3.2.47 \(\int \frac {1}{x^4 (d+e x) (d^2-e^2 x^2)^{5/2}} \, dx\) [147]

3.2.47.1 Optimal result

Integrand size = 27, antiderivative size = 215 \[ \int \frac {1}{x^4 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {8 d-7 e x}{15 d^4 x^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{5 d^2 x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {48 d-35 e x}{15 d^6 x^3 \sqrt {d^2-e^2 x^2}}-\frac {64 \sqrt {d^2-e^2 x^2}}{15 d^7 x^3}+\frac {7 e \sqrt {d^2-e^2 x^2}}{2 d^8 x^2}-\frac {128 e^2 \sqrt {d^2-e^2 x^2}}{15 d^9 x}+\frac {7 e^3 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^9} \]

output

1/15*(-7*e*x+8*d)/d^4/x^3/(-e^2*x^2+d^2)^(3/2)+1/5/d^2/x^3/(e*x+d)/(-e^2*x 
^2+d^2)^(3/2)+7/2*e^3*arctanh((-e^2*x^2+d^2)^(1/2)/d)/d^9+1/15*(-35*e*x+48 
*d)/d^6/x^3/(-e^2*x^2+d^2)^(1/2)-64/15*(-e^2*x^2+d^2)^(1/2)/d^7/x^3+7/2*e* 
(-e^2*x^2+d^2)^(1/2)/d^8/x^2-128/15*e^2*(-e^2*x^2+d^2)^(1/2)/d^9/x
 

3.2.47.2 Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.80 \[ \int \frac {1}{x^4 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=-\frac {\frac {d \sqrt {d^2-e^2 x^2} \left (10 d^7-5 d^6 e x+75 d^5 e^2 x^2+236 d^4 e^3 x^3-244 d^3 e^4 x^4-489 d^2 e^5 x^5+151 d e^6 x^6+256 e^7 x^7\right )}{x^3 (d-e x)^2 (d+e x)^3}-105 \sqrt {d^2} e^3 \log (x)+105 \sqrt {d^2} e^3 \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right )}{30 d^{10}} \]

input

Integrate[1/(x^4*(d + e*x)*(d^2 - e^2*x^2)^(5/2)),x]
 

output

-1/30*((d*Sqrt[d^2 - e^2*x^2]*(10*d^7 - 5*d^6*e*x + 75*d^5*e^2*x^2 + 236*d 
^4*e^3*x^3 - 244*d^3*e^4*x^4 - 489*d^2*e^5*x^5 + 151*d*e^6*x^6 + 256*e^7*x 
^7))/(x^3*(d - e*x)^2*(d + e*x)^3) - 105*Sqrt[d^2]*e^3*Log[x] + 105*Sqrt[d 
^2]*e^3*Log[Sqrt[d^2] - Sqrt[d^2 - e^2*x^2]])/d^10
 

3.2.47.3 Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.11, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {569, 25, 532, 25, 2336, 27, 2338, 2338, 27, 534, 243, 73, 221}

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.

\(\displaystyle \int \frac {1}{x^4 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 569

\(\displaystyle \frac {1}{5 d^2 x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\int -\frac {8 d-7 e x}{x^4 \left (d^2-e^2 x^2\right )^{5/2}}dx}{5 d^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {8 d-7 e x}{x^4 \left (d^2-e^2 x^2\right )^{5/2}}dx}{5 d^2}+\frac {1}{5 d^2 x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 532

\(\displaystyle \frac {-\frac {\int -\frac {\frac {16 e^4 x^4}{d^3}-\frac {21 e^3 x^3}{d^2}+\frac {24 e^2 x^2}{d}-21 e x+24 d}{x^4 \left (d^2-e^2 x^2\right )^{3/2}}dx}{3 d^2}-\frac {e^3 (7 d-8 e x)}{3 d^5 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d^2}+\frac {1}{5 d^2 x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {\frac {16 e^4 x^4}{d^3}-\frac {21 e^3 x^3}{d^2}+\frac {24 e^2 x^2}{d}-21 e x+24 d}{x^4 \left (d^2-e^2 x^2\right )^{3/2}}dx}{3 d^2}-\frac {e^3 (7 d-8 e x)}{3 d^5 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d^2}+\frac {1}{5 d^2 x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 2336

\(\displaystyle \frac {\frac {-\frac {\int -\frac {3 \left (-\frac {14 e^3 x^3}{d^2}+\frac {16 e^2 x^2}{d}-7 e x+8 d\right )}{x^4 \sqrt {d^2-e^2 x^2}}dx}{d^2}-\frac {2 e^3 (21 d-32 e x)}{d^5 \sqrt {d^2-e^2 x^2}}}{3 d^2}-\frac {e^3 (7 d-8 e x)}{3 d^5 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d^2}+\frac {1}{5 d^2 x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {3 \int \frac {-\frac {14 e^3 x^3}{d^2}+\frac {16 e^2 x^2}{d}-7 e x+8 d}{x^4 \sqrt {d^2-e^2 x^2}}dx}{d^2}-\frac {2 e^3 (21 d-32 e x)}{d^5 \sqrt {d^2-e^2 x^2}}}{3 d^2}-\frac {e^3 (7 d-8 e x)}{3 d^5 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d^2}+\frac {1}{5 d^2 x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 2338

