3.2664 \(\int \frac{x^m}{\left (a+b x^n\right )^2} \, dx\)

Optimal. Leaf size=40 \[ \frac{x^{m+1} \, _2F_1\left (2,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a^2 (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.0341787, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{x^{m+1} \, _2F_1\left (2,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a^2 (m+1)} \]

Antiderivative was successfully verified.

[In]  Int[x^m/(a + b*x^n)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 4.28041, size = 29, normalized size = 0.72 \[ \frac{x^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 2, \frac{m + 1}{n} \\ \frac{m + n + 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a^{2} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0859673, size = 73, normalized size = 1.82 \[ \frac{x^{m+1} \left (a (m+1)-(m-n+1) \left (a+b x^n\right ) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )\right )}{a^2 (m+1) n \left (a+b x^n\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[x^m/(a + b*x^n)^2,x]

[Out]

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

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Maple [F]  time = 0.097, size = 0, normalized size = 0. \[ \int{\frac{{x}^{m}}{ \left ( a+b{x}^{n} \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m/(a+b*x^n)^2,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -{\left (m - n + 1\right )} \int \frac{x^{m}}{a b n x^{n} + a^{2} n}\,{d x} + \frac{x x^{m}}{a b n x^{n} + a^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{m}}{b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [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/(a+b*x**n)**2,x)

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^m/(b*x^n + a)^2, x)