\(\int \frac {x^2}{(c+d x)^3 (c^2-d^2 x^2)^{3/2}} \, dx\) [425]

Optimal result

Integrand size = 27, antiderivative size = 122 \[ \int \frac {x^2}{(c+d x)^3 \left (c^2-d^2 x^2\right )^{3/2}} \, dx=\frac {2 x}{21 c^3 d^2 \sqrt {c^2-d^2 x^2}}-\frac {c}{7 d^3 (c+d x)^3 \sqrt {c^2-d^2 x^2}}+\frac {2}{7 d^3 (c+d x)^2 \sqrt {c^2-d^2 x^2}}-\frac {1}{21 c d^3 (c+d x) \sqrt {c^2-d^2 x^2}} \] Output:

2/21*x/c^3/d^2/(-d^2*x^2+c^2)^(1/2)-1/7*c/d^3/(d*x+c)^3/(-d^2*x^2+c^2)^(1/ 
2)+2/7/d^3/(d*x+c)^2/(-d^2*x^2+c^2)^(1/2)-1/21/c/d^3/(d*x+c)/(-d^2*x^2+c^2 
)^(1/2)
 

Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.67 \[ \int \frac {x^2}{(c+d x)^3 \left (c^2-d^2 x^2\right )^{3/2}} \, dx=\frac {\sqrt {c^2-d^2 x^2} \left (2 c^4+6 c^3 d x+5 c^2 d^2 x^2+6 c d^3 x^3+2 d^4 x^4\right )}{21 c^3 d^3 (c-d x) (c+d x)^4} \] Input:

Integrate[x^2/((c + d*x)^3*(c^2 - d^2*x^2)^(3/2)),x]
 

Output:

(Sqrt[c^2 - d^2*x^2]*(2*c^4 + 6*c^3*d*x + 5*c^2*d^2*x^2 + 6*c*d^3*x^3 + 2* 
d^4*x^4))/(21*c^3*d^3*(c - d*x)*(c + d*x)^4)
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.42, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {581, 25, 27, 671, 461, 470, 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.

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

\(\Big \downarrow \) 581

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 671

\(\displaystyle \frac {c \left (\frac {5}{7} \int \frac {1}{(c+d x)^2 \left (c^2-d^2 x^2\right )^{3/2}}dx-\frac {3}{7 d (c+d x)^3 \sqrt {c^2-d^2 x^2}}\right )}{3 d^2}+\frac {1}{3 d^3 (c+d x)^2 \sqrt {c^2-d^2 x^2}}\)

\(\Big \downarrow \) 461

\(\displaystyle \frac {c \left (\frac {5}{7} \left (\frac {3 \int \frac {1}{(c+d x) \left (c^2-d^2 x^2\right )^{3/2}}dx}{5 c}-\frac {1}{5 c d (c+d x)^2 \sqrt {c^2-d^2 x^2}}\right )-\frac {3}{7 d (c+d x)^3 \sqrt {c^2-d^2 x^2}}\right )}{3 d^2}+\frac {1}{3 d^3 (c+d x)^2 \sqrt {c^2-d^2 x^2}}\)

\(\Big \downarrow \) 470

\(\displaystyle \frac {c \left (\frac {5}{7} \left (\frac {3 \left (\frac {2 \int \frac {1}{\left (c^2-d^2 x^2\right )^{3/2}}dx}{3 c}-\frac {1}{3 c d (c+d x) \sqrt {c^2-d^2 x^2}}\right )}{5 c}-\frac {1}{5 c d (c+d x)^2 \sqrt {c^2-d^2 x^2}}\right )-\frac {3}{7 d (c+d x)^3 \sqrt {c^2-d^2 x^2}}\right )}{3 d^2}+\frac {1}{3 d^3 (c+d x)^2 \sqrt {c^2-d^2 x^2}}\)

\(\Big \downarrow \) 208

\(\displaystyle \frac {1}{3 d^3 (c+d x)^2 \sqrt {c^2-d^2 x^2}}+\frac {c \left (\frac {5}{7} \left (\frac {3 \left (\frac {2 x}{3 c^3 \sqrt {c^2-d^2 x^2}}-\frac {1}{3 c d (c+d x) \sqrt {c^2-d^2 x^2}}\right )}{5 c}-\frac {1}{5 c d (c+d x)^2 \sqrt {c^2-d^2 x^2}}\right )-\frac {3}{7 d (c+d x)^3 \sqrt {c^2-d^2 x^2}}\right )}{3 d^2}\)

