3.3.15 \(\int \frac {1}{(d+e x)^4 (d^2-e^2 x^2)^{7/2}} \, dx\) [215]

3.3.15.1 Optimal result

Integrand size = 24, antiderivative size = 205 \[ \int \frac {1}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {48 x}{715 d^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {64 x}{715 d^8 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {128 x}{715 d^{10} \sqrt {d^2-e^2 x^2}} \]

output

48/715*x/d^6/(-e^2*x^2+d^2)^(5/2)-1/13/d/e/(e*x+d)^4/(-e^2*x^2+d^2)^(5/2)- 
9/143/d^2/e/(e*x+d)^3/(-e^2*x^2+d^2)^(5/2)-8/143/d^3/e/(e*x+d)^2/(-e^2*x^2 
+d^2)^(5/2)-8/143/d^4/e/(e*x+d)/(-e^2*x^2+d^2)^(5/2)+64/715*x/d^8/(-e^2*x^ 
2+d^2)^(3/2)+128/715*x/d^10/(-e^2*x^2+d^2)^(1/2)
 

3.3.15.2 Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.67 \[ \int \frac {1}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-180 d^9-5 d^8 e x+800 d^7 e^2 x^2+1080 d^6 e^3 x^3-320 d^5 e^4 x^4-1552 d^4 e^5 x^5-768 d^3 e^6 x^6+448 d^2 e^7 x^7+512 d e^8 x^8+128 e^9 x^9\right )}{715 d^{10} e (d-e x)^3 (d+e x)^7} \]

input

Integrate[1/((d + e*x)^4*(d^2 - e^2*x^2)^(7/2)),x]
 

output

(Sqrt[d^2 - e^2*x^2]*(-180*d^9 - 5*d^8*e*x + 800*d^7*e^2*x^2 + 1080*d^6*e^ 
3*x^3 - 320*d^5*e^4*x^4 - 1552*d^4*e^5*x^5 - 768*d^3*e^6*x^6 + 448*d^2*e^7 
*x^7 + 512*d*e^8*x^8 + 128*e^9*x^9))/(715*d^10*e*(d - e*x)^3*(d + e*x)^7)
 

3.3.15.3 Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.20, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {461, 461, 461, 470, 209, 209, 208}

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.

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

\(\Big \downarrow \) 461

\(\displaystyle \frac {9 \int \frac {1}{(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}}dx}{13 d}-\frac {1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 461

\(\displaystyle \frac {9 \left (\frac {8 \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}dx}{11 d}-\frac {1}{11 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}\right )}{13 d}-\frac {1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 461

\(\displaystyle \frac {9 \left (\frac {8 \left (\frac {7 \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}}dx}{9 d}-\frac {1}{9 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}\right )}{11 d}-\frac {1}{11 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}\right )}{13 d}-\frac {1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 470

\(\displaystyle \frac {9 \left (\frac {8 \left (\frac {7 \left (\frac {6 \int \frac {1}{\left (d^2-e^2 x^2\right )^{7/2}}dx}{7 d}-\frac {1}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}\right )}{9 d}-\frac {1}{9 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}\right )}{11 d}-\frac {1}{11 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}\right )}{13 d}-\frac {1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 209

\(\displaystyle \frac {9 \left (\frac {8 \left (\frac {7 \left (\frac {6 \left (\frac {4 \int \frac {1}{\left (d^2-e^2 x^2\right )^{5/2}}dx}{5 d^2}+\frac {x}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}\right )}{7 d}-\frac {1}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}\right )}{9 d}-\frac {1}{9 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}\right )}{11 d}-\frac {1}{11 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}\right )}{13 d}-\frac {1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 209

\(\displaystyle \frac {9 \left (\frac {8 \left (\frac {7 \left (\frac {6 \left (\frac {4 \left (\frac {2 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}}dx}{3 d^2}+\frac {x}{3 d^2 \left (d^2-e^2 x^2\right )^{3/2}}\right )}{5 d^2}+\frac {x}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}\right )}{7 d}-\frac {1}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}\right )}{9 d}-\frac {1}{9 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}\right )}{11 d}-\frac {1}{11 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}\right )}{13 d}-\frac {1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 208

