\(\int x^2 (c+d x)^2 (c^2-d^2 x^2)^{3/2} \, dx\) [184]

Optimal result

Integrand size = 27, antiderivative size = 195 \[ \int x^2 (c+d x)^2 \left (c^2-d^2 x^2\right )^{3/2} \, dx=\frac {11 c^6 x \sqrt {c^2-d^2 x^2}}{128 d^2}+\frac {11 c^4 x \left (c^2-d^2 x^2\right )^{3/2}}{192 d^2}+\frac {11 c^2 x \left (c^2-d^2 x^2\right )^{5/2}}{240 d^2}-\frac {c (c+d x)^2 \left (c^2-d^2 x^2\right )^{5/2}}{5 d^3}-\frac {11 c \left (c^2-d^2 x^2\right )^{7/2}}{280 d^3}+\frac {(c+d x) \left (c^2-d^2 x^2\right )^{7/2}}{8 d^3}+\frac {11 c^8 \arctan \left (\frac {d x}{\sqrt {c^2-d^2 x^2}}\right )}{128 d^3} \] Output:

11/128*c^6*x*(-d^2*x^2+c^2)^(1/2)/d^2+11/192*c^4*x*(-d^2*x^2+c^2)^(3/2)/d^ 
2+11/240*c^2*x*(-d^2*x^2+c^2)^(5/2)/d^2-1/5*c*(d*x+c)^2*(-d^2*x^2+c^2)^(5/ 
2)/d^3-11/280*c*(-d^2*x^2+c^2)^(7/2)/d^3+1/8*(d*x+c)*(-d^2*x^2+c^2)^(7/2)/ 
d^3+11/128*c^8*arctan(d*x/(-d^2*x^2+c^2)^(1/2))/d^3
 

Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.70 \[ \int x^2 (c+d x)^2 \left (c^2-d^2 x^2\right )^{3/2} \, dx=-\frac {\sqrt {c^2-d^2 x^2} \left (1536 c^7+1155 c^6 d x+768 c^5 d^2 x^2-3710 c^4 d^3 x^3-6144 c^3 d^4 x^4-280 c^2 d^5 x^5+3840 c d^6 x^6+1680 d^7 x^7\right )+2310 c^8 \arctan \left (\frac {d x}{\sqrt {c^2}-\sqrt {c^2-d^2 x^2}}\right )}{13440 d^3} \] Input:

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

Output:

-1/13440*(Sqrt[c^2 - d^2*x^2]*(1536*c^7 + 1155*c^6*d*x + 768*c^5*d^2*x^2 - 
 3710*c^4*d^3*x^3 - 6144*c^3*d^4*x^4 - 280*c^2*d^5*x^5 + 3840*c*d^6*x^6 + 
1680*d^7*x^7) + 2310*c^8*ArcTan[(d*x)/(Sqrt[c^2] - Sqrt[c^2 - d^2*x^2])])/ 
d^3
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.05, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {541, 25, 27, 533, 27, 533, 27, 455, 211, 211, 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.

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

\(\Big \downarrow \) 541

\(\displaystyle -\frac {\int -c d^2 x^2 (11 c+16 d x) \left (c^2-d^2 x^2\right )^{3/2}dx}{8 d^2}-\frac {1}{8} x^3 \left (c^2-d^2 x^2\right )^{5/2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int c d^2 x^2 (11 c+16 d x) \left (c^2-d^2 x^2\right )^{3/2}dx}{8 d^2}-\frac {1}{8} x^3 \left (c^2-d^2 x^2\right )^{5/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{8} c \int x^2 (11 c+16 d x) \left (c^2-d^2 x^2\right )^{3/2}dx-\frac {1}{8} x^3 \left (c^2-d^2 x^2\right )^{5/2}\)

