3.942 \(\int \frac{\left (a+b x^2+c x^4\right )^{3/2}}{x^5} \, dx\)

Optimal. Leaf size=151 \[ -\frac{3 \left (4 a c+b^2\right ) \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{16 \sqrt{a}}-\frac{\left (a+b x^2+c x^4\right )^{3/2}}{4 x^4}-\frac{3 \left (b-2 c x^2\right ) \sqrt{a+b x^2+c x^4}}{8 x^2}+\frac{3}{4} b \sqrt{c} \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right ) \]

[Out]

(-3*(b - 2*c*x^2)*Sqrt[a + b*x^2 + c*x^4])/(8*x^2) - (a + b*x^2 + c*x^4)^(3/2)/(
4*x^4) - (3*(b^2 + 4*a*c)*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^
4])])/(16*Sqrt[a]) + (3*b*Sqrt[c]*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^
2 + c*x^4])])/4

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Rubi [A]  time = 0.428448, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ -\frac{3 \left (4 a c+b^2\right ) \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{16 \sqrt{a}}-\frac{\left (a+b x^2+c x^4\right )^{3/2}}{4 x^4}-\frac{3 \left (b-2 c x^2\right ) \sqrt{a+b x^2+c x^4}}{8 x^2}+\frac{3}{4} b \sqrt{c} \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2 + c*x^4)^(3/2)/x^5,x]

[Out]

(-3*(b - 2*c*x^2)*Sqrt[a + b*x^2 + c*x^4])/(8*x^2) - (a + b*x^2 + c*x^4)^(3/2)/(
4*x^4) - (3*(b^2 + 4*a*c)*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^
4])])/(16*Sqrt[a]) + (3*b*Sqrt[c]*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^
2 + c*x^4])])/4

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Rubi in Sympy [A]  time = 35.9646, size = 141, normalized size = 0.93 \[ \frac{3 b \sqrt{c} \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{2 \sqrt{c} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{4} - \frac{3 \left (b - 2 c x^{2}\right ) \sqrt{a + b x^{2} + c x^{4}}}{8 x^{2}} - \frac{\left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{4 x^{4}} - \frac{3 \left (4 a c + b^{2}\right ) \operatorname{atanh}{\left (\frac{2 a + b x^{2}}{2 \sqrt{a} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{16 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+b*x**2+a)**(3/2)/x**5,x)

[Out]

3*b*sqrt(c)*atanh((b + 2*c*x**2)/(2*sqrt(c)*sqrt(a + b*x**2 + c*x**4)))/4 - 3*(b
- 2*c*x**2)*sqrt(a + b*x**2 + c*x**4)/(8*x**2) - (a + b*x**2 + c*x**4)**(3/2)/(
4*x**4) - 3*(4*a*c + b**2)*atanh((2*a + b*x**2)/(2*sqrt(a)*sqrt(a + b*x**2 + c*x
**4)))/(16*sqrt(a))

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Mathematica [A]  time = 0.631744, size = 153, normalized size = 1.01 \[ \frac{3 \left (\log \left (x^2\right ) \left (4 a c+b^2\right )-\left (4 a c+b^2\right ) \log \left (2 \sqrt{a} \sqrt{a+x^2 \left (b+c x^2\right )}+2 a+b x^2\right )+4 \sqrt{a} b \sqrt{c} \log \left (2 \sqrt{c} \sqrt{a+x^2 \left (b+c x^2\right )}+b+2 c x^2\right )\right )}{16 \sqrt{a}}+\sqrt{a+b x^2+c x^4} \left (-\frac{a}{4 x^4}-\frac{5 b}{8 x^2}+\frac{c}{2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2 + c*x^4)^(3/2)/x^5,x]

[Out]

(c/2 - a/(4*x^4) - (5*b)/(8*x^2))*Sqrt[a + b*x^2 + c*x^4] + (3*((b^2 + 4*a*c)*Lo
g[x^2] - (b^2 + 4*a*c)*Log[2*a + b*x^2 + 2*Sqrt[a]*Sqrt[a + x^2*(b + c*x^2)]] +
4*Sqrt[a]*b*Sqrt[c]*Log[b + 2*c*x^2 + 2*Sqrt[c]*Sqrt[a + x^2*(b + c*x^2)]]))/(16
*Sqrt[a])

