3.2745 \(\int \frac{x^m}{(a+b x^{2+2 m})^3} \, dx\)
Optimal. Leaf size=97 \[ \frac{3 x^{m+1}}{8 a^2 (m+1) \left (a+b x^{2 (m+1)}\right )}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{b} x^{m+1}}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b} (m+1)}+\frac{x^{m+1}}{4 a (m+1) \left (a+b x^{2 (m+1)}\right )^2} \]
[Out]
x^(1 + m)/(4*a*(1 + m)*(a + b*x^(2*(1 + m)))^2) + (3*x^(1 + m))/(8*a^2*(1 + m)*(a + b*x^(2*(1 + m)))) + (3*Arc
Tan[(Sqrt[b]*x^(1 + m))/Sqrt[a]])/(8*a^(5/2)*Sqrt[b]*(1 + m))
________________________________________________________________________________________
Rubi [A] time = 0.0380523, antiderivative size = 97, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used =
{345, 199, 205} \[ \frac{3 x^{m+1}}{8 a^2 (m+1) \left (a+b x^{2 (m+1)}\right )}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{b} x^{m+1}}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b} (m+1)}+\frac{x^{m+1}}{4 a (m+1) \left (a+b x^{2 (m+1)}\right )^2} \]
Antiderivative was successfully verified.
[In]
Int[x^m/(a + b*x^(2 + 2*m))^3,x]
[Out]
x^(1 + m)/(4*a*(1 + m)*(a + b*x^(2*(1 + m)))^2) + (3*x^(1 + m))/(8*a^2*(1 + m)*(a + b*x^(2*(1 + m)))) + (3*Arc
Tan[(Sqrt[b]*x^(1 + m))/Sqrt[a]])/(8*a^(5/2)*Sqrt[b]*(1 + m))
Rule 345
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/(m + 1), Subst[Int[(a + b*x^Simplify[n/(m +
1)])^p, x], x, x^(m + 1)], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[n/(m + 1)]] && !IntegerQ[n]
Rule 199
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)
Rule 205
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]
Rubi steps
\begin{align*} \int \frac{x^m}{\left (a+b x^{2+2 m}\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (a+b x^2\right )^3} \, dx,x,x^{1+m}\right )}{1+m}\\ &=\frac{x^{1+m}}{4 a (1+m) \left (a+b x^{2 (1+m)}\right )^2}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{\left (a+b x^2\right )^2} \, dx,x,x^{1+m}\right )}{4 a (1+m)}\\ &=\frac{x^{1+m}}{4 a (1+m) \left (a+b x^{2 (1+m)}\right )^2}+\frac{3 x^{1+m}}{8 a^2 (1+m) \left (a+b x^{2 (1+m)}\right )}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,x^{1+m}\right )}{8 a^2 (1+m)}\\ &=\frac{x^{1+m}}{4 a (1+m) \left (a+b x^{2 (1+m)}\right )^2}+\frac{3 x^{1+m}}{8 a^2 (1+m) \left (a+b x^{2 (1+m)}\right )}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{b} x^{1+m}}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b} (1+m)}\\ \end{align*}
Mathematica [C] time = 0.0076179, size = 35, normalized size = 0.36 \[ \frac{x^{m+1} \, _2F_1\left (\frac{1}{2},3;\frac{3}{2};-\frac{b x^{2 m+2}}{a}\right )}{a^3 (m+1)} \]
Antiderivative was successfully verified.