\(\displaystyle \frac {9 \left (\frac {8 \left (\frac {7 \left (\frac {6 \left (\frac {x}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {4 \left (\frac {x}{3 d^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 x}{3 d^4 \sqrt {d^2-e^2 x^2}}\right )}{5 d^2}\right )}{7 d}-\frac {1}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}\right )}{9 d}-\frac {1}{9 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}\right )}{11 d}-\frac {1}{11 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}\right )}{13 d}-\frac {1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}\)

input

Int[1/((d + e*x)^4*(d^2 - e^2*x^2)^(7/2)),x]
 

output

-1/13*1/(d*e*(d + e*x)^4*(d^2 - e^2*x^2)^(5/2)) + (9*(-1/11*1/(d*e*(d + e* 
x)^3*(d^2 - e^2*x^2)^(5/2)) + (8*(-1/9*1/(d*e*(d + e*x)^2*(d^2 - e^2*x^2)^ 
(5/2)) + (7*(-1/7*1/(d*e*(d + e*x)*(d^2 - e^2*x^2)^(5/2)) + (6*(x/(5*d^2*( 
d^2 - e^2*x^2)^(5/2)) + (4*(x/(3*d^2*(d^2 - e^2*x^2)^(3/2)) + (2*x)/(3*d^4 
*Sqrt[d^2 - e^2*x^2])))/(5*d^2)))/(7*d)))/(9*d)))/(11*d)))/(13*d)
 

3.3.15.3.1 Defintions of rubi rules used

rule 208

Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), 
x] /; FreeQ[{a, b}, x]
 

rule 209

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

rule 461

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

rule 470

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

3.3.15.4 Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.64

method	result	size
gosper	\(-\frac {\left (-e x +d \right ) \left (-128 e^{9} x^{9}-512 d \,e^{8} x^{8}-448 d^{2} e^{7} x^{7}+768 d^{3} e^{6} x^{6}+1552 d^{4} e^{5} x^{5}+320 d^{5} e^{4} x^{4}-1080 d^{6} e^{3} x^{3}-800 x^{2} d^{7} e^{2}+5 x \,d^{8} e +180 d^{9}\right )}{715 \left (e x +d \right )^{3} d^{10} e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}\)	\(132\)
trager	\(-\frac {\left (-128 e^{9} x^{9}-512 d \,e^{8} x^{8}-448 d^{2} e^{7} x^{7}+768 d^{3} e^{6} x^{6}+1552 d^{4} e^{5} x^{5}+320 d^{5} e^{4} x^{4}-1080 d^{6} e^{3} x^{3}-800 x^{2} d^{7} e^{2}+5 x \,d^{8} e +180 d^{9}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{715 d^{10} \left (e x +d \right )^{7} \left (-e x +d \right )^{3} e}\)	\(134\)
default	\(\frac {-\frac {1}{13 d e \left (x +\frac {d}{e}\right )^{4} \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {9 e \left (-\frac {1}{11 d e \left (x +\frac {d}{e}\right )^{3} \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {8 e \left (-\frac {1}{9 d e \left (x +\frac {d}{e}\right )^{2} \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {7 e \left (-\frac {1}{7 d e \left (x +\frac {d}{e}\right ) \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {6 e \left (-\frac {-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e}{10 d^{2} e^{2} \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {-\frac {2 \left (-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e \right )}{15 d^{2} e^{2} \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {4 \left (-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e \right )}{15 e^{2} d^{4} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}}{d^{2}}\right )}{7 d}\right )}{9 d}\right )}{11 d}\right )}{13 d}}{e^{4}}\)	\(379\)

input

int(1/(e*x+d)^4/(-e^2*x^2+d^2)^(7/2),x,method=_RETURNVERBOSE)
 

output

-1/715*(-e*x+d)*(-128*e^9*x^9-512*d*e^8*x^8-448*d^2*e^7*x^7+768*d^3*e^6*x^ 
6+1552*d^4*e^5*x^5+320*d^5*e^4*x^4-1080*d^6*e^3*x^3-800*d^7*e^2*x^2+5*d^8* 
e*x+180*d^9)/(e*x+d)^3/d^10/e/(-e^2*x^2+d^2)^(7/2)
 

