3.2743 \(\int \frac{x^m}{\left (a+b x^{2+2 m}\right )^{3/2}} \, dx\)

Optimal. Leaf size=29 \[ \frac{x^{m+1}}{a (m+1) \sqrt{a+b x^{2 (m+1)}}} \]

[Out]

x^(1 + m)/(a*(1 + m)*Sqrt[a + b*x^(2*(1 + m))])

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Rubi [A]  time = 0.0321369, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ \frac{x^{m+1}}{a (m+1) \sqrt{a+b x^{2 (m+1)}}} \]

Antiderivative was successfully verified.

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

[Out]

x^(1 + m)/(a*(1 + m)*Sqrt[a + b*x^(2*(1 + m))])

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Rubi in Sympy [A]  time = 3.31619, size = 22, normalized size = 0.76 \[ \frac{x^{m + 1}}{a \sqrt{a + b x^{2 m + 2}} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**(m + 1)/(a*sqrt(a + b*x**(2*m + 2))*(m + 1))

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Mathematica [A]  time = 0.0440921, size = 29, normalized size = 1. \[ \frac{x^{m+1}}{a (m+1) \sqrt{a+b x^{2 m+2}}} \]

Antiderivative was successfully verified.

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

[Out]

x^(1 + m)/(a*(1 + m)*Sqrt[a + b*x^(2 + 2*m)])

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Maple [F]  time = 0.055, size = 0, normalized size = 0. \[ \int{{x}^{m} \left ( a+b{x}^{2+2\,m} \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.22842, size = 63, normalized size = 2.17 \[ \frac{\sqrt{b x^{2} x^{2 \, m} + a} x x^{m}}{{\left (a b m + a b\right )} x^{2} x^{2 \, m} + a^{2} m + a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(b*x^2*x^(2*m) + a)*x*x^m/((a*b*m + a*b)*x^2*x^(2*m) + a^2*m + a^2)

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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**m/(a+b*x**(2+2*m))**(3/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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