}^{4}}\sqrt{b{x}^{4}+a}}+{\frac{bc}{4}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{4}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{d}{3\,a{x}^{3}}\sqrt{b{x}^{4}+a}}-{\frac{bd}{3\,a}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{\frac{e}{2\,a{x}^{2}}\sqrt{b{x}^{4}+a}}-{\frac{f}{ax}\sqrt{b{x}^{4}+a}}+{if\sqrt{b}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{if\sqrt{b}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticE} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x^3+e*x^2+d*x+c)/x^5/(b*x^4+a)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

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^5),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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

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^5),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 7.1351, size = 158, normalized size = 0.46 \[ - \frac{\sqrt{b} c \sqrt{\frac{a}{b x^{4}} + 1}}{4 a x^{2}} - \frac{\sqrt{b} e \sqrt{\frac{a}{b x^{4}} + 1}}{2 a} + \frac{d \Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{1}{2} \\ \frac{1}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt{a} x^{3} \Gamma \left (\frac{1}{4}\right )} + \frac{f \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{2} \\ \frac{3}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt{a} x \Gamma \left (\frac{3}{4}\right )} + \frac{b c \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{4 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x**3+e*x**2+d*x+c)/x**5/(b*x**4+a)**(1/2),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{f x^{3} + e x^{2} + d x + c}{\sqrt{b x^{4} + a} x^{5}}\,{d x} \]

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^5),x, algorithm="giac")

[Out]

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