\(\int \frac {d+e x}{(d^2-e^2 x^2)^{11/2}} \, dx\) [158]

Optimal result

Integrand size = 22, antiderivative size = 128 \[ \int \frac {d+e x}{\left (d^2-e^2 x^2\right )^{11/2}} \, dx=\frac {d+e x}{9 d e \left (d^2-e^2 x^2\right )^{9/2}}+\frac {8 x}{63 d^3 \left (d^2-e^2 x^2\right )^{7/2}}+\frac {16 x}{105 d^5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {64 x}{315 d^7 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {128 x}{315 d^9 \sqrt {d^2-e^2 x^2}} \] Output:

1/9*(e*x+d)/d/e/(-e^2*x^2+d^2)^(9/2)+8/63*x/d^3/(-e^2*x^2+d^2)^(7/2)+16/10 
5*x/d^5/(-e^2*x^2+d^2)^(5/2)+64/315*x/d^7/(-e^2*x^2+d^2)^(3/2)+128/315*x/d 
^9/(-e^2*x^2+d^2)^(1/2)
 

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.98 \[ \int \frac {d+e x}{\left (d^2-e^2 x^2\right )^{11/2}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (35 d^8+280 d^7 e x-280 d^6 e^2 x^2-560 d^5 e^3 x^3+560 d^4 e^4 x^4+448 d^3 e^5 x^5-448 d^2 e^6 x^6-128 d e^7 x^7+128 e^8 x^8\right )}{315 d^9 e (d-e x)^5 (d+e x)^4} \] Input:

Integrate[(d + e*x)/(d^2 - e^2*x^2)^(11/2),x]
 

Output:

(Sqrt[d^2 - e^2*x^2]*(35*d^8 + 280*d^7*e*x - 280*d^6*e^2*x^2 - 560*d^5*e^3 
*x^3 + 560*d^4*e^4*x^4 + 448*d^3*e^5*x^5 - 448*d^2*e^6*x^6 - 128*d*e^7*x^7 
 + 128*e^8*x^8))/(315*d^9*e*(d - e*x)^5*(d + e*x)^4)
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.19, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {454, 209, 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.

\(\displaystyle \int \frac {d+e x}{\left (d^2-e^2 x^2\right )^{11/2}} \, dx\)

\(\Big \downarrow \) 454

\(\displaystyle \frac {8 \int \frac {1}{\left (d^2-e^2 x^2\right )^{9/2}}dx}{9 d}+\frac {d+e x}{9 d e \left (d^2-e^2 x^2\right )^{9/2}}\)

\(\Big \downarrow \) 209

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

\(\Big \downarrow \) 209

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

\(\Big \downarrow \) 209

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

\(\Big \downarrow \) 208

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

Input:

Int[(d + e*x)/(d^2 - e^2*x^2)^(11/2),x]
 

Output:

(d + e*x)/(9*d*e*(d^2 - e^2*x^2)^(9/2)) + (8*(x/(7*d^2*(d^2 - e^2*x^2)^(7/ 
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^2)))/(9*d)
 

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 454

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

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.95

method	result	size
gosper	\(\frac {\left (e x +d \right )^{2} \left (-e x +d \right ) \left (128 e^{8} x^{8}-128 d \,e^{7} x^{7}-448 d^{2} e^{6} x^{6}+448 d^{3} e^{5} x^{5}+560 d^{4} e^{4} x^{4}-560 d^{5} e^{3} x^{3}-280 d^{6} e^{2} x^{2}+280 d^{7} e x +35 d^{8}\right )}{315 d^{9} e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {11}{2}}}\)	\(121\)
orering	\(\frac {\left (e x +d \right )^{2} \left (-e x +d \right ) \left (128 e^{8} x^{8}-128 d \,e^{7} x^{7}-448 d^{2} e^{6} x^{6}+448 d^{3} e^{5} x^{5}+560 d^{4} e^{4} x^{4}-560 d^{5} e^{3} x^{3}-280 d^{6} e^{2} x^{2}+280 d^{7} e x +35 d^{8}\right )}{315 d^{9} e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {11}{2}}}\)	\(121\)
trager	\(\frac {\left (128 e^{8} x^{8}-128 d \,e^{7} x^{7}-448 d^{2} e^{6} x^{6}+448 d^{3} e^{5} x^{5}+560 d^{4} e^{4} x^{4}-560 d^{5} e^{3} x^{3}-280 d^{6} e^{2} x^{2}+280 d^{7} e x +35 d^{8}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{315 d^{9} \left (-e x +d \right )^{5} \left (e x +d \right )^{4} e}\)	\(123\)
default	\(d \left (\frac {x}{9 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {9}{2}}}+\frac {\frac {8 x}{63 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}+\frac {8 \left (\frac {6 x}{35 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}\right )}{7 d^{2}}\right )}{9 d^{2}}}{d^{2}}\right )+\frac {1}{9 e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {9}{2}}}\)	\(142\)

