\(\int \frac {(d+e x)^{12}}{(d^2-e^2 x^2)^{11/2}} \, dx\) [147]

Optimal result

Integrand size = 24, antiderivative size = 227 \[ \int \frac {(d+e x)^{12}}{\left (d^2-e^2 x^2\right )^{11/2}} \, dx=\frac {2 (d+e x)^{11}}{9 e \left (d^2-e^2 x^2\right )^{9/2}}-\frac {26 (d+e x)^9}{63 e \left (d^2-e^2 x^2\right )^{7/2}}+\frac {286 (d+e x)^7}{315 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {286 (d+e x)^5}{105 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1144 d^2 (d+e x)}{15 e \sqrt {d^2-e^2 x^2}}+\frac {572 d \sqrt {d^2-e^2 x^2}}{15 e}+\frac {143}{30} x \sqrt {d^2-e^2 x^2}-\frac {143 d^2 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e} \] Output:

2/9*(e*x+d)^11/e/(-e^2*x^2+d^2)^(9/2)-26/63*(e*x+d)^9/e/(-e^2*x^2+d^2)^(7/ 
2)+286/315*(e*x+d)^7/e/(-e^2*x^2+d^2)^(5/2)-286/105*(e*x+d)^5/e/(-e^2*x^2+ 
d^2)^(3/2)+1144/15*d^2*(e*x+d)/e/(-e^2*x^2+d^2)^(1/2)+572/15*d*(-e^2*x^2+d 
^2)^(1/2)/e+143/30*x*(-e^2*x^2+d^2)^(1/2)-143/2*d^2*arctan(e*x/(-e^2*x^2+d 
^2)^(1/2))/e
 

Mathematica [A] (verified)

Time = 0.73 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.59 \[ \int \frac {(d+e x)^{12}}{\left (d^2-e^2 x^2\right )^{11/2}} \, dx=\frac {\frac {\sqrt {d^2-e^2 x^2} \left (70808 d^6-308365 d^5 e x+518889 d^4 e^2 x^2-401890 d^3 e^3 x^3+135818 d^2 e^4 x^4-5985 d e^5 x^5-315 e^6 x^6\right )}{(d-e x)^5}+90090 d^2 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )}{630 e} \] Input:

Integrate[(d + e*x)^12/(d^2 - e^2*x^2)^(11/2),x]
 

Output:

((Sqrt[d^2 - e^2*x^2]*(70808*d^6 - 308365*d^5*e*x + 518889*d^4*e^2*x^2 - 4 
01890*d^3*e^3*x^3 + 135818*d^2*e^4*x^4 - 5985*d*e^5*x^5 - 315*e^6*x^6))/(d 
 - e*x)^5 + 90090*d^2*ArcTan[(e*x)/(Sqrt[d^2] - Sqrt[d^2 - e^2*x^2])])/(63 
0*e)
 

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.07, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {468, 468, 468, 468, 462, 2346, 25, 27, 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.

\(\displaystyle \int \frac {(d+e x)^{12}}{\left (d^2-e^2 x^2\right )^{11/2}} \, dx\)

\(\Big \downarrow \) 468

\(\displaystyle \frac {2 (d+e x)^{11}}{9 e \left (d^2-e^2 x^2\right )^{9/2}}-\frac {13}{9} \int \frac {(d+e x)^{10}}{\left (d^2-e^2 x^2\right )^{9/2}}dx\)

