\(\int \frac {(d+e x)^6}{(d^2-e^2 x^2)^{11/2}} \, dx\) [153]

Optimal result

Integrand size = 24, antiderivative size = 133 \[ \int \frac {(d+e x)^6}{\left (d^2-e^2 x^2\right )^{11/2}} \, dx=\frac {(d+e x)^6}{9 d e \left (d^2-e^2 x^2\right )^{9/2}}+\frac {(d+e x)^5}{21 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}+\frac {2 (d+e x)^4}{105 d^3 e \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 (d+e x)^3}{315 d^4 e \left (d^2-e^2 x^2\right )^{3/2}} \] Output:

1/9*(e*x+d)^6/d/e/(-e^2*x^2+d^2)^(9/2)+1/21*(e*x+d)^5/d^2/e/(-e^2*x^2+d^2) 
^(7/2)+2/105*(e*x+d)^4/d^3/e/(-e^2*x^2+d^2)^(5/2)+2/315*(e*x+d)^3/d^4/e/(- 
e^2*x^2+d^2)^(3/2)
 

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.56 \[ \int \frac {(d+e x)^6}{\left (d^2-e^2 x^2\right )^{11/2}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (58 d^4+25 d^3 e x-21 d^2 e^2 x^2+10 d e^3 x^3-2 e^4 x^4\right )}{315 d^4 e (d-e x)^5} \] Input:

Integrate[(d + e*x)^6/(d^2 - e^2*x^2)^(11/2),x]
 

Output:

(Sqrt[d^2 - e^2*x^2]*(58*d^4 + 25*d^3*e*x - 21*d^2*e^2*x^2 + 10*d*e^3*x^3 
- 2*e^4*x^4))/(315*d^4*e*(d - e*x)^5)
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.11, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {461, 461, 461, 460}

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.

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

\(\Big \downarrow \) 461

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

\(\Big \downarrow \) 461

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

\(\Big \downarrow \) 461

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

\(\Big \downarrow \) 460

\(\displaystyle \frac {(d+e x)^6}{3 d e \left (d^2-e^2 x^2\right )^{9/2}}-\frac {\frac {(d+e x)^7}{5 d e \left (d^2-e^2 x^2\right )^{9/2}}-\frac {2 \left (\frac {(d+e x)^8}{7 d e \left (d^2-e^2 x^2\right )^{9/2}}-\frac {(d+e x)^9}{63 d^2 e \left (d^2-e^2 x^2\right )^{9/2}}\right )}{5 d}}{d}\)

Input:

Int[(d + e*x)^6/(d^2 - e^2*x^2)^(11/2),x]
 

Output:

(d + e*x)^6/(3*d*e*(d^2 - e^2*x^2)^(9/2)) - ((d + e*x)^7/(5*d*e*(d^2 - e^2 
*x^2)^(9/2)) - (2*((d + e*x)^8/(7*d*e*(d^2 - e^2*x^2)^(9/2)) - (d + e*x)^9 
/(63*d^2*e*(d^2 - e^2*x^2)^(9/2))))/(5*d))/d
 

Defintions of rubi rules used

rule 460

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

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.50

method	result	size
gosper	\(\frac {\left (e x +d \right )^{7} \left (-e x +d \right ) \left (-2 e^{3} x^{3}+12 d \,e^{2} x^{2}-33 d^{2} e x +58 d^{3}\right )}{315 d^{4} e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {11}{2}}}\)	\(66\)
orering	\(\frac {\left (e x +d \right )^{7} \left (-e x +d \right ) \left (-2 e^{3} x^{3}+12 d \,e^{2} x^{2}-33 d^{2} e x +58 d^{3}\right )}{315 d^{4} e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {11}{2}}}\)	\(66\)
trager	\(\frac {\left (-2 e^{4} x^{4}+10 d \,e^{3} x^{3}-21 d^{2} e^{2} x^{2}+25 d^{3} e x +58 d^{4}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{315 d^{4} \left (-e x +d \right )^{5} e}\)	\(72\)
default	\(d^{6} \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^{6} \left (\frac {x^{5}}{4 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {9}{2}}}-\frac {5 d^{2} \left (\frac {x^{3}}{6 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {9}{2}}}-\frac {d^{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 )}{2 e^{2}}\right )}{4 e^{2}}\right )+6 d \,e^{5} \left (\frac {x^{4}}{5 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {9}{2}}}-\frac {4 d^{2} \left (\frac {x^{2}}{7 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {9}{2}}}-\frac {2 d^{2}}{63 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {9}{2}}}\right )}{5 e^{2}}\right )+15 d^{2} e^{4} \left (\frac {x^{3}}{6 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {9}{2}}}-\frac {d^{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 )}{2 e^{2}}\right )+20 d^{3} e^{3} \left (\frac {x^{2}}{7 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {9}{2}}}-\frac {2 d^{2}}{63 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {9}{2}}}\right )+15 d^{4} 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^{5}}{3 e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {9}{2}}}\)	\(839\)

