\(\int \frac {(d^2-e^2 x^2)^{7/2}}{(d+e x)^8} \, dx\) [70]

Optimal result

Integrand size = 24, antiderivative size = 143 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^8} \, dx=\frac {2 \sqrt {d^2-e^2 x^2}}{e (d+e x)}-\frac {2 \left (d^2-e^2 x^2\right )^{3/2}}{3 e (d+e x)^3}+\frac {2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e (d+e x)^5}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{7 e (d+e x)^7}+\frac {\arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e} \] Output:

2*(-e^2*x^2+d^2)^(1/2)/e/(e*x+d)-2/3*(-e^2*x^2+d^2)^(3/2)/e/(e*x+d)^3+2/5* 
(-e^2*x^2+d^2)^(5/2)/e/(e*x+d)^5-2/7*(-e^2*x^2+d^2)^(7/2)/e/(e*x+d)^7+arct 
an(e*x/(-e^2*x^2+d^2)^(1/2))/e
 

Mathematica [A] (verified)

Time = 0.56 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.69 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^8} \, dx=\frac {8 \sqrt {d^2-e^2 x^2} \left (19 d^3+76 d^2 e x+71 d e^2 x^2+44 e^3 x^3\right )}{105 e (d+e x)^4}-\frac {2 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )}{e} \] Input:

Integrate[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^8,x]
 

Output:

(8*Sqrt[d^2 - e^2*x^2]*(19*d^3 + 76*d^2*e*x + 71*d*e^2*x^2 + 44*e^3*x^3))/ 
(105*e*(d + e*x)^4) - (2*ArcTan[(e*x)/(Sqrt[d^2] - Sqrt[d^2 - e^2*x^2])])/ 
e
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {465, 465, 465, 463, 25, 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.

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

\(\Big \downarrow \) 465

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

\(\Big \downarrow \) 465

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

\(\Big \downarrow \) 465

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

\(\Big \downarrow \) 463

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 216

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

Input:

Int[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^8,x]
 

Output:

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

Defintions of rubi rules used

rule 25

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

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 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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(610\) vs. \(2(127)=254\).

Time = 1.19 (sec) , antiderivative size = 611, normalized size of antiderivative = 4.27

method	result	size
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}}}{7 d e \left (x +\frac {d}{e}\right )^{8}}-\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}}}{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}\right )}{7 d}}{e^{8}}\)	\(611\)

Input:

int((-e^2*x^2+d^2)^(7/2)/(e*x+d)^8,x,method=_RETURNVERBOSE)
 

Output:

1/e^8*(-1/7/d/e/(x+d/e)^8*(-e^2*(x+d/e)^2+2*d*e*(x+d/e))^(9/2)-1/7*e/d*(-1 
/5/d/e/(x+d/e)^7*(-e^2*(x+d/e)^2+2*d*e*(x+d/e))^(9/2)-2/5*e/d*(-1/3/d/e/(x 
+d/e)^6*(-e^2*(x+d/e)^2+2*d*e*(x+d/e))^(9/2)-e/d*(-1/d/e/(x+d/e)^5*(-e^2*( 
x+d/e)^2+2*d*e*(x+d/e))^(9/2)-4*e/d*(1/d/e/(x+d/e)^4*(-e^2*(x+d/e)^2+2*d*e 
*(x+d/e))^(9/2)+5*e/d*(1/3/d/e/(x+d/e)^3*(-e^2*(x+d/e)^2+2*d*e*(x+d/e))^(9 
/2)+2*e/d*(1/5/d/e/(x+d/e)^2*(-e^2*(x+d/e)^2+2*d*e*(x+d/e))^(9/2)+7/5*e/d* 
(1/7*(-e^2*(x+d/e)^2+2*d*e*(x+d/e))^(7/2)+d*e*(-1/12*(-2*e^2*(x+d/e)+2*d*e 
)/e^2*(-e^2*(x+d/e)^2+2*d*e*(x+d/e))^(5/2)+5/6*d^2*(-1/8*(-2*e^2*(x+d/e)+2 
*d*e)/e^2*(-e^2*(x+d/e)^2+2*d*e*(x+d/e))^(3/2)+3/4*d^2*(-1/4*(-2*e^2*(x+d/ 
e)+2*d*e)/e^2*(-e^2*(x+d/e)^2+2*d*e*(x+d/e))^(1/2)+1/2*d^2/(e^2)^(1/2)*arc 
tan((e^2)^(1/2)*x/(-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 = 200, normalized size of antiderivative = 1.40 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^8} \, dx=\frac {2 \, {\left (76 \, e^{4} x^{4} + 304 \, d e^{3} x^{3} + 456 \, d^{2} e^{2} x^{2} + 304 \, d^{3} e x + 76 \, d^{4} - 105 \, {\left (e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + 4 \, {\left (44 \, e^{3} x^{3} + 71 \, d e^{2} x^{2} + 76 \, d^{2} e x + 19 \, d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}\right )}}{105 \, {\left (e^{5} x^{4} + 4 \, d e^{4} x^{3} + 6 \, d^{2} e^{3} x^{2} + 4 \, d^{3} e^{2} x + d^{4} e\right )}} \] Input:

integrate((-e^2*x^2+d^2)^(7/2)/(e*x+d)^8,x, algorithm="fricas")
 

Output:

2/105*(76*e^4*x^4 + 304*d*e^3*x^3 + 456*d^2*e^2*x^2 + 304*d^3*e*x + 76*d^4 
 - 105*(e^4*x^4 + 4*d*e^3*x^3 + 6*d^2*e^2*x^2 + 4*d^3*e*x + d^4)*arctan(-( 
d - sqrt(-e^2*x^2 + d^2))/(e*x)) + 4*(44*e^3*x^3 + 71*d*e^2*x^2 + 76*d^2*e 
*x + 19*d^3)*sqrt(-e^2*x^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)
                                                                                    
                                                                                    
 

Sympy [F]

\[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^8} \, dx=\int \frac {\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}{\left (d + e x\right )^{8}}\, dx \] Input:

integrate((-e**2*x**2+d**2)**(7/2)/(e*x+d)**8,x)
 

Output:

Integral((-(-d + e*x)*(d + e*x))**(7/2)/(d + e*x)**8, x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 623 vs. \(2 (127) = 254\).

Time = 0.13 (sec) , antiderivative size = 623, normalized size of antiderivative = 4.36 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^8} \, dx=-\frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}}{7 \, {\left (e^{8} x^{7} + 7 \, d e^{7} x^{6} + 21 \, d^{2} e^{6} x^{5} + 35 \, d^{3} e^{5} x^{4} + 35 \, d^{4} e^{4} x^{3} + 21 \, d^{5} e^{3} x^{2} + 7 \, d^{6} e^{2} x + d^{7} e\right )}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d}{e^{7} x^{6} + 6 \, d e^{6} x^{5} + 15 \, d^{2} e^{5} x^{4} + 20 \, d^{3} e^{4} x^{3} + 15 \, d^{4} e^{3} x^{2} + 6 \, d^{5} e^{2} x + d^{6} e} + \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2}}{2 \, {\left (e^{6} x^{5} + 5 \, d e^{5} x^{4} + 10 \, d^{2} e^{4} x^{3} + 10 \, d^{3} e^{3} x^{2} + 5 \, d^{4} e^{2} x + d^{5} e\right )}} - \frac {15 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{3}}{7 \, {\left (e^{5} x^{4} + 4 \, d e^{4} x^{3} + 6 \, d^{2} e^{3} x^{2} + 4 \, d^{3} e^{2} x + d^{4} e\right )}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}}{5 \, {\left (e^{6} x^{5} + 5 \, d e^{5} x^{4} + 10 \, d^{2} e^{4} x^{3} + 10 \, d^{3} e^{3} x^{2} + 5 \, d^{4} e^{2} x + d^{5} e\right )}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d}{e^{5} x^{4} + 4 \, d e^{4} x^{3} + 6 \, d^{2} e^{3} x^{2} + 4 \, d^{3} e^{2} x + d^{4} e} - \frac {69 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2}}{70 \, {\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}}}{3 \, {\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} - \frac {34 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{105 \, {\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} + \frac {\arcsin \left (\frac {e x}{d}\right )}{e} + \frac {281 \, \sqrt {-e^{2} x^{2} + d^{2}}}{105 \, {\left (e^{2} x + d e\right )}} \] Input:

integrate((-e^2*x^2+d^2)^(7/2)/(e*x+d)^8,x, algorithm="maxima")
 

Output:

-1/7*(-e^2*x^2 + d^2)^(7/2)/(e^8*x^7 + 7*d*e^7*x^6 + 21*d^2*e^6*x^5 + 35*d 
^3*e^5*x^4 + 35*d^4*e^4*x^3 + 21*d^5*e^3*x^2 + 7*d^6*e^2*x + d^7*e) - (-e^ 
2*x^2 + d^2)^(5/2)*d/(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) + 5/2*(-e^2*x^2 + d^2)^(3/2)*d 
^2/(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) - 15/7*sqrt(-e^2*x^2 + d^2)*d^3/(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) + 1/5*(-e^2*x^2 + d^2)^(5/2)/(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) + (-e^2*x^ 
2 + d^2)^(3/2)*d/(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) - 69/70*sqrt(-e^2*x^2 + d^2)*d^2/(e^4*x^3 + 3*d*e^3*x^2 + 3*d^2*e^2*x 
 + d^3*e) - 1/3*(-e^2*x^2 + d^2)^(3/2)/(e^4*x^3 + 3*d*e^3*x^2 + 3*d^2*e^2* 
x + d^3*e) - 34/105*sqrt(-e^2*x^2 + d^2)*d/(e^3*x^2 + 2*d*e^2*x + d^2*e) + 
 arcsin(e*x/d)/e + 281/105*sqrt(-e^2*x^2 + d^2)/(e^2*x + d*e)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.47 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^8} \, dx=\frac {\arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{{\left | e \right |}} - \frac {16 \, {\left (\frac {133 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}}{e^{2} x} + \frac {294 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2}}{e^{4} x^{2}} + \frac {490 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3}}{e^{6} x^{3}} + \frac {175 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4}}{e^{8} x^{4}} + \frac {105 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{5}}{e^{10} x^{5}} + 19\right )}}{105 \, {\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} + 1\right )}^{7} {\left | e \right |}} \] Input:

integrate((-e^2*x^2+d^2)^(7/2)/(e*x+d)^8,x, algorithm="giac")
 

Output:

arcsin(e*x/d)*sgn(d)*sgn(e)/abs(e) - 16/105*(133*(d*e + sqrt(-e^2*x^2 + d^ 
2)*abs(e))/(e^2*x) + 294*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2/(e^4*x^2) + 
 490*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3/(e^6*x^3) + 175*(d*e + sqrt(-e^ 
2*x^2 + d^2)*abs(e))^4/(e^8*x^4) + 105*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e)) 
^5/(e^10*x^5) + 19)/(((d*e + sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*x) + 1)^7*a 
bs(e))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^8} \, dx=\int \frac {{\left (d^2-e^2\,x^2\right )}^{7/2}}{{\left (d+e\,x\right )}^8} \,d x \] Input:

int((d^2 - e^2*x^2)^(7/2)/(d + e*x)^8,x)
 

Output:

int((d^2 - e^2*x^2)^(7/2)/(d + e*x)^8, x)
 

Reduce [F]

\[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^8} \, dx=\int \frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{\left (e x +d \right )^{8}}d x \] Input:

int((-e^2*x^2+d^2)^(7/2)/(e*x+d)^8,x)
 

Output:

int((-e^2*x^2+d^2)^(7/2)/(e*x+d)^8,x)