\(\int \frac {(d^2-e^2 x^2)^{7/2}}{(d+e x)^7} \, dx\) [69]

Optimal result

Integrand size = 24, antiderivative size = 141 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^7} \, dx=-\frac {7 \sqrt {d^2-e^2 x^2}}{3 e}-\frac {28 d \sqrt {d^2-e^2 x^2}}{3 e (d+e x)}+\frac {14 \left (d^2-e^2 x^2\right )^{5/2}}{15 e (d+e x)^4}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{5 e (d+e x)^6}-\frac {7 d \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e} \] Output:

-7/3*(-e^2*x^2+d^2)^(1/2)/e-28/3*d*(-e^2*x^2+d^2)^(1/2)/e/(e*x+d)+14/15*(- 
e^2*x^2+d^2)^(5/2)/e/(e*x+d)^4-2/5*(-e^2*x^2+d^2)^(7/2)/e/(e*x+d)^6-7*d*ar 
ctan(e*x/(-e^2*x^2+d^2)^(1/2))/e
 

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.73 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^7} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-167 d^3-381 d^2 e x-277 d e^2 x^2-15 e^3 x^3\right )}{15 e (d+e x)^3}+\frac {7 d \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{\sqrt {-e^2}} \] Input:

Integrate[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^7,x]
 

Output:

(Sqrt[d^2 - e^2*x^2]*(-167*d^3 - 381*d^2*e*x - 277*d*e^2*x^2 - 15*e^3*x^3) 
)/(15*e*(d + e*x)^3) + (7*d*Log[-(Sqrt[-e^2]*x) + Sqrt[d^2 - e^2*x^2]])/Sq 
rt[-e^2]
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.04, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {465, 465, 463, 455, 224, 216}

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.

\(\displaystyle \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^7} \, dx\)

\(\Big \downarrow \) 465

\(\displaystyle -\frac {7}{5} \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^5}dx-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{5 e (d+e x)^6}\)

\(\Big \downarrow \) 465

\(\displaystyle -\frac {7}{5} \left (-\frac {5}{3} \int \frac {\left (d^2-e^2 x^2\right )^{3/2}}{(d+e x)^3}dx-\frac {2 \left (d^2-e^2 x^2\right )^{5/2}}{3 e (d+e x)^4}\right )-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{5 e (d+e x)^6}\)

\(\Big \downarrow \) 463

\(\displaystyle -\frac {7}{5} \left (-\frac {5}{3} \left (-\int \frac {3 d-e x}{\sqrt {d^2-e^2 x^2}}dx-\frac {4 d \sqrt {d^2-e^2 x^2}}{e (d+e x)}\right )-\frac {2 \left (d^2-e^2 x^2\right )^{5/2}}{3 e (d+e x)^4}\right )-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{5 e (d+e x)^6}\)

\(\Big \downarrow \) 455

\(\displaystyle -\frac {7}{5} \left (-\frac {5}{3} \left (-3 d \int \frac {1}{\sqrt {d^2-e^2 x^2}}dx-\frac {4 d \sqrt {d^2-e^2 x^2}}{e (d+e x)}-\frac {\sqrt {d^2-e^2 x^2}}{e}\right )-\frac {2 \left (d^2-e^2 x^2\right )^{5/2}}{3 e (d+e x)^4}\right )-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{5 e (d+e x)^6}\)

\(\Big \downarrow \) 224

\(\displaystyle -\frac {7}{5} \left (-\frac {5}{3} \left (-3 d \int \frac {1}{\frac {e^2 x^2}{d^2-e^2 x^2}+1}d\frac {x}{\sqrt {d^2-e^2 x^2}}-\frac {4 d \sqrt {d^2-e^2 x^2}}{e (d+e x)}-\frac {\sqrt {d^2-e^2 x^2}}{e}\right )-\frac {2 \left (d^2-e^2 x^2\right )^{5/2}}{3 e (d+e x)^4}\right )-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{5 e (d+e x)^6}\)

