3.236 \(\int \frac{(g x)^m}{(d^2-e^2 x^2)^{7/2}} \, dx\)

Optimal. Leaf size=80 \[ \frac{\sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^{m+1} \, _2F_1\left (\frac{7}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{d^6 g (m+1) \sqrt{d^2-e^2 x^2}} \]

[Out]

((g*x)^(1 + m)*Sqrt[1 - (e^2*x^2)/d^2]*Hypergeometric2F1[7/2, (1 + m)/2, (3 + m)/2, (e^2*x^2)/d^2])/(d^6*g*(1
+ m)*Sqrt[d^2 - e^2*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.0252806, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {365, 364} \[ \frac{\sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^{m+1} \, _2F_1\left (\frac{7}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{d^6 g (m+1) \sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(g*x)^m/(d^2 - e^2*x^2)^(7/2),x]

[Out]

((g*x)^(1 + m)*Sqrt[1 - (e^2*x^2)/d^2]*Hypergeometric2F1[7/2, (1 + m)/2, (3 + m)/2, (e^2*x^2)/d^2])/(d^6*g*(1
+ m)*Sqrt[d^2 - e^2*x^2])

Rule 365

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])
/(1 + (b*x^n)/a)^FracPart[p], Int[(c*x)^m*(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[
p, 0] &&  !(ILtQ[p, 0] || GtQ[a, 0])

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{(g x)^m}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac{\sqrt{1-\frac{e^2 x^2}{d^2}} \int \frac{(g x)^m}{\left (1-\frac{e^2 x^2}{d^2}\right )^{7/2}} \, dx}{d^6 \sqrt{d^2-e^2 x^2}}\\ &=\frac{(g x)^{1+m} \sqrt{1-\frac{e^2 x^2}{d^2}} \, _2F_1\left (\frac{7}{2},\frac{1+m}{2};\frac{3+m}{2};\frac{e^2 x^2}{d^2}\right )}{d^6 g (1+m) \sqrt{d^2-e^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0191127, size = 78, normalized size = 0.98 \[ \frac{x \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^m \, _2F_1\left (\frac{7}{2},\frac{m+1}{2};\frac{m+1}{2}+1;\frac{e^2 x^2}{d^2}\right )}{d^6 (m+1) \sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(g*x)^m/(d^2 - e^2*x^2)^(7/2),x]

[Out]

(x*(g*x)^m*Sqrt[1 - (e^2*x^2)/d^2]*Hypergeometric2F1[7/2, (1 + m)/2, 1 + (1 + m)/2, (e^2*x^2)/d^2])/(d^6*(1 +
m)*Sqrt[d^2 - e^2*x^2])

________________________________________________________________________________________

Maple [F]  time = 0.365, size = 0, normalized size = 0. \begin{align*} \int{ \left ( gx \right ) ^{m} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{7}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x)^m/(-e^2*x^2+d^2)^(7/2),x)

[Out]

int((g*x)^m/(-e^2*x^2+d^2)^(7/2),x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (g x\right )^{m}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x)^m/(-e^2*x^2+d^2)^(7/2),x, algorithm="maxima")

[Out]

integrate((g*x)^m/(-e^2*x^2 + d^2)^(7/2), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-e^{2} x^{2} + d^{2}} \left (g x\right )^{m}}{e^{8} x^{8} - 4 \, d^{2} e^{6} x^{6} + 6 \, d^{4} e^{4} x^{4} - 4 \, d^{6} e^{2} x^{2} + d^{8}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x)^m/(-e^2*x^2+d^2)^(7/2),x, algorithm="fricas")

[Out]

integral(sqrt(-e^2*x^2 + d^2)*(g*x)^m/(e^8*x^8 - 4*d^2*e^6*x^6 + 6*d^4*e^4*x^4 - 4*d^6*e^2*x^2 + d^8), x)

________________________________________________________________________________________

Sympy [C]  time = 63.9377, size = 60, normalized size = 0.75 \begin{align*} \frac{g^{m} x x^{m} \Gamma \left (\frac{m}{2} + \frac{1}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{7}{2}, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{\frac{e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 d^{7} \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x)**m/(-e**2*x**2+d**2)**(7/2),x)

[Out]

g**m*x*x**m*gamma(m/2 + 1/2)*hyper((7/2, m/2 + 1/2), (m/2 + 3/2,), e**2*x**2*exp_polar(2*I*pi)/d**2)/(2*d**7*g
amma(m/2 + 3/2))

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (g x\right )^{m}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x)^m/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

[Out]

integrate((g*x)^m/(-e^2*x^2 + d^2)^(7/2), x)