\(\int x^5 \sqrt {-c+d x} \sqrt {c+d x} (a+b x^2) \, dx\) [1]

Optimal result

Integrand size = 31, antiderivative size = 152 \[ \int x^5 \sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right ) \, dx=\frac {c^4 \left (b c^2+a d^2\right ) (-c+d x)^{3/2} (c+d x)^{3/2}}{3 d^8}+\frac {c^2 \left (3 b c^2+2 a d^2\right ) (-c+d x)^{5/2} (c+d x)^{5/2}}{5 d^8}+\frac {\left (3 b c^2+a d^2\right ) (-c+d x)^{7/2} (c+d x)^{7/2}}{7 d^8}+\frac {b (-c+d x)^{9/2} (c+d x)^{9/2}}{9 d^8} \] Output:

1/3*c^4*(a*d^2+b*c^2)*(d*x-c)^(3/2)*(d*x+c)^(3/2)/d^8+1/5*c^2*(2*a*d^2+3*b 
*c^2)*(d*x-c)^(5/2)*(d*x+c)^(5/2)/d^8+1/7*(a*d^2+3*b*c^2)*(d*x-c)^(7/2)*(d 
*x+c)^(7/2)/d^8+1/9*b*(d*x-c)^(9/2)*(d*x+c)^(9/2)/d^8
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.64 \[ \int x^5 \sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right ) \, dx=\frac {(-c+d x)^{3/2} (c+d x)^{3/2} \left (3 a d^2 \left (8 c^4+12 c^2 d^2 x^2+15 d^4 x^4\right )+b \left (16 c^6+24 c^4 d^2 x^2+30 c^2 d^4 x^4+35 d^6 x^6\right )\right )}{315 d^8} \] Input:

Integrate[x^5*Sqrt[-c + d*x]*Sqrt[c + d*x]*(a + b*x^2),x]
 

Output:

((-c + d*x)^(3/2)*(c + d*x)^(3/2)*(3*a*d^2*(8*c^4 + 12*c^2*d^2*x^2 + 15*d^ 
4*x^4) + b*(16*c^6 + 24*c^4*d^2*x^2 + 30*c^2*d^4*x^4 + 35*d^6*x^6)))/(315* 
d^8)
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {960, 111, 27, 111, 27, 83}

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.

\(\displaystyle \int x^5 \left (a+b x^2\right ) \sqrt {d x-c} \sqrt {c+d x} \, dx\)

\(\Big \downarrow \) 960

\(\displaystyle \frac {1}{3} \left (3 a+\frac {2 b c^2}{d^2}\right ) \int x^5 \sqrt {d x-c} \sqrt {c+d x}dx+\frac {b x^6 (d x-c)^{3/2} (c+d x)^{3/2}}{9 d^2}\)

\(\Big \downarrow \) 111

\(\displaystyle \frac {1}{3} \left (3 a+\frac {2 b c^2}{d^2}\right ) \left (\frac {\int 4 c^2 x^3 \sqrt {d x-c} \sqrt {c+d x}dx}{7 d^2}+\frac {x^4 (d x-c)^{3/2} (c+d x)^{3/2}}{7 d^2}\right )+\frac {b x^6 (d x-c)^{3/2} (c+d x)^{3/2}}{9 d^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \left (3 a+\frac {2 b c^2}{d^2}\right ) \left (\frac {4 c^2 \int x^3 \sqrt {d x-c} \sqrt {c+d x}dx}{7 d^2}+\frac {x^4 (d x-c)^{3/2} (c+d x)^{3/2}}{7 d^2}\right )+\frac {b x^6 (d x-c)^{3/2} (c+d x)^{3/2}}{9 d^2}\)

\(\Big \downarrow \) 111

\(\displaystyle \frac {1}{3} \left (3 a+\frac {2 b c^2}{d^2}\right ) \left (\frac {4 c^2 \left (\frac {\int 2 c^2 x \sqrt {d x-c} \sqrt {c+d x}dx}{5 d^2}+\frac {x^2 (d x-c)^{3/2} (c+d x)^{3/2}}{5 d^2}\right )}{7 d^2}+\frac {x^4 (d x-c)^{3/2} (c+d x)^{3/2}}{7 d^2}\right )+\frac {b x^6 (d x-c)^{3/2} (c+d x)^{3/2}}{9 d^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \left (3 a+\frac {2 b c^2}{d^2}\right ) \left (\frac {4 c^2 \left (\frac {2 c^2 \int x \sqrt {d x-c} \sqrt {c+d x}dx}{5 d^2}+\frac {x^2 (d x-c)^{3/2} (c+d x)^{3/2}}{5 d^2}\right )}{7 d^2}+\frac {x^4 (d x-c)^{3/2} (c+d x)^{3/2}}{7 d^2}\right )+\frac {b x^6 (d x-c)^{3/2} (c+d x)^{3/2}}{9 d^2}\)

