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

Optimal. Leaf size=71 \[ \frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right )}{16 \sqrt{c}}+\frac{3}{16} a x^2 \sqrt{a+c x^4}+\frac{1}{8} x^2 \left (a+c x^4\right )^{3/2} \]

[Out]

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

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Rubi [A]  time = 0.0791766, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right )}{16 \sqrt{c}}+\frac{3}{16} a x^2 \sqrt{a+c x^4}+\frac{1}{8} x^2 \left (a+c x^4\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[x*(a + c*x^4)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 6.5933, size = 65, normalized size = 0.92 \[ \frac{3 a^{2} \operatorname{atanh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a + c x^{4}}} \right )}}{16 \sqrt{c}} + \frac{3 a x^{2} \sqrt{a + c x^{4}}}{16} + \frac{x^{2} \left (a + c x^{4}\right )^{\frac{3}{2}}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0656864, size = 64, normalized size = 0.9 \[ \frac{1}{16} \left (\frac{3 a^2 \log \left (\sqrt{c} \sqrt{a+c x^4}+c x^2\right )}{\sqrt{c}}+x^2 \sqrt{a+c x^4} \left (5 a+2 c x^4\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x^2*Sqrt[a + c*x^4]*(5*a + 2*c*x^4) + (3*a^2*Log[c*x^2 + Sqrt[c]*Sqrt[a + c*x^4
]])/Sqrt[c])/16

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Maple [A]  time = 0.014, size = 58, normalized size = 0.8 \[{\frac{3\,{a}^{2}}{16}\ln \left ({x}^{2}\sqrt{c}+\sqrt{c{x}^{4}+a} \right ){\frac{1}{\sqrt{c}}}}+{\frac{c{x}^{6}}{8}\sqrt{c{x}^{4}+a}}+{\frac{5\,a{x}^{2}}{16}\sqrt{c{x}^{4}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(c*x^4+a)^(3/2),x)

[Out]

3/16*a^2*ln(x^2*c^(1/2)+(c*x^4+a)^(1/2))/c^(1/2)+1/8*c*x^6*(c*x^4+a)^(1/2)+5/16*
a*x^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,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.263374, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a^{2} \log \left (-2 \, \sqrt{c x^{4} + a} c x^{2} -{\left (2 \, c x^{4} + a\right )} \sqrt{c}\right ) + 2 \,{\left (2 \, c x^{6} + 5 \, a x^{2}\right )} \sqrt{c x^{4} + a} \sqrt{c}}{32 \, \sqrt{c}}, \frac{3 \, a^{2} \arctan \left (\frac{\sqrt{-c} x^{2}}{\sqrt{c x^{4} + a}}\right ) +{\left (2 \, c x^{6} + 5 \, a x^{2}\right )} \sqrt{c x^{4} + a} \sqrt{-c}}{16 \, \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 10.0555, size = 73, normalized size = 1.03 \[ \frac{5 a^{\frac{3}{2}} x^{2} \sqrt{1 + \frac{c x^{4}}{a}}}{16} + \frac{\sqrt{a} c x^{6} \sqrt{1 + \frac{c x^{4}}{a}}}{8} + \frac{3 a^{2} \operatorname{asinh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{16 \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*a**(3/2)*x**2*sqrt(1 + c*x**4/a)/16 + sqrt(a)*c*x**6*sqrt(1 + c*x**4/a)/8 + 3*
a**2*asinh(sqrt(c)*x**2/sqrt(a))/(16*sqrt(c))

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GIAC/XCAS [A]  time = 0.224441, size = 72, normalized size = 1.01 \[ \frac{1}{16} \,{\left (2 \, c x^{4} + 5 \, a\right )} \sqrt{c x^{4} + a} x^{2} - \frac{3 \, a^{2}{\rm ln}\left ({\left | -\sqrt{c} x^{2} + \sqrt{c x^{4} + a} \right |}\right )}{16 \, \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*(2*c*x^4 + 5*a)*sqrt(c*x^4 + a)*x^2 - 3/16*a^2*ln(abs(-sqrt(c)*x^2 + sqrt(c
*x^4 + a)))/sqrt(c)