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Maple [A]  time = 0.025, size = 174, normalized size = 1.2 \[{\frac{c}{2}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{3\,b}{4}\sqrt{c}\ln \left ({1 \left ({\frac{b}{2}}+c{x}^{2} \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) }-{\frac{a}{4\,{x}^{4}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{5\,b}{8\,{x}^{2}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{3\,{b}^{2}}{16}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}-{\frac{3\,c}{4}\sqrt{a}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2+a)^(3/2)/x^5,x)

[Out]

1/2*c*(c*x^4+b*x^2+a)^(1/2)+3/4*c^(1/2)*b*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+
a)^(1/2))-1/4*a/x^4*(c*x^4+b*x^2+a)^(1/2)-5/8*b/x^2*(c*x^4+b*x^2+a)^(1/2)-3/16/a
^(1/2)*b^2*ln((2*a+b*x^2+2*a^(1/2)*(c*x^4+b*x^2+a)^(1/2))/x^2)-3/4*a^(1/2)*c*ln(
(2*a+b*x^2+2*a^(1/2)*(c*x^4+b*x^2+a)^(1/2))/x^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((c*x^4 + b*x^2 + a)^(3/2)/x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.354103, size = 1, normalized size = 0.01 \[ \left [\frac{12 \, \sqrt{a} b \sqrt{c} x^{4} \log \left (-8 \, c^{2} x^{4} - 8 \, b c x^{2} - b^{2} - 4 \, \sqrt{c x^{4} + b x^{2} + a}{\left (2 \, c x^{2} + b\right )} \sqrt{c} - 4 \, a c\right ) + 3 \,{\left (b^{2} + 4 \, a c\right )} x^{4} \log \left (\frac{4 \, \sqrt{c x^{4} + b x^{2} + a}{\left (a b x^{2} + 2 \, a^{2}\right )} -{\left ({\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a b x^{2} + 8 \, a^{2}\right )} \sqrt{a}}{x^{4}}\right ) + 4 \,{\left (4 \, c x^{4} - 5 \, b x^{2} - 2 \, a\right )} \sqrt{c x^{4} + b x^{2} + a} \sqrt{a}}{32 \, \sqrt{a} x^{4}}, \frac{24 \, \sqrt{a} b \sqrt{-c} x^{4} \arctan \left (\frac{2 \, c x^{2} + b}{2 \, \sqrt{c x^{4} + b x^{2} + a} \sqrt{-c}}\right ) + 3 \,{\left (b^{2} + 4 \, a c\right )} x^{4} \log \left (\frac{4 \, \sqrt{c x^{4} + b x^{2} + a}{\left (a b x^{2} + 2 \, a^{2}\right )} -{\left ({\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a b x^{2} + 8 \, a^{2}\right )} \sqrt{a}}{x^{4}}\right ) + 4 \,{\left (4 \, c x^{4} - 5 \, b x^{2} - 2 \, a\right )} \sqrt{c x^{4} + b x^{2} + a} \sqrt{a}}{32 \, \sqrt{a} x^{4}}, \frac{6 \, \sqrt{-a} b \sqrt{c} x^{4} \log \left (-8 \, c^{2} x^{4} - 8 \, b c x^{2} - b^{2} - 4 \, \sqrt{c x^{4} + b x^{2} + a}{\left (2 \, c x^{2} + b\right )} \sqrt{c} - 4 \, a c\right ) - 3 \,{\left (b^{2} + 4 \, a c\right )} x^{4} \arctan \left (\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{-a}}{2 \, \sqrt{c x^{4} + b x^{2} + a} a}\right ) + 2 \,{\left (4 \, c x^{4} - 5 \, b x^{2} - 2 \, a\right )} \sqrt{c x^{4} + b x^{2} + a} \sqrt{-a}}{16 \, \sqrt{-a} x^{4}}, \frac{12 \, \sqrt{-a} b \sqrt{-c} x^{4} \arctan \left (\frac{2 \, c x^{2} + b}{2 \, \sqrt{c x^{4} + b x^{2} + a} \sqrt{-c}}\right ) - 3 \,{\left (b^{2} + 4 \, a c\right )} x^{4} \arctan \left (\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{-a}}{2 \, \sqrt{c x^{4} + b x^{2} + a} a}\right ) + 2 \,{\left (4 \, c x^{4} - 5 \, b x^{2} - 2 \, a\right )} \sqrt{c x^{4} + b x^{2} + a} \sqrt{-a}}{16 \, \sqrt{-a} x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^(3/2)/x^5,x, algorithm="fricas")

