\(\int \frac {(d+e x)^6}{(d^2-e^2 x^2)^{7/2}} \, dx\) [138]

Optimal result

Integrand size = 24, antiderivative size = 112 \[ \int \frac {(d+e x)^6}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {2 (d+e x)^5}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {2 (d+e x)^3}{3 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 (d+e x)}{e \sqrt {d^2-e^2 x^2}}-\frac {\arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e} \] Output:

2/5*(e*x+d)^5/e/(-e^2*x^2+d^2)^(5/2)-2/3*(e*x+d)^3/e/(-e^2*x^2+d^2)^(3/2)+ 
2*(e*x+d)/e/(-e^2*x^2+d^2)^(1/2)-arctan(e*x/(-e^2*x^2+d^2)^(1/2))/e
 

Mathematica [A] (verified)

Time = 0.46 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.77 \[ \int \frac {(d+e x)^6}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {2 \left (\frac {\sqrt {d^2-e^2 x^2} \left (13 d^2-24 d e x+23 e^2 x^2\right )}{(d-e x)^3}+15 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )\right )}{15 e} \] Input:

Integrate[(d + e*x)^6/(d^2 - e^2*x^2)^(7/2),x]
 

Output:

(2*((Sqrt[d^2 - e^2*x^2]*(13*d^2 - 24*d*e*x + 23*e^2*x^2))/(d - e*x)^3 + 1 
5*ArcTan[(e*x)/(Sqrt[d^2] - Sqrt[d^2 - e^2*x^2])]))/(15*e)
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {468, 468, 457, 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.

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

\(\Big \downarrow \) 468

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

\(\Big \downarrow \) 468

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

\(\Big \downarrow \) 457

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

\(\Big \downarrow \) 224

\(\displaystyle -\int \frac {1}{\frac {e^2 x^2}{d^2-e^2 x^2}+1}d\frac {x}{\sqrt {d^2-e^2 x^2}}+\frac {2 (d+e x)^5}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {2 (d+e x)^3}{3 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 (d+e x)}{e \sqrt {d^2-e^2 x^2}}\)

\(\Big \downarrow \) 216

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

Input:

Int[(d + e*x)^6/(d^2 - e^2*x^2)^(7/2),x]
 

Output:

(2*(d + e*x)^5)/(5*e*(d^2 - e^2*x^2)^(5/2)) - (2*(d + e*x)^3)/(3*e*(d^2 - 
e^2*x^2)^(3/2)) + (2*(d + e*x))/(e*Sqrt[d^2 - e^2*x^2]) - ArcTan[(e*x)/Sqr 
t[d^2 - e^2*x^2]]/e
 

Defintions of rubi rules used

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 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]
 

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(579\) vs. \(2(100)=200\).

Time = 0.40 (sec) , antiderivative size = 580, normalized size of antiderivative = 5.18

method	result	size
default	\(d^{6} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )+e^{6} \left (\frac {x^{5}}{5 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {\frac {x^{3}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {\frac {x}{e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{2} \sqrt {e^{2}}}}{e^{2}}}{e^{2}}\right )+6 d \,e^{5} \left (\frac {x^{4}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {4 d^{2} \left (\frac {x^{2}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\right )}{e^{2}}\right )+15 d^{2} e^{4} \left (\frac {x^{3}}{2 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {3 d^{2} \left (\frac {x}{4 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {d^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )}{4 e^{2}}\right )}{2 e^{2}}\right )+20 d^{3} e^{3} \left (\frac {x^{2}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\right )+15 d^{4} e^{2} \left (\frac {x}{4 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {d^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )}{4 e^{2}}\right )+\frac {6 d^{5}}{5 e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\)	\(580\)

Input:

int((e*x+d)^6/(-e^2*x^2+d^2)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

