\(\int \frac {(g x)^m (d^2-e^2 x^2)^p}{d+e x} \, dx\) [310]

Optimal result

Integrand size = 27, antiderivative size = 163 \[ \int \frac {(g x)^m \left (d^2-e^2 x^2\right )^p}{d+e x} \, dx=\frac {(g x)^{1+m} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},1-p,\frac {3+m}{2},\frac {e^2 x^2}{d^2}\right )}{d g (1+m)}-\frac {e (g x)^{2+m} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {2+m}{2},1-p,\frac {4+m}{2},\frac {e^2 x^2}{d^2}\right )}{d^2 g^2 (2+m)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {906, 83, 127, 372, 371} \[ \int \frac {(g x)^m \left (d^2-e^2 x^2\right )^p}{d+e x} \, dx=\frac {(g x)^{m+1} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {m+1}{2},1-p,\frac {m+3}{2},\frac {e^2 x^2}{d^2}\right )}{d g (m+1)}-\frac {e (g x)^{m+2} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {m+2}{2},1-p,\frac {m+4}{2},\frac {e^2 x^2}{d^2}\right )}{d^2 g^2 (m+2)} \]

[In]

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

[Out]

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

Rule 83

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

Rule 127

Int[((f_.)*(x_))^(p_.)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Dist[(a + b*x)^Frac
Part[m]*((c + d*x)^FracPart[m]/(a*c + b*d*x^2)^FracPart[m]), Int[(a*c + b*d*x^2)^m*(f*x)^p, x], x] /; FreeQ[{a
, b, c, d, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[n, m]

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 906

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

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.76 \[ \int \frac {(g x)^m \left (d^2-e^2 x^2\right )^p}{d+e x} \, dx=\frac {x (g x)^m \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \left (-e (1+m) x \operatorname {Hypergeometric2F1}\left (1+\frac {m}{2},1-p,2+\frac {m}{2},\frac {e^2 x^2}{d^2}\right )+d (2+m) \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},1-p,\frac {3+m}{2},\frac {e^2 x^2}{d^2}\right )\right )}{d^2 (1+m) (2+m)} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 5.23 (sec) , antiderivative size = 372, normalized size of antiderivative = 2.28 \[ \int \frac {(g x)^m \left (d^2-e^2 x^2\right )^p}{d+e x} \, dx=- \frac {0^{p} d^{1 - m} d^{m + 2 p} e^{- m - 1} e^{m - 1} g^{m} m x^{m - 1} \Phi \left (\frac {d^{2}}{e^{2} x^{2}}, 1, \frac {1}{2} - \frac {m}{2}\right ) \Gamma \left (\frac {1}{2} - \frac {m}{2}\right )}{4 \Gamma \left (\frac {3}{2} - \frac {m}{2}\right )} + \frac {0^{p} d^{1 - m} d^{m + 2 p} e^{- m - 1} e^{m - 1} g^{m} x^{m - 1} \Phi \left (\frac {d^{2}}{e^{2} x^{2}}, 1, \frac {1}{2} - \frac {m}{2}\right ) \Gamma \left (\frac {1}{2} - \frac {m}{2}\right )}{4 \Gamma \left (\frac {3}{2} - \frac {m}{2}\right )} + \frac {0^{p} d^{- m} d^{m + 2 p} e^{m} e^{- m - 1} g^{m} m x^{m} \Phi \left (\frac {d^{2}}{e^{2} x^{2}}, 1, \frac {m e^{i \pi }}{2}\right ) \Gamma \left (- \frac {m}{2}\right )}{4 \Gamma \left (1 - \frac {m}{2}\right )} + \frac {d e^{2 p - 2} g^{m} p x^{m + 2 p - 1} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (- \frac {m}{2} - p + \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, - \frac {m}{2} - p + \frac {1}{2} \\ - \frac {m}{2} - p + \frac {3}{2} \end {matrix}\middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{2 \Gamma \left (p + 1\right ) \Gamma \left (- \frac {m}{2} - p + \frac {3}{2}\right )} - \frac {e^{2 p - 1} g^{m} p x^{m + 2 p} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (- \frac {m}{2} - p\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, - \frac {m}{2} - p \\ - \frac {m}{2} - p + 1 \end {matrix}\middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{2 \Gamma \left (p + 1\right ) \Gamma \left (- \frac {m}{2} - p + 1\right )} \]

[In]

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

[Out]

-0**p*d**(1 - m)*d**(m + 2*p)*e**(-m - 1)*e**(m - 1)*g**m*m*x**(m - 1)*lerchphi(d**2/(e**2*x**2), 1, 1/2 - m/2
)*gamma(1/2 - m/2)/(4*gamma(3/2 - m/2)) + 0**p*d**(1 - m)*d**(m + 2*p)*e**(-m - 1)*e**(m - 1)*g**m*x**(m - 1)*
lerchphi(d**2/(e**2*x**2), 1, 1/2 - m/2)*gamma(1/2 - m/2)/(4*gamma(3/2 - m/2)) + 0**p*d**(m + 2*p)*e**m*e**(-m
- 1)*g**m*m*x**m*lerchphi(d**2/(e**2*x**2), 1, m*exp_polar(I*pi)/2)*gamma(-m/2)/(4*d**m*gamma(1 - m/2)) + d*e
**(2*p - 2)*g**m*p*x**(m + 2*p - 1)*exp(I*pi*p)*gamma(p)*gamma(-m/2 - p + 1/2)*hyper((1 - p, -m/2 - p + 1/2),
(-m/2 - p + 3/2,), d**2/(e**2*x**2))/(2*gamma(p + 1)*gamma(-m/2 - p + 3/2)) - e**(2*p - 1)*g**m*p*x**(m + 2*p)
*exp(I*pi*p)*gamma(p)*gamma(-m/2 - p)*hyper((1 - p, -m/2 - p), (-m/2 - p + 1,), d**2/(e**2*x**2))/(2*gamma(p +
1)*gamma(-m/2 - p + 1))

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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