\(\Big \downarrow \) 533

\(\displaystyle \frac {1}{8} c \left (\frac {\int c d x (32 c+77 d x) \left (c^2-d^2 x^2\right )^{3/2}dx}{7 d^2}-\frac {16 x^2 \left (c^2-d^2 x^2\right )^{5/2}}{7 d}\right )-\frac {1}{8} x^3 \left (c^2-d^2 x^2\right )^{5/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{8} c \left (\frac {c \int x (32 c+77 d x) \left (c^2-d^2 x^2\right )^{3/2}dx}{7 d}-\frac {16 x^2 \left (c^2-d^2 x^2\right )^{5/2}}{7 d}\right )-\frac {1}{8} x^3 \left (c^2-d^2 x^2\right )^{5/2}\)

\(\Big \downarrow \) 533

\(\displaystyle \frac {1}{8} c \left (\frac {c \left (\frac {\int c d (77 c+192 d x) \left (c^2-d^2 x^2\right )^{3/2}dx}{6 d^2}-\frac {77 x \left (c^2-d^2 x^2\right )^{5/2}}{6 d}\right )}{7 d}-\frac {16 x^2 \left (c^2-d^2 x^2\right )^{5/2}}{7 d}\right )-\frac {1}{8} x^3 \left (c^2-d^2 x^2\right )^{5/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{8} c \left (\frac {c \left (\frac {c \int (77 c+192 d x) \left (c^2-d^2 x^2\right )^{3/2}dx}{6 d}-\frac {77 x \left (c^2-d^2 x^2\right )^{5/2}}{6 d}\right )}{7 d}-\frac {16 x^2 \left (c^2-d^2 x^2\right )^{5/2}}{7 d}\right )-\frac {1}{8} x^3 \left (c^2-d^2 x^2\right )^{5/2}\)

\(\Big \downarrow \) 455

\(\displaystyle \frac {1}{8} c \left (\frac {c \left (\frac {c \left (77 c \int \left (c^2-d^2 x^2\right )^{3/2}dx-\frac {192 \left (c^2-d^2 x^2\right )^{5/2}}{5 d}\right )}{6 d}-\frac {77 x \left (c^2-d^2 x^2\right )^{5/2}}{6 d}\right )}{7 d}-\frac {16 x^2 \left (c^2-d^2 x^2\right )^{5/2}}{7 d}\right )-\frac {1}{8} x^3 \left (c^2-d^2 x^2\right )^{5/2}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {1}{8} c \left (\frac {c \left (\frac {c \left (77 c \left (\frac {3}{4} c^2 \int \sqrt {c^2-d^2 x^2}dx+\frac {1}{4} x \left (c^2-d^2 x^2\right )^{3/2}\right )-\frac {192 \left (c^2-d^2 x^2\right )^{5/2}}{5 d}\right )}{6 d}-\frac {77 x \left (c^2-d^2 x^2\right )^{5/2}}{6 d}\right )}{7 d}-\frac {16 x^2 \left (c^2-d^2 x^2\right )^{5/2}}{7 d}\right )-\frac {1}{8} x^3 \left (c^2-d^2 x^2\right )^{5/2}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {1}{8} c \left (\frac {c \left (\frac {c \left (77 c \left (\frac {3}{4} c^2 \left (\frac {1}{2} c^2 \int \frac {1}{\sqrt {c^2-d^2 x^2}}dx+\frac {1}{2} x \sqrt {c^2-d^2 x^2}\right )+\frac {1}{4} x \left (c^2-d^2 x^2\right )^{3/2}\right )-\frac {192 \left (c^2-d^2 x^2\right )^{5/2}}{5 d}\right )}{6 d}-\frac {77 x \left (c^2-d^2 x^2\right )^{5/2}}{6 d}\right )}{7 d}-\frac {16 x^2 \left (c^2-d^2 x^2\right )^{5/2}}{7 d}\right )-\frac {1}{8} x^3 \left (c^2-d^2 x^2\right )^{5/2}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {1}{8} c \left (\frac {c \left (\frac {c \left (77 c \left (\frac {3}{4} c^2 \left (\frac {1}{2} c^2 \int \frac {1}{\frac {d^2 x^2}{c^2-d^2 x^2}+1}d\frac {x}{\sqrt {c^2-d^2 x^2}}+\frac {1}{2} x \sqrt {c^2-d^2 x^2}\right )+\frac {1}{4} x \left (c^2-d^2 x^2\right )^{3/2}\right )-\frac {192 \left (c^2-d^2 x^2\right )^{5/2}}{5 d}\right )}{6 d}-\frac {77 x \left (c^2-d^2 x^2\right )^{5/2}}{6 d}\right )}{7 d}-\frac {16 x^2 \left (c^2-d^2 x^2\right )^{5/2}}{7 d}\right )-\frac {1}{8} x^3 \left (c^2-d^2 x^2\right )^{5/2}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {1}{8} c \left (\frac {c \left (\frac {c \left (77 c \left (\frac {3}{4} c^2 \left (\frac {c^2 \arctan \left (\frac {d x}{\sqrt {c^2-d^2 x^2}}\right )}{2 d}+\frac {1}{2} x \sqrt {c^2-d^2 x^2}\right )+\frac {1}{4} x \left (c^2-d^2 x^2\right )^{3/2}\right )-\frac {192 \left (c^2-d^2 x^2\right )^{5/2}}{5 d}\right )}{6 d}-\frac {77 x \left (c^2-d^2 x^2\right )^{5/2}}{6 d}\right )}{7 d}-\frac {16 x^2 \left (c^2-d^2 x^2\right )^{5/2}}{7 d}\right )-\frac {1}{8} x^3 \left (c^2-d^2 x^2\right )^{5/2}\)

