\(\int \frac {(g x)^m (d^2-e^2 x^2)^p}{(d+e x)^3} \, dx\) [312]

Optimal result

Integrand size = 27, antiderivative size = 275 \[ \int \frac {(g x)^m \left (d^2-e^2 x^2\right )^p}{(d+e x)^3} \, dx=\frac {3 d (g x)^{1+m} \left (d^2-e^2 x^2\right )^{-2+p}}{g (3-m-2 p)}-\frac {e (g x)^{2+m} \left (d^2-e^2 x^2\right )^{-2+p}}{g^2 (2-m-2 p)}-\frac {2 (2 m+p) (g x)^{1+m} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},3-p,\frac {3+m}{2},\frac {e^2 x^2}{d^2}\right )}{d^3 g (1+m) (3-m-2 p)}-\frac {2 e (2-2 m-3 p) (g x)^{2+m} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {2+m}{2},3-p,\frac {4+m}{2},\frac {e^2 x^2}{d^2}\right )}{d^4 g^2 (2+m) (2-m-2 p)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 275, 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 = {866, 1823, 822, 372, 371} \[ \int \frac {(g x)^m \left (d^2-e^2 x^2\right )^p}{(d+e x)^3} \, dx=-\frac {e (g x)^{m+2} \left (d^2-e^2 x^2\right )^{p-2}}{g^2 (-m-2 p+2)}+\frac {3 d (g x)^{m+1} \left (d^2-e^2 x^2\right )^{p-2}}{g (-m-2 p+3)}-\frac {2 e (-2 m-3 p+2) (g x)^{m+2} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {m+2}{2},3-p,\frac {m+4}{2},\frac {e^2 x^2}{d^2}\right )}{d^4 g^2 (m+2) (-m-2 p+2)}-\frac {2 (2 m+p) (g x)^{m+1} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {m+1}{2},3-p,\frac {m+3}{2},\frac {e^2 x^2}{d^2}\right )}{d^3 g (m+1) (-m-2 p+3)} \]

[In]

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

[Out]

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

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]

Rule 866

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 1823

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])

Rubi steps \begin{align*} \text {integral}& = \int (g x)^m (d-e x)^3 \left (d^2-e^2 x^2\right )^{-3+p} \, dx \\ & = -\frac {e (g x)^{2+m} \left (d^2-e^2 x^2\right )^{-2+p}}{g^2 (2-m-2 p)}+\frac {\int (g x)^m \left (d^2-e^2 x^2\right )^{-3+p} \left (d^3 e^2 (2-m-2 p)-2 d^2 e^3 (2-2 m-3 p) x+3 d e^4 (2-m-2 p) x^2\right ) \, dx}{e^2 (2-m-2 p)} \\ & = \frac {3 d (g x)^{1+m} \left (d^2-e^2 x^2\right )^{-2+p}}{g (3-m-2 p)}-\frac {e (g x)^{2+m} \left (d^2-e^2 x^2\right )^{-2+p}}{g^2 (2-m-2 p)}+\frac {\int (g x)^m \left (-2 d^3 e^4 (2-m-2 p) (2 m+p)-2 d^2 e^5 (2-2 m-3 p) (3-m-2 p) x\right ) \left (d^2-e^2 x^2\right )^{-3+p} \, dx}{e^4 (2-m-2 p) (3-m-2 p)} \\ & = \frac {3 d (g x)^{1+m} \left (d^2-e^2 x^2\right )^{-2+p}}{g (3-m-2 p)}-\frac {e (g x)^{2+m} \left (d^2-e^2 x^2\right )^{-2+p}}{g^2 (2-m-2 p)}-\frac {\left (2 d^2 e (2-2 m-3 p)\right ) \int (g x)^{1+m} \left (d^2-e^2 x^2\right )^{-3+p} \, dx}{g (2-m-2 p)}-\frac {\left (2 d^3 (2 m+p)\right ) \int (g x)^m \left (d^2-e^2 x^2\right )^{-3+p} \, dx}{3-m-2 p} \\ & = \frac {3 d (g x)^{1+m} \left (d^2-e^2 x^2\right )^{-2+p}}{g (3-m-2 p)}-\frac {e (g x)^{2+m} \left (d^2-e^2 x^2\right )^{-2+p}}{g^2 (2-m-2 p)}-\frac {\left (2 e (2-2 m-3 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)^{1+m} \left (1-\frac {e^2 x^2}{d^2}\right )^{-3+p} \, dx}{d^4 g (2-m-2 p)}-\frac {\left (2 (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 )^{-3+p} \, dx}{d^3 (3-m-2 p)} \\ & = \frac {3 d (g x)^{1+m} \left (d^2-e^2 x^2\right )^{-2+p}}{g (3-m-2 p)}-\frac {e (g x)^{2+m} \left (d^2-e^2 x^2\right )^{-2+p}}{g^2 (2-m-2 p)}-\frac {2 (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},3-p;\frac {3+m}{2};\frac {e^2 x^2}{d^2}\right )}{d^3 g (1+m) (3-m-2 p)}-\frac {2 e (2-2 m-3 p) (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},3-p;\frac {4+m}{2};\frac {e^2 x^2}{d^2}\right )}{d^4 g^2 (2+m) (2-m-2 p)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.75 \[ \int \frac {(g x)^m \left (d^2-e^2 x^2\right )^p}{(d+e x)^3} \, dx=\frac {x (g x)^m \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \left (\frac {d^3 \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},3-p,\frac {3+m}{2},\frac {e^2 x^2}{d^2}\right )}{1+m}+e x \left (-\frac {3 d^2 \operatorname {Hypergeometric2F1}\left (\frac {2+m}{2},3-p,\frac {4+m}{2},\frac {e^2 x^2}{d^2}\right )}{2+m}+e x \left (\frac {3 d \operatorname {Hypergeometric2F1}\left (\frac {3+m}{2},3-p,\frac {5+m}{2},\frac {e^2 x^2}{d^2}\right )}{3+m}-\frac {e x \operatorname {Hypergeometric2F1}\left (\frac {4+m}{2},3-p,\frac {6+m}{2},\frac {e^2 x^2}{d^2}\right )}{4+m}\right )\right )\right )}{d^6} \]

