\(\int \frac {x}{(c+d x) (c^2-d^2 x^2)^{7/2}} \, dx\) [492]

Optimal result

Integrand size = 25, antiderivative size = 112 \[ \int \frac {x}{(c+d x) \left (c^2-d^2 x^2\right )^{7/2}} \, dx=\frac {x}{35 c^2 d \left (c^2-d^2 x^2\right )^{5/2}}+\frac {1}{7 d^2 (c+d x) \left (c^2-d^2 x^2\right )^{5/2}}+\frac {4 x}{105 c^4 d \left (c^2-d^2 x^2\right )^{3/2}}+\frac {8 x}{105 c^6 d \sqrt {c^2-d^2 x^2}} \] Output:

1/35*x/c^2/d/(-d^2*x^2+c^2)^(5/2)+1/7/d^2/(d*x+c)/(-d^2*x^2+c^2)^(5/2)+4/1 
05*x/c^4/d/(-d^2*x^2+c^2)^(3/2)+8/105*x/c^6/d/(-d^2*x^2+c^2)^(1/2)
 

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.93 \[ \int \frac {x}{(c+d x) \left (c^2-d^2 x^2\right )^{7/2}} \, dx=\frac {\sqrt {c^2-d^2 x^2} \left (15 c^6+15 c^5 d x+15 c^4 d^2 x^2-20 c^3 d^3 x^3-20 c^2 d^4 x^4+8 c d^5 x^5+8 d^6 x^6\right )}{105 c^6 d^2 (c-d x)^3 (c+d x)^4} \] Input:

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

Output:

(Sqrt[c^2 - d^2*x^2]*(15*c^6 + 15*c^5*d*x + 15*c^4*d^2*x^2 - 20*c^3*d^3*x^ 
3 - 20*c^2*d^4*x^4 + 8*c*d^5*x^5 + 8*d^6*x^6))/(105*c^6*d^2*(c - d*x)^3*(c 
 + d*x)^4)
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.08, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {565, 245, 245, 242}

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.

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

\(\Big \downarrow \) 565

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

\(\Big \downarrow \) 245

\(\displaystyle \frac {c}{7 d^2 \left (c^2-d^2 x^2\right )^{7/2}}-d \left (\frac {x^3}{3 c^2 \left (c^2-d^2 x^2\right )^{7/2}}-\frac {4 d^2 \int \frac {x^4}{\left (c^2-d^2 x^2\right )^{9/2}}dx}{3 c^2}\right )\)

\(\Big \downarrow \) 245

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

\(\Big \downarrow \) 242

\(\displaystyle \frac {c}{7 d^2 \left (c^2-d^2 x^2\right )^{7/2}}-d \left (\frac {x^3}{3 c^2 \left (c^2-d^2 x^2\right )^{7/2}}-\frac {4 d^2 \left (\frac {x^5}{5 c^2 \left (c^2-d^2 x^2\right )^{7/2}}-\frac {2 d^2 x^7}{35 c^4 \left (c^2-d^2 x^2\right )^{7/2}}\right )}{3 c^2}\right )\)

Input:

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

Output:

c/(7*d^2*(c^2 - d^2*x^2)^(7/2)) - d*(x^3/(3*c^2*(c^2 - d^2*x^2)^(7/2)) - ( 
4*d^2*(x^5/(5*c^2*(c^2 - d^2*x^2)^(7/2)) - (2*d^2*x^7)/(35*c^4*(c^2 - d^2* 
x^2)^(7/2))))/(3*c^2))
 

Defintions of rubi rules used

rule 242

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ 
(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, p}, x 
] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
 

rule 245

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + 
 b*x^2)^(p + 1)/(a*(m + 1))), x] - Simp[b*((m + 2*(p + 1) + 1)/(a*(m + 1))) 
   Int[x^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, m, p}, x] && ILtQ[Si 
mplify[(m + 1)/2 + p + 1], 0] && NeQ[m, -1]
 

