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

Optimal. Leaf size=124 \[ \frac{e (g x)^{m+2} \, _2F_1\left (1,\frac{m-3}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right )}{d^2 g^2 (m+2) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{(g x)^{m+1} \, _2F_1\left (1,\frac{m-4}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{d g (m+1) \left (d^2-e^2 x^2\right )^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.0819333, antiderivative size = 162, normalized size of antiderivative = 1.31, number of steps used = 5, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {808, 365, 364} \[ \frac{e \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^{m+2} \, _2F_1\left (\frac{7}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right )}{d^6 g^2 (m+2) \sqrt{d^2-e^2 x^2}}+\frac{\sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^{m+1} \, _2F_1\left (\frac{7}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{d^5 g (m+1) \sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[((g*x)^m*(d + e*x))/(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^5*g*(1
+ m)*Sqrt[d^2 - e^2*x^2]) + (e*(g*x)^(2 + m)*Sqrt[1 - (e^2*x^2)/d^2]*Hypergeometric2F1[7/2, (2 + m)/2, (4 + m)
/2, (e^2*x^2)/d^2])/(d^6*g^2*(2 + m)*Sqrt[d^2 - e^2*x^2])

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)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=d \int \frac{(g x)^m}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx+\frac{e \int \frac{(g x)^{1+m}}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx}{g}\\ &=\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^5 \sqrt{d^2-e^2 x^2}}+\frac{\left (e \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 )^{7/2}} \, dx}{d^6 g \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^5 g (1+m) \sqrt{d^2-e^2 x^2}}+\frac{e (g x)^{2+m} \sqrt{1-\frac{e^2 x^2}{d^2}} \, _2F_1\left (\frac{7}{2},\frac{2+m}{2};\frac{4+m}{2};\frac{e^2 x^2}{d^2}\right )}{d^6 g^2 (2+m) \sqrt{d^2-e^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0566058, size = 121, normalized size = 0.98 \[ \frac{x \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^m \left (d (m+2) \, _2F_1\left (\frac{7}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )+e (m+1) x \, _2F_1\left (\frac{7}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right )\right )}{d^6 (m+1) (m+2) \sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(g*x)^m*Sqrt[1 - (e^2*x^2)/d^2]*(d*(2 + m)*Hypergeometric2F1[7/2, (1 + m)/2, (3 + m)/2, (e^2*x^2)/d^2] + e*
(1 + m)*x*Hypergeometric2F1[7/2, (2 + m)/2, (4 + m)/2, (e^2*x^2)/d^2]))/(d^6*(1 + m)*(2 + m)*Sqrt[d^2 - e^2*x^
2])

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Maple [F]  time = 0.352, size = 0, normalized size = 0. \begin{align*} \int{ \left ( gx \right ) ^{m} \left ( ex+d \right ) \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)/(-e^2*x^2+d^2)^(7/2),x)

[Out]

int((g*x)^m*(e*x+d)/(-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 (e x + d\right )} \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*x+d)/(-e^2*x^2+d^2)^(7/2),x, algorithm="maxima")

[Out]

integrate((e*x + d)*(g*x)^m/(-e^2*x^2 + d^2)^(7/2), 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^{7} x^{7} - d e^{6} x^{6} - 3 \, d^{2} e^{5} x^{5} + 3 \, d^{3} e^{4} x^{4} + 3 \, d^{4} e^{3} x^{3} - 3 \, d^{5} e^{2} x^{2} - d^{6} e x + d^{7}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(-e^2*x^2 + d^2)*(g*x)^m/(e^7*x^7 - d*e^6*x^6 - 3*d^2*e^5*x^5 + 3*d^3*e^4*x^4 + 3*d^4*e^3*x^3 - 3
*d^5*e^2*x^2 - d^6*e*x + d^7), 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)/(-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*} \int \frac{{\left (e x + d\right )} \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*x+d)/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

[Out]

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