\(\Big \downarrow \) 83

\(\displaystyle \frac {1}{3} \left (\frac {4 c^2 \left (\frac {2 c^2 (d x-c)^{3/2} (c+d x)^{3/2}}{15 d^4}+\frac {x^2 (d x-c)^{3/2} (c+d x)^{3/2}}{5 d^2}\right )}{7 d^2}+\frac {x^4 (d x-c)^{3/2} (c+d x)^{3/2}}{7 d^2}\right ) \left (3 a+\frac {2 b c^2}{d^2}\right )+\frac {b x^6 (d x-c)^{3/2} (c+d x)^{3/2}}{9 d^2}\)

Input:

Int[x^5*Sqrt[-c + d*x]*Sqrt[c + d*x]*(a + b*x^2),x]
 

Output:

(b*x^6*(-c + d*x)^(3/2)*(c + d*x)^(3/2))/(9*d^2) + ((3*a + (2*b*c^2)/d^2)* 
((x^4*(-c + d*x)^(3/2)*(c + d*x)^(3/2))/(7*d^2) + (4*c^2*((2*c^2*(-c + d*x 
)^(3/2)*(c + d*x)^(3/2))/(15*d^4) + (x^2*(-c + d*x)^(3/2)*(c + d*x)^(3/2)) 
/(5*d^2)))/(7*d^2)))/3
 

Defintions of rubi rules used

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 83

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

rule 111

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1))   Int[(a + b*x) 
^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m 
 - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m 
 + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] & 
& GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]
 

rule 960

Int[((e_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.) 
*(x_)^(non2_.))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*(e*x)^( 
m + 1)*(a1 + b1*x^(n/2))^(p + 1)*((a2 + b2*x^(n/2))^(p + 1)/(b1*b2*e*(m + n 
*(p + 1) + 1))), x] - Simp[(a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1))/ 
(b1*b2*(m + n*(p + 1) + 1))   Int[(e*x)^m*(a1 + b1*x^(n/2))^p*(a2 + b2*x^(n 
/2))^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, e, m, n, p}, x] && EqQ[non2, 
 n/2] && EqQ[a2*b1 + a1*b2, 0] && NeQ[m + n*(p + 1) + 1, 0]
 

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.61

method	result	size
gosper	\(\frac {\left (d x -c \right )^{\frac {3}{2}} \left (d x +c \right )^{\frac {3}{2}} \left (35 b \,x^{6} d^{6}+45 a \,d^{6} x^{4}+30 b \,c^{2} d^{4} x^{4}+36 a \,c^{2} d^{4} x^{2}+24 b \,c^{4} d^{2} x^{2}+24 a \,c^{4} d^{2}+16 b \,c^{6}\right )}{315 d^{8}}\)	\(92\)
orering	\(-\frac {\left (d x +c \right )^{\frac {3}{2}} \left (-d x +c \right ) \left (35 b \,x^{6} d^{6}+45 a \,d^{6} x^{4}+30 b \,c^{2} d^{4} x^{4}+36 a \,c^{2} d^{4} x^{2}+24 b \,c^{4} d^{2} x^{2}+24 a \,c^{4} d^{2}+16 b \,c^{6}\right ) \sqrt {d x -c}}{315 d^{8}}\)	\(98\)
default	\(-\frac {\sqrt {d x -c}\, \sqrt {d x +c}\, \left (-d^{2} x^{2}+c^{2}\right ) \left (35 b \,x^{6} d^{6}+45 a \,d^{6} x^{4}+30 b \,c^{2} d^{4} x^{4}+36 a \,c^{2} d^{4} x^{2}+24 b \,c^{4} d^{2} x^{2}+24 a \,c^{4} d^{2}+16 b \,c^{6}\right )}{315 d^{8}}\)	\(104\)
risch	\(\frac {\sqrt {d x +c}\, \left (-35 b \,d^{8} x^{8}-45 a \,d^{8} x^{6}+5 b \,c^{2} d^{6} x^{6}+9 a \,c^{2} d^{6} x^{4}+6 b \,c^{4} d^{4} x^{4}+12 a \,c^{4} d^{4} x^{2}+8 b \,c^{6} d^{2} x^{2}+24 a \,c^{6} d^{2}+16 b \,c^{8}\right ) \left (-d x +c \right )}{315 \sqrt {d x -c}\, d^{8}}\)	\(122\)

Input:

int(x^5*(d*x-c)^(1/2)*(d*x+c)^(1/2)*(b*x^2+a),x,method=_RETURNVERBOSE)
 

Output:

1/315/d^8*(d*x-c)^(3/2)*(d*x+c)^(3/2)*(35*b*d^6*x^6+45*a*d^6*x^4+30*b*c^2* 
d^4*x^4+36*a*c^2*d^4*x^2+24*b*c^4*d^2*x^2+24*a*c^4*d^2+16*b*c^6)
 

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.75 \[ \int x^5 \sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right ) \, dx=\frac {{\left (35 \, b d^{8} x^{8} - 16 \, b c^{8} - 24 \, a c^{6} d^{2} - 5 \, {\left (b c^{2} d^{6} - 9 \, a d^{8}\right )} x^{6} - 3 \, {\left (2 \, b c^{4} d^{4} + 3 \, a c^{2} d^{6}\right )} x^{4} - 4 \, {\left (2 \, b c^{6} d^{2} + 3 \, a c^{4} d^{4}\right )} x^{2}\right )} \sqrt {d x + c} \sqrt {d x - c}}{315 \, d^{8}} \] Input:

integrate(x^5*(d*x-c)^(1/2)*(d*x+c)^(1/2)*(b*x^2+a),x, algorithm="fricas")
 

Output:

1/315*(35*b*d^8*x^8 - 16*b*c^8 - 24*a*c^6*d^2 - 5*(b*c^2*d^6 - 9*a*d^8)*x^ 
6 - 3*(2*b*c^4*d^4 + 3*a*c^2*d^6)*x^4 - 4*(2*b*c^6*d^2 + 3*a*c^4*d^4)*x^2) 
*sqrt(d*x + c)*sqrt(d*x - c)/d^8
 

Sympy [F]

\[ \int x^5 \sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right ) \, dx=\int x^{5} \left (a + b x^{2}\right ) \sqrt {- c + d x} \sqrt {c + d x}\, dx \] Input:

integrate(x**5*(d*x-c)**(1/2)*(d*x+c)**(1/2)*(b*x**2+a),x)
 

Output:

Integral(x**5*(a + b*x**2)*sqrt(-c + d*x)*sqrt(c + d*x), x)
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.17 \[ \int x^5 \sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right ) \, dx=\frac {{\left (d^{2} x^{2} - c^{2}\right )}^{\frac {3}{2}} b x^{6}}{9 \, d^{2}} + \frac {2 \, {\left (d^{2} x^{2} - c^{2}\right )}^{\frac {3}{2}} b c^{2} x^{4}}{21 \, d^{4}} + \frac {{\left (d^{2} x^{2} - c^{2}\right )}^{\frac {3}{2}} a x^{4}}{7 \, d^{2}} + \frac {8 \, {\left (d^{2} x^{2} - c^{2}\right )}^{\frac {3}{2}} b c^{4} x^{2}}{105 \, d^{6}} + \frac {4 \, {\left (d^{2} x^{2} - c^{2}\right )}^{\frac {3}{2}} a c^{2} x^{2}}{35 \, d^{4}} + \frac {16 \, {\left (d^{2} x^{2} - c^{2}\right )}^{\frac {3}{2}} b c^{6}}{315 \, d^{8}} + \frac {8 \, {\left (d^{2} x^{2} - c^{2}\right )}^{\frac {3}{2}} a c^{4}}{105 \, d^{6}} \] Input:

integrate(x^5*(d*x-c)^(1/2)*(d*x+c)^(1/2)*(b*x^2+a),x, algorithm="maxima")
 

Output:

1/9*(d^2*x^2 - c^2)^(3/2)*b*x^6/d^2 + 2/21*(d^2*x^2 - c^2)^(3/2)*b*c^2*x^4 
/d^4 + 1/7*(d^2*x^2 - c^2)^(3/2)*a*x^4/d^2 + 8/105*(d^2*x^2 - c^2)^(3/2)*b 
*c^4*x^2/d^6 + 4/35*(d^2*x^2 - c^2)^(3/2)*a*c^2*x^2/d^4 + 16/315*(d^2*x^2 
- c^2)^(3/2)*b*c^6/d^8 + 8/105*(d^2*x^2 - c^2)^(3/2)*a*c^4/d^6
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 569 vs. \(2 (128) = 256\).

