\(\int x^5 \sqrt {-c+d x} \sqrt {c+d x} (a+b x^2) \, dx\) [337]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {471, 102, 12, 75} \[ \int x^5 \sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right ) \, dx=\frac {4 c^2 x^2 (d x-c)^{3/2} (c+d x)^{3/2} \left (3 a d^2+2 b c^2\right )}{105 d^6}+\frac {x^4 (d x-c)^{3/2} (c+d x)^{3/2} \left (3 a d^2+2 b c^2\right )}{21 d^4}+\frac {8 c^4 (d x-c)^{3/2} (c+d x)^{3/2} \left (3 a d^2+2 b c^2\right )}{315 d^8}+\frac {b x^6 (d x-c)^{3/2} (c+d x)^{3/2}}{9 d^2} \]

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 75

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> 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 102

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(m + n + p + 1))), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(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 471

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] - Dist[(a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1))/(b1*b2*(m + n*(p + 1) + 1)), I
nt[(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]

Rubi steps \begin{align*} \text {integral}& = \frac {b x^6 (-c+d x)^{3/2} (c+d x)^{3/2}}{9 d^2}+\frac {1}{3} \left (3 a+\frac {2 b c^2}{d^2}\right ) \int x^5 \sqrt {-c+d x} \sqrt {c+d x} \, dx \\ & = \frac {\left (2 b c^2+3 a d^2\right ) x^4 (-c+d x)^{3/2} (c+d x)^{3/2}}{21 d^4}+\frac {b x^6 (-c+d x)^{3/2} (c+d x)^{3/2}}{9 d^2}+\frac {\left (2 b c^2+3 a d^2\right ) \int 4 c^2 x^3 \sqrt {-c+d x} \sqrt {c+d x} \, dx}{21 d^4} \\ & = \frac {\left (2 b c^2+3 a d^2\right ) x^4 (-c+d x)^{3/2} (c+d x)^{3/2}}{21 d^4}+\frac {b x^6 (-c+d x)^{3/2} (c+d x)^{3/2}}{9 d^2}+\frac {\left (4 c^2 \left (2 b c^2+3 a d^2\right )\right ) \int x^3 \sqrt {-c+d x} \sqrt {c+d x} \, dx}{21 d^4} \\ & = \frac {4 c^2 \left (2 b c^2+3 a d^2\right ) x^2 (-c+d x)^{3/2} (c+d x)^{3/2}}{105 d^6}+\frac {\left (2 b c^2+3 a d^2\right ) x^4 (-c+d x)^{3/2} (c+d x)^{3/2}}{21 d^4}+\frac {b x^6 (-c+d x)^{3/2} (c+d x)^{3/2}}{9 d^2}+\frac {\left (4 c^2 \left (2 b c^2+3 a d^2\right )\right ) \int 2 c^2 x \sqrt {-c+d x} \sqrt {c+d x} \, dx}{105 d^6} \\ & = \frac {4 c^2 \left (2 b c^2+3 a d^2\right ) x^2 (-c+d x)^{3/2} (c+d x)^{3/2}}{105 d^6}+\frac {\left (2 b c^2+3 a d^2\right ) x^4 (-c+d x)^{3/2} (c+d x)^{3/2}}{21 d^4}+\frac {b x^6 (-c+d x)^{3/2} (c+d x)^{3/2}}{9 d^2}+\frac {\left (8 c^4 \left (2 b c^2+3 a d^2\right )\right ) \int x \sqrt {-c+d x} \sqrt {c+d x} \, dx}{105 d^6} \\ & = \frac {8 c^4 \left (2 b c^2+3 a d^2\right ) (-c+d x)^{3/2} (c+d x)^{3/2}}{315 d^8}+\frac {4 c^2 \left (2 b c^2+3 a d^2\right ) x^2 (-c+d x)^{3/2} (c+d x)^{3/2}}{105 d^6}+\frac {\left (2 b c^2+3 a d^2\right ) x^4 (-c+d x)^{3/2} (c+d x)^{3/2}}{21 d^4}+\frac {b x^6 (-c+d x)^{3/2} (c+d x)^{3/2}}{9 d^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.64 \[ \int x^5 \sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right ) \, dx=\frac {(-c+d x)^{3/2} (c+d x)^{3/2} \left (3 a d^2 \left (8 c^4+12 c^2 d^2 x^2+15 d^4 x^4\right )+b \left (16 c^6+24 c^4 d^2 x^2+30 c^2 d^4 x^4+35 d^6 x^6\right )\right )}{315 d^8} \]