\(\displaystyle \frac {\frac {\frac {3 \left (-\frac {\int \frac {42 x^2 e^3-64 d x e^2+21 d^2 e}{x^3 \sqrt {d^2-e^2 x^2}}dx}{3 d^2}-\frac {8 \sqrt {d^2-e^2 x^2}}{3 d x^3}\right )}{d^2}-\frac {2 e^3 (21 d-32 e x)}{d^5 \sqrt {d^2-e^2 x^2}}}{3 d^2}-\frac {e^3 (7 d-8 e x)}{3 d^5 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d^2}+\frac {1}{5 d^2 x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 2338

\(\displaystyle \frac {\frac {\frac {3 \left (-\frac {-\frac {\int \frac {d^2 e^2 (128 d-105 e x)}{x^2 \sqrt {d^2-e^2 x^2}}dx}{2 d^2}-\frac {21 e \sqrt {d^2-e^2 x^2}}{2 x^2}}{3 d^2}-\frac {8 \sqrt {d^2-e^2 x^2}}{3 d x^3}\right )}{d^2}-\frac {2 e^3 (21 d-32 e x)}{d^5 \sqrt {d^2-e^2 x^2}}}{3 d^2}-\frac {e^3 (7 d-8 e x)}{3 d^5 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d^2}+\frac {1}{5 d^2 x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {3 \left (-\frac {-\frac {1}{2} e^2 \int \frac {128 d-105 e x}{x^2 \sqrt {d^2-e^2 x^2}}dx-\frac {21 e \sqrt {d^2-e^2 x^2}}{2 x^2}}{3 d^2}-\frac {8 \sqrt {d^2-e^2 x^2}}{3 d x^3}\right )}{d^2}-\frac {2 e^3 (21 d-32 e x)}{d^5 \sqrt {d^2-e^2 x^2}}}{3 d^2}-\frac {e^3 (7 d-8 e x)}{3 d^5 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d^2}+\frac {1}{5 d^2 x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 534

\(\displaystyle \frac {\frac {\frac {3 \left (-\frac {-\frac {1}{2} e^2 \left (-105 e \int \frac {1}{x \sqrt {d^2-e^2 x^2}}dx-\frac {128 \sqrt {d^2-e^2 x^2}}{d x}\right )-\frac {21 e \sqrt {d^2-e^2 x^2}}{2 x^2}}{3 d^2}-\frac {8 \sqrt {d^2-e^2 x^2}}{3 d x^3}\right )}{d^2}-\frac {2 e^3 (21 d-32 e x)}{d^5 \sqrt {d^2-e^2 x^2}}}{3 d^2}-\frac {e^3 (7 d-8 e x)}{3 d^5 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d^2}+\frac {1}{5 d^2 x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {\frac {\frac {3 \left (-\frac {-\frac {1}{2} e^2 \left (-\frac {105}{2} e \int \frac {1}{x^2 \sqrt {d^2-e^2 x^2}}dx^2-\frac {128 \sqrt {d^2-e^2 x^2}}{d x}\right )-\frac {21 e \sqrt {d^2-e^2 x^2}}{2 x^2}}{3 d^2}-\frac {8 \sqrt {d^2-e^2 x^2}}{3 d x^3}\right )}{d^2}-\frac {2 e^3 (21 d-32 e x)}{d^5 \sqrt {d^2-e^2 x^2}}}{3 d^2}-\frac {e^3 (7 d-8 e x)}{3 d^5 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d^2}+\frac {1}{5 d^2 x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {\frac {3 \left (-\frac {-\frac {1}{2} e^2 \left (\frac {105 \int \frac {1}{\frac {d^2}{e^2}-\frac {x^4}{e^2}}d\sqrt {d^2-e^2 x^2}}{e}-\frac {128 \sqrt {d^2-e^2 x^2}}{d x}\right )-\frac {21 e \sqrt {d^2-e^2 x^2}}{2 x^2}}{3 d^2}-\frac {8 \sqrt {d^2-e^2 x^2}}{3 d x^3}\right )}{d^2}-\frac {2 e^3 (21 d-32 e x)}{d^5 \sqrt {d^2-e^2 x^2}}}{3 d^2}-\frac {e^3 (7 d-8 e x)}{3 d^5 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d^2}+\frac {1}{5 d^2 x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\frac {3 \left (-\frac {-\frac {1}{2} e^2 \left (\frac {105 e \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d}-\frac {128 \sqrt {d^2-e^2 x^2}}{d x}\right )-\frac {21 e \sqrt {d^2-e^2 x^2}}{2 x^2}}{3 d^2}-\frac {8 \sqrt {d^2-e^2 x^2}}{3 d x^3}\right )}{d^2}-\frac {2 e^3 (21 d-32 e x)}{d^5 \sqrt {d^2-e^2 x^2}}}{3 d^2}-\frac {e^3 (7 d-8 e x)}{3 d^5 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d^2}+\frac {1}{5 d^2 x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}\)