\(\Big \downarrow \) 468

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

\(\Big \downarrow \) 468

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

\(\Big \downarrow \) 468

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

\(\Big \downarrow \) 462

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

\(\Big \downarrow \) 2346

\(\displaystyle \frac {2 (d+e x)^{11}}{9 e \left (d^2-e^2 x^2\right )^{9/2}}-\frac {13}{9} \left (\frac {2 (d+e x)^9}{7 e \left (d^2-e^2 x^2\right )^{7/2}}-\frac {11}{7} \left (\frac {2 (d+e x)^7}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {9}{5} \left (\frac {2 (d+e x)^5}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac {7}{3} \left (\frac {\int -\frac {d e^2 (15 d+8 e x)}{\sqrt {d^2-e^2 x^2}}dx}{2 e^2}+\frac {8 d^2 (d+e x)}{e \sqrt {d^2-e^2 x^2}}+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )\right )\right )\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 (d+e x)^{11}}{9 e \left (d^2-e^2 x^2\right )^{9/2}}-\frac {13}{9} \left (\frac {2 (d+e x)^9}{7 e \left (d^2-e^2 x^2\right )^{7/2}}-\frac {11}{7} \left (\frac {2 (d+e x)^7}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {9}{5} \left (\frac {2 (d+e x)^5}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac {7}{3} \left (-\frac {\int \frac {d e^2 (15 d+8 e x)}{\sqrt {d^2-e^2 x^2}}dx}{2 e^2}+\frac {8 d^2 (d+e x)}{e \sqrt {d^2-e^2 x^2}}+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )\right )\right )\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 (d+e x)^{11}}{9 e \left (d^2-e^2 x^2\right )^{9/2}}-\frac {13}{9} \left (\frac {2 (d+e x)^9}{7 e \left (d^2-e^2 x^2\right )^{7/2}}-\frac {11}{7} \left (\frac {2 (d+e x)^7}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {9}{5} \left (\frac {2 (d+e x)^5}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac {7}{3} \left (-\frac {1}{2} d \int \frac {15 d+8 e x}{\sqrt {d^2-e^2 x^2}}dx+\frac {8 d^2 (d+e x)}{e \sqrt {d^2-e^2 x^2}}+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )\right )\right )\right )\)

\(\Big \downarrow \) 455

\(\displaystyle \frac {2 (d+e x)^{11}}{9 e \left (d^2-e^2 x^2\right )^{9/2}}-\frac {13}{9} \left (\frac {2 (d+e x)^9}{7 e \left (d^2-e^2 x^2\right )^{7/2}}-\frac {11}{7} \left (\frac {2 (d+e x)^7}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {9}{5} \left (\frac {2 (d+e x)^5}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac {7}{3} \left (-\frac {1}{2} d \left (15 d \int \frac {1}{\sqrt {d^2-e^2 x^2}}dx-\frac {8 \sqrt {d^2-e^2 x^2}}{e}\right )+\frac {8 d^2 (d+e x)}{e \sqrt {d^2-e^2 x^2}}+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )\right )\right )\right )\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {2 (d+e x)^{11}}{9 e \left (d^2-e^2 x^2\right )^{9/2}}-\frac {13}{9} \left (\frac {2 (d+e x)^9}{7 e \left (d^2-e^2 x^2\right )^{7/2}}-\frac {11}{7} \left (\frac {2 (d+e x)^7}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {9}{5} \left (\frac {2 (d+e x)^5}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac {7}{3} \left (-\frac {1}{2} d \left (15 d \int \frac {1}{\frac {e^2 x^2}{d^2-e^2 x^2}+1}d\frac {x}{\sqrt {d^2-e^2 x^2}}-\frac {8 \sqrt {d^2-e^2 x^2}}{e}\right )+\frac {8 d^2 (d+e x)}{e \sqrt {d^2-e^2 x^2}}+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )\right )\right )\right )\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {2 (d+e x)^{11}}{9 e \left (d^2-e^2 x^2\right )^{9/2}}-\frac {13}{9} \left (\frac {2 (d+e x)^9}{7 e \left (d^2-e^2 x^2\right )^{7/2}}-\frac {11}{7} \left (\frac {2 (d+e x)^7}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {9}{5} \left (\frac {2 (d+e x)^5}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac {7}{3} \left (-\frac {1}{2} d \left (\frac {15 d \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e}-\frac {8 \sqrt {d^2-e^2 x^2}}{e}\right )+\frac {8 d^2 (d+e x)}{e \sqrt {d^2-e^2 x^2}}+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )\right )\right )\right )\)

Input:

Int[(d + e*x)^12/(d^2 - e^2*x^2)^(11/2),x]
 

Output:

