3.790 \(\int \frac{\left (a+c x^4\right )^{3/2}}{x^7} \, dx\)

Optimal. Leaf size=68 \[ \frac{1}{2} c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right )-\frac{\left (a+c x^4\right )^{3/2}}{6 x^6}-\frac{c \sqrt{a+c x^4}}{2 x^2} \]

[Out]

-(c*Sqrt[a + c*x^4])/(2*x^2) - (a + c*x^4)^(3/2)/(6*x^6) + (c^(3/2)*ArcTanh[(Sqr
t[c]*x^2)/Sqrt[a + c*x^4]])/2

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Rubi [A]  time = 0.100201, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{1}{2} c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right )-\frac{\left (a+c x^4\right )^{3/2}}{6 x^6}-\frac{c \sqrt{a+c x^4}}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^4)^(3/2)/x^7,x]

[Out]

-(c*Sqrt[a + c*x^4])/(2*x^2) - (a + c*x^4)^(3/2)/(6*x^6) + (c^(3/2)*ArcTanh[(Sqr
t[c]*x^2)/Sqrt[a + c*x^4]])/2

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Rubi in Sympy [A]  time = 9.87035, size = 58, normalized size = 0.85 \[ \frac{c^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a + c x^{4}}} \right )}}{2} - \frac{c \sqrt{a + c x^{4}}}{2 x^{2}} - \frac{\left (a + c x^{4}\right )^{\frac{3}{2}}}{6 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+a)**(3/2)/x**7,x)

[Out]

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

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Mathematica [A]  time = 0.0707399, size = 57, normalized size = 0.84 \[ \frac{1}{2} c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right )-\frac{\sqrt{a+c x^4} \left (a+4 c x^4\right )}{6 x^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + c*x^4)^(3/2)/x^7,x]

[Out]

-(Sqrt[a + c*x^4]*(a + 4*c*x^4))/(6*x^6) + (c^(3/2)*ArcTanh[(Sqrt[c]*x^2)/Sqrt[a
+ c*x^4]])/2

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Maple [A]  time = 0.022, size = 55, normalized size = 0.8 \[{\frac{1}{2}{c}^{{\frac{3}{2}}}\ln \left ({x}^{2}\sqrt{c}+\sqrt{c{x}^{4}+a} \right ) }-{\frac{a}{6\,{x}^{6}}\sqrt{c{x}^{4}+a}}-{\frac{2\,c}{3\,{x}^{2}}\sqrt{c{x}^{4}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+a)^(3/2)/x^7,x)

[Out]

1/2*c^(3/2)*ln(x^2*c^(1/2)+(c*x^4+a)^(1/2))-1/6*a/x^6*(c*x^4+a)^(1/2)-2/3*c*(c*x
^4+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 + a)^(3/2)/x^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.276377, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, c^{\frac{3}{2}} x^{6} \log \left (-2 \, c x^{4} - 2 \, \sqrt{c x^{4} + a} \sqrt{c} x^{2} - a\right ) - 2 \,{\left (4 \, c x^{4} + a\right )} \sqrt{c x^{4} + a}}{12 \, x^{6}}, \frac{3 \, \sqrt{-c} c x^{6} \arctan \left (\frac{c x^{2}}{\sqrt{c x^{4} + a} \sqrt{-c}}\right ) -{\left (4 \, c x^{4} + a\right )} \sqrt{c x^{4} + a}}{6 \, x^{6}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + a)^(3/2)/x^7,x, algorithm="fricas")

[Out]

[1/12*(3*c^(3/2)*x^6*log(-2*c*x^4 - 2*sqrt(c*x^4 + a)*sqrt(c)*x^2 - a) - 2*(4*c*
x^4 + a)*sqrt(c*x^4 + a))/x^6, 1/6*(3*sqrt(-c)*c*x^6*arctan(c*x^2/(sqrt(c*x^4 +
a)*sqrt(-c))) - (4*c*x^4 + a)*sqrt(c*x^4 + a))/x^6]

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Sympy [A]  time = 8.7653, size = 80, normalized size = 1.18 \[ - \frac{a \sqrt{c} \sqrt{\frac{a}{c x^{4}} + 1}}{6 x^{4}} - \frac{2 c^{\frac{3}{2}} \sqrt{\frac{a}{c x^{4}} + 1}}{3} - \frac{c^{\frac{3}{2}} \log{\left (\frac{a}{c x^{4}} \right )}}{4} + \frac{c^{\frac{3}{2}} \log{\left (\sqrt{\frac{a}{c x^{4}} + 1} + 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+a)**(3/2)/x**7,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.219101, size = 68, normalized size = 1. \[ -\frac{c^{2} \arctan \left (\frac{\sqrt{c + \frac{a}{x^{4}}}}{\sqrt{-c}}\right )}{2 \, \sqrt{-c}} - \frac{1}{6} \,{\left (c + \frac{a}{x^{4}}\right )}^{\frac{3}{2}} - \frac{1}{2} \, \sqrt{c + \frac{a}{x^{4}}} c \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + a)^(3/2)/x^7,x, algorithm="giac")

[Out]

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