\(\Big \downarrow \) 216

\(\displaystyle -\frac {7}{5} \left (-\frac {5}{3} \left (-\frac {3 d \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e}-\frac {4 d \sqrt {d^2-e^2 x^2}}{e (d+e x)}-\frac {\sqrt {d^2-e^2 x^2}}{e}\right )-\frac {2 \left (d^2-e^2 x^2\right )^{5/2}}{3 e (d+e x)^4}\right )-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{5 e (d+e x)^6}\)

Input:

Int[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^7,x]
 

Output:

(-2*(d^2 - e^2*x^2)^(7/2))/(5*e*(d + e*x)^6) - (7*((-2*(d^2 - e^2*x^2)^(5/ 
2))/(3*e*(d + e*x)^4) - (5*(-(Sqrt[d^2 - e^2*x^2]/e) - (4*d*Sqrt[d^2 - e^2 
*x^2])/(e*(d + e*x)) - (3*d*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e))/3))/5
 

Defintions of rubi rules used

rule 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 224

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

rule 455

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

rule 463

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
(-(-c)^(-n - 2))*d^(2*n + 3)*(Sqrt[a + b*x^2]/(2^(n + 1)*b^(n + 2)*(c + d*x 
))), x] - Simp[d^(2*n + 2)/b^(n + 1)   Int[(1/Sqrt[a + b*x^2])*ExpandToSum[ 
(2^(-n - 1)*(-c)^(-n - 1) - (-c + d*x)^(-n - 1))/(c + d*x), x], x], x] /; F 
reeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0] && ILtQ[n, 0] && EqQ[n + p, 
-3/2]
 

rule 465

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

Maple [A] (verified)

Time = 0.87 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.30

method	result	size
risch	\(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{e}-\frac {7 d \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{\sqrt {e^{2}}}-\frac {232 d \sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}{15 e^{2} \left (x +\frac {d}{e}\right )}+\frac {128 d^{2} \sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}{15 e^{3} \left (x +\frac {d}{e}\right )^{2}}-\frac {16 d^{3} \sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}{5 e^{4} \left (x +\frac {d}{e}\right )^{3}}\)	\(184\)
default	\(\frac {-\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{5 d e \left (x +\frac {d}{e}\right )^{7}}-\frac {2 e \left (-\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{3 d e \left (x +\frac {d}{e}\right )^{6}}-\frac {e \left (-\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{d e \left (x +\frac {d}{e}\right )^{5}}-\frac {4 e \left (\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{d e \left (x +\frac {d}{e}\right )^{4}}+\frac {5 e \left (\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{3 d e \left (x +\frac {d}{e}\right )^{3}}+\frac {2 e \left (\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{5 d e \left (x +\frac {d}{e}\right )^{2}}+\frac {7 e \left (\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{7}+d e \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{12 e^{2}}+\frac {5 d^{2} \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{8 e^{2}}+\frac {3 d^{2} \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}{4 e^{2}}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )\right )}{5 d}\right )}{d}\right )}{d}\right )}{d}\right )}{d}\right )}{5 d}}{e^{7}}\)	\(559\)

Input:

int((-e^2*x^2+d^2)^(7/2)/(e*x+d)^7,x,method=_RETURNVERBOSE)
 

Output:

-(-e^2*x^2+d^2)^(1/2)/e-7*d/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2 
)^(1/2))-232/15*d/e^2/(x+d/e)*(-e^2*(x+d/e)^2+2*d*e*(x+d/e))^(1/2)+128/15* 
d^2/e^3/(x+d/e)^2*(-e^2*(x+d/e)^2+2*d*e*(x+d/e))^(1/2)-16/5*d^3/e^4/(x+d/e 
)^3*(-e^2*(x+d/e)^2+2*d*e*(x+d/e))^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.22 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^7} \, dx=-\frac {167 \, d e^{3} x^{3} + 501 \, d^{2} e^{2} x^{2} + 501 \, d^{3} e x + 167 \, d^{4} - 210 \, {\left (d e^{3} x^{3} + 3 \, d^{2} e^{2} x^{2} + 3 \, d^{3} e x + d^{4}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (15 \, e^{3} x^{3} + 277 \, d e^{2} x^{2} + 381 \, d^{2} e x + 167 \, d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} \] Input:

integrate((-e^2*x^2+d^2)^(7/2)/(e*x+d)^7,x, algorithm="fricas")
 

