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

Optimal. Leaf size=217 \[ -\frac{2 e (7-m) \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^{m+2} \, _2F_1\left (\frac{9}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right )}{9 d^9 g^2 (m+2) \sqrt{d^2-e^2 x^2}}+\frac{(7-2 m) \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^{m+1} \, _2F_1\left (\frac{9}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{9 d^8 g (m+1) \sqrt{d^2-e^2 x^2}}+\frac{2 (d-e x) (g x)^{m+1}}{9 d g \left (d^2-e^2 x^2\right )^{9/2}} \]

[Out]

(2*(g*x)^(1 + m)*(d - e*x))/(9*d*g*(d^2 - e^2*x^2)^(9/2)) + ((7 - 2*m)*(g*x)^(1 + m)*Sqrt[1 - (e^2*x^2)/d^2]*H
ypergeometric2F1[9/2, (1 + m)/2, (3 + m)/2, (e^2*x^2)/d^2])/(9*d^8*g*(1 + m)*Sqrt[d^2 - e^2*x^2]) - (2*e*(7 -
m)*(g*x)^(2 + m)*Sqrt[1 - (e^2*x^2)/d^2]*Hypergeometric2F1[9/2, (2 + m)/2, (4 + m)/2, (e^2*x^2)/d^2])/(9*d^9*g
^2*(2 + m)*Sqrt[d^2 - e^2*x^2])

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Rubi [A]  time = 0.231118, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {852, 1806, 808, 365, 364} \[ -\frac{2 e (7-m) \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^{m+2} \, _2F_1\left (\frac{9}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right )}{9 d^9 g^2 (m+2) \sqrt{d^2-e^2 x^2}}+\frac{(7-2 m) \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^{m+1} \, _2F_1\left (\frac{9}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{9 d^8 g (m+1) \sqrt{d^2-e^2 x^2}}+\frac{2 (d-e x) (g x)^{m+1}}{9 d g \left (d^2-e^2 x^2\right )^{9/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(g*x)^(1 + m)*(d - e*x))/(9*d*g*(d^2 - e^2*x^2)^(9/2)) + ((7 - 2*m)*(g*x)^(1 + m)*Sqrt[1 - (e^2*x^2)/d^2]*H
ypergeometric2F1[9/2, (1 + m)/2, (3 + m)/2, (e^2*x^2)/d^2])/(9*d^8*g*(1 + m)*Sqrt[d^2 - e^2*x^2]) - (2*e*(7 -
m)*(g*x)^(2 + m)*Sqrt[1 - (e^2*x^2)/d^2]*Hypergeometric2F1[9/2, (2 + m)/2, (4 + m)/2, (e^2*x^2)/d^2])/(9*d^9*g
^2*(2 + m)*Sqrt[d^2 - e^2*x^2])

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 1806

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x
^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x
], x, 1]}, -Simp[((c*x)^(m + 1)*(f + g*x)*(a + b*x^2)^(p + 1))/(2*a*c*(p + 1)), x] + Dist[1/(2*a*(p + 1)), Int
[(c*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(m + 2*p + 3) + g*(m + 2*p + 4)*x, x], x], x]] /; F
reeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && LtQ[p, -1] &&  !GtQ[m, 0]

