3.527 \(\int \frac{c+d x+e x^2+f x^3}{x^6 \sqrt{a+b x^4}} \, dx\)

Optimal. Leaf size=377 \[ -\frac{b^{3/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (5 \sqrt{a} e+9 \sqrt{b} c\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{30 a^{7/4} \sqrt{a+b x^4}}+\frac{3 b^{5/4} c \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 a^{7/4} \sqrt{a+b x^4}}+\frac{b d \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{4 a^{3/2}}-\frac{3 b^{3/2} c x \sqrt{a+b x^4}}{5 a^2 \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{3 b c \sqrt{a+b x^4}}{5 a^2 x}-\frac{c \sqrt{a+b x^4}}{5 a x^5}-\frac{d \sqrt{a+b x^4}}{4 a x^4}-\frac{e \sqrt{a+b x^4}}{3 a x^3}-\frac{f \sqrt{a+b x^4}}{2 a x^2} \]

[Out]

-(c*Sqrt[a + b*x^4])/(5*a*x^5) - (d*Sqrt[a + b*x^4])/(4*a*x^4) - (e*Sqrt[a + b*x
^4])/(3*a*x^3) - (f*Sqrt[a + b*x^4])/(2*a*x^2) + (3*b*c*Sqrt[a + b*x^4])/(5*a^2*
x) - (3*b^(3/2)*c*x*Sqrt[a + b*x^4])/(5*a^2*(Sqrt[a] + Sqrt[b]*x^2)) + (b*d*ArcT
anh[Sqrt[a + b*x^4]/Sqrt[a]])/(4*a^(3/2)) + (3*b^(5/4)*c*(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])/(5*a^(7/4)*Sqrt[a + b*x^4]) - (b^(3/4)*(9*Sqrt[b]*c + 5*Sqrt[a]*e)*(
Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticF[2*A
rcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(30*a^(7/4)*Sqrt[a + b*x^4])

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Rubi [A]  time = 0.864639, antiderivative size = 377, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.367 \[ -\frac{b^{3/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (5 \sqrt{a} e+9 \sqrt{b} c\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{30 a^{7/4} \sqrt{a+b x^4}}+\frac{3 b^{5/4} c \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 a^{7/4} \sqrt{a+b x^4}}+\frac{b d \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{4 a^{3/2}}-\frac{3 b^{3/2} c x \sqrt{a+b x^4}}{5 a^2 \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{3 b c \sqrt{a+b x^4}}{5 a^2 x}-\frac{c \sqrt{a+b x^4}}{5 a x^5}-\frac{d \sqrt{a+b x^4}}{4 a x^4}-\frac{e \sqrt{a+b x^4}}{3 a x^3}-\frac{f \sqrt{a+b x^4}}{2 a x^2} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x + e*x^2 + f*x^3)/(x^6*Sqrt[a + b*x^4]),x]

[Out]

-(c*Sqrt[a + b*x^4])/(5*a*x^5) - (d*Sqrt[a + b*x^4])/(4*a*x^4) - (e*Sqrt[a + b*x
^4])/(3*a*x^3) - (f*Sqrt[a + b*x^4])/(2*a*x^2) + (3*b*c*Sqrt[a + b*x^4])/(5*a^2*
x) - (3*b^(3/2)*c*x*Sqrt[a + b*x^4])/(5*a^2*(Sqrt[a] + Sqrt[b]*x^2)) + (b*d*ArcT
anh[Sqrt[a + b*x^4]/Sqrt[a]])/(4*a^(3/2)) + (3*b^(5/4)*c*(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])/(5*a^(7/4)*Sqrt[a + b*x^4]) - (b^(3/4)*(9*Sqrt[b]*c + 5*Sqrt[a]*e)*(
Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticF[2*A
rcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(30*a^(7/4)*Sqrt[a + b*x^4])

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Rubi in Sympy [A]  time = 98.6803, size = 342, normalized size = 0.91 \[ - \frac{c \sqrt{a + b x^{4}}}{5 a x^{5}} - \frac{d \sqrt{a + b x^{4}}}{4 a x^{4}} - \frac{e \sqrt{a + b x^{4}}}{3 a x^{3}} - \frac{f \sqrt{a + b x^{4}}}{2 a x^{2}} - \frac{3 b^{\frac{3}{2}} c x \sqrt{a + b x^{4}}}{5 a^{2} \left (\sqrt{a} + \sqrt{b} x^{2}\right )} + \frac{3 b c \sqrt{a + b x^{4}}}{5 a^{2} x} + \frac{b d \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{4}}}{\sqrt{a}} \right )}}{4 a^{\frac{3}{2}}} + \frac{3 b^{\frac{5}{4}} c \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{5 a^{\frac{7}{4}} \sqrt{a + b x^{4}}} - \frac{b^{\frac{3}{4}} \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) \left (5 \sqrt{a} e + 9 \sqrt{b} c\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{30 a^{\frac{7}{4}} \sqrt{a + b x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((f*x**3+e*x**2+d*x+c)/x**6/(b*x**4+a)**(1/2),x)

[Out]

-c*sqrt(a + b*x**4)/(5*a*x**5) - d*sqrt(a + b*x**4)/(4*a*x**4) - e*sqrt(a + b*x*
*4)/(3*a*x**3) - f*sqrt(a + b*x**4)/(2*a*x**2) - 3*b**(3/2)*c*x*sqrt(a + b*x**4)
/(5*a**2*(sqrt(a) + sqrt(b)*x**2)) + 3*b*c*sqrt(a + b*x**4)/(5*a**2*x) + b*d*ata
nh(sqrt(a + b*x**4)/sqrt(a))/(4*a**(3/2)) + 3*b**(5/4)*c*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)/(5*a**(7/4)*sqrt(a + b*x**4)) - b**(3/4)*sqrt((a + b*x**4)/(sqrt(a
) + sqrt(b)*x**2)**2)*(sqrt(a) + sqrt(b)*x**2)*(5*sqrt(a)*e + 9*sqrt(b)*c)*ellip
tic_f(2*atan(b**(1/4)*x/a**(1/4)), 1/2)/(30*a**(7/4)*sqrt(a + b*x**4))