Output:

-1/15*(167*d*e^3*x^3 + 501*d^2*e^2*x^2 + 501*d^3*e*x + 167*d^4 - 210*(d*e^ 
3*x^3 + 3*d^2*e^2*x^2 + 3*d^3*e*x + d^4)*arctan(-(d - sqrt(-e^2*x^2 + d^2) 
)/(e*x)) + (15*e^3*x^3 + 277*d*e^2*x^2 + 381*d^2*e*x + 167*d^3)*sqrt(-e^2* 
x^2 + d^2))/(e^4*x^3 + 3*d*e^3*x^2 + 3*d^2*e^2*x + d^3*e)
 

Sympy [F]

\[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^7} \, dx=\int \frac {\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}{\left (d + e x\right )^{7}}\, dx \] Input:

integrate((-e**2*x**2+d**2)**(7/2)/(e*x+d)**7,x)
 

Output:

Integral((-(-d + e*x)*(d + e*x))**(7/2)/(d + e*x)**7, x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 401 vs. \(2 (123) = 246\).

Time = 0.14 (sec) , antiderivative size = 401, normalized size of antiderivative = 2.84 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^7} \, dx=\frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}}{e^{7} x^{6} + 6 \, d e^{6} x^{5} + 15 \, d^{2} e^{5} x^{4} + 20 \, d^{3} e^{4} x^{3} + 15 \, d^{4} e^{3} x^{2} + 6 \, d^{5} e^{2} x + d^{6} e} - \frac {7 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d}{5 \, {\left (e^{6} x^{5} + 5 \, d e^{5} x^{4} + 10 \, d^{2} e^{4} x^{3} + 10 \, d^{3} e^{3} x^{2} + 5 \, d^{4} e^{2} x + d^{5} e\right )}} - \frac {7 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2}}{e^{5} x^{4} + 4 \, d e^{4} x^{3} + 6 \, d^{2} e^{3} x^{2} + 4 \, d^{3} e^{2} x + d^{4} e} + \frac {42 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{3}}{5 \, {\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} + \frac {7 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d}{3 \, {\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} + \frac {49 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2}}{15 \, {\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} - \frac {7 \, d \arcsin \left (\frac {e x}{d}\right )}{e} - \frac {266 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{15 \, {\left (e^{2} x + d e\right )}} \] Input:

integrate((-e^2*x^2+d^2)^(7/2)/(e*x+d)^7,x, algorithm="maxima")
 

Output:

(-e^2*x^2 + d^2)^(7/2)/(e^7*x^6 + 6*d*e^6*x^5 + 15*d^2*e^5*x^4 + 20*d^3*e^ 
4*x^3 + 15*d^4*e^3*x^2 + 6*d^5*e^2*x + d^6*e) - 7/5*(-e^2*x^2 + d^2)^(5/2) 
*d/(e^6*x^5 + 5*d*e^5*x^4 + 10*d^2*e^4*x^3 + 10*d^3*e^3*x^2 + 5*d^4*e^2*x 
+ d^5*e) - 7*(-e^2*x^2 + d^2)^(3/2)*d^2/(e^5*x^4 + 4*d*e^4*x^3 + 6*d^2*e^3 
*x^2 + 4*d^3*e^2*x + d^4*e) + 42/5*sqrt(-e^2*x^2 + d^2)*d^3/(e^4*x^3 + 3*d 
*e^3*x^2 + 3*d^2*e^2*x + d^3*e) + 7/3*(-e^2*x^2 + d^2)^(3/2)*d/(e^4*x^3 + 
3*d*e^3*x^2 + 3*d^2*e^2*x + d^3*e) + 49/15*sqrt(-e^2*x^2 + d^2)*d^2/(e^3*x 
^2 + 2*d*e^2*x + d^2*e) - 7*d*arcsin(e*x/d)/e - 266/15*sqrt(-e^2*x^2 + d^2 
)*d/(e^2*x + d*e)
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.46 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^7} \, dx=-\frac {7 \, d \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{{\left | e \right |}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}}}{e} + \frac {16 \, {\left (19 \, d + \frac {80 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d}{e^{2} x} + \frac {130 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d}{e^{4} x^{2}} + \frac {60 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d}{e^{6} x^{3}} + \frac {15 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} d}{e^{8} x^{4}}\right )}}{15 \, {\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} + 1\right )}^{5} {\left | e \right |}} \] Input:

integrate((-e^2*x^2+d^2)^(7/2)/(e*x+d)^7,x, algorithm="giac")
 

