3.4.11 \(\int \frac {(g x)^m (d^2-e^2 x^2)^p}{(d+e x)^2} \, dx\) [311]

3.4.11.1 Optimal result

Integrand size = 27, antiderivative size = 214 \[ \int \frac {(g x)^m \left (d^2-e^2 x^2\right )^p}{(d+e x)^2} \, dx=\frac {(g x)^{1+m} \left (d^2-e^2 x^2\right )^{-1+p}}{g (1-m-2 p)}-\frac {2 (m+p) (g x)^{1+m} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},2-p,\frac {3+m}{2},\frac {e^2 x^2}{d^2}\right )}{d^2 g (1+m) (1-m-2 p)}-\frac {2 e (g x)^{2+m} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {2+m}{2},2-p,\frac {4+m}{2},\frac {e^2 x^2}{d^2}\right )}{d^3 g^2 (2+m)} \]

output

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

3.4.11.2 Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.84 \[ \int \frac {(g x)^m \left (d^2-e^2 x^2\right )^p}{(d+e x)^2} \, dx=\frac {x (g x)^m \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \left (d^2 \left (6+5 m+m^2\right ) \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},2-p,\frac {3+m}{2},\frac {e^2 x^2}{d^2}\right )-e (1+m) x \left (2 d (3+m) \operatorname {Hypergeometric2F1}\left (\frac {2+m}{2},2-p,\frac {4+m}{2},\frac {e^2 x^2}{d^2}\right )-e (2+m) x \operatorname {Hypergeometric2F1}\left (\frac {3+m}{2},2-p,\frac {5+m}{2},\frac {e^2 x^2}{d^2}\right )\right )\right )}{d^4 (1+m) (2+m) (3+m)} \]

input

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

output

(x*(g*x)^m*(d^2 - e^2*x^2)^p*(d^2*(6 + 5*m + m^2)*Hypergeometric2F1[(1 + m 
)/2, 2 - p, (3 + m)/2, (e^2*x^2)/d^2] - e*(1 + m)*x*(2*d*(3 + m)*Hypergeom 
etric2F1[(2 + m)/2, 2 - p, (4 + m)/2, (e^2*x^2)/d^2] - e*(2 + m)*x*Hyperge 
ometric2F1[(3 + m)/2, 2 - p, (5 + m)/2, (e^2*x^2)/d^2])))/(d^4*(1 + m)*(2 
+ m)*(3 + m)*(1 - (e^2*x^2)/d^2)^p)
 

3.4.11.3 Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {570, 559, 27, 557, 279, 278}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 570

\(\displaystyle \int (d-e x)^2 (g x)^m \left (d^2-e^2 x^2\right )^{p-2}dx\)

\(\Big \downarrow \) 559

\(\displaystyle \frac {\int -2 d e^2 (g x)^m (d (m+p)+e (-m-2 p+1) x) \left (d^2-e^2 x^2\right )^{p-2}dx}{e^2 (-m-2 p+1)}+\frac {(g x)^{m+1} \left (d^2-e^2 x^2\right )^{p-1}}{g (-m-2 p+1)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(g x)^{m+1} \left (d^2-e^2 x^2\right )^{p-1}}{g (-m-2 p+1)}-\frac {2 d \int (g x)^m (d (m+p)+e (-m-2 p+1) x) \left (d^2-e^2 x^2\right )^{p-2}dx}{-m-2 p+1}\)

\(\Big \downarrow \) 557

\(\displaystyle \frac {(g x)^{m+1} \left (d^2-e^2 x^2\right )^{p-1}}{g (-m-2 p+1)}-\frac {2 d \left (d (m+p) \int (g x)^m \left (d^2-e^2 x^2\right )^{p-2}dx+\frac {e (-m-2 p+1) \int (g x)^{m+1} \left (d^2-e^2 x^2\right )^{p-2}dx}{g}\right )}{-m-2 p+1}\)

\(\Big \downarrow \) 279

\(\displaystyle \frac {(g x)^{m+1} \left (d^2-e^2 x^2\right )^{p-1}}{g (-m-2 p+1)}-\frac {2 d \left (\frac {e (-m-2 p+1) \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \int (g x)^{m+1} \left (1-\frac {e^2 x^2}{d^2}\right )^{p-2}dx}{d^4 g}+\frac {(m+p) \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \int (g x)^m \left (1-\frac {e^2 x^2}{d^2}\right )^{p-2}dx}{d^3}\right )}{-m-2 p+1}\)

\(\Big \downarrow \) 278

\(\displaystyle \frac {(g x)^{m+1} \left (d^2-e^2 x^2\right )^{p-1}}{g (-m-2 p+1)}-\frac {2 d \left (\frac {e (-m-2 p+1) (g x)^{m+2} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {m+2}{2},2-p,\frac {m+4}{2},\frac {e^2 x^2}{d^2}\right )}{d^4 g^2 (m+2)}+\frac {(m+p) (g x)^{m+1} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {m+1}{2},2-p,\frac {m+3}{2},\frac {e^2 x^2}{d^2}\right )}{d^3 g (m+1)}\right )}{-m-2 p+1}\)

input

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

output

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

3.4.11.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 278

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( 
c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( 
-b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && (ILtQ[p, 0 
] || GtQ[a, 0])
 

rule 279

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^IntP 
art[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^FracPart[p])   Int[(c*x)^m* 
(1 + b*(x^2/a))^p, x], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && 
!(ILtQ[p, 0] || GtQ[a, 0])
 

rule 557

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym 
bol] :> Simp[c   Int[(e*x)^m*(a + b*x^2)^p, x], x] + Simp[d/e   Int[(e*x)^( 
m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x]
 

rule 559

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

rule 570

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), 
x_Symbol] :> Simp[c^(2*n)/a^n   Int[(e*x)^m*((a + b*x^2)^(n + p)/(c - d*x)^ 
n), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b*c^2 + a*d^2, 0] && I 
LtQ[n, -1] &&  !(IGtQ[m, 0] && ILtQ[m + n, 0] &&  !GtQ[p, 1])
 

3.4.11.4 Maple [F]

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

input

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

output

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

3.4.11.5 Fricas [F]

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

input

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

output

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

3.4.11.6 Sympy [F]

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

input

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

output

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

3.4.11.7 Maxima [F]

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

input

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

output

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

3.4.11.8 Giac [F]

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

input

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

output

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

3.4.11.9 Mupad [F(-1)]

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

input

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

output

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