3.311 \(\int \frac{(g x)^m (d^2-e^2 x^2)^p}{(d+e x)^2} \, dx\)

Optimal. Leaf size=214 \[ -\frac{2 e (g x)^{m+2} \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{m+2}{2},2-p;\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right )}{d^3 g^2 (m+2)}-\frac{2 (m+p) (g x)^{m+1} \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{m+1}{2},2-p;\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{d^2 g (m+1) (-m-2 p+1)}+\frac{(g x)^{m+1} \left (d^2-e^2 x^2\right )^{p-1}}{g (-m-2 p+1)} \]

[Out]

((g*x)^(1 + m)*(d^2 - e^2*x^2)^(-1 + p))/(g*(1 - m - 2*p)) - (2*(m + p)*(g*x)^(1 + m)*(d^2 - e^2*x^2)^p*Hyperg
eometric2F1[(1 + m)/2, 2 - p, (3 + m)/2, (e^2*x^2)/d^2])/(d^2*g*(1 + m)*(1 - m - 2*p)*(1 - (e^2*x^2)/d^2)^p) -
(2*e*(g*x)^(2 + m)*(d^2 - e^2*x^2)^p*Hypergeometric2F1[(2 + m)/2, 2 - p, (4 + m)/2, (e^2*x^2)/d^2])/(d^3*g^2*
(2 + m)*(1 - (e^2*x^2)/d^2)^p)

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Rubi [A]  time = 0.223438, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {852, 1809, 808, 365, 364} \[ -\frac{2 e (g x)^{m+2} \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{m+2}{2},2-p;\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right )}{d^3 g^2 (m+2)}-\frac{2 (m+p) (g x)^{m+1} \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{m+1}{2},2-p;\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{d^2 g (m+1) (-m-2 p+1)}+\frac{(g x)^{m+1} \left (d^2-e^2 x^2\right )^{p-1}}{g (-m-2 p+1)} \]

Antiderivative was successfully verified.

[In]

Int[((g*x)^m*(d^2 - e^2*x^2)^p)/(d + e*x)^2,x]

[Out]

((g*x)^(1 + m)*(d^2 - e^2*x^2)^(-1 + p))/(g*(1 - m - 2*p)) - (2*(m + p)*(g*x)^(1 + m)*(d^2 - e^2*x^2)^p*Hyperg
eometric2F1[(1 + m)/2, 2 - p, (3 + m)/2, (e^2*x^2)/d^2])/(d^2*g*(1 + m)*(1 - m - 2*p)*(1 - (e^2*x^2)/d^2)^p) -
(2*e*(g*x)^(2 + m)*(d^2 - e^2*x^2)^p*Hypergeometric2F1[(2 + m)/2, 2 - p, (4 + m)/2, (e^2*x^2)/d^2])/(d^3*g^2*
(2 + m)*(1 - (e^2*x^2)/d^2)^p)

Rule 852

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[d^(2*m)/a
^m, Int[((f + g*x)^n*(a + c*x^2)^(m + p))/(d - e*x)^m, x], x] /; FreeQ[{a, c, d, e, f, g, n, p}, x] && NeQ[e*f
- d*g, 0] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[f, 0] && ILtQ[m, -1] &&  !(IGtQ[n, 0] && ILtQ[m +
n, 0] &&  !GtQ[p, 1])

Rule 1809

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x,
Expon[Pq, x]]}, Simp[(f*(c*x)^(m + q - 1)*(a + b*x^2)^(p + 1))/(b*c^(q - 1)*(m + q + 2*p + 1)), x] + Dist[1/(
b*(m + q + 2*p + 1)), Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*x^q
- a*f*(m + q - 1)*x^(q - 2), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x]
&& PolyQ[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])

Rule 808

Int[((e_.)*(x_))^(m_)*((f_) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[f, Int[(e*x)^m*(a + c*
x^2)^p, x], x] + Dist[g/e, Int[(e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, e, f, g, p}, x] &&  !Ration
alQ[m] &&  !IGtQ[p, 0]