[In]
Integrate[x^m/(a + b*x^(2 + 2*m))^3,x]
[Out]
(x^(1 + m)*Hypergeometric2F1[1/2, 3, 3/2, -((b*x^(2 + 2*m))/a)])/(a^3*(1 + m))
________________________________________________________________________________________
Maple [A] time = 0.037, size = 110, normalized size = 1.1 \begin{align*}{\frac{ \left ( 3\,b{x}^{2} \left ({x}^{m} \right ) ^{2}+5\,a \right ){x}^{m}x}{ \left ( 8+8\,m \right ){a}^{2} \left ( a+b{x}^{2} \left ({x}^{m} \right ) ^{2} \right ) ^{2}}}-{\frac{3}{ \left ( 16+16\,m \right ){a}^{2}}\ln \left ({x}^{m}-{\frac{a}{x}{\frac{1}{\sqrt{-ab}}}} \right ){\frac{1}{\sqrt{-ab}}}}+{\frac{3}{ \left ( 16+16\,m \right ){a}^{2}}\ln \left ({x}^{m}+{\frac{a}{x}{\frac{1}{\sqrt{-ab}}}} \right ){\frac{1}{\sqrt{-ab}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^m/(a+b*x^(2+2*m))^3,x)
[Out]
1/8*x*x^m*(3*b*x^2*(x^m)^2+5*a)/(1+m)/a^2/(a+b*x^2*(x^m)^2)^2-3/16/(-a*b)^(1/2)/(1+m)/a^2*ln(x^m-a/x/(-a*b)^(1
/2))+3/16/(-a*b)^(1/2)/(1+m)/a^2*ln(x^m+a/x/(-a*b)^(1/2))
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{3 \, b x^{3} x^{3 \, m} + 5 \, a x x^{m}}{8 \,{\left (a^{2} b^{2}{\left (m + 1\right )} x^{4} x^{4 \, m} + 2 \, a^{3} b{\left (m + 1\right )} x^{2} x^{2 \, m} + a^{4}{\left (m + 1\right )}\right )}} + 3 \, \int \frac{x^{m}}{8 \,{\left (a^{2} b x^{2} x^{2 \, m} + a^{3}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^m/(a+b*x^(2+2*m))^3,x, algorithm="maxima")
[Out]
1/8*(3*b*x^3*x^(3*m) + 5*a*x*x^m)/(a^2*b^2*(m + 1)*x^4*x^(4*m) + 2*a^3*b*(m + 1)*x^2*x^(2*m) + a^4*(m + 1)) +
3*integrate(1/8*x^m/(a^2*b*x^2*x^(2*m) + a^3), x)
________________________________________________________________________________________
Fricas [A] time = 1.45475, size = 714, normalized size = 7.36 \begin{align*} \left [\frac{6 \, a b^{2} x^{3} x^{3 \, m} + 10 \, a^{2} b x x^{m} - 3 \,{\left (\sqrt{-a b} b^{2} x^{4} x^{4 \, m} + 2 \, \sqrt{-a b} a b x^{2} x^{2 \, m} + \sqrt{-a b} a^{2}\right )} \log \left (\frac{b x^{2} x^{2 \, m} - 2 \, \sqrt{-a b} x x^{m} - a}{b x^{2} x^{2 \, m} + a}\right )}{16 \,{\left (a^{5} b m + a^{5} b +{\left (a^{3} b^{3} m + a^{3} b^{3}\right )} x^{4} x^{4 \, m} + 2 \,{\left (a^{4} b^{2} m + a^{4} b^{2}\right )} x^{2} x^{2 \, m}\right )}}, \frac{3 \, a b^{2} x^{3} x^{3 \, m} + 5 \, a^{2} b x x^{m} - 3 \,{\left (\sqrt{a b} b^{2} x^{4} x^{4 \, m} + 2 \, \sqrt{a b} a b x^{2} x^{2 \, m} + \sqrt{a b} a^{2}\right )} \arctan \left (\frac{\sqrt{a b}}{b x x^{m}}\right )}{8 \,{\left (a^{5} b m + a^{5} b +{\left (a^{3} b^{3} m + a^{3} b^{3}\right )} x^{4} x^{4 \, m} + 2 \,{\left (a^{4} b^{2} m + a^{4} b^{2}\right )} x^{2} x^{2 \, m}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^m/(a+b*x^(2+2*m))^3,x, algorithm="fricas")
[Out]
[1/16*(6*a*b^2*x^3*x^(3*m) + 10*a^2*b*x*x^m - 3*(sqrt(-a*b)*b^2*x^4*x^(4*m) + 2*sqrt(-a*b)*a*b*x^2*x^(2*m) + s
qrt(-a*b)*a^2)*log((b*x^2*x^(2*m) - 2*sqrt(-a*b)*x*x^m - a)/(b*x^2*x^(2*m) + a)))/(a^5*b*m + a^5*b + (a^3*b^3*
m + a^3*b^3)*x^4*x^(4*m) + 2*(a^4*b^2*m + a^4*b^2)*x^2*x^(2*m)), 1/8*(3*a*b^2*x^3*x^(3*m) + 5*a^2*b*x*x^m - 3*