[In]

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

[Out]

((-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)

Maple [A] (verified)

Time = 4.04 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.61

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

[In]

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

[Out]

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)

none

Time = 0.30 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.75 \[ \int x^5 \sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right ) \, dx=\frac {{\left (35 \, b d^{8} x^{8} - 16 \, b c^{8} - 24 \, a c^{6} d^{2} - 5 \, {\left (b c^{2} d^{6} - 9 \, a d^{8}\right )} x^{6} - 3 \, {\left (2 \, b c^{4} d^{4} + 3 \, a c^{2} d^{6}\right )} x^{4} - 4 \, {\left (2 \, b c^{6} d^{2} + 3 \, a c^{4} d^{4}\right )} x^{2}\right )} \sqrt {d x + c} \sqrt {d x - c}}{315 \, d^{8}} \]

[In]

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

[Out]

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]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.17 \[ \int x^5 \sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right ) \, dx=\frac {{\left (d^{2} x^{2} - c^{2}\right )}^{\frac {3}{2}} b x^{6}}{9 \, d^{2}} + \frac {2 \, {\left (d^{2} x^{2} - c^{2}\right )}^{\frac {3}{2}} b c^{2} x^{4}}{21 \, d^{4}} + \frac {{\left (d^{2} x^{2} - c^{2}\right )}^{\frac {3}{2}} a x^{4}}{7 \, d^{2}} + \frac {8 \, {\left (d^{2} x^{2} - c^{2}\right )}^{\frac {3}{2}} b c^{4} x^{2}}{105 \, d^{6}} + \frac {4 \, {\left (d^{2} x^{2} - c^{2}\right )}^{\frac {3}{2}} a c^{2} x^{2}}{35 \, d^{4}} + \frac {16 \, {\left (d^{2} x^{2} - c^{2}\right )}^{\frac {3}{2}} b c^{6}}{315 \, d^{8}} + \frac {8 \, {\left (d^{2} x^{2} - c^{2}\right )}^{\frac {3}{2}} a c^{4}}{105 \, d^{6}} \]

[In]

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

[Out]

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. 621 vs. \(2 (128) = 256\).

Time = 0.45 (sec) , antiderivative size = 621, normalized size of antiderivative = 4.09 \[ \int x^5 \sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right ) \, dx=\frac {168 \, {\left ({\left ({\left (2 \, {\left ({\left (d x + c\right )} {\left (4 \, {\left (d x + c\right )} {\left (\frac {5 \, {\left (d x + c\right )}}{d^{5}} - \frac {31 \, c}{d^{5}}\right )} + \frac {321 \, c^{2}}{d^{5}}\right )} - \frac {451 \, c^{3}}{d^{5}}\right )} {\left (d x + c\right )} + \frac {745 \, c^{4}}{d^{5}}\right )} {\left (d x + c\right )} - \frac {405 \, c^{5}}{d^{5}}\right )} \sqrt {d x + c} \sqrt {d x - c} - \frac {150 \, c^{6} \log \left ({\left | -\sqrt {d x + c} + \sqrt {d x - c} \right |}\right )}{d^{5}}\right )} a c + 3 \, {\left ({\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, {\left (d x + c\right )} {\left (6 \, {\left (d x + c\right )} {\left (\frac {7 \, {\left (d x + c\right )}}{d^{7}} - \frac {57 \, c}{d^{7}}\right )} + \frac {1219 \, c^{2}}{d^{7}}\right )} - \frac {12463 \, c^{3}}{d^{7}}\right )} {\left (d x + c\right )} + \frac {64233 \, c^{4}}{d^{7}}\right )} {\left (d x + c\right )} - \frac {53963 \, c^{5}}{d^{7}}\right )} {\left (d x + c\right )} + \frac {59465 \, c^{6}}{d^{7}}\right )} {\left (d x + c\right )} - \frac {23205 \, c^{7}}{d^{7}}\right )} \sqrt {d x + c} \sqrt {d x - c} - \frac {7350 \, c^{8} \log \left ({\left | -\sqrt {d x + c} + \sqrt {d x - c} \right |}\right )}{d^{7}}\right )} b c + 24 \, {\left ({\left ({\left (2 \, {\left ({\left (4 \, {\left (d x + c\right )} {\left (5 \, {\left (d x + c\right )} {\left (\frac {6 \, {\left (d x + c\right )}}{d^{6}} - \frac {43 \, c}{d^{6}}\right )} + \frac {661 \, c^{2}}{d^{6}}\right )} - \frac {4551 \, c^{3}}{d^{6}}\right )} {\left (d x + c\right )} + \frac {4781 \, c^{4}}{d^{6}}\right )} {\left (d x + c\right )} - \frac {6335 \, c^{5}}{d^{6}}\right )} {\left (d x + c\right )} + \frac {2835 \, c^{6}}{d^{6}}\right )} \sqrt {d x + c} \sqrt {d x - c} + \frac {1050 \, c^{7} \log \left ({\left | -\sqrt {d x + c} + \sqrt {d x - c} \right |}\right )}{d^{6}}\right )} a d + {\left ({\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, {\left (2 \, {\left (d x + c\right )} {\left (7 \, {\left (d x + c\right )} {\left (\frac {8 \, {\left (d x + c\right )}}{d^{8}} - \frac {73 \, c}{d^{8}}\right )} + \frac {2073 \, c^{2}}{d^{8}}\right )} - \frac {9833 \, c^{3}}{d^{8}}\right )} {\left (d x + c\right )} + \frac {75293 \, c^{4}}{d^{8}}\right )} {\left (d x + c\right )} - \frac {310203 \, c^{5}}{d^{8}}\right )} {\left (d x + c\right )} + \frac {216993 \, c^{6}}{d^{8}}\right )} {\left (d x + c\right )} - \frac {205275 \, c^{7}}{d^{8}}\right )} {\left (d x + c\right )} + \frac {69615 \, c^{8}}{d^{8}}\right )} \sqrt {d x + c} \sqrt {d x - c} + \frac {22050 \, c^{9} \log \left ({\left | -\sqrt {d x + c} + \sqrt {d x - c} \right |}\right )}{d^{8}}\right )} b d}{40320 \, d} \]