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 )^p}{(d+e x)^2} \, dx &=\int (g x)^m (d-e x)^2 \left (d^2-e^2 x^2\right )^{-2+p} \, dx\\ &=\frac{(g x)^{1+m} \left (d^2-e^2 x^2\right )^{-1+p}}{g (1-m-2 p)}+\frac{\int (g x)^m \left (-2 d^2 e^2 (m+p)-2 d e^3 (1-m-2 p) x\right ) \left (d^2-e^2 x^2\right )^{-2+p} \, dx}{e^2 (1-m-2 p)}\\ &=\frac{(g x)^{1+m} \left (d^2-e^2 x^2\right )^{-1+p}}{g (1-m-2 p)}-\frac{(2 d e) \int (g x)^{1+m} \left (d^2-e^2 x^2\right )^{-2+p} \, dx}{g}-\frac{\left (2 d^2 (m+p)\right ) \int (g x)^m \left (d^2-e^2 x^2\right )^{-2+p} \, dx}{1-m-2 p}\\ &=\frac{(g x)^{1+m} \left (d^2-e^2 x^2\right )^{-1+p}}{g (1-m-2 p)}-\frac{\left (2 e \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p}\right ) \int (g x)^{1+m} \left (1-\frac{e^2 x^2}{d^2}\right )^{-2+p} \, dx}{d^3 g}-\frac{\left (2 (m+p) \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p}\right ) \int (g x)^m \left (1-\frac{e^2 x^2}{d^2}\right )^{-2+p} \, dx}{d^2 (1-m-2 p)}\\ &=\frac{(g x)^{1+m} \left (d^2-e^2 x^2\right )^{-1+p}}{g (1-m-2 p)}-\frac{2 (m+p) (g x)^{1+m} \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{1+m}{2},2-p;\frac{3+m}{2};\frac{e^2 x^2}{d^2}\right )}{d^2 g (1+m) (1-m-2 p)}-\frac{2 e (g x)^{2+m} \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{2+m}{2},2-p;\frac{4+m}{2};\frac{e^2 x^2}{d^2}\right )}{d^3 g^2 (2+m)}\\ \end{align*}

Mathematica [A]  time = 0.110775, size = 180, normalized size = 0.84 \[ \frac{x (g x)^m \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d^2 \left (m^2+5 m+6\right ) \, _2F_1\left (\frac{m+1}{2},2-p;\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )-e (m+1) x \left (2 d (m+3) \, _2F_1\left (\frac{m+2}{2},2-p;\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right )-e (m+2) x \, _2F_1\left (\frac{m+3}{2},2-p;\frac{m+5}{2};\frac{e^2 x^2}{d^2}\right )\right )\right )}{d^4 (m+1) (m+2) (m+3)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(g*x)^m*(d^2 - e^2*x^2)^p*(d^2*(6 + 5*m + m^2)*Hypergeometric2F1[(1 + m)/2, 2 - p, (3 + m)/2, (e^2*x^2)/d^2
] - e*(1 + m)*x*(2*d*(3 + m)*Hypergeometric2F1[(2 + m)/2, 2 - p, (4 + m)/2, (e^2*x^2)/d^2] - e*(2 + m)*x*Hyper
geometric2F1[(3 + m)/2, 2 - p, (5 + m)/2, (e^2*x^2)/d^2])))/(d^4*(1 + m)*(2 + m)*(3 + m)*(1 - (e^2*x^2)/d^2)^p
)

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Maple [F]  time = 0.694, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( gx \right ) ^{m} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{p}}{ \left ( ex+d \right ) ^{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)^p/(e*x+d)^2,x)

[Out]

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

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p} \left (g x\right )^{m}}{e^{2} x^{2} + 2 \, d e x + d^{2}}, 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)^p/(e*x+d)^2,x, algorithm="fricas")

[Out]

integral((-e^2*x^2 + d^2)^p*(g*x)^m/(e^2*x^2 + 2*d*e*x + d^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((g*x)**m*(-(-d + e*x)*(d + e*x))**p/(d + e*x)**2, x)

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

[Out]

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