Input:

int((e*x+d)/(-e^2*x^2+d^2)^(11/2),x,method=_RETURNVERBOSE)
 

Output:

1/315*(e*x+d)^2*(-e*x+d)*(128*e^8*x^8-128*d*e^7*x^7-448*d^2*e^6*x^6+448*d^ 
3*e^5*x^5+560*d^4*e^4*x^4-560*d^5*e^3*x^3-280*d^6*e^2*x^2+280*d^7*e*x+35*d 
^8)/d^9/e/(-e^2*x^2+d^2)^(11/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 303 vs. \(2 (108) = 216\).

Time = 0.37 (sec) , antiderivative size = 303, normalized size of antiderivative = 2.37 \[ \int \frac {d+e x}{\left (d^2-e^2 x^2\right )^{11/2}} \, dx=\frac {35 \, e^{9} x^{9} - 35 \, d e^{8} x^{8} - 140 \, d^{2} e^{7} x^{7} + 140 \, d^{3} e^{6} x^{6} + 210 \, d^{4} e^{5} x^{5} - 210 \, d^{5} e^{4} x^{4} - 140 \, d^{6} e^{3} x^{3} + 140 \, d^{7} e^{2} x^{2} + 35 \, d^{8} e x - 35 \, d^{9} - {\left (128 \, e^{8} x^{8} - 128 \, d e^{7} x^{7} - 448 \, d^{2} e^{6} x^{6} + 448 \, d^{3} e^{5} x^{5} + 560 \, d^{4} e^{4} x^{4} - 560 \, d^{5} e^{3} x^{3} - 280 \, d^{6} e^{2} x^{2} + 280 \, d^{7} e x + 35 \, d^{8}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{315 \, {\left (d^{9} e^{10} x^{9} - d^{10} e^{9} x^{8} - 4 \, d^{11} e^{8} x^{7} + 4 \, d^{12} e^{7} x^{6} + 6 \, d^{13} e^{6} x^{5} - 6 \, d^{14} e^{5} x^{4} - 4 \, d^{15} e^{4} x^{3} + 4 \, d^{16} e^{3} x^{2} + d^{17} e^{2} x - d^{18} e\right )}} \] Input:

integrate((e*x+d)/(-e^2*x^2+d^2)^(11/2),x, algorithm="fricas")
 

Output:

1/315*(35*e^9*x^9 - 35*d*e^8*x^8 - 140*d^2*e^7*x^7 + 140*d^3*e^6*x^6 + 210 
*d^4*e^5*x^5 - 210*d^5*e^4*x^4 - 140*d^6*e^3*x^3 + 140*d^7*e^2*x^2 + 35*d^ 
8*e*x - 35*d^9 - (128*e^8*x^8 - 128*d*e^7*x^7 - 448*d^2*e^6*x^6 + 448*d^3* 
e^5*x^5 + 560*d^4*e^4*x^4 - 560*d^5*e^3*x^3 - 280*d^6*e^2*x^2 + 280*d^7*e* 
x + 35*d^8)*sqrt(-e^2*x^2 + d^2))/(d^9*e^10*x^9 - d^10*e^9*x^8 - 4*d^11*e^ 
8*x^7 + 4*d^12*e^7*x^6 + 6*d^13*e^6*x^5 - 6*d^14*e^5*x^4 - 4*d^15*e^4*x^3 
+ 4*d^16*e^3*x^2 + d^17*e^2*x - d^18*e)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 17.02 (sec) , antiderivative size = 1549, normalized size of antiderivative = 12.10 \[ \int \frac {d+e x}{\left (d^2-e^2 x^2\right )^{11/2}} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)/(-e**2*x**2+d**2)**(11/2),x)
 