(sqrt(a*b)*b^2*x^4*x^(4*m) + 2*sqrt(a*b)*a*b*x^2*x^(2*m) + sqrt(a*b)*a^2)*arctan(sqrt(a*b)/(b*x*x^m)))/(a^5*b*
m + a^5*b + (a^3*b^3*m + a^3*b^3)*x^4*x^(4*m) + 2*(a^4*b^2*m + a^4*b^2)*x^2*x^(2*m))]
________________________________________________________________________________________
Sympy [C] time = 140.938, size = 3611, normalized size = 37.23 \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**(2+2*m))**3,x)
[Out]
10*sqrt(pi)*a**(3/2)*a**(-m/(2*(m + 1)))*a**(-1/(2*(m + 1)))*sqrt(b)*x*x**m/(32*a**4*sqrt(b)*m*gamma(m/(2*(m +
1)) + 1 + 1/(2*(m + 1))) + 32*a**4*sqrt(b)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 96*a**3*b**(3/2)*m*x**2
*x**(2*m)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 96*a**3*b**(3/2)*x**2*x**(2*m)*gamma(m/(2*(m + 1)) + 1 +
1/(2*(m + 1))) + 96*a**2*b**(5/2)*m*x**4*x**(4*m)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 96*a**2*b**(5/2)*
x**4*x**(4*m)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 32*a*b**(7/2)*m*x**6*x**(6*m)*gamma(m/(2*(m + 1)) + 1
+ 1/(2*(m + 1))) + 32*a*b**(7/2)*x**6*x**(6*m)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1)))) + 16*sqrt(pi)*sqrt(a
)*a**(-m/(2*(m + 1)))*a**(-1/(2*(m + 1)))*b**(3/2)*x**3*x**(3*m)/(32*a**4*sqrt(b)*m*gamma(m/(2*(m + 1)) + 1 +
1/(2*(m + 1))) + 32*a**4*sqrt(b)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 96*a**3*b**(3/2)*m*x**2*x**(2*m)*g
amma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 96*a**3*b**(3/2)*x**2*x**(2*m)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1
))) + 96*a**2*b**(5/2)*m*x**4*x**(4*m)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 96*a**2*b**(5/2)*x**4*x**(4*
m)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 32*a*b**(7/2)*m*x**6*x**(6*m)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m
+ 1))) + 32*a*b**(7/2)*x**6*x**(6*m)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1)))) + 3*I*sqrt(pi)*a**2*a**(-m/(2*(
m + 1)))*a**(-1/(2*(m + 1)))*log(1 - sqrt(b)*x*x**m*exp_polar(I*pi/2)/sqrt(a))/(32*a**4*sqrt(b)*m*gamma(m/(2*(
m + 1)) + 1 + 1/(2*(m + 1))) + 32*a**4*sqrt(b)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 96*a**3*b**(3/2)*m*x
**2*x**(2*m)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 96*a**3*b**(3/2)*x**2*x**(2*m)*gamma(m/(2*(m + 1)) + 1
+ 1/(2*(m + 1))) + 96*a**2*b**(5/2)*m*x**4*x**(4*m)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 96*a**2*b**(5/
2)*x**4*x**(4*m)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 32*a*b**(7/2)*m*x**6*x**(6*m)*gamma(m/(2*(m + 1))
+ 1 + 1/(2*(m + 1))) + 32*a*b**(7/2)*x**6*x**(6*m)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1)))) - 3*I*sqrt(pi)*a*
*2*a**(-m/(2*(m + 1)))*a**(-1/(2*(m + 1)))*log(1 - sqrt(b)*x*x**m*exp_polar(3*I*pi/2)/sqrt(a))/(32*a**4*sqrt(b
)*m*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 32*a**4*sqrt(b)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 96*a