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Mathematica [C]  time = 0.905436, size = 268, normalized size = 0.71 \[ \frac{-36 \sqrt{a} b^{3/2} c x^5 \sqrt{\frac{b x^4}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )+\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \left (15 \sqrt{a} b d x^5 \sqrt{a+b x^4} \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )-\left (a+b x^4\right ) \left (12 a c+5 a x \left (3 d+4 e x+6 f x^2\right )-36 b c x^4\right )\right )+4 \sqrt{a} b x^5 \sqrt{\frac{b x^4}{a}+1} \left (9 \sqrt{b} c+5 i \sqrt{a} e\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )}{60 a^2 x^5 \sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x + e*x^2 + f*x^3)/(x^6*Sqrt[a + b*x^4]),x]

[Out]

(Sqrt[(I*Sqrt[b])/Sqrt[a]]*(-((a + b*x^4)*(12*a*c - 36*b*c*x^4 + 5*a*x*(3*d + 4*
e*x + 6*f*x^2))) + 15*Sqrt[a]*b*d*x^5*Sqrt[a + b*x^4]*ArcTanh[Sqrt[a + b*x^4]/Sq
rt[a]]) - 36*Sqrt[a]*b^(3/2)*c*x^5*Sqrt[1 + (b*x^4)/a]*EllipticE[I*ArcSinh[Sqrt[
(I*Sqrt[b])/Sqrt[a]]*x], -1] + 4*Sqrt[a]*b*(9*Sqrt[b]*c + (5*I)*Sqrt[a]*e)*x^5*S
qrt[1 + (b*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -1])/(60*a^
2*Sqrt[(I*Sqrt[b])/Sqrt[a]]*x^5*Sqrt[a + b*x^4])

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Maple [C]  time = 0.015, size = 354, normalized size = 0.9 \[ -{\frac{c}{5\,a{x}^{5}}\sqrt{b{x}^{4}+a}}+{\frac{3\,bc}{5\,{a}^{2}x}\sqrt{b{x}^{4}+a}}-{{\frac{3\,i}{5}}c{b}^{{\frac{3}{2}}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){a}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}+{{\frac{3\,i}{5}}c{b}^{{\frac{3}{2}}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticE} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){a}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{\frac{d}{4\,a{x}^{4}}\sqrt{b{x}^{4}+a}}+{\frac{bd}{4}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{4}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{e}{3\,a{x}^{3}}\sqrt{b{x}^{4}+a}}-{\frac{be}{3\,a}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{\frac{f}{2\,a{x}^{2}}\sqrt{b{x}^{4}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x^3+e*x^2+d*x+c)/x^6/(b*x^4+a)^(1/2),x)

[Out]

-1/5*c*(b*x^4+a)^(1/2)/a/x^5+3/5*b*c*(b*x^4+a)^(1/2)/a^2/x-3/5*I*c/a^(3/2)*b^(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)+3/5*I*c/a^
(3/2)*b^(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)
-1/4*d*(b*x^4+a)^(1/2)/a/x^4+1/4*d*b/a^(3/2)*ln((2*a+2*a^(1/2)*(b*x^4+a)^(1/2))/
x^2)-1/3*e*(b*x^4+a)^(1/2)/a/x^3-1/3*e*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*f*(b*x^4+a)^(1/2)/a/x^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{f x^{3} + e x^{2} + d x + c}{\sqrt{b x^{4} + a} x^{6}}\,{d x} \]

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^6),x, algorithm="maxima")

[Out]

integrate((f*x^3 + e*x^2 + d*x + c)/(sqrt(b*x^4 + a)*x^6), x)

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

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^6),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 7.77864, size = 163, normalized size = 0.43 \[ - \frac{\sqrt{b} d \sqrt{\frac{a}{b x^{4}} + 1}}{4 a x^{2}} - \frac{\sqrt{b} f \sqrt{\frac{a}{b x^{4}} + 1}}{2 a} + \frac{c \Gamma \left (- \frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, \frac{1}{2} \\ - \frac{1}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt{a} x^{5} \Gamma \left (- \frac{1}{4}\right )} + \frac{e \Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{1}{2} \\ \frac{1}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt{a} x^{3} \Gamma \left (\frac{1}{4}\right )} + \frac{b d \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{4 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x**3+e*x**2+d*x+c)/x**6/(b*x**4+a)**(1/2),x)

[Out]

-sqrt(b)*d*sqrt(a/(b*x**4) + 1)/(4*a*x**2) - sqrt(b)*f*sqrt(a/(b*x**4) + 1)/(2*a
) + c*gamma(-5/4)*hyper((-5/4, 1/2), (-1/4,), b*x**4*exp_polar(I*pi)/a)/(4*sqrt(
a)*x**5*gamma(-1/4)) + e*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)) + b*d*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. \[ \int \frac{f x^{3} + e x^{2} + d x + c}{\sqrt{b x^{4} + a} x^{6}}\,{d x} \]

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^6),x, algorithm="giac")

[Out]

integrate((f*x^3 + e*x^2 + d*x + c)/(sqrt(b*x^4 + a)*x^6), x)