Input:

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

Output:

-1/8*(x^3*(c^2 - d^2*x^2)^(5/2)) + (c*((-16*x^2*(c^2 - d^2*x^2)^(5/2))/(7* 
d) + (c*((-77*x*(c^2 - d^2*x^2)^(5/2))/(6*d) + (c*((-192*(c^2 - d^2*x^2)^( 
5/2))/(5*d) + 77*c*((x*(c^2 - d^2*x^2)^(3/2))/4 + (3*c^2*((x*Sqrt[c^2 - d^ 
2*x^2])/2 + (c^2*ArcTan[(d*x)/Sqrt[c^2 - d^2*x^2]])/(2*d)))/4)))/(6*d)))/( 
7*d)))/8
 

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 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

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 533

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

rule 541

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> Simp[d^n*x^(m + n - 1)*((a + b*x^2)^(p + 1)/(b*(m + n + 2*p + 1))), x 
] + Simp[1/(b*(m + n + 2*p + 1))   Int[x^m*(a + b*x^2)^p*ExpandToSum[b*(m + 
 n + 2*p + 1)*(c + d*x)^n - b*d^n*(m + n + 2*p + 1)*x^n - a*d^n*(m + n - 1) 
*x^(n - 2), x], x], x] /; FreeQ[{a, b, c, d, m, p}, x] && IGtQ[n, 1] && IGt 
Q[m, -2] && GtQ[p, -1] && IntegerQ[2*p]
 

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.67

method	result	size
risch	\(-\frac {\left (1680 d^{7} x^{7}+3840 c \,d^{6} x^{6}-280 c^{2} d^{5} x^{5}-6144 d^{4} c^{3} x^{4}-3710 d^{3} c^{4} x^{3}+768 c^{5} d^{2} x^{2}+1155 c^{6} d x +1536 c^{7}\right ) \sqrt {-d^{2} x^{2}+c^{2}}}{13440 d^{3}}+\frac {11 c^{8} \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+c^{2}}}\right )}{128 d^{2} \sqrt {d^{2}}}\)	\(130\)
default	\(c^{2} \left (-\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{6 d^{2}}+\frac {c^{2} \left (\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 c^{2} \left (\frac {x \sqrt {-d^{2} x^{2}+c^{2}}}{2}+\frac {c^{2} \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+c^{2}}}\right )}{2 \sqrt {d^{2}}}\right )}{4}\right )}{6 d^{2}}\right )+d^{2} \left (-\frac {x^{3} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{8 d^{2}}+\frac {3 c^{2} \left (-\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{6 d^{2}}+\frac {c^{2} \left (\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 c^{2} \left (\frac {x \sqrt {-d^{2} x^{2}+c^{2}}}{2}+\frac {c^{2} \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+c^{2}}}\right )}{2 \sqrt {d^{2}}}\right )}{4}\right )}{6 d^{2}}\right )}{8 d^{2}}\right )+2 c d \left (-\frac {x^{2} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{7 d^{2}}-\frac {2 c^{2} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{35 d^{4}}\right )\)	\(294\)