Output:

d*Piecewise((-315*I*d**8*x/(315*d**19*sqrt(-1 + e**2*x**2/d**2) - 1260*d** 
17*e**2*x**2*sqrt(-1 + e**2*x**2/d**2) + 1890*d**15*e**4*x**4*sqrt(-1 + e* 
*2*x**2/d**2) - 1260*d**13*e**6*x**6*sqrt(-1 + e**2*x**2/d**2) + 315*d**11 
*e**8*x**8*sqrt(-1 + e**2*x**2/d**2)) + 840*I*d**6*e**2*x**3/(315*d**19*sq 
rt(-1 + e**2*x**2/d**2) - 1260*d**17*e**2*x**2*sqrt(-1 + e**2*x**2/d**2) + 
 1890*d**15*e**4*x**4*sqrt(-1 + e**2*x**2/d**2) - 1260*d**13*e**6*x**6*sqr 
t(-1 + e**2*x**2/d**2) + 315*d**11*e**8*x**8*sqrt(-1 + e**2*x**2/d**2)) - 
1008*I*d**4*e**4*x**5/(315*d**19*sqrt(-1 + e**2*x**2/d**2) - 1260*d**17*e* 
*2*x**2*sqrt(-1 + e**2*x**2/d**2) + 1890*d**15*e**4*x**4*sqrt(-1 + e**2*x* 
*2/d**2) - 1260*d**13*e**6*x**6*sqrt(-1 + e**2*x**2/d**2) + 315*d**11*e**8 
*x**8*sqrt(-1 + e**2*x**2/d**2)) + 576*I*d**2*e**6*x**7/(315*d**19*sqrt(-1 
 + e**2*x**2/d**2) - 1260*d**17*e**2*x**2*sqrt(-1 + e**2*x**2/d**2) + 1890 
*d**15*e**4*x**4*sqrt(-1 + e**2*x**2/d**2) - 1260*d**13*e**6*x**6*sqrt(-1 
+ e**2*x**2/d**2) + 315*d**11*e**8*x**8*sqrt(-1 + e**2*x**2/d**2)) - 128*I 
*e**8*x**9/(315*d**19*sqrt(-1 + e**2*x**2/d**2) - 1260*d**17*e**2*x**2*sqr 
t(-1 + e**2*x**2/d**2) + 1890*d**15*e**4*x**4*sqrt(-1 + e**2*x**2/d**2) - 
1260*d**13*e**6*x**6*sqrt(-1 + e**2*x**2/d**2) + 315*d**11*e**8*x**8*sqrt( 
-1 + e**2*x**2/d**2)), Abs(e**2*x**2/d**2) > 1), (315*d**8*x/(315*d**19*sq 
rt(1 - e**2*x**2/d**2) - 1260*d**17*e**2*x**2*sqrt(1 - e**2*x**2/d**2) + 1 
890*d**15*e**4*x**4*sqrt(1 - e**2*x**2/d**2) - 1260*d**13*e**6*x**6*sqr...
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.94 \[ \int \frac {d+e x}{\left (d^2-e^2 x^2\right )^{11/2}} \, dx=\frac {x}{9 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {9}{2}} d} + \frac {1}{9 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {9}{2}} e} + \frac {8 \, x}{63 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{3}} + \frac {16 \, x}{105 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5}} + \frac {64 \, x}{315 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{7}} + \frac {128 \, x}{315 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{9}} \] Input:

integrate((e*x+d)/(-e^2*x^2+d^2)^(11/2),x, algorithm="maxima")
 

Output:

1/9*x/((-e^2*x^2 + d^2)^(9/2)*d) + 1/9/((-e^2*x^2 + d^2)^(9/2)*e) + 8/63*x 
/((-e^2*x^2 + d^2)^(7/2)*d^3) + 16/105*x/((-e^2*x^2 + d^2)^(5/2)*d^5) + 64 
/315*x/((-e^2*x^2 + d^2)^(3/2)*d^7) + 128/315*x/(sqrt(-e^2*x^2 + d^2)*d^9)
 

Giac [F]

\[ \int \frac {d+e x}{\left (d^2-e^2 x^2\right )^{11/2}} \, dx=\int { \frac {e x + d}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {11}{2}}} \,d x } \] Input:

integrate((e*x+d)/(-e^2*x^2+d^2)^(11/2),x, algorithm="giac")
 

Output:

integrate((e*x + d)/(-e^2*x^2 + d^2)^(11/2), x)
 

Mupad [B] (verification not implemented)

