\(\int \frac {(g x)^m (d+e x)}{(d^2-e^2 x^2)^{7/2}} \, dx\) [235]

Optimal result

Integrand size = 27, antiderivative size = 124 \[ \int \frac {(g x)^m (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {(g x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (-4+m),\frac {3+m}{2},\frac {e^2 x^2}{d^2}\right )}{d g (1+m) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e (g x)^{2+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (-3+m),\frac {4+m}{2},\frac {e^2 x^2}{d^2}\right )}{d^2 g^2 (2+m) \left (d^2-e^2 x^2\right )^{5/2}} \]

[Out]

(g*x)^(1+m)*hypergeom([1, -2+1/2*m],[3/2+1/2*m],e^2*x^2/d^2)/d/g/(1+m)/(-e^2*x^2+d^2)^(5/2)+e*(g*x)^(2+m)*hype
rgeom([1, -3/2+1/2*m],[2+1/2*m],e^2*x^2/d^2)/d^2/g^2/(2+m)/(-e^2*x^2+d^2)^(5/2)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.31, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {822, 372, 371} \[ \int \frac {(g x)^m (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {e \sqrt {1-\frac {e^2 x^2}{d^2}} (g x)^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {7}{2},\frac {m+2}{2},\frac {m+4}{2},\frac {e^2 x^2}{d^2}\right )}{d^6 g^2 (m+2) \sqrt {d^2-e^2 x^2}}+\frac {\sqrt {1-\frac {e^2 x^2}{d^2}} (g x)^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {7}{2},\frac {m+1}{2},\frac {m+3}{2},\frac {e^2 x^2}{d^2}\right )}{d^5 g (m+1) \sqrt {d^2-e^2 x^2}} \]

[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 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 372

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 822

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]

Rubi steps \begin{align*} \text {integral}& = 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] (verified)

Time = 0.78 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.98 \[ \int \frac {(g x)^m (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {x (g x)^m \sqrt {1-\frac {e^2 x^2}{d^2}} \left (d (2+m) \operatorname {Hypergeometric2F1}\left (\frac {7}{2},\frac {1+m}{2},\frac {3+m}{2},\frac {e^2 x^2}{d^2}\right )+e (1+m) x \operatorname {Hypergeometric2F1}\left (\frac {7}{2},\frac {2+m}{2},\frac {4+m}{2},\frac {e^2 x^2}{d^2}\right )\right )}{d^6 (1+m) (2+m) \sqrt {d^2-e^2 x^2}} \]

[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])

Maple [F]

\[\int \frac {\left (g x \right )^{m} \left (e x +d \right )}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}d x\]

[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)

Fricas [F]

\[ \int \frac {(g x)^m (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int { \frac {{\left (e x + d\right )} \left (g x\right )^{m}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}} \,d x } \]

[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)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 26.22 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.94 \[ \int \frac {(g x)^m (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {g^{m} x^{m + 1} \Gamma \left (\frac {m}{2} + \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {7}{2}, \frac {m}{2} + \frac {1}{2} \\ \frac {m}{2} + \frac {3}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 d^{6} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {e g^{m} x^{m + 2} \Gamma \left (\frac {m}{2} + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {7}{2}, \frac {m}{2} + 1 \\ \frac {m}{2} + 2 \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 d^{7} \Gamma \left (\frac {m}{2} + 2\right )} \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {(g x)^m (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int { \frac {{\left (e x + d\right )} \left (g x\right )^{m}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {(g x)^m (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int { \frac {{\left (e x + d\right )} \left (g x\right )^{m}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {(g x)^m (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {{\left (g\,x\right )}^m\,\left (d+e\,x\right )}{{\left (d^2-e^2\,x^2\right )}^{7/2}} \,d x \]

[In]

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

[Out]

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