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

Optimal. Leaf size=118 \[ \frac{\sqrt{a+b x^2} (4 b c-5 a d)}{15 a^2 x^3}-\frac{\sqrt{a+b x^2} \left (15 a^2 e-10 a b d+8 b^2 c\right )}{15 a^3 x}-\frac{c \sqrt{a+b x^2}}{5 a x^5}+\frac{f \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{\sqrt{b}} \]

[Out]

-(c*Sqrt[a + b*x^2])/(5*a*x^5) + ((4*b*c - 5*a*d)*Sqrt[a + b*x^2])/(15*a^2*x^3)
- ((8*b^2*c - 10*a*b*d + 15*a^2*e)*Sqrt[a + b*x^2])/(15*a^3*x) + (f*ArcTanh[(Sqr
t[b]*x)/Sqrt[a + b*x^2]])/Sqrt[b]

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Rubi [A]  time = 0.359818, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \frac{\sqrt{a+b x^2} (4 b c-5 a d)}{15 a^2 x^3}-\frac{\sqrt{a+b x^2} \left (15 a^2 e-10 a b d+8 b^2 c\right )}{15 a^3 x}-\frac{c \sqrt{a+b x^2}}{5 a x^5}+\frac{f \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

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

[Out]

-(c*Sqrt[a + b*x^2])/(5*a*x^5) + ((4*b*c - 5*a*d)*Sqrt[a + b*x^2])/(15*a^2*x^3)
- ((8*b^2*c - 10*a*b*d + 15*a^2*e)*Sqrt[a + b*x^2])/(15*a^3*x) + (f*ArcTanh[(Sqr
t[b]*x)/Sqrt[a + b*x^2]])/Sqrt[b]

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Rubi in Sympy [A]  time = 79.0095, size = 117, normalized size = 0.99 \[ \frac{f \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{\sqrt{b}} - \frac{c \sqrt{a + b x^{2}}}{5 a x^{5}} - \frac{e \sqrt{a + b x^{2}}}{a x} - \frac{\sqrt{a + b x^{2}} \left (5 a d - 4 b c\right )}{15 a^{2} x^{3}} + \frac{2 b \sqrt{a + b x^{2}} \left (5 a d - 4 b c\right )}{15 a^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

f*atanh(sqrt(b)*x/sqrt(a + b*x**2))/sqrt(b) - c*sqrt(a + b*x**2)/(5*a*x**5) - e*
sqrt(a + b*x**2)/(a*x) - sqrt(a + b*x**2)*(5*a*d - 4*b*c)/(15*a**2*x**3) + 2*b*s
qrt(a + b*x**2)*(5*a*d - 4*b*c)/(15*a**3*x)

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Mathematica [A]  time = 0.215144, size = 98, normalized size = 0.83 \[ \frac{f \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{\sqrt{b}}-\frac{\sqrt{a+b x^2} \left (a^2 \left (3 c+5 d x^2+15 e x^4\right )-2 a b x^2 \left (2 c+5 d x^2\right )+8 b^2 c x^4\right )}{15 a^3 x^5} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[a + b*x^2]*(8*b^2*c*x^4 - 2*a*b*x^2*(2*c + 5*d*x^2) + a^2*(3*c + 5*d*x^2
+ 15*e*x^4)))/(15*a^3*x^5) + (f*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/Sqrt[b]

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Maple [A]  time = 0.015, size = 136, normalized size = 1.2 \[{f\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){\frac{1}{\sqrt{b}}}}-{\frac{c}{5\,a{x}^{5}}\sqrt{b{x}^{2}+a}}+{\frac{4\,bc}{15\,{x}^{3}{a}^{2}}\sqrt{b{x}^{2}+a}}-{\frac{8\,{b}^{2}c}{15\,{a}^{3}x}\sqrt{b{x}^{2}+a}}-{\frac{d}{3\,a{x}^{3}}\sqrt{b{x}^{2}+a}}+{\frac{2\,bd}{3\,x{a}^{2}}\sqrt{b{x}^{2}+a}}-{\frac{e}{ax}\sqrt{b{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

