3.448 \(\int \sqrt{x^{2 (-1+n)} \left (a+b x^n\right )} \, dx\)

Optimal. Leaf size=44 \[ \frac{2 x^{3 (1-n)} \left (a x^{-2 (1-n)}+b x^{3 n-2}\right )^{3/2}}{3 b n} \]

[Out]

(2*x^(3*(1 - n))*(a/x^(2*(1 - n)) + b*x^(-2 + 3*n))^(3/2))/(3*b*n)

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Rubi [A]  time = 0.0392088, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{2 x^{3 (1-n)} \left (a x^{-2 (1-n)}+b x^{3 n-2}\right )^{3/2}}{3 b n} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x^(2*(-1 + n))*(a + b*x^n)],x]

[Out]

(2*x^(3*(1 - n))*(a/x^(2*(1 - n)) + b*x^(-2 + 3*n))^(3/2))/(3*b*n)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**(-2+2*n)*(a+b*x**n))**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0608137, size = 36, normalized size = 0.82 \[ \frac{2 x^{3-3 n} \left (x^{2 n-2} \left (a+b x^n\right )\right )^{3/2}}{3 b n} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[x^(2*(-1 + n))*(a + b*x^n)],x]

[Out]

(2*x^(3 - 3*n)*(x^(-2 + 2*n)*(a + b*x^n))^(3/2))/(3*b*n)

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Maple [A]  time = 0.051, size = 40, normalized size = 0.9 \[{\frac{ \left ( 2\,a+2\,b{x}^{n} \right ) x}{3\,b{x}^{n}n}\sqrt{{\frac{ \left ({x}^{n} \right ) ^{2} \left ( a+b{x}^{n} \right ) }{{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^(-2+2*n)*(a+b*x^n))^(1/2),x)

[Out]

2/3*(1/x^2*(x^n)^2*(a+b*x^n))^(1/2)*(a+b*x^n)/(x^n)*x/b/n

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Maxima [A]  time = 1.40278, size = 23, normalized size = 0.52 \[ \frac{2 \,{\left (b x^{n} + a\right )}^{\frac{3}{2}}}{3 \, b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x^n + a)*x^(2*n - 2)),x, algorithm="maxima")

[Out]

2/3*(b*x^n + a)^(3/2)/(b*n)

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Fricas [A]  time = 0.24123, size = 59, normalized size = 1.34 \[ \frac{2 \,{\left (b x x^{n} + a x\right )} \sqrt{\frac{b x^{3 \, n} + a x^{2 \, n}}{x^{2}}}}{3 \, b n x^{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x^n + a)*x^(2*n - 2)),x, algorithm="fricas")

[Out]

2/3*(b*x*x^n + a*x)*sqrt((b*x^(3*n) + a*x^(2*n))/x^2)/(b*n*x^n)

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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((x**(-2+2*n)*(a+b*x**n))**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x^n + a)*x^(2*n - 2)),x, algorithm="giac")

[Out]

integrate(sqrt((b*x^n + a)*x^(2*n - 2)), x)