input

Int[1/(x^4*(d + e*x)*(d^2 - e^2*x^2)^(5/2)),x]
 

output

1/(5*d^2*x^3*(d + e*x)*(d^2 - e^2*x^2)^(3/2)) + (-1/3*(e^3*(7*d - 8*e*x))/ 
(d^5*(d^2 - e^2*x^2)^(3/2)) + ((-2*e^3*(21*d - 32*e*x))/(d^5*Sqrt[d^2 - e^ 
2*x^2]) + (3*((-8*Sqrt[d^2 - e^2*x^2])/(3*d*x^3) - ((-21*e*Sqrt[d^2 - e^2* 
x^2])/(2*x^2) - (e^2*((-128*Sqrt[d^2 - e^2*x^2])/(d*x) + (105*e*ArcTanh[Sq 
rt[d^2 - e^2*x^2]/d])/d))/2)/(3*d^2)))/d^2)/(3*d^2))/(5*d^2)
 

3.2.47.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 532

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Coe 
ff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Pol 
ynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x) 
*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   Int[x^m 
*(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*(Qx/x^m) + e*((2*p + 3)/x^m), 
x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && ILtQ[m, 0] && LtQ[p, 
 -1] && IntegerQ[2*p]
 

rule 534

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

rule 569

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

rule 2336

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ 
{Q = PolynomialQuotient[(c*x)^m*Pq, a + b*x^2, x], f = Coeff[PolynomialRema 
inder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[(c*x) 
^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b*f*x)*((a + b*x^2)^(p + 1)/(2*a* 
b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   Int[(c*x)^m*(a + b*x^2)^(p + 1)*Ex 
pandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x], x]] /; F 
reeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]
 

rule 2338

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{ 
Q = PolynomialQuotient[Pq, c*x, x], R = PolynomialRemainder[Pq, c*x, x]}, S 
imp[R*(c*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] + Simp[1/(a*c*( 
m + 1))   Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*( 
m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && Lt 
Q[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])
 

3.2.47.4 Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.45

method	result	size
risch	\(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (22 e^{2} x^{2}-3 d e x +2 d^{2}\right )}{6 d^{9} x^{3}}+\frac {7 e^{3} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 d^{8} \sqrt {d^{2}}}-\frac {331 e^{2} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{80 d^{9} \left (x +\frac {d}{e}\right )}-\frac {35 e^{2} \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{48 d^{9} \left (x -\frac {d}{e}\right )}-\frac {9 e \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{20 d^{8} \left (x +\frac {d}{e}\right )^{2}}-\frac {\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{20 d^{7} \left (x +\frac {d}{e}\right )^{3}}+\frac {e \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{24 d^{8} \left (x -\frac {d}{e}\right )^{2}}\)	\(312\)
default	\(\frac {-\frac {1}{3 d^{2} x^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {2 e^{2} \left (-\frac {1}{d^{2} x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {4 e^{2} \left (\frac {x}{3 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}\right )}{d^{2}}\right )}{d^{2}}}{d}+\frac {e^{2} \left (-\frac {1}{d^{2} x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {4 e^{2} \left (\frac {x}{3 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}\right )}{d^{2}}\right )}{d^{3}}-\frac {e^{3} \left (\frac {1}{3 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {\frac {1}{d^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{d^{2} \sqrt {d^{2}}}}{d^{2}}\right )}{d^{4}}-\frac {e \left (-\frac {1}{2 d^{2} x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {5 e^{2} \left (\frac {1}{3 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {\frac {1}{d^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{d^{2} \sqrt {d^{2}}}}{d^{2}}\right )}{2 d^{2}}\right )}{d^{2}}+\frac {e^{3} \left (-\frac {1}{5 d e \left (x +\frac {d}{e}\right ) \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}+\frac {4 e \left (-\frac {-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e}{6 d^{2} e^{2} \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e}{3 e^{2} d^{4} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{5 d}\right )}{d^{4}}\)	\(569\)

input

int(1/x^4/(e*x+d)/(-e^2*x^2+d^2)^(5/2),x,method=_RETURNVERBOSE)
 

output

-1/6*(-e^2*x^2+d^2)^(1/2)*(22*e^2*x^2-3*d*e*x+2*d^2)/d^9/x^3+7/2*e^3/d^8/( 
d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)-331/80*e^2/d^9 
/(x+d/e)*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)-35/48*e^2/d^9/(x-d/e)*(-(x-d 
/e)^2*e^2-2*d*e*(x-d/e))^(1/2)-9/20/d^8*e/(x+d/e)^2*(-(x+d/e)^2*e^2+2*d*e* 
(x+d/e))^(1/2)-1/20/d^7/(x+d/e)^3*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)+1/2 
4/d^8*e/(x-d/e)^2*(-(x-d/e)^2*e^2-2*d*e*(x-d/e))^(1/2)
 