Time = 0.22 (sec) , antiderivative size = 569, normalized size of antiderivative = 3.74 \[ \int x^5 \sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right ) \, dx =\text {Too large to display} \] Input:

integrate(x^5*(d*x-c)^(1/2)*(d*x+c)^(1/2)*(b*x^2+a),x, algorithm="giac")
 

Output:

1/40320*(168*(((2*((d*x + c)*(4*(d*x + c)*(5*(d*x + c)/d^5 - 31*c/d^5) + 3 
21*c^2/d^5) - 451*c^3/d^5)*(d*x + c) + 745*c^4/d^5)*(d*x + c) - 405*c^5/d^ 
5)*sqrt(d*x + c)*sqrt(d*x - c) - 150*c^6*log(abs(-sqrt(d*x + c) + sqrt(d*x 
 - c)))/d^5)*a*c + 3*(((2*((4*(5*(d*x + c)*(6*(d*x + c)*(7*(d*x + c)/d^7 - 
 57*c/d^7) + 1219*c^2/d^7) - 12463*c^3/d^7)*(d*x + c) + 64233*c^4/d^7)*(d* 
x + c) - 53963*c^5/d^7)*(d*x + c) + 59465*c^6/d^7)*(d*x + c) - 23205*c^7/d 
^7)*sqrt(d*x + c)*sqrt(d*x - c) - 7350*c^8*log(abs(-sqrt(d*x + c) + sqrt(d 
*x - c)))/d^7)*b*c + 24*(1050*c^7*log(abs(-sqrt(d*x + c) + sqrt(d*x - c))) 
 + (2835*c^6 - (6335*c^5 - 2*(4781*c^4 - (4551*c^3 - 4*(5*(6*d*x - 37*c)*( 
d*x + c) + 661*c^2)*(d*x + c))*(d*x + c))*(d*x + c))*(d*x + c))*sqrt(d*x + 
 c)*sqrt(d*x - c))*a/d^5 + (22050*c^9*log(abs(-sqrt(d*x + c) + sqrt(d*x - 
c))) + (69615*c^8 - (205275*c^7 - 2*(216993*c^6 - (310203*c^5 - 4*(75293*c 
^4 - 5*(9833*c^3 - 2*(7*(8*d*x - 65*c)*(d*x + c) + 2073*c^2)*(d*x + c))*(d 
*x + c))*(d*x + c))*(d*x + c))*(d*x + c))*(d*x + c))*sqrt(d*x + c)*sqrt(d* 
x - c))*b/d^7)/d
 

Mupad [B] (verification not implemented)

Time = 5.17 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00 \[ \int x^5 \sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right ) \, dx=-\sqrt {d\,x-c}\,\left (\frac {\left (16\,b\,c^8+24\,a\,c^6\,d^2\right )\,\sqrt {c+d\,x}}{315\,d^8}-\frac {b\,x^8\,\sqrt {c+d\,x}}{9}+\frac {x^4\,\left (6\,b\,c^4\,d^4+9\,a\,c^2\,d^6\right )\,\sqrt {c+d\,x}}{315\,d^8}+\frac {x^2\,\left (8\,b\,c^6\,d^2+12\,a\,c^4\,d^4\right )\,\sqrt {c+d\,x}}{315\,d^8}-\frac {x^6\,\left (45\,a\,d^8-5\,b\,c^2\,d^6\right )\,\sqrt {c+d\,x}}{315\,d^8}\right ) \] Input:

int(x^5*(a + b*x^2)*(c + d*x)^(1/2)*(d*x - c)^(1/2),x)
 

Output:

-(d*x - c)^(1/2)*(((16*b*c^8 + 24*a*c^6*d^2)*(c + d*x)^(1/2))/(315*d^8) - 
(b*x^8*(c + d*x)^(1/2))/9 + (x^4*(9*a*c^2*d^6 + 6*b*c^4*d^4)*(c + d*x)^(1/ 
2))/(315*d^8) + (x^2*(12*a*c^4*d^4 + 8*b*c^6*d^2)*(c + d*x)^(1/2))/(315*d^ 
8) - (x^6*(45*a*d^8 - 5*b*c^2*d^6)*(c + d*x)^(1/2))/(315*d^8))
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.74 \[ \int x^5 \sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right ) \, dx=\frac {\sqrt {d x +c}\, \sqrt {d x -c}\, \left (35 b \,d^{8} x^{8}+45 a \,d^{8} x^{6}-5 b \,c^{2} d^{6} x^{6}-9 a \,c^{2} d^{6} x^{4}-6 b \,c^{4} d^{4} x^{4}-12 a \,c^{4} d^{4} x^{2}-8 b \,c^{6} d^{2} x^{2}-24 a \,c^{6} d^{2}-16 b \,c^{8}\right )}{315 d^{8}} \] Input:

int(x^5*(d*x-c)^(1/2)*(d*x+c)^(1/2)*(b*x^2+a),x)
 

Output:

(sqrt(c + d*x)*sqrt( - c + d*x)*( - 24*a*c**6*d**2 - 12*a*c**4*d**4*x**2 - 
 9*a*c**2*d**6*x**4 + 45*a*d**8*x**6 - 16*b*c**8 - 8*b*c**6*d**2*x**2 - 6* 
b*c**4*d**4*x**4 - 5*b*c**2*d**6*x**6 + 35*b*d**8*x**8))/(315*d**8)