\(\int \frac {(d+e x)^2}{(d^2-e^2 x^2)^{11/2}} \, dx\) [157]

Optimal result

Integrand size = 24, antiderivative size = 125 \[ \int \frac {(d+e x)^2}{\left (d^2-e^2 x^2\right )^{11/2}} \, dx=\frac {2 (d+e x)}{9 e \left (d^2-e^2 x^2\right )^{9/2}}+\frac {x}{9 d^2 \left (d^2-e^2 x^2\right )^{7/2}}+\frac {2 x}{15 d^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {8 x}{45 d^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {16 x}{45 d^8 \sqrt {d^2-e^2 x^2}} \] Output:

2/9*(e*x+d)/e/(-e^2*x^2+d^2)^(9/2)+1/9*x/d^2/(-e^2*x^2+d^2)^(7/2)+2/15*x/d 
^4/(-e^2*x^2+d^2)^(5/2)+8/45*x/d^6/(-e^2*x^2+d^2)^(3/2)+16/45*x/d^8/(-e^2* 
x^2+d^2)^(1/2)
 

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.92 \[ \int \frac {(d+e x)^2}{\left (d^2-e^2 x^2\right )^{11/2}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (10 d^7+25 d^6 e x-60 d^5 e^2 x^2-10 d^4 e^3 x^3+80 d^3 e^4 x^4-24 d^2 e^5 x^5-32 d e^6 x^6+16 e^7 x^7\right )}{45 d^8 e (d-e x)^5 (d+e x)^3} \] Input:

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

Output:

(Sqrt[d^2 - e^2*x^2]*(10*d^7 + 25*d^6*e*x - 60*d^5*e^2*x^2 - 10*d^4*e^3*x^ 
3 + 80*d^3*e^4*x^4 - 24*d^2*e^5*x^5 - 32*d*e^6*x^6 + 16*e^7*x^7))/(45*d^8* 
e*(d - e*x)^5*(d + e*x)^3)
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.17, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {457, 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.

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

\(\Big \downarrow \) 457

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

\(\Big \downarrow \) 209

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

\(\Big \downarrow \) 209

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

\(\Big \downarrow \) 209

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

\(\Big \downarrow \) 208

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

Input:

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

Output:

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

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 457

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

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.88

method	result	size
gosper	\(\frac {\left (e x +d \right )^{3} \left (-e x +d \right ) \left (16 e^{7} x^{7}-32 d \,e^{6} x^{6}-24 d^{2} e^{5} x^{5}+80 d^{3} e^{4} x^{4}-10 d^{4} e^{3} x^{3}-60 d^{5} e^{2} x^{2}+25 d^{6} e x +10 d^{7}\right )}{45 d^{8} e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {11}{2}}}\)	\(110\)
orering	\(\frac {\left (e x +d \right )^{3} \left (-e x +d \right ) \left (16 e^{7} x^{7}-32 d \,e^{6} x^{6}-24 d^{2} e^{5} x^{5}+80 d^{3} e^{4} x^{4}-10 d^{4} e^{3} x^{3}-60 d^{5} e^{2} x^{2}+25 d^{6} e x +10 d^{7}\right )}{45 d^{8} e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {11}{2}}}\)	\(110\)
trager	\(\frac {\left (16 e^{7} x^{7}-32 d \,e^{6} x^{6}-24 d^{2} e^{5} x^{5}+80 d^{3} e^{4} x^{4}-10 d^{4} e^{3} x^{3}-60 d^{5} e^{2} x^{2}+25 d^{6} e x +10 d^{7}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{45 d^{8} \left (-e x +d \right )^{5} \left (e x +d \right )^{3} e}\)	\(112\)
default	\(d^{2} \left (\frac {x}{9 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {9}{2}}}+\frac {\frac {8 x}{63 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}+\frac {8 \left (\frac {6 x}{35 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}\right )}{7 d^{2}}\right )}{9 d^{2}}}{d^{2}}\right )+e^{2} \left (\frac {x}{8 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {9}{2}}}-\frac {d^{2} \left (\frac {x}{9 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {9}{2}}}+\frac {\frac {8 x}{63 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}+\frac {8 \left (\frac {6 x}{35 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}\right )}{7 d^{2}}\right )}{9 d^{2}}}{d^{2}}\right )}{8 e^{2}}\right )+\frac {2 d}{9 e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {9}{2}}}\)	\(297\)