Input:

int((e*x+d)^6/(-e^2*x^2+d^2)^(11/2),x,method=_RETURNVERBOSE)
 

Output:

1/315*(e*x+d)^7*(-e*x+d)*(-2*e^3*x^3+12*d*e^2*x^2-33*d^2*e*x+58*d^3)/d^4/e 
/(-e^2*x^2+d^2)^(11/2)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.29 \[ \int \frac {(d+e x)^6}{\left (d^2-e^2 x^2\right )^{11/2}} \, dx=\frac {58 \, e^{5} x^{5} - 290 \, d e^{4} x^{4} + 580 \, d^{2} e^{3} x^{3} - 580 \, d^{3} e^{2} x^{2} + 290 \, d^{4} e x - 58 \, d^{5} + {\left (2 \, e^{4} x^{4} - 10 \, d e^{3} x^{3} + 21 \, d^{2} e^{2} x^{2} - 25 \, d^{3} e x - 58 \, d^{4}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{315 \, {\left (d^{4} e^{6} x^{5} - 5 \, d^{5} e^{5} x^{4} + 10 \, d^{6} e^{4} x^{3} - 10 \, d^{7} e^{3} x^{2} + 5 \, d^{8} e^{2} x - d^{9} e\right )}} \] Input:

integrate((e*x+d)^6/(-e^2*x^2+d^2)^(11/2),x, algorithm="fricas")
 

Output:

1/315*(58*e^5*x^5 - 290*d*e^4*x^4 + 580*d^2*e^3*x^3 - 580*d^3*e^2*x^2 + 29 
0*d^4*e*x - 58*d^5 + (2*e^4*x^4 - 10*d*e^3*x^3 + 21*d^2*e^2*x^2 - 25*d^3*e 
*x - 58*d^4)*sqrt(-e^2*x^2 + d^2))/(d^4*e^6*x^5 - 5*d^5*e^5*x^4 + 10*d^6*e 
^4*x^3 - 10*d^7*e^3*x^2 + 5*d^8*e^2*x - d^9*e)
 

Sympy [F]

\[ \int \frac {(d+e x)^6}{\left (d^2-e^2 x^2\right )^{11/2}} \, dx=\int \frac {\left (d + e x\right )^{6}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {11}{2}}}\, dx \] Input:

integrate((e*x+d)**6/(-e**2*x**2+d**2)**(11/2),x)
 

Output:

Integral((d + e*x)**6/(-(-d + e*x)*(d + e*x))**(11/2), x)
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.60 \[ \int \frac {(d+e x)^6}{\left (d^2-e^2 x^2\right )^{11/2}} \, dx=\frac {e^{4} x^{5}}{4 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {9}{2}}} + \frac {6 \, d e^{3} x^{4}}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {9}{2}}} + \frac {55 \, d^{2} e^{2} x^{3}}{24 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {9}{2}}} + \frac {76 \, d^{3} e x^{2}}{35 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {9}{2}}} + \frac {73 \, d^{4} x}{72 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {9}{2}}} + \frac {58 \, d^{5}}{315 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {9}{2}} e} - \frac {d^{2} x}{504 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}} - \frac {x}{420 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {x}{315 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2}} - \frac {2 \, x}{315 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{4}} \] Input:

integrate((e*x+d)^6/(-e^2*x^2+d^2)^(11/2),x, algorithm="maxima")
 

Output:

1/4*e^4*x^5/(-e^2*x^2 + d^2)^(9/2) + 6/5*d*e^3*x^4/(-e^2*x^2 + d^2)^(9/2) 
+ 55/24*d^2*e^2*x^3/(-e^2*x^2 + d^2)^(9/2) + 76/35*d^3*e*x^2/(-e^2*x^2 + d 
^2)^(9/2) + 73/72*d^4*x/(-e^2*x^2 + d^2)^(9/2) + 58/315*d^5/((-e^2*x^2 + d 
^2)^(9/2)*e) - 1/504*d^2*x/(-e^2*x^2 + d^2)^(7/2) - 1/420*x/(-e^2*x^2 + d^ 
2)^(5/2) - 1/315*x/((-e^2*x^2 + d^2)^(3/2)*d^2) - 2/315*x/(sqrt(-e^2*x^2 + 
 d^2)*d^4)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 289 vs. \(2 (117) = 234\).

Time = 0.15 (sec) , antiderivative size = 289, normalized size of antiderivative = 2.17 \[ \int \frac {(d+e x)^6}{\left (d^2-e^2 x^2\right )^{11/2}} \, dx=-\frac {2 \, {\left (\frac {207 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}}{e^{2} x} - \frac {1143 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2}}{e^{4} x^{2}} + \frac {2247 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3}}{e^{6} x^{3}} - \frac {3843 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4}}{e^{8} x^{4}} + \frac {3465 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{5}}{e^{10} x^{5}} - \frac {2625 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{6}}{e^{12} x^{6}} + \frac {945 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{7}}{e^{14} x^{7}} - \frac {315 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{8}}{e^{16} x^{8}} - 58\right )}}{315 \, d^{4} {\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} - 1\right )}^{9} {\left | e \right |}} \] Input:

integrate((e*x+d)^6/(-e^2*x^2+d^2)^(11/2),x, algorithm="giac")
 

Output:

-2/315*(207*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*x) - 1143*(d*e + sqrt 
(-e^2*x^2 + d^2)*abs(e))^2/(e^4*x^2) + 2247*(d*e + sqrt(-e^2*x^2 + d^2)*ab 
s(e))^3/(e^6*x^3) - 3843*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4/(e^8*x^4) + 
 3465*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^5/(e^10*x^5) - 2625*(d*e + sqrt( 
-e^2*x^2 + d^2)*abs(e))^6/(e^12*x^6) + 945*(d*e + sqrt(-e^2*x^2 + d^2)*abs 
(e))^7/(e^14*x^7) - 315*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^8/(e^16*x^8) - 
 58)/(d^4*((d*e + sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*x) - 1)^9*abs(e))
 

Mupad [B] (verification not implemented)

Time = 7.70 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.11 \[ \int \frac {(d+e x)^6}{\left (d^2-e^2 x^2\right )^{11/2}} \, dx=\frac {2\,\sqrt {d^2-e^2\,x^2}}{9\,e\,{\left (d-e\,x\right )}^5}-\frac {\sqrt {d^2-e^2\,x^2}}{63\,d\,e\,{\left (d-e\,x\right )}^4}-\frac {\sqrt {d^2-e^2\,x^2}}{105\,d^2\,e\,{\left (d-e\,x\right )}^3}-\frac {2\,\sqrt {d^2-e^2\,x^2}}{315\,d^3\,e\,{\left (d-e\,x\right )}^2}-\frac {2\,\sqrt {d^2-e^2\,x^2}}{315\,d^4\,e\,\left (d-e\,x\right )} \] Input:

int((d + e*x)^6/(d^2 - e^2*x^2)^(11/2),x)
 

Output:

(2*(d^2 - e^2*x^2)^(1/2))/(9*e*(d - e*x)^5) - (d^2 - e^2*x^2)^(1/2)/(63*d* 
e*(d - e*x)^4) - (d^2 - e^2*x^2)^(1/2)/(105*d^2*e*(d - e*x)^3) - (2*(d^2 - 
 e^2*x^2)^(1/2))/(315*d^3*e*(d - e*x)^2) - (2*(d^2 - e^2*x^2)^(1/2))/(315* 
d^4*e*(d - e*x))
 

Reduce [F]

\[ \int \frac {(d+e x)^6}{\left (d^2-e^2 x^2\right )^{11/2}} \, dx=\int \frac {\left (e x +d \right )^{6}}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {11}{2}}}d x \] Input:

int((e*x+d)^6/(-e^2*x^2+d^2)^(11/2),x)
 

Output:

int((e*x+d)^6/(-e^2*x^2+d^2)^(11/2),x)