**3*b**(3/2)*m*x**2*x**(2*m)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 96*a**3*b**(3/2)*x**2*x**(2*m)*gamma(m
/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 96*a**2*b**(5/2)*m*x**4*x**(4*m)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1)))
+ 96*a**2*b**(5/2)*x**4*x**(4*m)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 32*a*b**(7/2)*m*x**6*x**(6*m)*gamm
a(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 32*a*b**(7/2)*x**6*x**(6*m)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1)))) +
9*I*sqrt(pi)*a*a**(-m/(2*(m + 1)))*a**(-1/(2*(m + 1)))*b*x**2*x**(2*m)*log(1 - sqrt(b)*x*x**m*exp_polar(I*pi/
2)/sqrt(a))/(32*a**4*sqrt(b)*m*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 32*a**4*sqrt(b)*gamma(m/(2*(m + 1))
+ 1 + 1/(2*(m + 1))) + 96*a**3*b**(3/2)*m*x**2*x**(2*m)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 96*a**3*b**
(3/2)*x**2*x**(2*m)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 96*a**2*b**(5/2)*m*x**4*x**(4*m)*gamma(m/(2*(m
+ 1)) + 1 + 1/(2*(m + 1))) + 96*a**2*b**(5/2)*x**4*x**(4*m)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 32*a*b*
*(7/2)*m*x**6*x**(6*m)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 32*a*b**(7/2)*x**6*x**(6*m)*gamma(m/(2*(m +
1)) + 1 + 1/(2*(m + 1)))) - 9*I*sqrt(pi)*a*a**(-m/(2*(m + 1)))*a**(-1/(2*(m + 1)))*b*x**2*x**(2*m)*log(1 - sqr
t(b)*x*x**m*exp_polar(3*I*pi/2)/sqrt(a))/(32*a**4*sqrt(b)*m*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 32*a**4
*sqrt(b)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 96*a**3*b**(3/2)*m*x**2*x**(2*m)*gamma(m/(2*(m + 1)) + 1 +
1/(2*(m + 1))) + 96*a**3*b**(3/2)*x**2*x**(2*m)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 96*a**2*b**(5/2)*m
*x**4*x**(4*m)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 96*a**2*b**(5/2)*x**4*x**(4*m)*gamma(m/(2*(m + 1)) +
1 + 1/(2*(m + 1))) + 32*a*b**(7/2)*m*x**6*x**(6*m)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 32*a*b**(7/2)*x
**6*x**(6*m)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1)))) + 9*I*sqrt(pi)*a**(-m/(2*(m + 1)))*a**(-1/(2*(m + 1)))*
b**2*x**4*x**(4*m)*log(1 - sqrt(b)*x*x**m*exp_polar(I*pi/2)/sqrt(a))/(32*a**4*sqrt(b)*m*gamma(m/(2*(m + 1)) +
1 + 1/(2*(m + 1))) + 32*a**4*sqrt(b)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 96*a**3*b**(3/2)*m*x**2*x**(2*
m)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 96*a**3*b**(3/2)*x**2*x**(2*m)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m
+ 1))) + 96*a**2*b**(5/2)*m*x**4*x**(4*m)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 96*a**2*b**(5/2)*x**4*x*
*(4*m)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 32*a*b**(7/2)*m*x**6*x**(6*m)*gamma(m/(2*(m + 1)) + 1 + 1/(2
*(m + 1))) + 32*a*b**(7/2)*x**6*x**(6*m)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1)))) - 9*I*sqrt(pi)*a**(-m/(2*(m
+ 1)))*a**(-1/(2*(m + 1)))*b**2*x**4*x**(4*m)*log(1 - sqrt(b)*x*x**m*exp_polar(3*I*pi/2)/sqrt(a))/(32*a**4*sq
rt(b)*m*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 32*a**4*sqrt(b)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) +