Input:

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

Output:

1/45*(e*x+d)^3*(-e*x+d)*(16*e^7*x^7-32*d*e^6*x^6-24*d^2*e^5*x^5+80*d^3*e^4 
*x^4-10*d^4*e^3*x^3-60*d^5*e^2*x^2+25*d^6*e*x+10*d^7)/d^8/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. 249 vs. \(2 (105) = 210\).

Time = 0.22 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.99 \[ \int \frac {(d+e x)^2}{\left (d^2-e^2 x^2\right )^{11/2}} \, dx=\frac {10 \, e^{8} x^{8} - 20 \, d e^{7} x^{7} - 20 \, d^{2} e^{6} x^{6} + 60 \, d^{3} e^{5} x^{5} - 60 \, d^{5} e^{3} x^{3} + 20 \, d^{6} e^{2} x^{2} + 20 \, d^{7} e x - 10 \, d^{8} - {\left (16 \, e^{7} x^{7} - 32 \, d e^{6} x^{6} - 24 \, d^{2} e^{5} x^{5} + 80 \, d^{3} e^{4} x^{4} - 10 \, d^{4} e^{3} x^{3} - 60 \, d^{5} e^{2} x^{2} + 25 \, d^{6} e x + 10 \, d^{7}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{45 \, {\left (d^{8} e^{9} x^{8} - 2 \, d^{9} e^{8} x^{7} - 2 \, d^{10} e^{7} x^{6} + 6 \, d^{11} e^{6} x^{5} - 6 \, d^{13} e^{4} x^{3} + 2 \, d^{14} e^{3} x^{2} + 2 \, d^{15} e^{2} x - d^{16} e\right )}} \] Input:

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

Output:

1/45*(10*e^8*x^8 - 20*d*e^7*x^7 - 20*d^2*e^6*x^6 + 60*d^3*e^5*x^5 - 60*d^5 
*e^3*x^3 + 20*d^6*e^2*x^2 + 20*d^7*e*x - 10*d^8 - (16*e^7*x^7 - 32*d*e^6*x 
^6 - 24*d^2*e^5*x^5 + 80*d^3*e^4*x^4 - 10*d^4*e^3*x^3 - 60*d^5*e^2*x^2 + 2 
5*d^6*e*x + 10*d^7)*sqrt(-e^2*x^2 + d^2))/(d^8*e^9*x^8 - 2*d^9*e^8*x^7 - 2 
*d^10*e^7*x^6 + 6*d^11*e^6*x^5 - 6*d^13*e^4*x^3 + 2*d^14*e^3*x^2 + 2*d^15* 
e^2*x - d^16*e)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.94 \[ \int \frac {(d+e x)^2}{\left (d^2-e^2 x^2\right )^{11/2}} \, dx=\frac {2 \, x}{9 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {9}{2}}} + \frac {2 \, d}{9 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {9}{2}} e} + \frac {x}{9 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{2}} + \frac {2 \, x}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4}} + \frac {8 \, x}{45 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{6}} + \frac {16 \, x}{45 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{8}} \] Input:

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

Output:

2/9*x/(-e^2*x^2 + d^2)^(9/2) + 2/9*d/((-e^2*x^2 + d^2)^(9/2)*e) + 1/9*x/(( 
-e^2*x^2 + d^2)^(7/2)*d^2) + 2/15*x/((-e^2*x^2 + d^2)^(5/2)*d^4) + 8/45*x/ 
((-e^2*x^2 + d^2)^(3/2)*d^6) + 16/45*x/(sqrt(-e^2*x^2 + d^2)*d^8)
 

