3.2671 \(\int \frac{x^m}{(a+b x^n)^2} \, dx\)
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 = 0.0100506, 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, Rules used =
{364} \[ \frac{x^{m+1} \, _2F_1\left (2,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a^2 (m+1)} \]
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))
Rule 364
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])
Rubi steps
\begin{align*} \int \frac{x^m}{\left (a+b x^n\right )^2} \, dx &=\frac{x^{1+m} \, _2F_1\left (2,\frac{1+m}{n};\frac{1+m+n}{n};-\frac{b x^n}{a}\right )}{a^2 (1+m)}\\ \end{align*}
Mathematica [A] time = 0.0067243, size = 41, normalized size = 1.02 \[ \frac{x^{m+1} \, _2F_1\left (2,\frac{m+1}{n};\frac{m+1}{n}+1;-\frac{b x^n}{a}\right )}{a^2 (m+1)} \]
Antiderivative was successfully verified.
[In]
Integrate[x^m/(a + b*x^n)^2,x]
[Out]
(x^(1 + m)*Hypergeometric2F1[2, (1 + m)/n, 1 + (1 + m)/n, -((b*x^n)/a)])/(a^2*(1 + m))
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Maple [F] time = 0.079, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{m}}{ \left ( a+b{x}^{n} \right ) ^{2}}}\, dx \end{align*}
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. \begin{align*} -{\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^m/(a+b*x^n)^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. \begin{align*}{\rm integral}\left (\frac{x^{m}}{b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^m/(a+b*x^n)^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 [C] time = 1.67494, size = 840, normalized size = 21. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**m/(a+b*x**n)**2,x)
[Out]
-m**2*x*x**m*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(a*(a*n**3*gamma(m/n + 1 + 1/n)
+ b*n**3*x**n*gamma(m/n + 1 + 1/n))) + m*n*x*x**m*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n
+ 1/n)/(a*(a*n**3*gamma(m/n + 1 + 1/n) + b*n**3*x**n*gamma(m/n + 1 + 1/n))) + m*n*x*x**m*gamma(m/n + 1/n)/(a*(
a*n**3*gamma(m/n + 1 + 1/n) + b*n**3*x**n*gamma(m/n + 1 + 1/n))) - 2*m*x*x**m*lerchphi(b*x**n*exp_polar(I*pi)/
a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(a*(a*n**3*gamma(m/n + 1 + 1/n) + b*n**3*x**n*gamma(m/n + 1 + 1/n))) + n*x*x
**m*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(a*(a*n**3*gamma(m/n + 1 + 1/n) + b*n**3
*x**n*gamma(m/n + 1 + 1/n))) + n*x*x**m*gamma(m/n + 1/n)/(a*(a*n**3*gamma(m/n + 1 + 1/n) + b*n**3*x**n*gamma(m
/n + 1 + 1/n))) - x*x**m*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(a*(a*n**3*gamma(m/
n + 1 + 1/n) + b*n**3*x**n*gamma(m/n + 1 + 1/n))) - b*m**2*x*x**m*x**n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m
/n + 1/n)*gamma(m/n + 1/n)/(a**2*(a*n**3*gamma(m/n + 1 + 1/n) + b*n**3*x**n*gamma(m/n + 1 + 1/n))) + b*m*n*x*x
**m*x**n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(a**2*(a*n**3*gamma(m/n + 1 + 1/n)
+ b*n**3*x**n*gamma(m/n + 1 + 1/n))) - 2*b*m*x*x**m*x**n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1/n)*gamm
a(m/n + 1/n)/(a**2*(a*n**3*gamma(m/n + 1 + 1/n) + b*n**3*x**n*gamma(m/n + 1 + 1/n))) + b*n*x*x**m*x**n*lerchph
i(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(a**2*(a*n**3*gamma(m/n + 1 + 1/n) + b*n**3*x**n*ga
mma(m/n + 1 + 1/n))) - b*x*x**m*x**n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(a**2*(
a*n**3*gamma(m/n + 1 + 1/n) + b*n**3*x**n*gamma(m/n + 1 + 1/n)))
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m}}{{\left (b x^{n} + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^m/(a+b*x^n)^2,x, algorithm="giac")
[Out]
integrate(x^m/(b*x^n + a)^2, x)