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

Optimal. Leaf size=98 \[ -\frac{2 a^{3/2} c^2 \tanh ^{-1}\left (\frac{\sqrt{a}}{x \sqrt{\frac{a}{x^2}+b x^n}}\right )}{n+2}+\frac{2 a c^2 x \sqrt{\frac{a}{x^2}+b x^n}}{n+2}+\frac{2 c^2 x^3 \left (\frac{a}{x^2}+b x^n\right )^{3/2}}{3 (n+2)} \]

[Out]

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

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Rubi [A]  time = 0.268372, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{2 a^{3/2} c^2 \tanh ^{-1}\left (\frac{\sqrt{a}}{x \sqrt{\frac{a}{x^2}+b x^n}}\right )}{n+2}+\frac{2 a c^2 x \sqrt{\frac{a}{x^2}+b x^n}}{n+2}+\frac{2 c^2 x^3 \left (\frac{a}{x^2}+b x^n\right )^{3/2}}{3 (n+2)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 21.755, size = 87, normalized size = 0.89 \[ - \frac{2 a^{\frac{3}{2}} c^{2} \operatorname{atanh}{\left (\frac{\sqrt{a}}{x \sqrt{\frac{a}{x^{2}} + b x^{n}}} \right )}}{n + 2} + \frac{2 a c^{2} x \sqrt{\frac{a}{x^{2}} + b x^{n}}}{n + 2} + \frac{2 c^{2} x^{3} \left (\frac{a}{x^{2}} + b x^{n}\right )^{\frac{3}{2}}}{3 \left (n + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**(3/2)*c**2*atanh(sqrt(a)/(x*sqrt(a/x**2 + b*x**n)))/(n + 2) + 2*a*c**2*x*s
qrt(a/x**2 + b*x**n)/(n + 2) + 2*c**2*x**3*(a/x**2 + b*x**n)**(3/2)/(3*(n + 2))

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Mathematica [A]  time = 0.225331, size = 113, normalized size = 1.15 \[ \frac{2 c^2 x \sqrt{\frac{a}{x^2}+b x^n} \left (-3 a^{3/2} \log \left (\sqrt{a} \sqrt{a+b x^{n+2}}+a\right )+3 a^{3/2} \log \left (x^{\frac{n+2}{2}}\right )+\sqrt{a+b x^{n+2}} \left (4 a+b x^{n+2}\right )\right )}{3 (n+2) \sqrt{a+b x^{n+2}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.052, size = 0, normalized size = 0. \[ \int{c}^{2}{x}^{2} \left ({\frac{a}{{x}^{2}}}+b{x}^{n} \right ) ^{{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c^2*integrate((b*x^n + a/x^2)^(3/2)*x^2, 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^2)^(3/2)*c^2*x^2,x, algorithm="fricas")

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c**2*(Integral(a*sqrt(a/x**2 + b*x**n), x) + Integral(b*x**2*x**n*sqrt(a/x**2 +
b*x**n), x))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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