3.2.47.5 Fricas [A] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.38 \[ \int \frac {1}{x^4 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=-\frac {116 \, e^{8} x^{8} + 116 \, d e^{7} x^{7} - 232 \, d^{2} e^{6} x^{6} - 232 \, d^{3} e^{5} x^{5} + 116 \, d^{4} e^{4} x^{4} + 116 \, d^{5} e^{3} x^{3} + 105 \, {\left (e^{8} x^{8} + d e^{7} x^{7} - 2 \, d^{2} e^{6} x^{6} - 2 \, d^{3} e^{5} x^{5} + d^{4} e^{4} x^{4} + d^{5} e^{3} x^{3}\right )} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + {\left (256 \, e^{7} x^{7} + 151 \, d e^{6} x^{6} - 489 \, d^{2} e^{5} x^{5} - 244 \, d^{3} e^{4} x^{4} + 236 \, d^{4} e^{3} x^{3} + 75 \, d^{5} e^{2} x^{2} - 5 \, d^{6} e x + 10 \, d^{7}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{30 \, {\left (d^{9} e^{5} x^{8} + d^{10} e^{4} x^{7} - 2 \, d^{11} e^{3} x^{6} - 2 \, d^{12} e^{2} x^{5} + d^{13} e x^{4} + d^{14} x^{3}\right )}} \]

input

integrate(1/x^4/(e*x+d)/(-e^2*x^2+d^2)^(5/2),x, algorithm="fricas")
 

output

-1/30*(116*e^8*x^8 + 116*d*e^7*x^7 - 232*d^2*e^6*x^6 - 232*d^3*e^5*x^5 + 1 
16*d^4*e^4*x^4 + 116*d^5*e^3*x^3 + 105*(e^8*x^8 + d*e^7*x^7 - 2*d^2*e^6*x^ 
6 - 2*d^3*e^5*x^5 + d^4*e^4*x^4 + d^5*e^3*x^3)*log(-(d - sqrt(-e^2*x^2 + d 
^2))/x) + (256*e^7*x^7 + 151*d*e^6*x^6 - 489*d^2*e^5*x^5 - 244*d^3*e^4*x^4 
 + 236*d^4*e^3*x^3 + 75*d^5*e^2*x^2 - 5*d^6*e*x + 10*d^7)*sqrt(-e^2*x^2 + 
d^2))/(d^9*e^5*x^8 + d^10*e^4*x^7 - 2*d^11*e^3*x^6 - 2*d^12*e^2*x^5 + d^13 
*e*x^4 + d^14*x^3)
 

3.2.47.6 Sympy [F]

\[ \int \frac {1}{x^4 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\int \frac {1}{x^{4} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {5}{2}} \left (d + e x\right )}\, dx \]

input

integrate(1/x**4/(e*x+d)/(-e**2*x**2+d**2)**(5/2),x)
 

output

Integral(1/(x**4*(-(-d + e*x)*(d + e*x))**(5/2)*(d + e*x)), x)
 

3.2.47.7 Maxima [F]

\[ \int \frac {1}{x^4 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} {\left (e x + d\right )} x^{4}} \,d x } \]

input

integrate(1/x^4/(e*x+d)/(-e^2*x^2+d^2)^(5/2),x, algorithm="maxima")
 

output

integrate(1/((-e^2*x^2 + d^2)^(5/2)*(e*x + d)*x^4), x)
 

3.2.47.8 Giac [F]

\[ \int \frac {1}{x^4 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} {\left (e x + d\right )} x^{4}} \,d x } \]

input

integrate(1/x^4/(e*x+d)/(-e^2*x^2+d^2)^(5/2),x, algorithm="giac")
 

output

integrate(1/((-e^2*x^2 + d^2)^(5/2)*(e*x + d)*x^4), x)
 

3.2.47.9 Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^4 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\int \frac {1}{x^4\,{\left (d^2-e^2\,x^2\right )}^{5/2}\,\left (d+e\,x\right )} \,d x \]

input

int(1/(x^4*(d^2 - e^2*x^2)^(5/2)*(d + e*x)),x)
 

output

int(1/(x^4*(d^2 - e^2*x^2)^(5/2)*(d + e*x)), x)