Output:

-7*d*arcsin(e*x/d)*sgn(d)*sgn(e)/abs(e) - sqrt(-e^2*x^2 + d^2)/e + 16/15*( 
19*d + 80*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*d/(e^2*x) + 130*(d*e + sqrt( 
-e^2*x^2 + d^2)*abs(e))^2*d/(e^4*x^2) + 60*(d*e + sqrt(-e^2*x^2 + d^2)*abs 
(e))^3*d/(e^6*x^3) + 15*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4*d/(e^8*x^4)) 
/(((d*e + sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*x) + 1)^5*abs(e))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^7} \, dx=\int \frac {{\left (d^2-e^2\,x^2\right )}^{7/2}}{{\left (d+e\,x\right )}^7} \,d x \] Input:

int((d^2 - e^2*x^2)^(7/2)/(d + e*x)^7,x)
 

Output:

int((d^2 - e^2*x^2)^(7/2)/(d + e*x)^7, x)
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 384, normalized size of antiderivative = 2.72 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^7} \, dx=\frac {105 \mathit {atan} \left (\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, d^{2}-2 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{2} x^{2}}{-2 e^{3} x^{3}+2 d^{2} e x}\right ) d^{4}+315 \mathit {atan} \left (\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, d^{2}-2 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{2} x^{2}}{-2 e^{3} x^{3}+2 d^{2} e x}\right ) d^{3} e x +315 \mathit {atan} \left (\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, d^{2}-2 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{2} x^{2}}{-2 e^{3} x^{3}+2 d^{2} e x}\right ) d^{2} e^{2} x^{2}+105 \mathit {atan} \left (\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, d^{2}-2 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{2} x^{2}}{-2 e^{3} x^{3}+2 d^{2} e x}\right ) d \,e^{3} x^{3}-334 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{3}-762 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{2} e x -554 \sqrt {-e^{2} x^{2}+d^{2}}\, d \,e^{2} x^{2}-30 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{3} x^{3}}{30 e \left (e^{3} x^{3}+3 d \,e^{2} x^{2}+3 d^{2} e x +d^{3}\right )} \] Input:

int((-e^2*x^2+d^2)^(7/2)/(e*x+d)^7,x)
 

Output:

(105*atan((sqrt(d**2 - e**2*x**2)*d**2 - 2*sqrt(d**2 - e**2*x**2)*e**2*x** 
2)/(2*d**2*e*x - 2*e**3*x**3))*d**4 + 315*atan((sqrt(d**2 - e**2*x**2)*d** 
2 - 2*sqrt(d**2 - e**2*x**2)*e**2*x**2)/(2*d**2*e*x - 2*e**3*x**3))*d**3*e 
*x + 315*atan((sqrt(d**2 - e**2*x**2)*d**2 - 2*sqrt(d**2 - e**2*x**2)*e**2 
*x**2)/(2*d**2*e*x - 2*e**3*x**3))*d**2*e**2*x**2 + 105*atan((sqrt(d**2 - 
e**2*x**2)*d**2 - 2*sqrt(d**2 - e**2*x**2)*e**2*x**2)/(2*d**2*e*x - 2*e**3 
*x**3))*d*e**3*x**3 - 334*sqrt(d**2 - e**2*x**2)*d**3 - 762*sqrt(d**2 - e* 
*2*x**2)*d**2*e*x - 554*sqrt(d**2 - e**2*x**2)*d*e**2*x**2 - 30*sqrt(d**2 
- e**2*x**2)*e**3*x**3)/(30*e*(d**3 + 3*d**2*e*x + 3*d*e**2*x**2 + e**3*x* 
*3))