(2*(d + e*x)^11)/(9*e*(d^2 - e^2*x^2)^(9/2)) - (13*((2*(d + e*x)^9)/(7*e*( 
d^2 - e^2*x^2)^(7/2)) - (11*((2*(d + e*x)^7)/(5*e*(d^2 - e^2*x^2)^(5/2)) - 
 (9*((2*(d + e*x)^5)/(3*e*(d^2 - e^2*x^2)^(3/2)) - (7*((8*d^2*(d + e*x))/( 
e*Sqrt[d^2 - e^2*x^2]) + (x*Sqrt[d^2 - e^2*x^2])/2 - (d*((-8*Sqrt[d^2 - e^ 
2*x^2])/e + (15*d*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e))/2))/3))/5))/7))/9
 

Defintions of rubi rules used

rule 25

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

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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 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 462

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

rule 468

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

rule 2346

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], 
e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x^2)^(p + 1)/(b*( 
q + 2*p + 1))), x] + Simp[1/(b*(q + 2*p + 1))   Int[(a + b*x^2)^p*ExpandToS 
um[b*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + 2*p + 1)*x^q, x], 
x], x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] &&  !LeQ[p, -1]
 

Maple [A] (verified)

Time = 2.41 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.32

method	result	size
risch	\(\frac {\left (e x +24 d \right ) \sqrt {-e^{2} x^{2}+d^{2}}}{2 e}-\frac {143 d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}-\frac {50584 d^{2} \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{315 e^{2} \left (x -\frac {d}{e}\right )}-\frac {37616 d^{3} \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{315 e^{3} \left (x -\frac {d}{e}\right )^{2}}-\frac {10592 d^{4} \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{105 e^{4} \left (x -\frac {d}{e}\right )^{3}}-\frac {3520 d^{5} \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{63 e^{5} \left (x -\frac {d}{e}\right )^{4}}-\frac {128 d^{6} \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{9 e^{6} \left (x -\frac {d}{e}\right )^{5}}\)	\(300\)
default	\(\text {Expression too large to display}\)	\(1899\)

Input:

int((e*x+d)^12/(-e^2*x^2+d^2)^(11/2),x,method=_RETURNVERBOSE)
 

Output:

1/2*(e*x+24*d)/e*(-e^2*x^2+d^2)^(1/2)-143/2*d^2/(e^2)^(1/2)*arctan((e^2)^( 
1/2)*x/(-e^2*x^2+d^2)^(1/2))-50584/315*d^2/e^2/(x-d/e)*(-(x-d/e)^2*e^2-2*d 
*e*(x-d/e))^(1/2)-37616/315*d^3/e^3/(x-d/e)^2*(-(x-d/e)^2*e^2-2*d*e*(x-d/e 
))^(1/2)-10592/105*d^4/e^4/(x-d/e)^3*(-(x-d/e)^2*e^2-2*d*e*(x-d/e))^(1/2)- 
3520/63*d^5/e^5/(x-d/e)^4*(-(x-d/e)^2*e^2-2*d*e*(x-d/e))^(1/2)-128/9*d^6/e 
^6/(x-d/e)^5*(-(x-d/e)^2*e^2-2*d*e*(x-d/e))^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.22 \[ \int \frac {(d+e x)^{12}}{\left (d^2-e^2 x^2\right )^{11/2}} \, dx=\frac {70808 \, d^{2} e^{5} x^{5} - 354040 \, d^{3} e^{4} x^{4} + 708080 \, d^{4} e^{3} x^{3} - 708080 \, d^{5} e^{2} x^{2} + 354040 \, d^{6} e x - 70808 \, d^{7} + 90090 \, {\left (d^{2} e^{5} x^{5} - 5 \, d^{3} e^{4} x^{4} + 10 \, d^{4} e^{3} x^{3} - 10 \, d^{5} e^{2} x^{2} + 5 \, d^{6} e x - d^{7}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (315 \, e^{6} x^{6} + 5985 \, d e^{5} x^{5} - 135818 \, d^{2} e^{4} x^{4} + 401890 \, d^{3} e^{3} x^{3} - 518889 \, d^{4} e^{2} x^{2} + 308365 \, d^{5} e x - 70808 \, d^{6}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{630 \, {\left (e^{6} x^{5} - 5 \, d e^{5} x^{4} + 10 \, d^{2} e^{4} x^{3} - 10 \, d^{3} e^{3} x^{2} + 5 \, d^{4} e^{2} x - d^{5} e\right )}} \] Input:

integrate((e*x+d)^12/(-e^2*x^2+d^2)^(11/2),x, algorithm="fricas")
 