f*ln(x*b^(1/2)+(b*x^2+a)^(1/2))/b^(1/2)-1/5*c*(b*x^2+a)^(1/2)/a/x^5+4/15*c*b/a^2
/x^3*(b*x^2+a)^(1/2)-8/15*c*b^2/a^3/x*(b*x^2+a)^(1/2)-1/3*d/a/x^3*(b*x^2+a)^(1/2
)+2/3*d*b/a^2/x*(b*x^2+a)^(1/2)-e/a/x*(b*x^2+a)^(1/2)

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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^6 + e*x^4 + d*x^2 + c)/(sqrt(b*x^2 + a)*x^6),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.277702, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{3} f x^{5} \log \left (-2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right ) - 2 \,{\left ({\left (8 \, b^{2} c - 10 \, a b d + 15 \, a^{2} e\right )} x^{4} + 3 \, a^{2} c -{\left (4 \, a b c - 5 \, a^{2} d\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{b}}{30 \, a^{3} \sqrt{b} x^{5}}, \frac{15 \, a^{3} f x^{5} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) -{\left ({\left (8 \, b^{2} c - 10 \, a b d + 15 \, a^{2} e\right )} x^{4} + 3 \, a^{2} c -{\left (4 \, a b c - 5 \, a^{2} d\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{-b}}{15 \, a^{3} \sqrt{-b} x^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^6 + e*x^4 + d*x^2 + c)/(sqrt(b*x^2 + a)*x^6),x, algorithm="fricas")

[Out]

[1/30*(15*a^3*f*x^5*log(-2*sqrt(b*x^2 + a)*b*x - (2*b*x^2 + a)*sqrt(b)) - 2*((8*
b^2*c - 10*a*b*d + 15*a^2*e)*x^4 + 3*a^2*c - (4*a*b*c - 5*a^2*d)*x^2)*sqrt(b*x^2
+ a)*sqrt(b))/(a^3*sqrt(b)*x^5), 1/15*(15*a^3*f*x^5*arctan(sqrt(-b)*x/sqrt(b*x^
2 + a)) - ((8*b^2*c - 10*a*b*d + 15*a^2*e)*x^4 + 3*a^2*c - (4*a*b*c - 5*a^2*d)*x
^2)*sqrt(b*x^2 + a)*sqrt(-b))/(a^3*sqrt(-b)*x^5)]