rule 565

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

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.82

method	result	size
gosper	\(\frac {\left (-d x +c \right ) \left (8 d^{6} x^{6}+8 c \,d^{5} x^{5}-20 c^{2} d^{4} x^{4}-20 d^{3} c^{3} x^{3}+15 c^{4} d^{2} x^{2}+15 c^{5} d x +15 c^{6}\right )}{105 c^{6} d^{2} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {7}{2}}}\)	\(92\)
orering	\(\frac {\left (-d x +c \right ) \left (8 d^{6} x^{6}+8 c \,d^{5} x^{5}-20 c^{2} d^{4} x^{4}-20 d^{3} c^{3} x^{3}+15 c^{4} d^{2} x^{2}+15 c^{5} d x +15 c^{6}\right )}{105 c^{6} d^{2} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {7}{2}}}\)	\(92\)
trager	\(\frac {\left (8 d^{6} x^{6}+8 c \,d^{5} x^{5}-20 c^{2} d^{4} x^{4}-20 d^{3} c^{3} x^{3}+15 c^{4} d^{2} x^{2}+15 c^{5} d x +15 c^{6}\right ) \sqrt {-d^{2} x^{2}+c^{2}}}{105 c^{6} \left (d x +c \right )^{4} \left (-d x +c \right )^{3} d^{2}}\)	\(101\)
default	\(\frac {\frac {x}{5 c^{2} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 c^{2} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 c^{4} \sqrt {-d^{2} x^{2}+c^{2}}}}{c^{2}}}{d}-\frac {c \left (-\frac {1}{7 c d \left (x +\frac {c}{d}\right ) \left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {5}{2}}}+\frac {6 d \left (-\frac {-2 d^{2} \left (x +\frac {c}{d}\right )+2 c d}{10 c^{2} d^{2} \left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {5}{2}}}+\frac {-\frac {2 \left (-2 d^{2} \left (x +\frac {c}{d}\right )+2 c d \right )}{15 c^{2} d^{2} \left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {3}{2}}}-\frac {4 \left (-2 d^{2} \left (x +\frac {c}{d}\right )+2 c d \right )}{15 d^{2} c^{4} \sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}}{c^{2}}\right )}{7 c}\right )}{d^{2}}\)	\(297\)

Input:

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

Output:

1/105*(-d*x+c)*(8*d^6*x^6+8*c*d^5*x^5-20*c^2*d^4*x^4-20*c^3*d^3*x^3+15*c^4 
*d^2*x^2+15*c^5*d*x+15*c^6)/c^6/d^2/(-d^2*x^2+c^2)^(7/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 239 vs. \(2 (96) = 192\).

Time = 0.18 (sec) , antiderivative size = 239, normalized size of antiderivative = 2.13 \[ \int \frac {x}{(c+d x) \left (c^2-d^2 x^2\right )^{7/2}} \, dx=\frac {15 \, d^{7} x^{7} + 15 \, c d^{6} x^{6} - 45 \, c^{2} d^{5} x^{5} - 45 \, c^{3} d^{4} x^{4} + 45 \, c^{4} d^{3} x^{3} + 45 \, c^{5} d^{2} x^{2} - 15 \, c^{6} d x - 15 \, c^{7} - {\left (8 \, d^{6} x^{6} + 8 \, c d^{5} x^{5} - 20 \, c^{2} d^{4} x^{4} - 20 \, c^{3} d^{3} x^{3} + 15 \, c^{4} d^{2} x^{2} + 15 \, c^{5} d x + 15 \, c^{6}\right )} \sqrt {-d^{2} x^{2} + c^{2}}}{105 \, {\left (c^{6} d^{9} x^{7} + c^{7} d^{8} x^{6} - 3 \, c^{8} d^{7} x^{5} - 3 \, c^{9} d^{6} x^{4} + 3 \, c^{10} d^{5} x^{3} + 3 \, c^{11} d^{4} x^{2} - c^{12} d^{3} x - c^{13} d^{2}\right )}} \] Input:

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

Output:

1/105*(15*d^7*x^7 + 15*c*d^6*x^6 - 45*c^2*d^5*x^5 - 45*c^3*d^4*x^4 + 45*c^ 
4*d^3*x^3 + 45*c^5*d^2*x^2 - 15*c^6*d*x - 15*c^7 - (8*d^6*x^6 + 8*c*d^5*x^ 
5 - 20*c^2*d^4*x^4 - 20*c^3*d^3*x^3 + 15*c^4*d^2*x^2 + 15*c^5*d*x + 15*c^6 
)*sqrt(-d^2*x^2 + c^2))/(c^6*d^9*x^7 + c^7*d^8*x^6 - 3*c^8*d^7*x^5 - 3*c^9 
*d^6*x^4 + 3*c^10*d^5*x^3 + 3*c^11*d^4*x^2 - c^12*d^3*x - c^13*d^2)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.01 \[ \int \frac {x}{(c+d x) \left (c^2-d^2 x^2\right )^{7/2}} \, dx=\frac {1}{7 \, {\left ({\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} d^{3} x + {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} c d^{2}\right )}} + \frac {x}{35 \, {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} c^{2} d} + \frac {4 \, x}{105 \, {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} c^{4} d} + \frac {8 \, x}{105 \, \sqrt {-d^{2} x^{2} + c^{2}} c^{6} d} \] Input:

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

Output:

1/7/((-d^2*x^2 + c^2)^(5/2)*d^3*x + (-d^2*x^2 + c^2)^(5/2)*c*d^2) + 1/35*x 
/((-d^2*x^2 + c^2)^(5/2)*c^2*d) + 4/105*x/((-d^2*x^2 + c^2)^(3/2)*c^4*d) + 
 8/105*x/(sqrt(-d^2*x^2 + c^2)*c^6*d)
 

Giac [F]

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

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

Output:

integrate(x/((-d^2*x^2 + c^2)^(7/2)*(d*x + c)), x)
 

Mupad [B] (verification not implemented)

Time = 6.96 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.47 \[ \int \frac {x}{(c+d x) \left (c^2-d^2 x^2\right )^{7/2}} \, dx=\frac {\sqrt {c^2-d^2\,x^2}\,\left (\frac {1}{7\,c\,d^2}-\frac {3\,x}{70\,c^2\,d}\right )}{{\left (c+d\,x\right )}^3\,{\left (c-d\,x\right )}^3}-\frac {\sqrt {c^2-d^2\,x^2}\,\left (\frac {1}{56\,c^3\,d^2}-\frac {4\,x}{105\,c^4\,d}\right )}{{\left (c+d\,x\right )}^2\,{\left (c-d\,x\right )}^2}+\frac {\sqrt {c^2-d^2\,x^2}}{56\,c^3\,d^2\,{\left (c+d\,x\right )}^4}+\frac {8\,x\,\sqrt {c^2-d^2\,x^2}}{105\,c^6\,d\,\left (c+d\,x\right )\,\left (c-d\,x\right )} \] Input:

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

Output:

((c^2 - d^2*x^2)^(1/2)*(1/(7*c*d^2) - (3*x)/(70*c^2*d)))/((c + d*x)^3*(c - 
 d*x)^3) - ((c^2 - d^2*x^2)^(1/2)*(1/(56*c^3*d^2) - (4*x)/(105*c^4*d)))/(( 
c + d*x)^2*(c - d*x)^2) + (c^2 - d^2*x^2)^(1/2)/(56*c^3*d^2*(c + d*x)^4) + 
 (8*x*(c^2 - d^2*x^2)^(1/2))/(105*c^6*d*(c + d*x)*(c - d*x))
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 264, normalized size of antiderivative = 2.36 \[ \int \frac {x}{(c+d x) \left (c^2-d^2 x^2\right )^{7/2}} \, dx=\frac {15 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{5}+15 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{4} d x -30 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{3} d^{2} x^{2}-30 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{2} d^{3} x^{3}+15 \sqrt {-d^{2} x^{2}+c^{2}}\, c \,d^{4} x^{4}+15 \sqrt {-d^{2} x^{2}+c^{2}}\, d^{5} x^{5}+15 c^{6}+15 c^{5} d x +15 c^{4} d^{2} x^{2}-20 c^{3} d^{3} x^{3}-20 c^{2} d^{4} x^{4}+8 c \,d^{5} x^{5}+8 d^{6} x^{6}}{105 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{6} d^{2} \left (d^{5} x^{5}+c \,d^{4} x^{4}-2 c^{2} d^{3} x^{3}-2 c^{3} d^{2} x^{2}+c^{4} d x +c^{5}\right )} \] Input:

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

Output:

(15*sqrt(c**2 - d**2*x**2)*c**5 + 15*sqrt(c**2 - d**2*x**2)*c**4*d*x - 30* 
sqrt(c**2 - d**2*x**2)*c**3*d**2*x**2 - 30*sqrt(c**2 - d**2*x**2)*c**2*d** 
3*x**3 + 15*sqrt(c**2 - d**2*x**2)*c*d**4*x**4 + 15*sqrt(c**2 - d**2*x**2) 
*d**5*x**5 + 15*c**6 + 15*c**5*d*x + 15*c**4*d**2*x**2 - 20*c**3*d**3*x**3 
 - 20*c**2*d**4*x**4 + 8*c*d**5*x**5 + 8*d**6*x**6)/(105*sqrt(c**2 - d**2* 
x**2)*c**6*d**2*(c**5 + c**4*d*x - 2*c**3*d**2*x**2 - 2*c**2*d**3*x**3 + c 
*d**4*x**4 + d**5*x**5))