3.222 \(\int \frac{x^5}{\sqrt{a+b x^3+c x^6}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.114341, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{\sqrt{a+b x^3+c x^6}}{3 c}-\frac{b \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{6 c^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^5/Sqrt[a + b*x^3 + c*x^6],x]

[Out]

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

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Rubi in Sympy [A]  time = 12.9748, size = 58, normalized size = 0.85 \[ - \frac{b \operatorname{atanh}{\left (\frac{b + 2 c x^{3}}{2 \sqrt{c} \sqrt{a + b x^{3} + c x^{6}}} \right )}}{6 c^{\frac{3}{2}}} + \frac{\sqrt{a + b x^{3} + c x^{6}}}{3 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(c*x**6+b*x**3+a)**(1/2),x)

[Out]

-b*atanh((b + 2*c*x**3)/(2*sqrt(c)*sqrt(a + b*x**3 + c*x**6)))/(6*c**(3/2)) + sq
rt(a + b*x**3 + c*x**6)/(3*c)

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Mathematica [A]  time = 0.0381865, size = 66, normalized size = 0.97 \[ \frac{\sqrt{a+b x^3+c x^6}}{3 c}-\frac{b \log \left (2 \sqrt{c} \sqrt{a+b x^3+c x^6}+b+2 c x^3\right )}{6 c^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/Sqrt[a + b*x^3 + c*x^6],x]

[Out]

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

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Maple [F]  time = 0.032, size = 0, normalized size = 0. \[ \int{{x}^{5}{\frac{1}{\sqrt{c{x}^{6}+b{x}^{3}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(c*x^6+b*x^3+a)^(1/2),x)

[Out]

int(x^5/(c*x^6+b*x^3+a)^(1/2),x)

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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(x^5/sqrt(c*x^6 + b*x^3 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.290517, size = 1, normalized size = 0.01 \[ \left [\frac{b \log \left (4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (2 \, c^{2} x^{3} + b c\right )} -{\left (8 \, c^{2} x^{6} + 8 \, b c x^{3} + b^{2} + 4 \, a c\right )} \sqrt{c}\right ) + 4 \, \sqrt{c x^{6} + b x^{3} + a} \sqrt{c}}{12 \, c^{\frac{3}{2}}}, -\frac{b \arctan \left (\frac{{\left (2 \, c x^{3} + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{6} + b x^{3} + a} c}\right ) - 2 \, \sqrt{c x^{6} + b x^{3} + a} \sqrt{-c}}{6 \, \sqrt{-c} c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/sqrt(c*x^6 + b*x^3 + a),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5}}{\sqrt{a + b x^{3} + c x^{6}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(c*x**6+b*x**3+a)**(1/2),x)

[Out]

Integral(x**5/sqrt(a + b*x**3 + c*x**6), x)

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GIAC/XCAS [A]  time = 0.291017, size = 82, normalized size = 1.21 \[ \frac{b{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x^{3} - \sqrt{c x^{6} + b x^{3} + a}\right )} \sqrt{c} - b \right |}\right )}{6 \, c^{\frac{3}{2}}} + \frac{\sqrt{c x^{6} + b x^{3} + a}}{3 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/sqrt(c*x^6 + b*x^3 + a),x, algorithm="giac")

[Out]

1/6*b*ln(abs(-2*(sqrt(c)*x^3 - sqrt(c*x^6 + b*x^3 + a))*sqrt(c) - b))/c^(3/2) +
1/3*sqrt(c*x^6 + b*x^3 + a)/c