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Sympy [A]  time = 4.53568, size = 456, normalized size = 3.86 \[ - \frac{3 a^{4} b^{\frac{9}{2}} c \sqrt{\frac{a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} - \frac{2 a^{3} b^{\frac{11}{2}} c x^{2} \sqrt{\frac{a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} - \frac{3 a^{2} b^{\frac{13}{2}} c x^{4} \sqrt{\frac{a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} - \frac{12 a b^{\frac{15}{2}} c x^{6} \sqrt{\frac{a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} - \frac{8 b^{\frac{17}{2}} c x^{8} \sqrt{\frac{a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} + f \left (\begin{cases} \frac{\sqrt{- \frac{a}{b}} \operatorname{asin}{\left (x \sqrt{- \frac{b}{a}} \right )}}{\sqrt{a}} & \text{for}\: a > 0 \wedge b < 0 \\\frac{\sqrt{\frac{a}{b}} \operatorname{asinh}{\left (x \sqrt{\frac{b}{a}} \right )}}{\sqrt{a}} & \text{for}\: a > 0 \wedge b > 0 \\\frac{\sqrt{- \frac{a}{b}} \operatorname{acosh}{\left (x \sqrt{- \frac{b}{a}} \right )}}{\sqrt{- a}} & \text{for}\: b > 0 \wedge a < 0 \end{cases}\right ) - \frac{\sqrt{b} d \sqrt{\frac{a}{b x^{2}} + 1}}{3 a x^{2}} - \frac{\sqrt{b} e \sqrt{\frac{a}{b x^{2}} + 1}}{a} + \frac{2 b^{\frac{3}{2}} d \sqrt{\frac{a}{b x^{2}} + 1}}{3 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*a**4*b**(9/2)*c*sqrt(a/(b*x**2) + 1)/(15*a**5*b**4*x**4 + 30*a**4*b**5*x**6 +
15*a**3*b**6*x**8) - 2*a**3*b**(11/2)*c*x**2*sqrt(a/(b*x**2) + 1)/(15*a**5*b**4
*x**4 + 30*a**4*b**5*x**6 + 15*a**3*b**6*x**8) - 3*a**2*b**(13/2)*c*x**4*sqrt(a/
(b*x**2) + 1)/(15*a**5*b**4*x**4 + 30*a**4*b**5*x**6 + 15*a**3*b**6*x**8) - 12*a
*b**(15/2)*c*x**6*sqrt(a/(b*x**2) + 1)/(15*a**5*b**4*x**4 + 30*a**4*b**5*x**6 +
15*a**3*b**6*x**8) - 8*b**(17/2)*c*x**8*sqrt(a/(b*x**2) + 1)/(15*a**5*b**4*x**4
+ 30*a**4*b**5*x**6 + 15*a**3*b**6*x**8) + f*Piecewise((sqrt(-a/b)*asin(x*sqrt(-
b/a))/sqrt(a), (a > 0) & (b < 0)), (sqrt(a/b)*asinh(x*sqrt(b/a))/sqrt(a), (a > 0
) & (b > 0)), (sqrt(-a/b)*acosh(x*sqrt(-b/a))/sqrt(-a), (b > 0) & (a < 0))) - sq
rt(b)*d*sqrt(a/(b*x**2) + 1)/(3*a*x**2) - sqrt(b)*e*sqrt(a/(b*x**2) + 1)/a + 2*b
**(3/2)*d*sqrt(a/(b*x**2) + 1)/(3*a**2)

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GIAC/XCAS [A]  time = 0.229113, size = 437, normalized size = 3.7 \[ -\frac{f{\rm ln}\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2}\right )}{2 \, \sqrt{b}} + \frac{2 \,{\left (15 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} \sqrt{b} e + 30 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} b^{\frac{3}{2}} d - 60 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} a \sqrt{b} e + 80 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} b^{\frac{5}{2}} c - 70 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a b^{\frac{3}{2}} d + 90 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a^{2} \sqrt{b} e - 40 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a b^{\frac{5}{2}} c + 50 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{2} b^{\frac{3}{2}} d - 60 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{3} \sqrt{b} e + 8 \, a^{2} b^{\frac{5}{2}} c - 10 \, a^{3} b^{\frac{3}{2}} d + 15 \, a^{4} \sqrt{b} e\right )}}{15 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^6 + e*x^4 + d*x^2 + c)/(sqrt(b*x^2 + a)*x^6),x, algorithm="giac")

[Out]

-1/2*f*ln((sqrt(b)*x - sqrt(b*x^2 + a))^2)/sqrt(b) + 2/15*(15*(sqrt(b)*x - sqrt(
b*x^2 + a))^8*sqrt(b)*e + 30*(sqrt(b)*x - sqrt(b*x^2 + a))^6*b^(3/2)*d - 60*(sqr
t(b)*x - sqrt(b*x^2 + a))^6*a*sqrt(b)*e + 80*(sqrt(b)*x - sqrt(b*x^2 + a))^4*b^(
5/2)*c - 70*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a*b^(3/2)*d + 90*(sqrt(b)*x - sqrt(b
*x^2 + a))^4*a^2*sqrt(b)*e - 40*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a*b^(5/2)*c + 50
*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^2*b^(3/2)*d - 60*(sqrt(b)*x - sqrt(b*x^2 + a)
)^2*a^3*sqrt(b)*e + 8*a^2*b^(5/2)*c - 10*a^3*b^(3/2)*d + 15*a^4*sqrt(b)*e)/((sqr
t(b)*x - sqrt(b*x^2 + a))^2 - a)^5