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

Optimal. Leaf size=69 \[ \frac{3}{4} c x^2 \sqrt{a+c x^4}-\frac{\left (a+c x^4\right )^{3/2}}{2 x^2}+\frac{3}{4} a \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0926946, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{3}{4} c x^2 \sqrt{a+c x^4}-\frac{\left (a+c x^4\right )^{3/2}}{2 x^2}+\frac{3}{4} a \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^4)^(3/2)/x^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 8.17768, size = 63, normalized size = 0.91 \[ \frac{3 a \sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a + c x^{4}}} \right )}}{4} + \frac{3 c x^{2} \sqrt{a + c x^{4}}}{4} - \frac{\left (a + c x^{4}\right )^{\frac{3}{2}}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0922338, size = 58, normalized size = 0.84 \[ \frac{1}{4} \left (\frac{\sqrt{a+c x^4} \left (c x^4-2 a\right )}{x^2}+3 a \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.023, size = 56, normalized size = 0.8 \[ -{\frac{a}{2\,{x}^{2}}\sqrt{c{x}^{4}+a}}+{\frac{c{x}^{2}}{4}\sqrt{c{x}^{4}+a}}+{\frac{3\,a}{4}\sqrt{c}\ln \left ({x}^{2}\sqrt{c}+\sqrt{c{x}^{4}+a} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+a)^(3/2)/x^3,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.267508, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a \sqrt{c} x^{2} \log \left (-2 \, c x^{4} - 2 \, \sqrt{c x^{4} + a} \sqrt{c} x^{2} - a\right ) + 2 \, \sqrt{c x^{4} + a}{\left (c x^{4} - 2 \, a\right )}}{8 \, x^{2}}, \frac{3 \, a \sqrt{-c} x^{2} \arctan \left (\frac{c x^{2}}{\sqrt{c x^{4} + a} \sqrt{-c}}\right ) + \sqrt{c x^{4} + a}{\left (c x^{4} - 2 \, a\right )}}{4 \, x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*(3*a*sqrt(c)*x^2*log(-2*c*x^4 - 2*sqrt(c*x^4 + a)*sqrt(c)*x^2 - a) + 2*sqrt
(c*x^4 + a)*(c*x^4 - 2*a))/x^2, 1/4*(3*a*sqrt(-c)*x^2*arctan(c*x^2/(sqrt(c*x^4 +
a)*sqrt(-c))) + sqrt(c*x^4 + a)*(c*x^4 - 2*a))/x^2]

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Sympy [A]  time = 8.37831, size = 95, normalized size = 1.38 \[ - \frac{a^{\frac{3}{2}}}{2 x^{2} \sqrt{1 + \frac{c x^{4}}{a}}} - \frac{\sqrt{a} c x^{2}}{4 \sqrt{1 + \frac{c x^{4}}{a}}} + \frac{3 a \sqrt{c} \operatorname{asinh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{4} + \frac{c^{2} x^{6}}{4 \sqrt{a} \sqrt{1 + \frac{c x^{4}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**(3/2)/(2*x**2*sqrt(1 + c*x**4/a)) - sqrt(a)*c*x**2/(4*sqrt(1 + c*x**4/a)) +
3*a*sqrt(c)*asinh(sqrt(c)*x**2/sqrt(a))/4 + c**2*x**6/(4*sqrt(a)*sqrt(1 + c*x**4
/a))

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GIAC/XCAS [A]  time = 0.243277, size = 72, normalized size = 1.04 \[ \frac{1}{4} \, \sqrt{c x^{4} + a} c x^{2} - \frac{3 \, a c \arctan \left (\frac{\sqrt{c + \frac{a}{x^{4}}}}{\sqrt{-c}}\right )}{4 \, \sqrt{-c}} - \frac{1}{2} \, a \sqrt{c + \frac{a}{x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*sqrt(c*x^4 + a)*c*x^2 - 3/4*a*c*arctan(sqrt(c + a/x^4)/sqrt(-c))/sqrt(-c) -
1/2*a*sqrt(c + a/x^4)