3.148 \(\int \frac{c+d x^2+e x^4+f x^6}{x^5 \sqrt{a+b x^2}} \, dx\)

Optimal. Leaf size=114 \[ \frac{\sqrt{a+b x^2} (3 b c-4 a d)}{8 a^2 x^2}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right ) \left (8 a^2 e-4 a b d+3 b^2 c\right )}{8 a^{5/2}}-\frac{c \sqrt{a+b x^2}}{4 a x^4}+\frac{f \sqrt{a+b x^2}}{b} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.534256, antiderivative size = 114, 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 \[ \frac{\sqrt{a+b x^2} (3 b c-4 a d)}{8 a^2 x^2}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right ) \left (8 a^2 e-4 a b d+3 b^2 c\right )}{8 a^{5/2}}-\frac{c \sqrt{a+b x^2}}{4 a x^4}+\frac{f \sqrt{a+b x^2}}{b} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^2 + e*x^4 + f*x^6)/(x^5*Sqrt[a + b*x^2]),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 147.611, size = 104, normalized size = 0.91 \[ \frac{f \sqrt{a + b x^{2}}}{b} - \frac{c \sqrt{a + b x^{2}}}{4 a x^{4}} - \frac{\sqrt{a + b x^{2}} \left (4 a d - 3 b c\right )}{8 a^{2} x^{2}} - \frac{\left (8 a^{2} e - 4 a b d + 3 b^{2} c\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{8 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((f*x**6+e*x**4+d*x**2+c)/x**5/(b*x**2+a)**(1/2),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.328466, size = 125, normalized size = 1.1 \[ \sqrt{a+b x^2} \left (\frac{3 b c-4 a d}{8 a^2 x^2}-\frac{c}{4 a x^4}+\frac{f}{b}\right )-\frac{\log \left (\sqrt{a} \sqrt{a+b x^2}+a\right ) \left (8 a^2 e-4 a b d+3 b^2 c\right )}{8 a^{5/2}}+\frac{\log (x) \left (8 a^2 e-4 a b d+3 b^2 c\right )}{8 a^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(x^5*Sqrt[a + b*x^2]),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.015, size = 162, normalized size = 1.4 \[{\frac{f}{b}\sqrt{b{x}^{2}+a}}-{\frac{c}{4\,a{x}^{4}}\sqrt{b{x}^{2}+a}}+{\frac{3\,bc}{8\,{a}^{2}{x}^{2}}\sqrt{b{x}^{2}+a}}-{\frac{3\,{b}^{2}c}{8}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}-{\frac{d}{2\,a{x}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{bd}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{e\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x^6+e*x^4+d*x^2+c)/x^5/(b*x^2+a)^(1/2),x)

[Out]

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

_______________________________________________________________________________________

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^5),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.274824, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (3 \, b^{3} c - 4 \, a b^{2} d + 8 \, a^{2} b e\right )} x^{4} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right ) + 2 \,{\left (8 \, a^{2} f x^{4} - 2 \, a b c +{\left (3 \, b^{2} c - 4 \, a b d\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{a}}{16 \, a^{\frac{5}{2}} b x^{4}}, -\frac{{\left (3 \, b^{3} c - 4 \, a b^{2} d + 8 \, a^{2} b e\right )} x^{4} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) -{\left (8 \, a^{2} f x^{4} - 2 \, a b c +{\left (3 \, b^{2} c - 4 \, a b d\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{-a}}{8 \, \sqrt{-a} a^{2} b x^{4}}\right ] \]

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^5),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 62.4029, size = 194, normalized size = 1.7 \[ f \left (\begin{cases} \frac{x^{2}}{2 \sqrt{a}} & \text{for}\: b = 0 \\\frac{\sqrt{a + b x^{2}}}{b} & \text{otherwise} \end{cases}\right ) - \frac{c}{4 \sqrt{b} x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{\sqrt{b} c}{8 a x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{\sqrt{b} d \sqrt{\frac{a}{b x^{2}} + 1}}{2 a x} + \frac{3 b^{\frac{3}{2}} c}{8 a^{2} x \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{e \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{\sqrt{a}} + \frac{b d \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2 a^{\frac{3}{2}}} - \frac{3 b^{2} c \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{8 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x**6+e*x**4+d*x**2+c)/x**5/(b*x**2+a)**(1/2),x)

[Out]

f*Piecewise((x**2/(2*sqrt(a)), Eq(b, 0)), (sqrt(a + b*x**2)/b, True)) - c/(4*sqr
t(b)*x**5*sqrt(a/(b*x**2) + 1)) + sqrt(b)*c/(8*a*x**3*sqrt(a/(b*x**2) + 1)) - sq
rt(b)*d*sqrt(a/(b*x**2) + 1)/(2*a*x) + 3*b**(3/2)*c/(8*a**2*x*sqrt(a/(b*x**2) +
1)) - e*asinh(sqrt(a)/(sqrt(b)*x))/sqrt(a) + b*d*asinh(sqrt(a)/(sqrt(b)*x))/(2*a
**(3/2)) - 3*b**2*c*asinh(sqrt(a)/(sqrt(b)*x))/(8*a**(5/2))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.223875, size = 190, normalized size = 1.67 \[ \frac{8 \, \sqrt{b x^{2} + a} f + \frac{{\left (3 \, b^{3} c - 4 \, a b^{2} d + 8 \, a^{2} b e\right )} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} b^{3} c - 5 \, \sqrt{b x^{2} + a} a b^{3} c - 4 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a b^{2} d + 4 \, \sqrt{b x^{2} + a} a^{2} b^{2} d}{a^{2} b^{2} x^{4}}}{8 \, b} \]

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^5),x, algorithm="giac")

[Out]

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