96*a**3*b**(3/2)*m*x**2*x**(2*m)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 96*a**3*b**(3/2)*x**2*x**(2*m)*gam
ma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 96*a**2*b**(5/2)*m*x**4*x**(4*m)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1
))) + 96*a**2*b**(5/2)*x**4*x**(4*m)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 32*a*b**(7/2)*m*x**6*x**(6*m)*
gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 32*a*b**(7/2)*x**6*x**(6*m)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))
)) + 3*I*sqrt(pi)*a**(-m/(2*(m + 1)))*a**(-1/(2*(m + 1)))*b**3*x**6*x**(6*m)*log(1 - sqrt(b)*x*x**m*exp_polar(
I*pi/2)/sqrt(a))/(a*(32*a**4*sqrt(b)*m*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 32*a**4*sqrt(b)*gamma(m/(2*(
m + 1)) + 1 + 1/(2*(m + 1))) + 96*a**3*b**(3/2)*m*x**2*x**(2*m)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 96*
a**3*b**(3/2)*x**2*x**(2*m)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 96*a**2*b**(5/2)*m*x**4*x**(4*m)*gamma(
m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 96*a**2*b**(5/2)*x**4*x**(4*m)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) +
32*a*b**(7/2)*m*x**6*x**(6*m)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 32*a*b**(7/2)*x**6*x**(6*m)*gamma(m/
(2*(m + 1)) + 1 + 1/(2*(m + 1))))) - 3*I*sqrt(pi)*a**(-m/(2*(m + 1)))*a**(-1/(2*(m + 1)))*b**3*x**6*x**(6*m)*l
og(1 - sqrt(b)*x*x**m*exp_polar(3*I*pi/2)/sqrt(a))/(a*(32*a**4*sqrt(b)*m*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1
))) + 32*a**4*sqrt(b)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 96*a**3*b**(3/2)*m*x**2*x**(2*m)*gamma(m/(2*(
m + 1)) + 1 + 1/(2*(m + 1))) + 96*a**3*b**(3/2)*x**2*x**(2*m)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 96*a*
*2*b**(5/2)*m*x**4*x**(4*m)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 96*a**2*b**(5/2)*x**4*x**(4*m)*gamma(m/
(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 32*a*b**(7/2)*m*x**6*x**(6*m)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 32
*a*b**(7/2)*x**6*x**(6*m)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))))) + 6*sqrt(pi)*a**(-m/(2*(m + 1)))*a**(-1/(
2*(m + 1)))*b**(5/2)*x**5*x**(5*m)/(sqrt(a)*(32*a**4*sqrt(b)*m*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 32*a
**4*sqrt(b)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 96*a**3*b**(3/2)*m*x**2*x**(2*m)*gamma(m/(2*(m + 1)) +
1 + 1/(2*(m + 1))) + 96*a**3*b**(3/2)*x**2*x**(2*m)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 96*a**2*b**(5/2
)*m*x**4*x**(4*m)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 96*a**2*b**(5/2)*x**4*x**(4*m)*gamma(m/(2*(m + 1)
) + 1 + 1/(2*(m + 1))) + 32*a*b**(7/2)*m*x**6*x**(6*m)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1))) + 32*a*b**(7/2
)*x**6*x**(6*m)*gamma(m/(2*(m + 1)) + 1 + 1/(2*(m + 1)))))
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m}}{{\left (b x^{2 \, m + 2} + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^m/(a+b*x^(2+2*m))^3,x, algorithm="giac")
[Out]
integrate(x^m/(b*x^(2*m + 2) + a)^3, x)