3.2670 \(\int \frac{x^m}{\left (a+b x^n\right )^{5/2}} \, dx\)

Optimal. Leaf size=55 \[ \frac{x^{m+1} \, _2F_1\left (1,\frac{m+1}{n}-\frac{3}{2};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a (m+1) \left (a+b x^n\right )^{3/2}} \]

[Out]

(x^(1 + m)*Hypergeometric2F1[1, -3/2 + (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/
(a*(1 + m)*(a + b*x^n)^(3/2))

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Rubi [A]  time = 0.0686236, antiderivative size = 67, normalized size of antiderivative = 1.22, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{x^{m+1} \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{5}{2},\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a^2 (m+1) \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

[In]  Int[x^m/(a + b*x^n)^(5/2),x]

[Out]

(x^(1 + m)*Sqrt[1 + (b*x^n)/a]*Hypergeometric2F1[5/2, (1 + m)/n, (1 + m + n)/n,
-((b*x^n)/a)])/(a^2*(1 + m)*Sqrt[a + b*x^n])

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Rubi in Sympy [A]  time = 8.49707, size = 53, normalized size = 0.96 \[ \frac{x^{m + 1} \sqrt{a + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} \frac{5}{2}, \frac{m + 1}{n} \\ \frac{m + n + 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a^{3} \sqrt{1 + \frac{b x^{n}}{a}} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**(m + 1)*sqrt(a + b*x**n)*hyper((5/2, (m + 1)/n), ((m + n + 1)/n,), -b*x**n/a)
/(a**3*sqrt(1 + b*x**n/a)*(m + 1))

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Mathematica [B]  time = 0.269432, size = 129, normalized size = 2.35 \[ \frac{x^{m+1} \left (\left (4 m^2-8 m (n-1)+3 n^2-8 n+4\right ) \left (a+b x^n\right ) \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )+2 (m+1) \left (a n-(2 m-3 n+2) \left (a+b x^n\right )\right )\right )}{3 a^2 (m+1) n^2 \left (a+b x^n\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(x^(1 + m)*(2*(1 + m)*(a*n - (2 + 2*m - 3*n)*(a + b*x^n)) + (4 + 4*m^2 - 8*m*(-1
+ n) - 8*n + 3*n^2)*(a + b*x^n)*Sqrt[1 + (b*x^n)/a]*Hypergeometric2F1[1/2, (1 +
m)/n, (1 + m + n)/n, -((b*x^n)/a)]))/(3*a^2*(1 + m)*n^2*(a + b*x^n)^(3/2))

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Maple [F]  time = 0.051, size = 0, normalized size = 0. \[ \int{{x}^{m} \left ( a+b{x}^{n} \right ) ^{-{\frac{5}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m/(a+b*x^n)^(5/2),x)

[Out]

int(x^m/(a+b*x^n)^(5/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^m/(b*x^n + a)^(5/2), 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(x^m/(b*x^n + a)^(5/2),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(x**m/(a+b*x**n)**(5/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^m/(b*x^n + a)^(5/2), x)