d^6*(1/5*x/d^2/(-e^2*x^2+d^2)^(5/2)+4/5/d^2*(1/3*x/d^2/(-e^2*x^2+d^2)^(3/2 
)+2/3*x/d^4/(-e^2*x^2+d^2)^(1/2)))+e^6*(1/5*x^5/e^2/(-e^2*x^2+d^2)^(5/2)-1 
/e^2*(1/3*x^3/e^2/(-e^2*x^2+d^2)^(3/2)-1/e^2*(x/e^2/(-e^2*x^2+d^2)^(1/2)-1 
/e^2/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2)))))+6*d*e^5*(x^ 
4/e^2/(-e^2*x^2+d^2)^(5/2)-4*d^2/e^2*(1/3*x^2/e^2/(-e^2*x^2+d^2)^(5/2)-2/1 
5*d^2/e^4/(-e^2*x^2+d^2)^(5/2)))+15*d^2*e^4*(1/2*x^3/e^2/(-e^2*x^2+d^2)^(5 
/2)-3/2*d^2/e^2*(1/4*x/e^2/(-e^2*x^2+d^2)^(5/2)-1/4*d^2/e^2*(1/5*x/d^2/(-e 
^2*x^2+d^2)^(5/2)+4/5/d^2*(1/3*x/d^2/(-e^2*x^2+d^2)^(3/2)+2/3*x/d^4/(-e^2* 
x^2+d^2)^(1/2)))))+20*d^3*e^3*(1/3*x^2/e^2/(-e^2*x^2+d^2)^(5/2)-2/15*d^2/e 
^4/(-e^2*x^2+d^2)^(5/2))+15*d^4*e^2*(1/4*x/e^2/(-e^2*x^2+d^2)^(5/2)-1/4*d^ 
2/e^2*(1/5*x/d^2/(-e^2*x^2+d^2)^(5/2)+4/5/d^2*(1/3*x/d^2/(-e^2*x^2+d^2)^(3 
/2)+2/3*x/d^4/(-e^2*x^2+d^2)^(1/2))))+6/5*d^5/e/(-e^2*x^2+d^2)^(5/2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.42 \[ \int \frac {(d+e x)^6}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {2 \, {\left (13 \, e^{3} x^{3} - 39 \, d e^{2} x^{2} + 39 \, d^{2} e x - 13 \, d^{3} + 15 \, {\left (e^{3} x^{3} - 3 \, d e^{2} x^{2} + 3 \, d^{2} e x - d^{3}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - {\left (23 \, e^{2} x^{2} - 24 \, d e x + 13 \, d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}\right )}}{15 \, {\left (e^{4} x^{3} - 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x - d^{3} e\right )}} \] Input:

integrate((e*x+d)^6/(-e^2*x^2+d^2)^(7/2),x, algorithm="fricas")
 

Output:

2/15*(13*e^3*x^3 - 39*d*e^2*x^2 + 39*d^2*e*x - 13*d^3 + 15*(e^3*x^3 - 3*d* 
e^2*x^2 + 3*d^2*e*x - d^3)*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - (23 
*e^2*x^2 - 24*d*e*x + 13*d^2)*sqrt(-e^2*x^2 + d^2))/(e^4*x^3 - 3*d*e^3*x^2 
 + 3*d^2*e^2*x - d^3*e)
 

Sympy [F]

\[ \int \frac {(d+e x)^6}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {\left (d + e x\right )^{6}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}\, dx \] Input:

integrate((e*x+d)**6/(-e**2*x**2+d**2)**(7/2),x)
 

Output:

Integral((d + e*x)**6/(-(-d + e*x)*(d + e*x))**(7/2), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 299 vs. \(2 (100) = 200\).

Time = 0.13 (sec) , antiderivative size = 299, normalized size of antiderivative = 2.67 \[ \int \frac {(d+e x)^6}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {1}{15} \, e^{6} x {\left (\frac {15 \, x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} - \frac {20 \, d^{2} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}} + \frac {8 \, d^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{6}}\right )} - \frac {1}{3} \, e^{4} x {\left (\frac {3 \, x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{2}} - \frac {2 \, d^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{4}}\right )} + \frac {6 \, d e^{3} x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {15 \, d^{2} e^{2} x^{3}}{2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {4 \, d^{3} e x^{2}}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {13 \, d^{4} x}{10 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {26 \, d^{5}}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e} + \frac {31 \, d^{2} x}{30 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}}} + \frac {16 \, x}{15 \, \sqrt {-e^{2} x^{2} + d^{2}}} - \frac {\arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{\sqrt {e^{2}}} \] Input:

integrate((e*x+d)^6/(-e^2*x^2+d^2)^(7/2),x, algorithm="maxima")
 

Output:

1/15*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)) - 1/3*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)) + 6*d*e^ 
3*x^4/(-e^2*x^2 + d^2)^(5/2) + 15/2*d^2*e^2*x^3/(-e^2*x^2 + d^2)^(5/2) - 4 
/3*d^3*e*x^2/(-e^2*x^2 + d^2)^(5/2) - 13/10*d^4*x/(-e^2*x^2 + d^2)^(5/2) + 
 26/15*d^5/((-e^2*x^2 + d^2)^(5/2)*e) + 31/30*d^2*x/(-e^2*x^2 + d^2)^(3/2) 
 + 16/15*x/sqrt(-e^2*x^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 = 180, normalized size of antiderivative = 1.61 \[ \int \frac {(d+e x)^6}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {\arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{{\left | e \right |}} - \frac {4 \, {\left (\frac {50 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}}{e^{2} x} - \frac {100 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2}}{e^{4} x^{2}} + \frac {30 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3}}{e^{6} x^{3}} - \frac {15 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4}}{e^{8} x^{4}} - 13\right )}}{15 \, {\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} - 1\right )}^{5} {\left | e \right |}} \] Input:

integrate((e*x+d)^6/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")
 

