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

Optimal. Leaf size=100 \[ -\frac{2 a^{3/2} c^5 \tanh ^{-1}\left (\frac{\sqrt{a}}{x^2 \sqrt{\frac{a}{x^4}+b x^n}}\right )}{n+4}+\frac{2 c^5 x^6 \left (\frac{a}{x^4}+b x^n\right )^{3/2}}{3 (n+4)}+\frac{2 a c^5 x^2 \sqrt{\frac{a}{x^4}+b x^n}}{n+4} \]

[Out]

(2*a*c^5*x^2*Sqrt[a/x^4 + b*x^n])/(4 + n) + (2*c^5*x^6*(a/x^4 + b*x^n)^(3/2))/(3
*(4 + n)) - (2*a^(3/2)*c^5*ArcTanh[Sqrt[a]/(x^2*Sqrt[a/x^4 + b*x^n])])/(4 + n)

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Rubi [A]  time = 0.325684, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{2 a^{3/2} c^5 \tanh ^{-1}\left (\frac{\sqrt{a}}{x^2 \sqrt{\frac{a}{x^4}+b x^n}}\right )}{n+4}+\frac{2 c^5 x^6 \left (\frac{a}{x^4}+b x^n\right )^{3/2}}{3 (n+4)}+\frac{2 a c^5 x^2 \sqrt{\frac{a}{x^4}+b x^n}}{n+4} \]

Antiderivative was successfully verified.

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

[Out]

(2*a*c^5*x^2*Sqrt[a/x^4 + b*x^n])/(4 + n) + (2*c^5*x^6*(a/x^4 + b*x^n)^(3/2))/(3
*(4 + n)) - (2*a^(3/2)*c^5*ArcTanh[Sqrt[a]/(x^2*Sqrt[a/x^4 + b*x^n])])/(4 + n)

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Rubi in Sympy [A]  time = 26.0652, size = 90, normalized size = 0.9 \[ - \frac{2 a^{\frac{3}{2}} c^{5} \operatorname{atanh}{\left (\frac{\sqrt{a}}{x^{2} \sqrt{\frac{a}{x^{4}} + b x^{n}}} \right )}}{n + 4} + \frac{2 a c^{5} x^{2} \sqrt{\frac{a}{x^{4}} + b x^{n}}}{n + 4} + \frac{2 c^{5} x^{6} \left (\frac{a}{x^{4}} + b x^{n}\right )^{\frac{3}{2}}}{3 \left (n + 4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**(3/2)*c**5*atanh(sqrt(a)/(x**2*sqrt(a/x**4 + b*x**n)))/(n + 4) + 2*a*c**5*
x**2*sqrt(a/x**4 + b*x**n)/(n + 4) + 2*c**5*x**6*(a/x**4 + b*x**n)**(3/2)/(3*(n
+ 4))

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Mathematica [A]  time = 0.230669, size = 115, normalized size = 1.15 \[ \frac{2 c^5 x^2 \sqrt{\frac{a}{x^4}+b x^n} \left (-3 a^{3/2} \log \left (\sqrt{a} \sqrt{a+b x^{n+4}}+a\right )+3 a^{3/2} \log \left (x^{\frac{n+4}{2}}\right )+\sqrt{a+b x^{n+4}} \left (4 a+b x^{n+4}\right )\right )}{3 (n+4) \sqrt{a+b x^{n+4}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.069, size = 0, normalized size = 0. \[ \int{c}^{5}{x}^{5} \left ({\frac{a}{{x}^{4}}}+b{x}^{n} \right ) ^{{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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