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

Optimal. Leaf size=117 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right ) \left (3 a^2 f-4 a b e+8 b^2 d\right )}{8 b^{5/2}}+\frac{x \sqrt{a+b x^2} (4 b e-3 a f)}{8 b^2}-\frac{c \sqrt{a+b x^2}}{a x}+\frac{f x^3 \sqrt{a+b x^2}}{4 b} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.289503, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right ) \left (3 a^2 f-4 a b e+8 b^2 d\right )}{8 b^{5/2}}+\frac{x \sqrt{a+b x^2} (4 b e-3 a f)}{8 b^2}-\frac{c \sqrt{a+b x^2}}{a x}+\frac{f x^3 \sqrt{a+b x^2}}{4 b} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 52.7855, size = 104, normalized size = 0.89 \[ \frac{f x^{3} \sqrt{a + b x^{2}}}{4 b} - \frac{x \sqrt{a + b x^{2}} \left (3 a f - 4 b e\right )}{8 b^{2}} + \frac{\left (a \left (3 a f - 4 b e\right ) + 8 b^{2} d\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{8 b^{\frac{5}{2}}} - \frac{c \sqrt{a + b x^{2}}}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

f*x**3*sqrt(a + b*x**2)/(4*b) - x*sqrt(a + b*x**2)*(3*a*f - 4*b*e)/(8*b**2) + (a
*(3*a*f - 4*b*e) + 8*b**2*d)*atanh(sqrt(b)*x/sqrt(a + b*x**2))/(8*b**(5/2)) - c*
sqrt(a + b*x**2)/(a*x)

_______________________________________________________________________________________

Mathematica [A]  time = 0.23801, size = 100, normalized size = 0.85 \[ \frac{\log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right ) \left (3 a^2 f-4 a b e+8 b^2 d\right )}{8 b^{5/2}}+\sqrt{a+b x^2} \left (-\frac{x (3 a f-4 b e)}{8 b^2}-\frac{c}{a x}+\frac{f x^3}{4 b}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.014, size = 140, normalized size = 1.2 \[{d\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){\frac{1}{\sqrt{b}}}}-{\frac{c}{ax}\sqrt{b{x}^{2}+a}}+{\frac{ex}{2\,b}\sqrt{b{x}^{2}+a}}-{\frac{ae}{2}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}}+{\frac{f{x}^{3}}{4\,b}\sqrt{b{x}^{2}+a}}-{\frac{3\,afx}{8\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{3\,{a}^{2}f}{8}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.261276, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (8 \, a b^{2} d - 4 \, a^{2} b e + 3 \, a^{3} f\right )} x \log \left (-2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right ) + 2 \,{\left (2 \, a b f x^{4} - 8 \, b^{2} c +{\left (4 \, a b e - 3 \, a^{2} f\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{b}}{16 \, a b^{\frac{5}{2}} x}, \frac{{\left (8 \, a b^{2} d - 4 \, a^{2} b e + 3 \, a^{3} f\right )} x \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) +{\left (2 \, a b f x^{4} - 8 \, b^{2} c +{\left (4 \, a b e - 3 \, a^{2} f\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{-b}}{8 \, a \sqrt{-b} b^{2} x}\right ] \]

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^2),x, algorithm="fricas")

[Out]

[1/16*((8*a*b^2*d - 4*a^2*b*e + 3*a^3*f)*x*log(-2*sqrt(b*x^2 + a)*b*x - (2*b*x^2
+ a)*sqrt(b)) + 2*(2*a*b*f*x^4 - 8*b^2*c + (4*a*b*e - 3*a^2*f)*x^2)*sqrt(b*x^2
+ a)*sqrt(b))/(a*b^(5/2)*x), 1/8*((8*a*b^2*d - 4*a^2*b*e + 3*a^3*f)*x*arctan(sqr
t(-b)*x/sqrt(b*x^2 + a)) + (2*a*b*f*x^4 - 8*b^2*c + (4*a*b*e - 3*a^2*f)*x^2)*sqr
t(b*x^2 + a)*sqrt(-b))/(a*sqrt(-b)*b^2*x)]

_______________________________________________________________________________________

Sympy [A]  time = 10.7989, size = 250, normalized size = 2.14 \[ - \frac{3 a^{\frac{3}{2}} f x}{8 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{\sqrt{a} e x \sqrt{1 + \frac{b x^{2}}{a}}}{2 b} - \frac{\sqrt{a} f x^{3}}{8 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 a^{2} f \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 b^{\frac{5}{2}}} - \frac{a e \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 b^{\frac{3}{2}}} + d \left (\begin{cases} \frac{\sqrt{- \frac{a}{b}} \operatorname{asin}{\left (x \sqrt{- \frac{b}{a}} \right )}}{\sqrt{a}} & \text{for}\: a > 0 \wedge b < 0 \\\frac{\sqrt{\frac{a}{b}} \operatorname{asinh}{\left (x \sqrt{\frac{b}{a}} \right )}}{\sqrt{a}} & \text{for}\: a > 0 \wedge b > 0 \\\frac{\sqrt{- \frac{a}{b}} \operatorname{acosh}{\left (x \sqrt{- \frac{b}{a}} \right )}}{\sqrt{- a}} & \text{for}\: b > 0 \wedge a < 0 \end{cases}\right ) - \frac{\sqrt{b} c \sqrt{\frac{a}{b x^{2}} + 1}}{a} + \frac{f x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*a**(3/2)*f*x/(8*b**2*sqrt(1 + b*x**2/a)) + sqrt(a)*e*x*sqrt(1 + b*x**2/a)/(2*
b) - sqrt(a)*f*x**3/(8*b*sqrt(1 + b*x**2/a)) + 3*a**2*f*asinh(sqrt(b)*x/sqrt(a))
/(8*b**(5/2)) - a*e*asinh(sqrt(b)*x/sqrt(a))/(2*b**(3/2)) + d*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))) - sqrt(b)*c*sqrt(a/(b*x**2) + 1)/a + f*x**5/(4*sqrt(a)*sqrt(1 + b*x
**2/a))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.227348, size = 163, normalized size = 1.39 \[ \frac{1}{8} \, \sqrt{b x^{2} + a}{\left (\frac{2 \, f x^{2}}{b} - \frac{3 \, a b f - 4 \, b^{2} e}{b^{3}}\right )} x + \frac{2 \, \sqrt{b} c}{{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a} - \frac{{\left (8 \, b^{\frac{5}{2}} d + 3 \, a^{2} \sqrt{b} f - 4 \, a b^{\frac{3}{2}} e\right )}{\rm ln}\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2}\right )}{16 \, b^{3}} \]

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^2),x, algorithm="giac")

[Out]

1/8*sqrt(b*x^2 + a)*(2*f*x^2/b - (3*a*b*f - 4*b^2*e)/b^3)*x + 2*sqrt(b)*c/((sqrt
(b)*x - sqrt(b*x^2 + a))^2 - a) - 1/16*(8*b^(5/2)*d + 3*a^2*sqrt(b)*f - 4*a*b^(3
/2)*e)*ln((sqrt(b)*x - sqrt(b*x^2 + a))^2)/b^3