Output:

-arcsin(e*x/d)*sgn(d)*sgn(e)/abs(e) - 4/15*(50*(d*e + sqrt(-e^2*x^2 + d^2) 
*abs(e))/(e^2*x) - 100*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2/(e^4*x^2) + 3 
0*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3/(e^6*x^3) - 15*(d*e + sqrt(-e^2*x^ 
2 + d^2)*abs(e))^4/(e^8*x^4) - 13)/(((d*e + sqrt(-e^2*x^2 + d^2)*abs(e))/( 
e^2*x) - 1)^5*abs(e))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^6}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^6}{{\left (d^2-e^2\,x^2\right )}^{7/2}} \,d x \] Input:

int((d + e*x)^6/(d^2 - e^2*x^2)^(7/2),x)
 

Output:

int((d + e*x)^6/(d^2 - e^2*x^2)^(7/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 229, normalized size of antiderivative = 2.04 \[ \int \frac {(d+e x)^6}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {-15 \mathit {asin} \left (\frac {e x}{d}\right ) \tan \left (\frac {\mathit {asin} \left (\frac {e x}{d}\right )}{2}\right )^{5}+75 \mathit {asin} \left (\frac {e x}{d}\right ) \tan \left (\frac {\mathit {asin} \left (\frac {e x}{d}\right )}{2}\right )^{4}-150 \mathit {asin} \left (\frac {e x}{d}\right ) \tan \left (\frac {\mathit {asin} \left (\frac {e x}{d}\right )}{2}\right )^{3}+150 \mathit {asin} \left (\frac {e x}{d}\right ) \tan \left (\frac {\mathit {asin} \left (\frac {e x}{d}\right )}{2}\right )^{2}-75 \mathit {asin} \left (\frac {e x}{d}\right ) \tan \left (\frac {\mathit {asin} \left (\frac {e x}{d}\right )}{2}\right )+15 \mathit {asin} \left (\frac {e x}{d}\right )-12 \tan \left (\frac {\mathit {asin} \left (\frac {e x}{d}\right )}{2}\right )^{5}-280 \tan \left (\frac {\mathit {asin} \left (\frac {e x}{d}\right )}{2}\right )^{2}+140 \tan \left (\frac {\mathit {asin} \left (\frac {e x}{d}\right )}{2}\right )-40}{15 e \left (\tan \left (\frac {\mathit {asin} \left (\frac {e x}{d}\right )}{2}\right )^{5}-5 \tan \left (\frac {\mathit {asin} \left (\frac {e x}{d}\right )}{2}\right )^{4}+10 \tan \left (\frac {\mathit {asin} \left (\frac {e x}{d}\right )}{2}\right )^{3}-10 \tan \left (\frac {\mathit {asin} \left (\frac {e x}{d}\right )}{2}\right )^{2}+5 \tan \left (\frac {\mathit {asin} \left (\frac {e x}{d}\right )}{2}\right )-1\right )} \] Input:

int((e*x+d)^6/(-e^2*x^2+d^2)^(7/2),x)
 

Output:

( - 15*asin((e*x)/d)*tan(asin((e*x)/d)/2)**5 + 75*asin((e*x)/d)*tan(asin(( 
e*x)/d)/2)**4 - 150*asin((e*x)/d)*tan(asin((e*x)/d)/2)**3 + 150*asin((e*x) 
/d)*tan(asin((e*x)/d)/2)**2 - 75*asin((e*x)/d)*tan(asin((e*x)/d)/2) + 15*a 
sin((e*x)/d) - 12*tan(asin((e*x)/d)/2)**5 - 280*tan(asin((e*x)/d)/2)**2 + 
140*tan(asin((e*x)/d)/2) - 40)/(15*e*(tan(asin((e*x)/d)/2)**5 - 5*tan(asin 
((e*x)/d)/2)**4 + 10*tan(asin((e*x)/d)/2)**3 - 10*tan(asin((e*x)/d)/2)**2 
+ 5*tan(asin((e*x)/d)/2) - 1))