[Out]

[1/32*(12*sqrt(a)*b*sqrt(c)*x^4*log(-8*c^2*x^4 - 8*b*c*x^2 - b^2 - 4*sqrt(c*x^4
+ b*x^2 + a)*(2*c*x^2 + b)*sqrt(c) - 4*a*c) + 3*(b^2 + 4*a*c)*x^4*log((4*sqrt(c*
x^4 + b*x^2 + a)*(a*b*x^2 + 2*a^2) - ((b^2 + 4*a*c)*x^4 + 8*a*b*x^2 + 8*a^2)*sqr
t(a))/x^4) + 4*(4*c*x^4 - 5*b*x^2 - 2*a)*sqrt(c*x^4 + b*x^2 + a)*sqrt(a))/(sqrt(
a)*x^4), 1/32*(24*sqrt(a)*b*sqrt(-c)*x^4*arctan(1/2*(2*c*x^2 + b)/(sqrt(c*x^4 +
b*x^2 + a)*sqrt(-c))) + 3*(b^2 + 4*a*c)*x^4*log((4*sqrt(c*x^4 + b*x^2 + a)*(a*b*
x^2 + 2*a^2) - ((b^2 + 4*a*c)*x^4 + 8*a*b*x^2 + 8*a^2)*sqrt(a))/x^4) + 4*(4*c*x^
4 - 5*b*x^2 - 2*a)*sqrt(c*x^4 + b*x^2 + a)*sqrt(a))/(sqrt(a)*x^4), 1/16*(6*sqrt(
-a)*b*sqrt(c)*x^4*log(-8*c^2*x^4 - 8*b*c*x^2 - b^2 - 4*sqrt(c*x^4 + b*x^2 + a)*(
2*c*x^2 + b)*sqrt(c) - 4*a*c) - 3*(b^2 + 4*a*c)*x^4*arctan(1/2*(b*x^2 + 2*a)*sqr
t(-a)/(sqrt(c*x^4 + b*x^2 + a)*a)) + 2*(4*c*x^4 - 5*b*x^2 - 2*a)*sqrt(c*x^4 + b*
x^2 + a)*sqrt(-a))/(sqrt(-a)*x^4), 1/16*(12*sqrt(-a)*b*sqrt(-c)*x^4*arctan(1/2*(
2*c*x^2 + b)/(sqrt(c*x^4 + b*x^2 + a)*sqrt(-c))) - 3*(b^2 + 4*a*c)*x^4*arctan(1/
2*(b*x^2 + 2*a)*sqrt(-a)/(sqrt(c*x^4 + b*x^2 + a)*a)) + 2*(4*c*x^4 - 5*b*x^2 - 2
*a)*sqrt(c*x^4 + b*x^2 + a)*sqrt(-a))/(sqrt(-a)*x^4)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{x^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+b*x**2+a)**(3/2)/x**5,x)

[Out]

Integral((a + b*x**2 + c*x**4)**(3/2)/x**5, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{4} + b x^{2} + a\right )}^{\frac{3}{2}}}{x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^(3/2)/x^5,x, algorithm="giac")

[Out]

integrate((c*x^4 + b*x^2 + a)^(3/2)/x^5, x)