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

Optimal. Leaf size=100 \[ \frac{(b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 a^{3/2}}+\frac{\sqrt{a+b x^2} (b e-a f)}{b^2}+\frac{f \left (a+b x^2\right )^{3/2}}{3 b^2}-\frac{c \sqrt{a+b x^2}}{2 a x^2} \]

[Out]

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

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Rubi [A]  time = 0.4077, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156 \[ \frac{(b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 a^{3/2}}+\frac{\sqrt{a+b x^2} (b e-a f)}{b^2}+\frac{f \left (a+b x^2\right )^{3/2}}{3 b^2}-\frac{c \sqrt{a+b x^2}}{2 a x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 156.743, size = 87, normalized size = 0.87 \[ \frac{f \left (a + b x^{2}\right )^{\frac{3}{2}}}{3 b^{2}} - \frac{\sqrt{a + b x^{2}} \left (a f - b e\right )}{b^{2}} - \frac{c \sqrt{a + b x^{2}}}{2 a x^{2}} - \frac{\left (2 a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{2 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.302753, size = 101, normalized size = 1.01 \[ \frac{1}{6} \left (\frac{3 (b c-2 a d) \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )}{a^{3/2}}+\frac{3 \log (x) (2 a d-b c)}{a^{3/2}}+\sqrt{a+b x^2} \left (-\frac{4 a f}{b^2}-\frac{3 c}{a x^2}+\frac{6 e+2 f x^2}{b}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.013, size = 127, normalized size = 1.3 \[ -{\frac{c}{2\,a{x}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{bc}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{d\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}+{\frac{e}{b}\sqrt{b{x}^{2}+a}}+{\frac{f{x}^{2}}{3\,b}\sqrt{b{x}^{2}+a}}-{\frac{2\,af}{3\,{b}^{2}}\sqrt{b{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*c*(b*x^2+a)^(1/2)/a/x^2+1/2*c*b/a^(3/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/
x)-d/a^(1/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)+e/b*(b*x^2+a)^(1/2)+1/3*f*x^2
/b*(b*x^2+a)^(1/2)-2/3*f*a/b^2*(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^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.260359, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (b^{3} c - 2 \, a b^{2} d\right )} x^{2} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right ) - 2 \,{\left (2 \, a b f x^{4} - 3 \, b^{2} c + 2 \,{\left (3 \, a b e - 2 \, a^{2} f\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{a}}{12 \, a^{\frac{3}{2}} b^{2} x^{2}}, \frac{3 \,{\left (b^{3} c - 2 \, a b^{2} d\right )} x^{2} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) +{\left (2 \, a b f x^{4} - 3 \, b^{2} c + 2 \,{\left (3 \, a b e - 2 \, a^{2} f\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{-a}}{6 \, \sqrt{-a} a b^{2} x^{2}}\right ] \]

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^3),x, algorithm="fricas")

[Out]

[-1/12*(3*(b^3*c - 2*a*b^2*d)*x^2*log(-((b*x^2 + 2*a)*sqrt(a) - 2*sqrt(b*x^2 + a
)*a)/x^2) - 2*(2*a*b*f*x^4 - 3*b^2*c + 2*(3*a*b*e - 2*a^2*f)*x^2)*sqrt(b*x^2 + a
)*sqrt(a))/(a^(3/2)*b^2*x^2), 1/6*(3*(b^3*c - 2*a*b^2*d)*x^2*arctan(sqrt(-a)/sqr
t(b*x^2 + a)) + (2*a*b*f*x^4 - 3*b^2*c + 2*(3*a*b*e - 2*a^2*f)*x^2)*sqrt(b*x^2 +
a)*sqrt(-a))/(sqrt(-a)*a*b^2*x^2)]

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Sympy [A]  time = 37.2329, size = 138, normalized size = 1.38 \[ e \left (\begin{cases} \frac{x^{2}}{2 \sqrt{a}} & \text{for}\: b = 0 \\\frac{\sqrt{a + b x^{2}}}{b} & \text{otherwise} \end{cases}\right ) + f \left (\begin{cases} - \frac{2 a \sqrt{a + b x^{2}}}{3 b^{2}} + \frac{x^{2} \sqrt{a + b x^{2}}}{3 b} & \text{for}\: b \neq 0 \\\frac{x^{4}}{4 \sqrt{a}} & \text{otherwise} \end{cases}\right ) - \frac{\sqrt{b} c \sqrt{\frac{a}{b x^{2}} + 1}}{2 a x} - \frac{d \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{\sqrt{a}} + \frac{b c \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e*Piecewise((x**2/(2*sqrt(a)), Eq(b, 0)), (sqrt(a + b*x**2)/b, True)) + f*Piecew
ise((-2*a*sqrt(a + b*x**2)/(3*b**2) + x**2*sqrt(a + b*x**2)/(3*b), Ne(b, 0)), (x
**4/(4*sqrt(a)), True)) - sqrt(b)*c*sqrt(a/(b*x**2) + 1)/(2*a*x) - d*asinh(sqrt(
a)/(sqrt(b)*x))/sqrt(a) + b*c*asinh(sqrt(a)/(sqrt(b)*x))/(2*a**(3/2))

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GIAC/XCAS [A]  time = 0.221151, size = 154, normalized size = 1.54 \[ -\frac{\frac{3 \,{\left (b^{2} c - 2 \, a b d\right )} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} + \frac{3 \, \sqrt{b x^{2} + a} b c}{a x^{2}} - \frac{2 \,{\left ({\left (b x^{2} + a\right )}^{\frac{3}{2}} b^{2} f - 3 \, \sqrt{b x^{2} + a} a b^{2} f + 3 \, \sqrt{b x^{2} + a} b^{3} e\right )}}{b^{3}}}{6 \, b} \]

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^3),x, algorithm="giac")

[Out]

-1/6*(3*(b^2*c - 2*a*b*d)*arctan(sqrt(b*x^2 + a)/sqrt(-a))/(sqrt(-a)*a) + 3*sqrt
(b*x^2 + a)*b*c/(a*x^2) - 2*((b*x^2 + a)^(3/2)*b^2*f - 3*sqrt(b*x^2 + a)*a*b^2*f
+ 3*sqrt(b*x^2 + a)*b^3*e)/b^3)/b