3.3.15.5 Fricas [A] (verification not implemented)

Time = 0.73 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.53 \[ \int \frac {1}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {180 \, e^{10} x^{10} + 720 \, d e^{9} x^{9} + 540 \, d^{2} e^{8} x^{8} - 1440 \, d^{3} e^{7} x^{7} - 2520 \, d^{4} e^{6} x^{6} + 2520 \, d^{6} e^{4} x^{4} + 1440 \, d^{7} e^{3} x^{3} - 540 \, d^{8} e^{2} x^{2} - 720 \, d^{9} e x - 180 \, d^{10} + {\left (128 \, e^{9} x^{9} + 512 \, d e^{8} x^{8} + 448 \, d^{2} e^{7} x^{7} - 768 \, d^{3} e^{6} x^{6} - 1552 \, d^{4} e^{5} x^{5} - 320 \, d^{5} e^{4} x^{4} + 1080 \, d^{6} e^{3} x^{3} + 800 \, d^{7} e^{2} x^{2} - 5 \, d^{8} e x - 180 \, d^{9}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{715 \, {\left (d^{10} e^{11} x^{10} + 4 \, d^{11} e^{10} x^{9} + 3 \, d^{12} e^{9} x^{8} - 8 \, d^{13} e^{8} x^{7} - 14 \, d^{14} e^{7} x^{6} + 14 \, d^{16} e^{5} x^{4} + 8 \, d^{17} e^{4} x^{3} - 3 \, d^{18} e^{3} x^{2} - 4 \, d^{19} e^{2} x - d^{20} e\right )}} \]

input

integrate(1/(e*x+d)^4/(-e^2*x^2+d^2)^(7/2),x, algorithm="fricas")
 

output

-1/715*(180*e^10*x^10 + 720*d*e^9*x^9 + 540*d^2*e^8*x^8 - 1440*d^3*e^7*x^7 
 - 2520*d^4*e^6*x^6 + 2520*d^6*e^4*x^4 + 1440*d^7*e^3*x^3 - 540*d^8*e^2*x^ 
2 - 720*d^9*e*x - 180*d^10 + (128*e^9*x^9 + 512*d*e^8*x^8 + 448*d^2*e^7*x^ 
7 - 768*d^3*e^6*x^6 - 1552*d^4*e^5*x^5 - 320*d^5*e^4*x^4 + 1080*d^6*e^3*x^ 
3 + 800*d^7*e^2*x^2 - 5*d^8*e*x - 180*d^9)*sqrt(-e^2*x^2 + d^2))/(d^10*e^1 
1*x^10 + 4*d^11*e^10*x^9 + 3*d^12*e^9*x^8 - 8*d^13*e^8*x^7 - 14*d^14*e^7*x 
^6 + 14*d^16*e^5*x^4 + 8*d^17*e^4*x^3 - 3*d^18*e^3*x^2 - 4*d^19*e^2*x - d^ 
20*e)
 

3.3.15.6 Sympy [F]

\[ \int \frac {1}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {1}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}} \left (d + e x\right )^{4}}\, dx \]

input

integrate(1/(e*x+d)**4/(-e**2*x**2+d**2)**(7/2),x)
 

output

Integral(1/((-(-d + e*x)*(d + e*x))**(7/2)*(d + e*x)**4), x)
 

3.3.15.7 Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 393 vs. \(2 (177) = 354\).

Time = 0.20 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.92 \[ \int \frac {1}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {1}{13 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d e^{5} x^{4} + 4 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e^{4} x^{3} + 6 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e^{3} x^{2} + 4 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5} e\right )}} - \frac {9}{143 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e^{4} x^{3} + 3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e^{3} x^{2} + 3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5} e\right )}} - \frac {8}{143 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e^{3} x^{2} + 2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5} e\right )}} - \frac {8}{143 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5} e\right )}} + \frac {48 \, x}{715 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{6}} + \frac {64 \, x}{715 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{8}} + \frac {128 \, x}{715 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{10}} \]

input

integrate(1/(e*x+d)^4/(-e^2*x^2+d^2)^(7/2),x, algorithm="maxima")
 