[In]

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

[Out]

1/40320*(168*(((2*((d*x + c)*(4*(d*x + c)*(5*(d*x + c)/d^5 - 31*c/d^5) + 321*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*(((2*((4
*(d*x + c)*(5*(d*x + c)*(6*(d*x + c)/d^6 - 43*c/d^6) + 661*c^2/d^6) - 4551*c^3/d^6)*(d*x + c) + 4781*c^4/d^6)*
(d*x + c) - 6335*c^5/d^6)*(d*x + c) + 2835*c^6/d^6)*sqrt(d*x + c)*sqrt(d*x - c) + 1050*c^7*log(abs(-sqrt(d*x +
c) + sqrt(d*x - c)))/d^6)*a*d + (((2*((4*(5*(2*(d*x + c)*(7*(d*x + c)*(8*(d*x + c)/d^8 - 73*c/d^8) + 2073*c^2
/d^8) - 9833*c^3/d^8)*(d*x + c) + 75293*c^4/d^8)*(d*x + c) - 310203*c^5/d^8)*(d*x + c) + 216993*c^6/d^8)*(d*x
+ c) - 205275*c^7/d^8)*(d*x + c) + 69615*c^8/d^8)*sqrt(d*x + c)*sqrt(d*x - c) + 22050*c^9*log(abs(-sqrt(d*x +
c) + sqrt(d*x - c)))/d^8)*b*d)/d

Mupad [B] (verification not implemented)

Time = 5.92 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00 \[ \int x^5 \sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right ) \, dx=-\sqrt {d\,x-c}\,\left (\frac {\left (16\,b\,c^8+24\,a\,c^6\,d^2\right )\,\sqrt {c+d\,x}}{315\,d^8}-\frac {b\,x^8\,\sqrt {c+d\,x}}{9}+\frac {x^4\,\left (6\,b\,c^4\,d^4+9\,a\,c^2\,d^6\right )\,\sqrt {c+d\,x}}{315\,d^8}+\frac {x^2\,\left (8\,b\,c^6\,d^2+12\,a\,c^4\,d^4\right )\,\sqrt {c+d\,x}}{315\,d^8}-\frac {x^6\,\left (45\,a\,d^8-5\,b\,c^2\,d^6\right )\,\sqrt {c+d\,x}}{315\,d^8}\right ) \]

[In]

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

[Out]

-(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))