Input:

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

Output:

-1/13440*(1680*d^7*x^7+3840*c*d^6*x^6-280*c^2*d^5*x^5-6144*c^3*d^4*x^4-371 
0*c^4*d^3*x^3+768*c^5*d^2*x^2+1155*c^6*d*x+1536*c^7)/d^3*(-d^2*x^2+c^2)^(1 
/2)+11/128*c^8/d^2/(d^2)^(1/2)*arctan((d^2)^(1/2)*x/(-d^2*x^2+c^2)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.65 \[ \int x^2 (c+d x)^2 \left (c^2-d^2 x^2\right )^{3/2} \, dx=-\frac {2310 \, c^{8} \arctan \left (-\frac {c - \sqrt {-d^{2} x^{2} + c^{2}}}{d x}\right ) + {\left (1680 \, d^{7} x^{7} + 3840 \, c d^{6} x^{6} - 280 \, c^{2} d^{5} x^{5} - 6144 \, c^{3} d^{4} x^{4} - 3710 \, c^{4} d^{3} x^{3} + 768 \, c^{5} d^{2} x^{2} + 1155 \, c^{6} d x + 1536 \, c^{7}\right )} \sqrt {-d^{2} x^{2} + c^{2}}}{13440 \, d^{3}} \] Input:

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

Output:

-1/13440*(2310*c^8*arctan(-(c - sqrt(-d^2*x^2 + c^2))/(d*x)) + (1680*d^7*x 
^7 + 3840*c*d^6*x^6 - 280*c^2*d^5*x^5 - 6144*c^3*d^4*x^4 - 3710*c^4*d^3*x^ 
3 + 768*c^5*d^2*x^2 + 1155*c^6*d*x + 1536*c^7)*sqrt(-d^2*x^2 + c^2))/d^3
 

Sympy [A] (verification not implemented)

Time = 0.51 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.05 \[ \int x^2 (c+d x)^2 \left (c^2-d^2 x^2\right )^{3/2} \, dx=\begin {cases} \frac {11 c^{8} \left (\begin {cases} \frac {\log {\left (- 2 d^{2} x + 2 \sqrt {- d^{2}} \sqrt {c^{2} - d^{2} x^{2}} \right )}}{\sqrt {- d^{2}}} & \text {for}\: c^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- d^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{128 d^{2}} + \sqrt {c^{2} - d^{2} x^{2}} \left (- \frac {4 c^{7}}{35 d^{3}} - \frac {11 c^{6} x}{128 d^{2}} - \frac {2 c^{5} x^{2}}{35 d} + \frac {53 c^{4} x^{3}}{192} + \frac {16 c^{3} d x^{4}}{35} + \frac {c^{2} d^{2} x^{5}}{48} - \frac {2 c d^{3} x^{6}}{7} - \frac {d^{4} x^{7}}{8}\right ) & \text {for}\: d^{2} \neq 0 \\\left (\frac {c^{2} x^{3}}{3} + \frac {c d x^{4}}{2} + \frac {d^{2} x^{5}}{5}\right ) \left (c^{2}\right )^{\frac {3}{2}} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((11*c**8*Piecewise((log(-2*d**2*x + 2*sqrt(-d**2)*sqrt(c**2 - d* 
*2*x**2))/sqrt(-d**2), Ne(c**2, 0)), (x*log(x)/sqrt(-d**2*x**2), True))/(1 
28*d**2) + sqrt(c**2 - d**2*x**2)*(-4*c**7/(35*d**3) - 11*c**6*x/(128*d**2 
) - 2*c**5*x**2/(35*d) + 53*c**4*x**3/192 + 16*c**3*d*x**4/35 + c**2*d**2* 
x**5/48 - 2*c*d**3*x**6/7 - d**4*x**7/8), Ne(d**2, 0)), ((c**2*x**3/3 + c* 
d*x**4/2 + d**2*x**5/5)*(c**2)**(3/2), True))
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.76 \[ \int x^2 (c+d x)^2 \left (c^2-d^2 x^2\right )^{3/2} \, dx=\frac {11 \, c^{8} \arcsin \left (\frac {d x}{c}\right )}{128 \, d^{3}} + \frac {11 \, \sqrt {-d^{2} x^{2} + c^{2}} c^{6} x}{128 \, d^{2}} - \frac {1}{8} \, {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} x^{3} + \frac {11 \, {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} c^{4} x}{192 \, d^{2}} - \frac {2 \, {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} c x^{2}}{7 \, d} - \frac {11 \, {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} c^{2} x}{48 \, d^{2}} - \frac {4 \, {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} c^{3}}{35 \, d^{3}} \] Input:

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

Output:

11/128*c^8*arcsin(d*x/c)/d^3 + 11/128*sqrt(-d^2*x^2 + c^2)*c^6*x/d^2 - 1/8 
*(-d^2*x^2 + c^2)^(5/2)*x^3 + 11/192*(-d^2*x^2 + c^2)^(3/2)*c^4*x/d^2 - 2/ 
7*(-d^2*x^2 + c^2)^(5/2)*c*x^2/d - 11/48*(-d^2*x^2 + c^2)^(5/2)*c^2*x/d^2 
- 4/35*(-d^2*x^2 + c^2)^(5/2)*c^3/d^3
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.61 \[ \int x^2 (c+d x)^2 \left (c^2-d^2 x^2\right )^{3/2} \, dx=\frac {11 \, c^{8} \arcsin \left (\frac {d x}{c}\right ) \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (d\right )}{128 \, d^{2} {\left | d \right |}} - \frac {1}{13440} \, {\left (\frac {1536 \, c^{7}}{d^{3}} + {\left (\frac {1155 \, c^{6}}{d^{2}} + 2 \, {\left (\frac {384 \, c^{5}}{d} - {\left (1855 \, c^{4} + 4 \, {\left (768 \, c^{3} d + 5 \, {\left (7 \, c^{2} d^{2} - 6 \, {\left (7 \, d^{4} x + 16 \, c d^{3}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}} \] Input:

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

Output:

11/128*c^8*arcsin(d*x/c)*sgn(c)*sgn(d)/(d^2*abs(d)) - 1/13440*(1536*c^7/d^ 
3 + (1155*c^6/d^2 + 2*(384*c^5/d - (1855*c^4 + 4*(768*c^3*d + 5*(7*c^2*d^2 
 - 6*(7*d^4*x + 16*c*d^3)*x)*x)*x)*x)*x)*x)*sqrt(-d^2*x^2 + c^2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.03 \[ \int x^2 (c+d x)^2 \left (c^2-d^2 x^2\right )^{3/2} \, dx=\frac {1155 \mathit {asin} \left (\frac {d x}{c}\right ) c^{8}-1536 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{7}-1155 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{6} d x -768 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{5} d^{2} x^{2}+3710 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{4} d^{3} x^{3}+6144 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{3} d^{4} x^{4}+280 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{2} d^{5} x^{5}-3840 \sqrt {-d^{2} x^{2}+c^{2}}\, c \,d^{6} x^{6}-1680 \sqrt {-d^{2} x^{2}+c^{2}}\, d^{7} x^{7}+1536 c^{8}}{13440 d^{3}} \] Input:

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

Output:

(1155*asin((d*x)/c)*c**8 - 1536*sqrt(c**2 - d**2*x**2)*c**7 - 1155*sqrt(c* 
*2 - d**2*x**2)*c**6*d*x - 768*sqrt(c**2 - d**2*x**2)*c**5*d**2*x**2 + 371 
0*sqrt(c**2 - d**2*x**2)*c**4*d**3*x**3 + 6144*sqrt(c**2 - d**2*x**2)*c**3 
*d**4*x**4 + 280*sqrt(c**2 - d**2*x**2)*c**2*d**5*x**5 - 3840*sqrt(c**2 - 
d**2*x**2)*c*d**6*x**6 - 1680*sqrt(c**2 - d**2*x**2)*d**7*x**7 + 1536*c**8 
)/(13440*d**3)