Time = 6.78 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.49 \[ \int \frac {d+e x}{\left (d^2-e^2 x^2\right )^{11/2}} \, dx=\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {53\,x}{252\,d^3}+\frac {5}{36\,d^2\,e}\right )}{{\left (d+e\,x\right )}^4\,{\left (d-e\,x\right )}^4}+\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {733\,x}{5040\,d^5}-\frac {5}{144\,d^4\,e}\right )}{{\left (d+e\,x\right )}^3\,{\left (d-e\,x\right )}^3}+\frac {\sqrt {d^2-e^2\,x^2}}{144\,d^5\,e\,{\left (d-e\,x\right )}^5}+\frac {64\,x\,\sqrt {d^2-e^2\,x^2}}{315\,d^7\,{\left (d+e\,x\right )}^2\,{\left (d-e\,x\right )}^2}+\frac {128\,x\,\sqrt {d^2-e^2\,x^2}}{315\,d^9\,\left (d+e\,x\right )\,\left (d-e\,x\right )} \] Input:

int((d + e*x)/(d^2 - e^2*x^2)^(11/2),x)
 

Output:

((d^2 - e^2*x^2)^(1/2)*((53*x)/(252*d^3) + 5/(36*d^2*e)))/((d + e*x)^4*(d 
- e*x)^4) + ((d^2 - e^2*x^2)^(1/2)*((733*x)/(5040*d^5) - 5/(144*d^4*e)))/( 
(d + e*x)^3*(d - e*x)^3) + (d^2 - e^2*x^2)^(1/2)/(144*d^5*e*(d - e*x)^5) + 
 (64*x*(d^2 - e^2*x^2)^(1/2))/(315*d^7*(d + e*x)^2*(d - e*x)^2) + (128*x*( 
d^2 - e^2*x^2)^(1/2))/(315*d^9*(d + e*x)*(d - e*x))
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 358, normalized size of antiderivative = 2.80 \[ \int \frac {d+e x}{\left (d^2-e^2 x^2\right )^{11/2}} \, dx=\frac {-280 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{7}+280 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{6} e x +840 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{5} e^{2} x^{2}-840 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{4} e^{3} x^{3}-840 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{3} e^{4} x^{4}+840 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{2} e^{5} x^{5}+280 \sqrt {-e^{2} x^{2}+d^{2}}\, d \,e^{6} x^{6}-280 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{7} x^{7}+35 d^{8}+280 d^{7} e x -280 d^{6} e^{2} x^{2}-560 d^{5} e^{3} x^{3}+560 d^{4} e^{4} x^{4}+448 d^{3} e^{5} x^{5}-448 d^{2} e^{6} x^{6}-128 d \,e^{7} x^{7}+128 e^{8} x^{8}}{315 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{9} e \left (e^{7} x^{7}-d \,e^{6} x^{6}-3 d^{2} e^{5} x^{5}+3 d^{3} e^{4} x^{4}+3 d^{4} e^{3} x^{3}-3 d^{5} e^{2} x^{2}-d^{6} e x +d^{7}\right )} \] Input:

int((e*x+d)/(-e^2*x^2+d^2)^(11/2),x)
 

Output:

( - 280*sqrt(d**2 - e**2*x**2)*d**7 + 280*sqrt(d**2 - e**2*x**2)*d**6*e*x 
+ 840*sqrt(d**2 - e**2*x**2)*d**5*e**2*x**2 - 840*sqrt(d**2 - e**2*x**2)*d 
**4*e**3*x**3 - 840*sqrt(d**2 - e**2*x**2)*d**3*e**4*x**4 + 840*sqrt(d**2 
- e**2*x**2)*d**2*e**5*x**5 + 280*sqrt(d**2 - e**2*x**2)*d*e**6*x**6 - 280 
*sqrt(d**2 - e**2*x**2)*e**7*x**7 + 35*d**8 + 280*d**7*e*x - 280*d**6*e**2 
*x**2 - 560*d**5*e**3*x**3 + 560*d**4*e**4*x**4 + 448*d**3*e**5*x**5 - 448 
*d**2*e**6*x**6 - 128*d*e**7*x**7 + 128*e**8*x**8)/(315*sqrt(d**2 - e**2*x 
**2)*d**9*e*(d**7 - d**6*e*x - 3*d**5*e**2*x**2 + 3*d**4*e**3*x**3 + 3*d** 
3*e**4*x**4 - 3*d**2*e**5*x**5 - d*e**6*x**6 + e**7*x**7))