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 6.79 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.49 \[ \int \frac {(d+e x)^2}{\left (d^2-e^2 x^2\right )^{11/2}} \, dx=\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {31\,x}{120\,d^4}+\frac {5}{24\,d^3\,e}\right )}{{\left (d+e\,x\right )}^3\,{\left (d-e\,x\right )}^3}+\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {8\,x}{45\,d^6}-\frac {5}{144\,d^5\,e}\right )}{{\left (d+e\,x\right )}^2\,{\left (d-e\,x\right )}^2}+\frac {\sqrt {d^2-e^2\,x^2}}{72\,d^4\,e\,{\left (d-e\,x\right )}^5}+\frac {5\,\sqrt {d^2-e^2\,x^2}}{144\,d^5\,e\,{\left (d-e\,x\right )}^4}+\frac {16\,x\,\sqrt {d^2-e^2\,x^2}}{45\,d^8\,\left (d+e\,x\right )\,\left (d-e\,x\right )} \] Input:

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

Output:

((d^2 - e^2*x^2)^(1/2)*((31*x)/(120*d^4) + 5/(24*d^3*e)))/((d + e*x)^3*(d 
- e*x)^3) + ((d^2 - e^2*x^2)^(1/2)*((8*x)/(45*d^6) - 5/(144*d^5*e)))/((d + 
 e*x)^2*(d - e*x)^2) + (d^2 - e^2*x^2)^(1/2)/(72*d^4*e*(d - e*x)^5) + (5*( 
d^2 - e^2*x^2)^(1/2))/(144*d^5*e*(d - e*x)^4) + (16*x*(d^2 - e^2*x^2)^(1/2 
))/(45*d^8*(d + e*x)*(d - e*x))
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 312, normalized size of antiderivative = 2.50 \[ \int \frac {(d+e x)^2}{\left (d^2-e^2 x^2\right )^{11/2}} \, dx=\frac {-25 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{6}+50 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{5} e x +25 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{4} e^{2} x^{2}-100 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{3} e^{3} x^{3}+25 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{2} e^{4} x^{4}+50 \sqrt {-e^{2} x^{2}+d^{2}}\, d \,e^{5} x^{5}-25 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{6} x^{6}+20 d^{7}+50 d^{6} e x -120 d^{5} e^{2} x^{2}-20 d^{4} e^{3} x^{3}+160 d^{3} e^{4} x^{4}-48 d^{2} e^{5} x^{5}-64 d \,e^{6} x^{6}+32 e^{7} x^{7}}{90 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{8} e \left (e^{6} x^{6}-2 d \,e^{5} x^{5}-d^{2} e^{4} x^{4}+4 d^{3} e^{3} x^{3}-d^{4} e^{2} x^{2}-2 d^{5} e x +d^{6}\right )} \] Input:

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

Output:

( - 25*sqrt(d**2 - e**2*x**2)*d**6 + 50*sqrt(d**2 - e**2*x**2)*d**5*e*x + 
25*sqrt(d**2 - e**2*x**2)*d**4*e**2*x**2 - 100*sqrt(d**2 - e**2*x**2)*d**3 
*e**3*x**3 + 25*sqrt(d**2 - e**2*x**2)*d**2*e**4*x**4 + 50*sqrt(d**2 - e** 
2*x**2)*d*e**5*x**5 - 25*sqrt(d**2 - e**2*x**2)*e**6*x**6 + 20*d**7 + 50*d 
**6*e*x - 120*d**5*e**2*x**2 - 20*d**4*e**3*x**3 + 160*d**3*e**4*x**4 - 48 
*d**2*e**5*x**5 - 64*d*e**6*x**6 + 32*e**7*x**7)/(90*sqrt(d**2 - e**2*x**2 
)*d**8*e*(d**6 - 2*d**5*e*x - d**4*e**2*x**2 + 4*d**3*e**3*x**3 - d**2*e** 
4*x**4 - 2*d*e**5*x**5 + e**6*x**6))