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

Optimal. Leaf size=103 \[ \frac{\sqrt{a+b x^2} \left (a^2 f-a b e+b^2 d\right )}{b^3}+\frac{\left (a+b x^2\right )^{3/2} (b e-2 a f)}{3 b^3}+\frac{f \left (a+b x^2\right )^{5/2}}{5 b^3}-\frac{c \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{\sqrt{a}} \]

[Out]

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

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Rubi [A]  time = 0.262866, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{\sqrt{a+b x^2} \left (a^2 f-a b e+b^2 d\right )}{b^3}+\frac{\left (a+b x^2\right )^{3/2} (b e-2 a f)}{3 b^3}+\frac{f \left (a+b x^2\right )^{5/2}}{5 b^3}-\frac{c \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 49.3283, size = 92, normalized size = 0.89 \[ \frac{f \left (a + b x^{2}\right )^{\frac{5}{2}}}{5 b^{3}} - \frac{\left (a + b x^{2}\right )^{\frac{3}{2}} \left (2 a f - b e\right )}{3 b^{3}} + \frac{\sqrt{a + b x^{2}} \left (a^{2} f - a b e + b^{2} d\right )}{b^{3}} - \frac{c \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{\sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.25925, size = 97, normalized size = 0.94 \[ \frac{\sqrt{a+b x^2} \left (8 a^2 f-2 a b \left (5 e+2 f x^2\right )+b^2 \left (15 d+5 e x^2+3 f x^4\right )\right )}{15 b^3}-\frac{c \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )}{\sqrt{a}}+\frac{c \log (x)}{\sqrt{a}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.013, size = 134, normalized size = 1.3 \[{\frac{d}{b}\sqrt{b{x}^{2}+a}}-{c\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}+{\frac{e{x}^{2}}{3\,b}\sqrt{b{x}^{2}+a}}-{\frac{2\,ae}{3\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{f{x}^{4}}{5\,b}\sqrt{b{x}^{2}+a}}-{\frac{4\,af{x}^{2}}{15\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{8\,{a}^{2}f}{15\,{b}^{3}}\sqrt{b{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.249379, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, b^{3} c \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right ) + 2 \,{\left (3 \, b^{2} f x^{4} + 15 \, b^{2} d - 10 \, a b e + 8 \, a^{2} f +{\left (5 \, b^{2} e - 4 \, a b f\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{a}}{30 \, \sqrt{a} b^{3}}, -\frac{15 \, b^{3} c \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) -{\left (3 \, b^{2} f x^{4} + 15 \, b^{2} d - 10 \, a b e + 8 \, a^{2} f +{\left (5 \, b^{2} e - 4 \, a b f\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{-a}}{15 \, \sqrt{-a} b^{3}}\right ] \]

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),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 23.5296, size = 192, normalized size = 1.86 \[ c \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{1}{\sqrt{- \frac{1}{a}} \sqrt{a + b x^{2}}} \right )}}{a \sqrt{- \frac{1}{a}}} & \text{for}\: - \frac{1}{a} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{1}{\sqrt{a + b x^{2}} \sqrt{\frac{1}{a}}} \right )}}{a \sqrt{\frac{1}{a}}} & \text{for}\: - \frac{1}{a} < 0 \wedge \frac{1}{a} < \frac{1}{a + b x^{2}} \\- \frac{\operatorname{atanh}{\left (\frac{1}{\sqrt{a + b x^{2}} \sqrt{\frac{1}{a}}} \right )}}{a \sqrt{\frac{1}{a}}} & \text{for}\: \frac{1}{a} > \frac{1}{a + b x^{2}} \wedge - \frac{1}{a} < 0 \end{cases}\right ) + \frac{f \left (a + b x^{2}\right )^{\frac{5}{2}}}{5 b^{3}} - \frac{\left (a + b x^{2}\right )^{\frac{3}{2}} \left (2 a f - b e\right )}{3 b^{3}} + \frac{\sqrt{a + b x^{2}} \left (a^{2} f - a b e + b^{2} d\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*Piecewise((atan(1/(sqrt(-1/a)*sqrt(a + b*x**2)))/(a*sqrt(-1/a)), -1/a > 0), (-
acoth(1/(sqrt(a + b*x**2)*sqrt(1/a)))/(a*sqrt(1/a)), (-1/a < 0) & (1/a < 1/(a +
b*x**2))), (-atanh(1/(sqrt(a + b*x**2)*sqrt(1/a)))/(a*sqrt(1/a)), (-1/a < 0) & (
1/a > 1/(a + b*x**2)))) + f*(a + b*x**2)**(5/2)/(5*b**3) - (a + b*x**2)**(3/2)*(
2*a*f - b*e)/(3*b**3) + sqrt(a + b*x**2)*(a**2*f - a*b*e + b**2*d)/b**3

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GIAC/XCAS [A]  time = 0.220671, size = 171, normalized size = 1.66 \[ \frac{c \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + \frac{15 \, \sqrt{b x^{2} + a} b^{14} d + 3 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} b^{12} f - 10 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a b^{12} f + 15 \, \sqrt{b x^{2} + a} a^{2} b^{12} f + 5 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} b^{13} e - 15 \, \sqrt{b x^{2} + a} a b^{13} e}{15 \, b^{15}} \]

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),x, algorithm="giac")

[Out]

c*arctan(sqrt(b*x^2 + a)/sqrt(-a))/sqrt(-a) + 1/15*(15*sqrt(b*x^2 + a)*b^14*d +
3*(b*x^2 + a)^(5/2)*b^12*f - 10*(b*x^2 + a)^(3/2)*a*b^12*f + 15*sqrt(b*x^2 + a)*
a^2*b^12*f + 5*(b*x^2 + a)^(3/2)*b^13*e - 15*sqrt(b*x^2 + a)*a*b^13*e)/b^15