[In]

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

[Out]

(x*(g*x)^m*(d^2 - e^2*x^2)^p*((d^3*Hypergeometric2F1[(1 + m)/2, 3 - p, (3 + m)/2, (e^2*x^2)/d^2])/(1 + m) + e*
x*((-3*d^2*Hypergeometric2F1[(2 + m)/2, 3 - p, (4 + m)/2, (e^2*x^2)/d^2])/(2 + m) + e*x*((3*d*Hypergeometric2F
1[(3 + m)/2, 3 - p, (5 + m)/2, (e^2*x^2)/d^2])/(3 + m) - (e*x*Hypergeometric2F1[(4 + m)/2, 3 - p, (6 + m)/2, (
e^2*x^2)/d^2])/(4 + m)))))/(d^6*(1 - (e^2*x^2)/d^2)^p)

Maple [F]

\[\int \frac {\left (g x \right )^{m} \left (-e^{2} x^{2}+d^{2}\right )^{p}}{\left (e x +d \right )^{3}}d x\]

[In]

int((g*x)^m*(-e^2*x^2+d^2)^p/(e*x+d)^3,x)

[Out]

int((g*x)^m*(-e^2*x^2+d^2)^p/(e*x+d)^3,x)

Fricas [F]

\[ \int \frac {(g x)^m \left (d^2-e^2 x^2\right )^p}{(d+e x)^3} \, dx=\int { \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p} \left (g x\right )^{m}}{{\left (e x + d\right )}^{3}} \,d x } \]

[In]

integrate((g*x)^m*(-e^2*x^2+d^2)^p/(e*x+d)^3,x, algorithm="fricas")

[Out]

integral((-e^2*x^2 + d^2)^p*(g*x)^m/(e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3), x)

Sympy [F]

\[ \int \frac {(g x)^m \left (d^2-e^2 x^2\right )^p}{(d+e x)^3} \, dx=\int \frac {\left (g x\right )^{m} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{p}}{\left (d + e x\right )^{3}}\, dx \]

[In]

integrate((g*x)**m*(-e**2*x**2+d**2)**p/(e*x+d)**3,x)

[Out]

Integral((g*x)**m*(-(-d + e*x)*(d + e*x))**p/(d + e*x)**3, x)

Maxima [F]

\[ \int \frac {(g x)^m \left (d^2-e^2 x^2\right )^p}{(d+e x)^3} \, dx=\int { \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p} \left (g x\right )^{m}}{{\left (e x + d\right )}^{3}} \,d x } \]

[In]

integrate((g*x)^m*(-e^2*x^2+d^2)^p/(e*x+d)^3,x, algorithm="maxima")

[Out]

integrate((-e^2*x^2 + d^2)^p*(g*x)^m/(e*x + d)^3, x)

Giac [F]

\[ \int \frac {(g x)^m \left (d^2-e^2 x^2\right )^p}{(d+e x)^3} \, dx=\int { \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p} \left (g x\right )^{m}}{{\left (e x + d\right )}^{3}} \,d x } \]

[In]

integrate((g*x)^m*(-e^2*x^2+d^2)^p/(e*x+d)^3,x, algorithm="giac")

[Out]

integrate((-e^2*x^2 + d^2)^p*(g*x)^m/(e*x + d)^3, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(g x)^m \left (d^2-e^2 x^2\right )^p}{(d+e x)^3} \, dx=\int \frac {{\left (d^2-e^2\,x^2\right )}^p\,{\left (g\,x\right )}^m}{{\left (d+e\,x\right )}^3} \,d x \]

[In]

int(((d^2 - e^2*x^2)^p*(g*x)^m)/(d + e*x)^3,x)

[Out]

int(((d^2 - e^2*x^2)^p*(g*x)^m)/(d + e*x)^3, x)