Input:

Int[x^2/((c + d*x)^3*(c^2 - d^2*x^2)^(3/2)),x]
 

Output:

1/(3*d^3*(c + d*x)^2*Sqrt[c^2 - d^2*x^2]) + (c*(-3/(7*d*(c + d*x)^3*Sqrt[c 
^2 - d^2*x^2]) + (5*(-1/5*1/(c*d*(c + d*x)^2*Sqrt[c^2 - d^2*x^2]) + (3*((2 
*x)/(3*c^3*Sqrt[c^2 - d^2*x^2]) - 1/(3*c*d*(c + d*x)*Sqrt[c^2 - d^2*x^2])) 
)/(5*c)))/7))/(3*d^2)
 

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 208

Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), 
x] /; FreeQ[{a, b}, x]
 

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])
 

rule 470

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[(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, p}, x] && EqQ[b*c^2 + a*d^2, 0] && LtQ[n, 0] && NeQ[n + 
 p + 1, 0] && IntegerQ[2*p]
 

rule 581

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

rule 671

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_ 
), x_Symbol] :> Simp[(d*g - e*f)*(d + e*x)^m*((a + c*x^2)^(p + 1)/(2*c*d*(m 
 + p + 1))), x] + Simp[(m*(g*c*d + c*e*f) + 2*e*c*f*(p + 1))/(e*(2*c*d)*(m 
+ p + 1))   Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, 
e, f, g, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && ((LtQ[m, -1] &&  !IGtQ[m + p 
 + 1, 0]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p 
 + 1, 0]
 

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.63

method	result	size
gosper	\(\frac {\left (-d x +c \right ) \left (2 d^{4} x^{4}+6 c \,d^{3} x^{3}+5 d^{2} c^{2} x^{2}+6 c^{3} d x +2 c^{4}\right )}{21 \left (d x +c \right )^{2} d^{3} c^{3} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}\)	\(77\)
orering	\(\frac {\left (-d x +c \right ) \left (2 d^{4} x^{4}+6 c \,d^{3} x^{3}+5 d^{2} c^{2} x^{2}+6 c^{3} d x +2 c^{4}\right )}{21 \left (d x +c \right )^{2} d^{3} c^{3} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}\)	\(77\)
trager	\(\frac {\left (2 d^{4} x^{4}+6 c \,d^{3} x^{3}+5 d^{2} c^{2} x^{2}+6 c^{3} d x +2 c^{4}\right ) \sqrt {-d^{2} x^{2}+c^{2}}}{21 c^{3} \left (d x +c \right )^{4} d^{3} \left (-d x +c \right )}\)	\(79\)
default	\(\frac {-\frac {1}{3 c d \left (x +\frac {c}{d}\right ) \sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}-\frac {-2 d^{2} \left (x +\frac {c}{d}\right )+2 c d}{3 d \,c^{3} \sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}}{d^{3}}+\frac {c^{2} \left (-\frac {1}{7 c d \left (x +\frac {c}{d}\right )^{3} \sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}+\frac {4 d \left (-\frac {1}{5 c d \left (x +\frac {c}{d}\right )^{2} \sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}+\frac {3 d \left (-\frac {1}{3 c d \left (x +\frac {c}{d}\right ) \sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}-\frac {-2 d^{2} \left (x +\frac {c}{d}\right )+2 c d}{3 d \,c^{3} \sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}\right )}{5 c}\right )}{7 c}\right )}{d^{5}}-\frac {2 c \left (-\frac {1}{5 c d \left (x +\frac {c}{d}\right )^{2} \sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}+\frac {3 d \left (-\frac {1}{3 c d \left (x +\frac {c}{d}\right ) \sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}-\frac {-2 d^{2} \left (x +\frac {c}{d}\right )+2 c d}{3 d \,c^{3} \sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}\right )}{5 c}\right )}{d^{4}}\)	\(472\)

