3.2665 \(\int \frac{x^m}{\left (a+b x^n\right )^3} \, dx\)

Optimal. Leaf size=40 \[ \frac{x^{m+1} \, _2F_1\left (3,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a^3 (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.033387, 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 \[ \frac{x^{m+1} \, _2F_1\left (3,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a^3 (m+1)} \]

Antiderivative was successfully verified.

[In]  Int[x^m/(a + b*x^n)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 4.32861, size = 29, normalized size = 0.72 \[ \frac{x^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 3, \frac{m + 1}{n} \\ \frac{m + n + 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a^{3} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.116015, size = 100, normalized size = 2.5 \[ \frac{x^{m+1} \left (\frac{a^2 n}{\left (a+b x^n\right )^2}+\frac{\left (m^2+m (2-3 n)+2 n^2-3 n+1\right ) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{m+1}-\frac{a (m-2 n+1)}{a+b x^n}\right )}{2 a^3 n^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^m/(a + b*x^n)^3,x]

[Out]

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

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Maple [F]  time = 0.115, size = 0, normalized size = 0. \[ \int{\frac{{x}^{m}}{ \left ( a+b{x}^{n} \right ) ^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m/(a+b*x^n)^3,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[{\left (m^{2} - m{\left (3 \, n - 2\right )} + 2 \, n^{2} - 3 \, n + 1\right )} \int \frac{x^{m}}{2 \,{\left (a^{2} b n^{2} x^{n} + a^{3} n^{2}\right )}}\,{d x} - \frac{a{\left (m - 3 \, n + 1\right )} x x^{m} + b{\left (m - 2 \, n + 1\right )} x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}}{2 \,{\left (a^{2} b^{2} n^{2} x^{2 \, n} + 2 \, a^{3} b n^{2} x^{n} + a^{4} n^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(m^2 - m*(3*n - 2) + 2*n^2 - 3*n + 1)*integrate(1/2*x^m/(a^2*b*n^2*x^n + a^3*n^2
), x) - 1/2*(a*(m - 3*n + 1)*x*x^m + b*(m - 2*n + 1)*x*e^(m*log(x) + n*log(x)))/
(a^2*b^2*n^2*x^(2*n) + 2*a^3*b*n^2*x^n + a^4*n^2)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{m}}{b^{3} x^{3 \, n} + 3 \, a b^{2} x^{2 \, n} + 3 \, a^{2} b x^{n} + a^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^m/(b*x^n + a)^3, x)