output

-1/13/((-e^2*x^2 + d^2)^(5/2)*d*e^5*x^4 + 4*(-e^2*x^2 + d^2)^(5/2)*d^2*e^4 
*x^3 + 6*(-e^2*x^2 + d^2)^(5/2)*d^3*e^3*x^2 + 4*(-e^2*x^2 + d^2)^(5/2)*d^4 
*e^2*x + (-e^2*x^2 + d^2)^(5/2)*d^5*e) - 9/143/((-e^2*x^2 + d^2)^(5/2)*d^2 
*e^4*x^3 + 3*(-e^2*x^2 + d^2)^(5/2)*d^3*e^3*x^2 + 3*(-e^2*x^2 + d^2)^(5/2) 
*d^4*e^2*x + (-e^2*x^2 + d^2)^(5/2)*d^5*e) - 8/143/((-e^2*x^2 + d^2)^(5/2) 
*d^3*e^3*x^2 + 2*(-e^2*x^2 + d^2)^(5/2)*d^4*e^2*x + (-e^2*x^2 + d^2)^(5/2) 
*d^5*e) - 8/143/((-e^2*x^2 + d^2)^(5/2)*d^4*e^2*x + (-e^2*x^2 + d^2)^(5/2) 
*d^5*e) + 48/715*x/((-e^2*x^2 + d^2)^(5/2)*d^6) + 64/715*x/((-e^2*x^2 + d^ 
2)^(3/2)*d^8) + 128/715*x/(sqrt(-e^2*x^2 + d^2)*d^10)
 

3.3.15.8 Giac [F]

\[ \int \frac {1}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int { \frac {1}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} {\left (e x + d\right )}^{4}} \,d x } \]

input

integrate(1/(e*x+d)^4/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")
 

output

integrate(1/((-e^2*x^2 + d^2)^(7/2)*(e*x + d)^4), x)
 

3.3.15.9 Mupad [B] (verification not implemented)

Time = 11.90 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.18 \[ \int \frac {1}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {64\,x}{715\,d^8}+\frac {189}{4576\,d^7\,e}\right )}{{\left (d+e\,x\right )}^2\,{\left (d-e\,x\right )}^2}+\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {1139\,x}{5720\,d^6}-\frac {427}{2288\,d^5\,e}\right )}{{\left (d+e\,x\right )}^3\,{\left (d-e\,x\right )}^3}-\frac {\sqrt {d^2-e^2\,x^2}}{104\,d^4\,e\,{\left (d+e\,x\right )}^7}-\frac {51\,\sqrt {d^2-e^2\,x^2}}{2288\,d^5\,e\,{\left (d+e\,x\right )}^6}-\frac {19\,\sqrt {d^2-e^2\,x^2}}{572\,d^6\,e\,{\left (d+e\,x\right )}^5}-\frac {189\,\sqrt {d^2-e^2\,x^2}}{4576\,d^7\,e\,{\left (d+e\,x\right )}^4}+\frac {128\,x\,\sqrt {d^2-e^2\,x^2}}{715\,d^{10}\,\left (d+e\,x\right )\,\left (d-e\,x\right )} \]

input

int(1/((d^2 - e^2*x^2)^(7/2)*(d + e*x)^4),x)
 

output

((d^2 - e^2*x^2)^(1/2)*((64*x)/(715*d^8) + 189/(4576*d^7*e)))/((d + e*x)^2 
*(d - e*x)^2) + ((d^2 - e^2*x^2)^(1/2)*((1139*x)/(5720*d^6) - 427/(2288*d^ 
5*e)))/((d + e*x)^3*(d - e*x)^3) - (d^2 - e^2*x^2)^(1/2)/(104*d^4*e*(d + e 
*x)^7) - (51*(d^2 - e^2*x^2)^(1/2))/(2288*d^5*e*(d + e*x)^6) - (19*(d^2 - 
e^2*x^2)^(1/2))/(572*d^6*e*(d + e*x)^5) - (189*(d^2 - e^2*x^2)^(1/2))/(457 
6*d^7*e*(d + e*x)^4) + (128*x*(d^2 - e^2*x^2)^(1/2))/(715*d^10*(d + e*x)*( 
d - e*x))