Input:

int(x^2/(d*x+c)^3/(-d^2*x^2+c^2)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

1/21*(-d*x+c)*(2*d^4*x^4+6*c*d^3*x^3+5*c^2*d^2*x^2+6*c^3*d*x+2*c^4)/(d*x+c 
)^2/d^3/c^3/(-d^2*x^2+c^2)^(3/2)
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.43 \[ \int \frac {x^2}{(c+d x)^3 \left (c^2-d^2 x^2\right )^{3/2}} \, dx=\frac {2 \, d^{5} x^{5} + 6 \, c d^{4} x^{4} + 4 \, c^{2} d^{3} x^{3} - 4 \, c^{3} d^{2} x^{2} - 6 \, c^{4} d x - 2 \, c^{5} - {\left (2 \, d^{4} x^{4} + 6 \, c d^{3} x^{3} + 5 \, c^{2} d^{2} x^{2} + 6 \, c^{3} d x + 2 \, c^{4}\right )} \sqrt {-d^{2} x^{2} + c^{2}}}{21 \, {\left (c^{3} d^{8} x^{5} + 3 \, c^{4} d^{7} x^{4} + 2 \, c^{5} d^{6} x^{3} - 2 \, c^{6} d^{5} x^{2} - 3 \, c^{7} d^{4} x - c^{8} d^{3}\right )}} \] Input:

integrate(x^2/(d*x+c)^3/(-d^2*x^2+c^2)^(3/2),x, algorithm="fricas")
 

Output:

1/21*(2*d^5*x^5 + 6*c*d^4*x^4 + 4*c^2*d^3*x^3 - 4*c^3*d^2*x^2 - 6*c^4*d*x 
- 2*c^5 - (2*d^4*x^4 + 6*c*d^3*x^3 + 5*c^2*d^2*x^2 + 6*c^3*d*x + 2*c^4)*sq 
rt(-d^2*x^2 + c^2))/(c^3*d^8*x^5 + 3*c^4*d^7*x^4 + 2*c^5*d^6*x^3 - 2*c^6*d 
^5*x^2 - 3*c^7*d^4*x - c^8*d^3)
 

Sympy [F]

\[ \int \frac {x^2}{(c+d x)^3 \left (c^2-d^2 x^2\right )^{3/2}} \, dx=\int \frac {x^{2}}{\left (- \left (- c + d x\right ) \left (c + d x\right )\right )^{\frac {3}{2}} \left (c + d x\right )^{3}}\, dx \] Input:

integrate(x**2/(d*x+c)**3/(-d**2*x**2+c**2)**(3/2),x)
 

Output:

Integral(x**2/((-(-c + d*x)*(c + d*x))**(3/2)*(c + d*x)**3), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (106) = 212\).

Time = 0.03 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.90 \[ \int \frac {x^2}{(c+d x)^3 \left (c^2-d^2 x^2\right )^{3/2}} \, dx=-\frac {c}{7 \, {\left (\sqrt {-d^{2} x^{2} + c^{2}} d^{6} x^{3} + 3 \, \sqrt {-d^{2} x^{2} + c^{2}} c d^{5} x^{2} + 3 \, \sqrt {-d^{2} x^{2} + c^{2}} c^{2} d^{4} x + \sqrt {-d^{2} x^{2} + c^{2}} c^{3} d^{3}\right )}} + \frac {2}{7 \, {\left (\sqrt {-d^{2} x^{2} + c^{2}} d^{5} x^{2} + 2 \, \sqrt {-d^{2} x^{2} + c^{2}} c d^{4} x + \sqrt {-d^{2} x^{2} + c^{2}} c^{2} d^{3}\right )}} - \frac {1}{21 \, {\left (\sqrt {-d^{2} x^{2} + c^{2}} c d^{4} x + \sqrt {-d^{2} x^{2} + c^{2}} c^{2} d^{3}\right )}} + \frac {2 \, x}{21 \, \sqrt {-d^{2} x^{2} + c^{2}} c^{3} d^{2}} \] Input:

integrate(x^2/(d*x+c)^3/(-d^2*x^2+c^2)^(3/2),x, algorithm="maxima")
 

Output:

-1/7*c/(sqrt(-d^2*x^2 + c^2)*d^6*x^3 + 3*sqrt(-d^2*x^2 + c^2)*c*d^5*x^2 + 
3*sqrt(-d^2*x^2 + c^2)*c^2*d^4*x + sqrt(-d^2*x^2 + c^2)*c^3*d^3) + 2/7/(sq 
rt(-d^2*x^2 + c^2)*d^5*x^2 + 2*sqrt(-d^2*x^2 + c^2)*c*d^4*x + sqrt(-d^2*x^ 
2 + c^2)*c^2*d^3) - 1/21/(sqrt(-d^2*x^2 + c^2)*c*d^4*x + sqrt(-d^2*x^2 + c 
^2)*c^2*d^3) + 2/21*x/(sqrt(-d^2*x^2 + c^2)*c^3*d^2)
 

Giac [F]

\[ \int \frac {x^2}{(c+d x)^3 \left (c^2-d^2 x^2\right )^{3/2}} \, dx=\int { \frac {x^{2}}{{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{3}} \,d x } \] Input:

integrate(x^2/(d*x+c)^3/(-d^2*x^2+c^2)^(3/2),x, algorithm="giac")
 

Output:

integrate(x^2/((-d^2*x^2 + c^2)^(3/2)*(d*x + c)^3), x)
 

Mupad [B] (verification not implemented)

Time = 7.05 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.09 \[ \int \frac {x^2}{(c+d x)^3 \left (c^2-d^2 x^2\right )^{3/2}} \, dx=\frac {\sqrt {c^2-d^2\,x^2}\,\left (\frac {5}{168\,c^2\,d^3}+\frac {2\,x}{21\,c^3\,d^2}\right )}{\left (c+d\,x\right )\,\left (c-d\,x\right )}-\frac {\sqrt {c^2-d^2\,x^2}}{14\,d^3\,{\left (c+d\,x\right )}^4}+\frac {3\,\sqrt {c^2-d^2\,x^2}}{28\,c\,d^3\,{\left (c+d\,x\right )}^3}+\frac {5\,\sqrt {c^2-d^2\,x^2}}{168\,c^2\,d^3\,{\left (c+d\,x\right )}^2} \] Input:

int(x^2/((c^2 - d^2*x^2)^(3/2)*(c + d*x)^3),x)
 

Output:

((c^2 - d^2*x^2)^(1/2)*(5/(168*c^2*d^3) + (2*x)/(21*c^3*d^2)))/((c + d*x)* 
(c - d*x)) - (c^2 - d^2*x^2)^(1/2)/(14*d^3*(c + d*x)^4) + (3*(c^2 - d^2*x^ 
2)^(1/2))/(28*c*d^3*(c + d*x)^3) + (5*(c^2 - d^2*x^2)^(1/2))/(168*c^2*d^3* 
(c + d*x)^2)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.43 \[ \int \frac {x^2}{(c+d x)^3 \left (c^2-d^2 x^2\right )^{3/2}} \, dx=\frac {2 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{3}+6 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{2} d x +6 \sqrt {-d^{2} x^{2}+c^{2}}\, c \,d^{2} x^{2}+2 \sqrt {-d^{2} x^{2}+c^{2}}\, d^{3} x^{3}+2 c^{4}+6 c^{3} d x +5 c^{2} d^{2} x^{2}+6 c \,d^{3} x^{3}+2 d^{4} x^{4}}{21 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{3} d^{3} \left (d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}\right )} \] Input:

int(x^2/(d*x+c)^3/(-d^2*x^2+c^2)^(3/2),x)
 

Output:

(2*sqrt(c**2 - d**2*x**2)*c**3 + 6*sqrt(c**2 - d**2*x**2)*c**2*d*x + 6*sqr 
t(c**2 - d**2*x**2)*c*d**2*x**2 + 2*sqrt(c**2 - d**2*x**2)*d**3*x**3 + 2*c 
**4 + 6*c**3*d*x + 5*c**2*d**2*x**2 + 6*c*d**3*x**3 + 2*d**4*x**4)/(21*sqr 
t(c**2 - d**2*x**2)*c**3*d**3*(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x* 
*3))