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}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\int \frac{(g x)^m (d-e x)^2}{\left (d^2-e^2 x^2\right )^{11/2}} \, dx\\ &=\frac{2 (g x)^{1+m} (d-e x)}{9 d g \left (d^2-e^2 x^2\right )^{9/2}}-\frac{\int \frac{(g x)^m \left (-d^2 (7-2 m)+2 d e (7-m) x\right )}{\left (d^2-e^2 x^2\right )^{9/2}} \, dx}{9 d^2}\\ &=\frac{2 (g x)^{1+m} (d-e x)}{9 d g \left (d^2-e^2 x^2\right )^{9/2}}-\frac{(2 e (7-m)) \int \frac{(g x)^{1+m}}{\left (d^2-e^2 x^2\right )^{9/2}} \, dx}{9 d g}-\frac{1}{9} (-7+2 m) \int \frac{(g x)^m}{\left (d^2-e^2 x^2\right )^{9/2}} \, dx\\ &=\frac{2 (g x)^{1+m} (d-e x)}{9 d g \left (d^2-e^2 x^2\right )^{9/2}}-\frac{\left (2 e (7-m) \sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \int \frac{(g x)^{1+m}}{\left (1-\frac{e^2 x^2}{d^2}\right )^{9/2}} \, dx}{9 d^9 g \sqrt{d^2-e^2 x^2}}-\frac{\left ((-7+2 m) \sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \int \frac{(g x)^m}{\left (1-\frac{e^2 x^2}{d^2}\right )^{9/2}} \, dx}{9 d^8 \sqrt{d^2-e^2 x^2}}\\ &=\frac{2 (g x)^{1+m} (d-e x)}{9 d g \left (d^2-e^2 x^2\right )^{9/2}}+\frac{(7-2 m) (g x)^{1+m} \sqrt{1-\frac{e^2 x^2}{d^2}} \, _2F_1\left (\frac{9}{2},\frac{1+m}{2};\frac{3+m}{2};\frac{e^2 x^2}{d^2}\right )}{9 d^8 g (1+m) \sqrt{d^2-e^2 x^2}}-\frac{2 e (7-m) (g x)^{2+m} \sqrt{1-\frac{e^2 x^2}{d^2}} \, _2F_1\left (\frac{9}{2},\frac{2+m}{2};\frac{4+m}{2};\frac{e^2 x^2}{d^2}\right )}{9 d^9 g^2 (2+m) \sqrt{d^2-e^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.13423, size = 176, normalized size = 0.81 \[ \frac{x \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^m \left (d^2 \left (m^2+5 m+6\right ) \, _2F_1\left (\frac{11}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )-e (m+1) x \left (2 d (m+3) \, _2F_1\left (\frac{11}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right )-e (m+2) x \, _2F_1\left (\frac{11}{2},\frac{m+3}{2};\frac{m+5}{2};\frac{e^2 x^2}{d^2}\right )\right )\right )}{d^{10} (m+1) (m+2) (m+3) \sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(g*x)^m*Sqrt[1 - (e^2*x^2)/d^2]*(d^2*(6 + 5*m + m^2)*Hypergeometric2F1[11/2, (1 + m)/2, (3 + m)/2, (e^2*x^2
)/d^2] - e*(1 + m)*x*(2*d*(3 + m)*Hypergeometric2F1[11/2, (2 + m)/2, (4 + m)/2, (e^2*x^2)/d^2] - e*(2 + m)*x*H
ypergeometric2F1[11/2, (3 + m)/2, (5 + m)/2, (e^2*x^2)/d^2])))/(d^10*(1 + m)*(2 + m)*(3 + m)*Sqrt[d^2 - e^2*x^
2])

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Maple [F]  time = 0.59, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( gx \right ) ^{m}}{ \left ( ex+d \right ) ^{2}} \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*x+d)^2/(-e^2*x^2+d^2)^(7/2),x)

[Out]

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

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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}}{\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*x+d)^2/(-e^2*x^2+d^2)^(7/2),x, algorithm="maxima")

[Out]

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

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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^{10} x^{10} + 2 \, d e^{9} x^{9} - 3 \, d^{2} e^{8} x^{8} - 8 \, d^{3} e^{7} x^{7} + 2 \, d^{4} e^{6} x^{6} + 12 \, d^{5} e^{5} x^{5} + 2 \, d^{6} e^{4} x^{4} - 8 \, d^{7} e^{3} x^{3} - 3 \, d^{8} e^{2} x^{2} + 2 \, d^{9} e x + d^{10}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(-e^2*x^2 + d^2)*(g*x)^m/(e^10*x^10 + 2*d*e^9*x^9 - 3*d^2*e^8*x^8 - 8*d^3*e^7*x^7 + 2*d^4*e^6*x^6
+ 12*d^5*e^5*x^5 + 2*d^6*e^4*x^4 - 8*d^7*e^3*x^3 - 3*d^8*e^2*x^2 + 2*d^9*e*x + d^10), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x