3.386 \(\int \frac{x^m}{(1-\frac{\sqrt{a} x}{\sqrt{-b}})^2 (1+\frac{\sqrt{a} x}{\sqrt{-b}})^2} \, dx\)

Optimal. Leaf size=36 \[ \frac{x^{m+1} \, _2F_1\left (2,\frac{m+1}{2};\frac{m+3}{2};-\frac{a x^2}{b}\right )}{m+1} \]

[Out]

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

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Rubi [A]  time = 0.0093259, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.049, Rules used = {73, 364} \[ \frac{x^{m+1} \, _2F_1\left (2,\frac{m+1}{2};\frac{m+3}{2};-\frac{a x^2}{b}\right )}{m+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 73

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[(a*c + b*
d*x^2)^m*(e + f*x)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[n, m] && Integer
Q[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 (1-\frac{\sqrt{a} x}{\sqrt{-b}}\right )^2 \left (1+\frac{\sqrt{a} x}{\sqrt{-b}}\right )^2} \, dx &=\int \frac{x^m}{\left (1+\frac{a x^2}{b}\right )^2} \, dx\\ &=\frac{x^{1+m} \, _2F_1\left (2,\frac{1+m}{2};\frac{3+m}{2};-\frac{a x^2}{b}\right )}{1+m}\\ \end{align*}

Mathematica [A]  time = 0.0054989, size = 38, normalized size = 1.06 \[ \frac{x^{m+1} \, _2F_1\left (2,\frac{m+1}{2};\frac{m+1}{2}+1;-\frac{a x^2}{b}\right )}{m+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.1, size = 0, normalized size = 0. \begin{align*} \int{{x}^{m} \left ( 1-{x\sqrt{a}{\frac{1}{\sqrt{-b}}}} \right ) ^{-2} \left ( 1+{x\sqrt{a}{\frac{1}{\sqrt{-b}}}} \right ) ^{-2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m/(1-x*a^(1/2)/(-b)^(1/2))^2/(1+x*a^(1/2)/(-b)^(1/2))^2,x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} x^{m}}{a^{2} x^{4} + 2 \, a b x^{2} + b^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m/(1-x*a^(1/2)/(-b)^(1/2))^2/(1+x*a^(1/2)/(-b)^(1/2))^2,x, algorithm="fricas")

[Out]

integral(b^2*x^m/(a^2*x^4 + 2*a*b*x^2 + b^2), x)

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Sympy [C]  time = 11.6521, size = 541, normalized size = 15.03 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*b**2*m**2*x**m*lerchphi(b*exp_polar(I*pi)/(a*x**2), 1, 3/2 - m/2)*gamma(3/2 - m/2)/(x*(8*a**3*x**2*gamma(5/2
- m/2) + 8*a**2*b*gamma(5/2 - m/2))) - 4*a*b**2*m*x**m*lerchphi(b*exp_polar(I*pi)/(a*x**2), 1, 3/2 - m/2)*gam
ma(3/2 - m/2)/(x*(8*a**3*x**2*gamma(5/2 - m/2) + 8*a**2*b*gamma(5/2 - m/2))) + 2*a*b**2*m*x**m*gamma(3/2 - m/2
)/(x*(8*a**3*x**2*gamma(5/2 - m/2) + 8*a**2*b*gamma(5/2 - m/2))) + 3*a*b**2*x**m*lerchphi(b*exp_polar(I*pi)/(a
*x**2), 1, 3/2 - m/2)*gamma(3/2 - m/2)/(x*(8*a**3*x**2*gamma(5/2 - m/2) + 8*a**2*b*gamma(5/2 - m/2))) - 6*a*b*
*2*x**m*gamma(3/2 - m/2)/(x*(8*a**3*x**2*gamma(5/2 - m/2) + 8*a**2*b*gamma(5/2 - m/2))) + b**3*m**2*x**m*lerch
phi(b*exp_polar(I*pi)/(a*x**2), 1, 3/2 - m/2)*gamma(3/2 - m/2)/(x**3*(8*a**3*x**2*gamma(5/2 - m/2) + 8*a**2*b*
gamma(5/2 - m/2))) - 4*b**3*m*x**m*lerchphi(b*exp_polar(I*pi)/(a*x**2), 1, 3/2 - m/2)*gamma(3/2 - m/2)/(x**3*(
8*a**3*x**2*gamma(5/2 - m/2) + 8*a**2*b*gamma(5/2 - m/2))) + 3*b**3*x**m*lerchphi(b*exp_polar(I*pi)/(a*x**2),
1, 3/2 - m/2)*gamma(3/2 - m/2)/(x**3*(8*a**3*x**2*gamma(5/2 - m/2) + 8*a**2*b*gamma(5/2 - m/2)))

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m/(1-x*a^(1/2)/(-b)^(1/2))^2/(1+x*a^(1/2)/(-b)^(1/2))^2,x, algorithm="giac")

[Out]

Exception raised: TypeError