3.3.91 \(\int \frac {(d^2-e^2 x^2)^p}{x (d+e x)^3} \, dx\) [291]

3.3.91.1 Optimal result

Integrand size = 25, antiderivative size = 175 \[ \int \frac {\left (d^2-e^2 x^2\right )^p}{x (d+e x)^3} \, dx=\frac {2 d \left (d^2-e^2 x^2\right )^{-2+p}}{2-p}-\frac {e x \left (d^2-e^2 x^2\right )^{-2+p}}{3-2 p}-\frac {2 e (4-3 p) x \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},3-p,\frac {3}{2},\frac {e^2 x^2}{d^2}\right )}{d^4 (3-2 p)}+\frac {\left (d^2-e^2 x^2\right )^{-1+p} \operatorname {Hypergeometric2F1}\left (1,-1+p,p,1-\frac {e^2 x^2}{d^2}\right )}{2 d (1-p)} \]

output

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

3.3.91.2 Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.87 \[ \int \frac {\left (d^2-e^2 x^2\right )^p}{x (d+e x)^3} \, dx=\frac {2^{-3+p} \left (1-\frac {d^2}{e^2 x^2}\right )^{-p} \left (1+\frac {e x}{d}\right )^{-p} \left (d^2-e^2 x^2\right )^p \left (4 p \left (1-\frac {d^2}{e^2 x^2}\right )^p (d-e x) \operatorname {Hypergeometric2F1}\left (1-p,1+p,2+p,\frac {d-e x}{2 d}\right )+2 p \left (1-\frac {d^2}{e^2 x^2}\right )^p (d-e x) \operatorname {Hypergeometric2F1}\left (2-p,1+p,2+p,\frac {d-e x}{2 d}\right )+d p \left (1-\frac {d^2}{e^2 x^2}\right )^p \operatorname {Hypergeometric2F1}\left (3-p,1+p,2+p,\frac {d-e x}{2 d}\right )-e p \left (1-\frac {d^2}{e^2 x^2}\right )^p x \operatorname {Hypergeometric2F1}\left (3-p,1+p,2+p,\frac {d-e x}{2 d}\right )+4 d \left (\frac {1}{2}+\frac {e x}{2 d}\right )^p \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,\frac {d^2}{e^2 x^2}\right )+4 d p \left (\frac {1}{2}+\frac {e x}{2 d}\right )^p \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,\frac {d^2}{e^2 x^2}\right )\right )}{d^4 p (1+p)} \]

input

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

output

(2^(-3 + p)*(d^2 - e^2*x^2)^p*(4*p*(1 - d^2/(e^2*x^2))^p*(d - e*x)*Hyperge 
ometric2F1[1 - p, 1 + p, 2 + p, (d - e*x)/(2*d)] + 2*p*(1 - d^2/(e^2*x^2)) 
^p*(d - e*x)*Hypergeometric2F1[2 - p, 1 + p, 2 + p, (d - e*x)/(2*d)] + d*p 
*(1 - d^2/(e^2*x^2))^p*Hypergeometric2F1[3 - p, 1 + p, 2 + p, (d - e*x)/(2 
*d)] - e*p*(1 - d^2/(e^2*x^2))^p*x*Hypergeometric2F1[3 - p, 1 + p, 2 + p, 
(d - e*x)/(2*d)] + 4*d*(1/2 + (e*x)/(2*d))^p*Hypergeometric2F1[-p, -p, 1 - 
 p, d^2/(e^2*x^2)] + 4*d*p*(1/2 + (e*x)/(2*d))^p*Hypergeometric2F1[-p, -p, 
 1 - p, d^2/(e^2*x^2)]))/(d^4*p*(1 + p)*(1 - d^2/(e^2*x^2))^p*(1 + (e*x)/d 
)^p)
 

3.3.91.3 Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.01, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {570, 543, 299, 238, 237, 354, 27, 88, 75}

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.

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

\(\Big \downarrow \) 570

\(\displaystyle \int \frac {(d-e x)^3 \left (d^2-e^2 x^2\right )^{p-3}}{x}dx\)

\(\Big \downarrow \) 543

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

\(\Big \downarrow \) 299

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

\(\Big \downarrow \) 238

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

\(\Big \downarrow \) 237

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

\(\Big \downarrow \) 354

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 88

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

\(\Big \downarrow \) 75

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

input

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

output

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

3.3.91.3.1 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 75

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x 
)^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))*Hypergeometric2F1[-m, n + 1, n + 2, 1 + 
 d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (IntegerQ[m] 
 || GtQ[-d/(b*c), 0])
 

rule 88

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^Simplify[p + 1], 
 x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] &&  !RationalQ[p] && SumSimpl 
erQ[p, 1]
 

rule 237

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

rule 238

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^2) 
^FracPart[p]/(1 + b*(x^2/a))^FracPart[p])   Int[(1 + b*(x^2/a))^p, x], x] / 
; FreeQ[{a, b, p}, x] &&  !IntegerQ[2*p] &&  !GtQ[a, 0]
 

rule 299

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x 
*((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 
*p + 3))   Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && NeQ[2*p + 3, 0]
 

rule 354

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S 
ymbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x 
, x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ 
[(m - 1)/2]
 

rule 543

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> Module[{k}, Int[x^m*Sum[Binomial[n, 2*k]*c^(n - 2*k)*d^(2*k)*x^(2*k), 
 {k, 0, n/2}]*(a + b*x^2)^p, x] + Int[x^(m + 1)*Sum[Binomial[n, 2*k + 1]*c^ 
(n - 2*k - 1)*d^(2*k + 1)*x^(2*k), {k, 0, (n - 1)/2}]*(a + b*x^2)^p, x]] /; 
 FreeQ[{a, b, c, d, p}, x] && IGtQ[n, 1] && IntegerQ[m] &&  !IntegerQ[2*p] 
&&  !(EqQ[m, 1] && EqQ[b*c^2 + a*d^2, 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])
 

3.3.91.4 Maple [F]

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

input

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

output

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

3.3.91.5 Fricas [F]

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

input

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

output

integral((-e^2*x^2 + d^2)^p/(e^3*x^4 + 3*d*e^2*x^3 + 3*d^2*e*x^2 + d^3*x), 
 x)
 

3.3.91.6 Sympy [F]

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

input

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

output

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

3.3.91.7 Maxima [F]

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

input

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

output

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

3.3.91.8 Giac [F]

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

input

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

output

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

3.3.91.9 Mupad [F(-1)]

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

input

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

output

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