Output:

1/630*(70808*d^2*e^5*x^5 - 354040*d^3*e^4*x^4 + 708080*d^4*e^3*x^3 - 70808 
0*d^5*e^2*x^2 + 354040*d^6*e*x - 70808*d^7 + 90090*(d^2*e^5*x^5 - 5*d^3*e^ 
4*x^4 + 10*d^4*e^3*x^3 - 10*d^5*e^2*x^2 + 5*d^6*e*x - d^7)*arctan(-(d - sq 
rt(-e^2*x^2 + d^2))/(e*x)) + (315*e^6*x^6 + 5985*d*e^5*x^5 - 135818*d^2*e^ 
4*x^4 + 401890*d^3*e^3*x^3 - 518889*d^4*e^2*x^2 + 308365*d^5*e*x - 70808*d 
^6)*sqrt(-e^2*x^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)
 

Sympy [F]

\[ \int \frac {(d+e x)^{12}}{\left (d^2-e^2 x^2\right )^{11/2}} \, dx=\int \frac {\left (d + e x\right )^{12}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {11}{2}}}\, dx \] Input:

integrate((e*x+d)**12/(-e**2*x**2+d**2)**(11/2),x)
 

Output:

Integral((d + e*x)**12/(-(-d + e*x)*(d + e*x))**(11/2), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 806 vs. \(2 (195) = 390\).

Time = 0.17 (sec) , antiderivative size = 806, normalized size of antiderivative = 3.55 \[ \int \frac {(d+e x)^{12}}{\left (d^2-e^2 x^2\right )^{11/2}} \, dx =\text {Too large to display} \] Input:

integrate((e*x+d)^12/(-e^2*x^2+d^2)^(11/2),x, algorithm="maxima")
 

Output:

-1/2*e^10*x^11/(-e^2*x^2 + d^2)^(9/2) - 12*d*e^9*x^10/(-e^2*x^2 + d^2)^(9/ 
2) + 143/630*(315*x^8/((-e^2*x^2 + d^2)^(9/2)*e^2) - 840*d^2*x^6/((-e^2*x^ 
2 + d^2)^(9/2)*e^4) + 1008*d^4*x^4/((-e^2*x^2 + d^2)^(9/2)*e^6) - 576*d^6* 
x^2/((-e^2*x^2 + d^2)^(9/2)*e^8) + 128*d^8/((-e^2*x^2 + d^2)^(9/2)*e^10))* 
d^2*e^10*x + 340*d^3*e^7*x^8/(-e^2*x^2 + d^2)^(9/2) + 495/2*d^4*e^6*x^7/(- 
e^2*x^2 + d^2)^(9/2) - 143/70*d^2*e^8*x*(35*x^6/((-e^2*x^2 + d^2)^(7/2)*e^ 
2) - 70*d^2*x^4/((-e^2*x^2 + d^2)^(7/2)*e^4) + 56*d^4*x^2/((-e^2*x^2 + d^2 
)^(7/2)*e^6) - 16*d^6/((-e^2*x^2 + d^2)^(7/2)*e^8)) - 1928/3*d^5*e^5*x^6/( 
-e^2*x^2 + d^2)^(9/2) - 1617/8*d^6*e^4*x^5/(-e^2*x^2 + d^2)^(9/2) + 143/30 
*d^2*e^6*x*(15*x^4/((-e^2*x^2 + d^2)^(5/2)*e^2) - 20*d^2*x^2/((-e^2*x^2 + 
d^2)^(5/2)*e^4) + 8*d^4/((-e^2*x^2 + d^2)^(5/2)*e^6)) + 4648/5*d^7*e^3*x^4 
/(-e^2*x^2 + d^2)^(9/2) + 286/3*d^4*e^4*x^5/(-e^2*x^2 + d^2)^(7/2) + 4015/ 
16*d^8*e^2*x^3/(-e^2*x^2 + d^2)^(9/2) - 143/6*d^2*e^4*x*(3*x^2/((-e^2*x^2 
+ d^2)^(3/2)*e^2) - 2*d^2/((-e^2*x^2 + d^2)^(3/2)*e^4)) - 17492/35*d^9*e*x 
^2/(-e^2*x^2 + d^2)^(9/2) - 5291/30*d^6*e^2*x^3/(-e^2*x^2 + d^2)^(7/2) - 1 
0973/144*d^10*x/(-e^2*x^2 + d^2)^(9/2) - 143/2*d^4*e^2*x^3/(-e^2*x^2 + d^2 
)^(5/2) + 35404/315*d^11/((-e^2*x^2 + d^2)^(9/2)*e) + 509753/5040*d^8*x/(- 
e^2*x^2 + d^2)^(7/2) + 40861/840*d^6*x/(-e^2*x^2 + d^2)^(5/2) - 8098/315*d 
^4*x/(-e^2*x^2 + d^2)^(3/2) - 77437/630*d^2*x/sqrt(-e^2*x^2 + d^2) - 143/2 
*d^2*arcsin(e^2*x/(d*sqrt(e^2)))/sqrt(e^2)
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.58 \[ \int \frac {(d+e x)^{12}}{\left (d^2-e^2 x^2\right )^{11/2}} \, dx=-\frac {143 \, d^{2} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{2 \, {\left | e \right |}} + \frac {1}{2} \, \sqrt {-e^{2} x^{2} + d^{2}} {\left (x + \frac {24 \, d}{e}\right )} + \frac {16 \, {\left (3953 \, d^{2} - \frac {32742 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{2}}{e^{2} x} + \frac {117738 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{2}}{e^{4} x^{2}} - \frac {235662 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d^{2}}{e^{6} x^{3}} + \frac {289548 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} d^{2}}{e^{8} x^{4}} - \frac {208530 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{5} d^{2}}{e^{10} x^{5}} + \frac {96390 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{6} d^{2}}{e^{12} x^{6}} - \frac {24570 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{7} d^{2}}{e^{14} x^{7}} + \frac {2835 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{8} d^{2}}{e^{16} x^{8}}\right )}}{315 \, {\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} - 1\right )}^{9} {\left | e \right |}} \] Input:

integrate((e*x+d)^12/(-e^2*x^2+d^2)^(11/2),x, algorithm="giac")
 

Output:

-143/2*d^2*arcsin(e*x/d)*sgn(d)*sgn(e)/abs(e) + 1/2*sqrt(-e^2*x^2 + d^2)*( 
x + 24*d/e) + 16/315*(3953*d^2 - 32742*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e)) 
*d^2/(e^2*x) + 117738*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d^2/(e^4*x^2) 
- 235662*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*d^2/(e^6*x^3) + 289548*(d*e 
 + sqrt(-e^2*x^2 + d^2)*abs(e))^4*d^2/(e^8*x^4) - 208530*(d*e + sqrt(-e^2* 
x^2 + d^2)*abs(e))^5*d^2/(e^10*x^5) + 96390*(d*e + sqrt(-e^2*x^2 + d^2)*ab 
s(e))^6*d^2/(e^12*x^6) - 24570*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^7*d^2/( 
e^14*x^7) + 2835*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^8*d^2/(e^16*x^8))/((( 
d*e + sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*x) - 1)^9*abs(e))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{12}}{\left (d^2-e^2 x^2\right )^{11/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^{12}}{{\left (d^2-e^2\,x^2\right )}^{11/2}} \,d x \] Input:

int((d + e*x)^12/(d^2 - e^2*x^2)^(11/2),x)
 

Output:

int((d + e*x)^12/(d^2 - e^2*x^2)^(11/2), x)
 

Reduce [F]

\[ \int \frac {(d+e x)^{12}}{\left (d^2-e^2 x^2\right )^{11/2}} \, dx=\int \frac {\left (e x +d \right )^{12}}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {11}{2}}}d x \] Input:

int((e*x+d)^12/(-e^2*x^2+d^2)^(11/2),x)
 

Output:

int((e*x+d)^12/(-e^2*x^2+d^2)^(11/2),x)