3.2738 \(\int \frac{x^m}{\left (a+b x^{2+2 m}\right )^3} \, dx\)

Optimal. Leaf size=97 \[ \frac{3 \tan ^{-1}\left (\frac{\sqrt{b} x^{m+1}}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b} (m+1)}+\frac{3 x^{m+1}}{8 a^2 (m+1) \left (a+b x^{2 (m+1)}\right )}+\frac{x^{m+1}}{4 a (m+1) \left (a+b x^{2 (m+1)}\right )^2} \]

[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*ArcTan[(Sqrt[b]*x^(1 + m))/Sqrt[a]])/(8*a^(5/2)*Sqrt[
b]*(1 + m))

_______________________________________________________________________________________

Rubi [A]  time = 0.097519, 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 \[ \frac{3 \tan ^{-1}\left (\frac{\sqrt{b} x^{m+1}}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b} (m+1)}+\frac{3 x^{m+1}}{8 a^2 (m+1) \left (a+b x^{2 (m+1)}\right )}+\frac{x^{m+1}}{4 a (m+1) \left (a+b x^{2 (m+1)}\right )^2} \]

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*ArcTan[(Sqrt[b]*x^(1 + m))/Sqrt[a]])/(8*a^(5/2)*Sqrt[
b]*(1 + m))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 4.93338, size = 27, normalized size = 0.28 \[ \frac{x^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 3, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{- \frac{b x^{2 m + 2}}{a}} \right )}}{a^{3} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m/(a+b*x**(2+2*m))**3,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.123544, size = 77, normalized size = 0.79 \[ \frac{\frac{\sqrt{a} x^{m+1} \left (5 a+3 b x^{2 m+2}\right )}{\left (a+b x^{2 m+2}\right )^2}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{b} x^{m+1}}{\sqrt{a}}\right )}{\sqrt{b}}}{8 a^{5/2} (m+1)} \]

Antiderivative was successfully verified.

[In]  Integrate[x^m/(a + b*x^(2 + 2*m))^3,x]

[Out]

((Sqrt[a]*x^(1 + m)*(5*a + 3*b*x^(2 + 2*m)))/(a + b*x^(2 + 2*m))^2 + (3*ArcTan[(
Sqrt[b]*x^(1 + m))/Sqrt[a]])/Sqrt[b])/(8*a^(5/2)*(1 + m))

_______________________________________________________________________________________

Maple [A]  time = 0.06, size = 110, normalized size = 1.1 \[{\frac{x{x}^{m} \left ( 3\,b{x}^{2} \left ({x}^{m} \right ) ^{2}+5\,a \right ) }{ \left ( 8+8\,m \right ){a}^{2} \left ( a+b{x}^{2} \left ({x}^{m} \right ) ^{2} \right ) ^{2}}}-{\frac{3}{ \left ( 16+16\,m \right ){a}^{2}}\ln \left ({x}^{m}-{\frac{a}{x}{\frac{1}{\sqrt{-ab}}}} \right ){\frac{1}{\sqrt{-ab}}}}+{\frac{3}{ \left ( 16+16\,m \right ){a}^{2}}\ln \left ({x}^{m}+{\frac{a}{x}{\frac{1}{\sqrt{-ab}}}} \right ){\frac{1}{\sqrt{-ab}}}} \]

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. \[ \frac{3 \, b x^{3} x^{3 \, m} + 5 \, a x x^{m}}{8 \,{\left (a^{2} b^{2}{\left (m + 1\right )} x^{4} x^{4 \, m} + 2 \, a^{3} b{\left (m + 1\right )} x^{2} x^{2 \, m} + a^{4}{\left (m + 1\right )}\right )}} + 3 \, \int \frac{x^{m}}{8 \,{\left (a^{2} b x^{2} x^{2 \, m} + a^{3}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/(b*x^(2*m + 2) + a)^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 = 0.240547, size = 1, normalized size = 0.01 \[ \left [\frac{6 \, \sqrt{-a b} b x^{3} x^{3 \, m} + 10 \, \sqrt{-a b} a x x^{m} + 3 \,{\left (b^{2} x^{4} x^{4 \, m} + 2 \, a b x^{2} x^{2 \, m} + a^{2}\right )} \log \left (\frac{\sqrt{-a b} b x^{2} x^{2 \, m} + 2 \, a b x x^{m} - \sqrt{-a b} a}{b x^{2} x^{2 \, m} + a}\right )}{16 \,{\left ({\left (a^{2} b^{2} m + a^{2} b^{2}\right )} \sqrt{-a b} x^{4} x^{4 \, m} + 2 \,{\left (a^{3} b m + a^{3} b\right )} \sqrt{-a b} x^{2} x^{2 \, m} +{\left (a^{4} m + a^{4}\right )} \sqrt{-a b}\right )}}, \frac{3 \, \sqrt{a b} b x^{3} x^{3 \, m} + 5 \, \sqrt{a b} a x x^{m} - 3 \,{\left (b^{2} x^{4} x^{4 \, m} + 2 \, a b x^{2} x^{2 \, m} + a^{2}\right )} \arctan \left (\frac{a}{\sqrt{a b} x x^{m}}\right )}{8 \,{\left ({\left (a^{2} b^{2} m + a^{2} b^{2}\right )} \sqrt{a b} x^{4} x^{4 \, m} + 2 \,{\left (a^{3} b m + a^{3} b\right )} \sqrt{a b} x^{2} x^{2 \, m} +{\left (a^{4} m + a^{4}\right )} \sqrt{a b}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/(b*x^(2*m + 2) + a)^3,x, algorithm="fricas")

[Out]

[1/16*(6*sqrt(-a*b)*b*x^3*x^(3*m) + 10*sqrt(-a*b)*a*x*x^m + 3*(b^2*x^4*x^(4*m) +
2*a*b*x^2*x^(2*m) + a^2)*log((sqrt(-a*b)*b*x^2*x^(2*m) + 2*a*b*x*x^m - sqrt(-a*
b)*a)/(b*x^2*x^(2*m) + a)))/((a^2*b^2*m + a^2*b^2)*sqrt(-a*b)*x^4*x^(4*m) + 2*(a
^3*b*m + a^3*b)*sqrt(-a*b)*x^2*x^(2*m) + (a^4*m + a^4)*sqrt(-a*b)), 1/8*(3*sqrt(
a*b)*b*x^3*x^(3*m) + 5*sqrt(a*b)*a*x*x^m - 3*(b^2*x^4*x^(4*m) + 2*a*b*x^2*x^(2*m
) + a^2)*arctan(a/(sqrt(a*b)*x*x^m)))/((a^2*b^2*m + a^2*b^2)*sqrt(a*b)*x^4*x^(4*
m) + 2*(a^3*b*m + a^3*b)*sqrt(a*b)*x^2*x^(2*m) + (a^4*m + a^4)*sqrt(a*b))]

_______________________________________________________________________________________

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**(2+2*m))**3,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{{\left (b x^{2 \, m + 2} + a\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/(b*x^(2*m + 2) + a)^3,x, algorithm="giac")

[Out]

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