Integrand size = 26, antiderivative size = 109 \[ \int x^5 \sqrt {a+b x} \sqrt {a c-b c x} \, dx=-\frac {8 a^4 (a+b x)^{3/2} (a c-b c x)^{3/2}}{105 b^6 c}-\frac {4 a^2 x^2 (a+b x)^{3/2} (a c-b c x)^{3/2}}{35 b^4 c}-\frac {x^4 (a+b x)^{3/2} (a c-b c x)^{3/2}}{7 b^2 c} \] Output:
-8/105*a^4*(b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2)/b^6/c-4/35*a^2*x^2*(b*x+a)^(3/ 2)*(-b*c*x+a*c)^(3/2)/b^4/c-1/7*x^4*(b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2)/b^2/c
Time = 0.14 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.51 \[ \int x^5 \sqrt {a+b x} \sqrt {a c-b c x} \, dx=-\frac {(c (a-b x))^{3/2} (a+b x)^{3/2} \left (8 a^4+12 a^2 b^2 x^2+15 b^4 x^4\right )}{105 b^6 c} \] Input:
Integrate[x^5*Sqrt[a + b*x]*Sqrt[a*c - b*c*x],x]
Output:
-1/105*((c*(a - b*x))^(3/2)*(a + b*x)^(3/2)*(8*a^4 + 12*a^2*b^2*x^2 + 15*b ^4*x^4))/(b^6*c)
Time = 0.20 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {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.
\(\displaystyle \int x^5 \sqrt {a+b x} \sqrt {a c-b c x} \, dx\) |
\(\Big \downarrow \) 111 |
\(\displaystyle -\frac {\int -4 a^2 c x^3 \sqrt {a+b x} \sqrt {a c-b c x}dx}{7 b^2 c}-\frac {x^4 (a+b x)^{3/2} (a c-b c x)^{3/2}}{7 b^2 c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {4 a^2 \int x^3 \sqrt {a+b x} \sqrt {a c-b c x}dx}{7 b^2}-\frac {x^4 (a+b x)^{3/2} (a c-b c x)^{3/2}}{7 b^2 c}\) |
\(\Big \downarrow \) 111 |
\(\displaystyle \frac {4 a^2 \left (-\frac {\int -2 a^2 c x \sqrt {a+b x} \sqrt {a c-b c x}dx}{5 b^2 c}-\frac {x^2 (a+b x)^{3/2} (a c-b c x)^{3/2}}{5 b^2 c}\right )}{7 b^2}-\frac {x^4 (a+b x)^{3/2} (a c-b c x)^{3/2}}{7 b^2 c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {4 a^2 \left (\frac {2 a^2 \int x \sqrt {a+b x} \sqrt {a c-b c x}dx}{5 b^2}-\frac {x^2 (a+b x)^{3/2} (a c-b c x)^{3/2}}{5 b^2 c}\right )}{7 b^2}-\frac {x^4 (a+b x)^{3/2} (a c-b c x)^{3/2}}{7 b^2 c}\) |
\(\Big \downarrow \) 83 |
\(\displaystyle \frac {4 a^2 \left (-\frac {2 a^2 (a+b x)^{3/2} (a c-b c x)^{3/2}}{15 b^4 c}-\frac {x^2 (a+b x)^{3/2} (a c-b c x)^{3/2}}{5 b^2 c}\right )}{7 b^2}-\frac {x^4 (a+b x)^{3/2} (a c-b c x)^{3/2}}{7 b^2 c}\) |
Input:
Int[x^5*Sqrt[a + b*x]*Sqrt[a*c - b*c*x],x]
Output:
-1/7*(x^4*(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2))/(b^2*c) + (4*a^2*((-2*a^2*( a + b*x)^(3/2)*(a*c - b*c*x)^(3/2))/(15*b^4*c) - (x^2*(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2))/(5*b^2*c)))/(7*b^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]
Time = 0.21 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.50
method | result | size |
gosper | \(-\frac {\left (-b x +a \right ) \left (b x +a \right )^{\frac {3}{2}} \left (15 b^{4} x^{4}+12 a^{2} b^{2} x^{2}+8 a^{4}\right ) \sqrt {-b c x +a c}}{105 b^{6}}\) | \(55\) |
orering | \(-\frac {\left (-b x +a \right ) \left (b x +a \right )^{\frac {3}{2}} \left (15 b^{4} x^{4}+12 a^{2} b^{2} x^{2}+8 a^{4}\right ) \sqrt {-b c x +a c}}{105 b^{6}}\) | \(55\) |
default | \(-\frac {\sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}\, \left (-b^{2} x^{2}+a^{2}\right ) \left (15 b^{4} x^{4}+12 a^{2} b^{2} x^{2}+8 a^{4}\right )}{105 b^{6}}\) | \(60\) |
risch | \(-\frac {\sqrt {b x +a}\, c \left (-15 b^{6} x^{6}+3 a^{2} x^{4} b^{4}+4 a^{4} x^{2} b^{2}+8 a^{6}\right ) \left (-b x +a \right )}{105 \sqrt {-c \left (b x -a \right )}\, b^{6}}\) | \(68\) |
Input:
int(x^5*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/105*(-b*x+a)*(b*x+a)^(3/2)*(15*b^4*x^4+12*a^2*b^2*x^2+8*a^4)*(-b*c*x+a* c)^(1/2)/b^6
Time = 0.10 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.54 \[ \int x^5 \sqrt {a+b x} \sqrt {a c-b c x} \, dx=\frac {{\left (15 \, b^{6} x^{6} - 3 \, a^{2} b^{4} x^{4} - 4 \, a^{4} b^{2} x^{2} - 8 \, a^{6}\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{105 \, b^{6}} \] Input:
integrate(x^5*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2),x, algorithm="fricas")
Output:
1/105*(15*b^6*x^6 - 3*a^2*b^4*x^4 - 4*a^4*b^2*x^2 - 8*a^6)*sqrt(-b*c*x + a *c)*sqrt(b*x + a)/b^6
\[ \int x^5 \sqrt {a+b x} \sqrt {a c-b c x} \, dx=\int x^{5} \sqrt {- c \left (- a + b x\right )} \sqrt {a + b x}\, dx \] Input:
integrate(x**5*(b*x+a)**(1/2)*(-b*c*x+a*c)**(1/2),x)
Output:
Integral(x**5*sqrt(-c*(-a + b*x))*sqrt(a + b*x), x)
Time = 0.15 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.81 \[ \int x^5 \sqrt {a+b x} \sqrt {a c-b c x} \, dx=-\frac {{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} x^{4}}{7 \, b^{2} c} - \frac {4 \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} a^{2} x^{2}}{35 \, b^{4} c} - \frac {8 \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} a^{4}}{105 \, b^{6} c} \] Input:
integrate(x^5*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2),x, algorithm="maxima")
Output:
-1/7*(-b^2*c*x^2 + a^2*c)^(3/2)*x^4/(b^2*c) - 4/35*(-b^2*c*x^2 + a^2*c)^(3 /2)*a^2*x^2/(b^4*c) - 8/105*(-b^2*c*x^2 + a^2*c)^(3/2)*a^4/(b^6*c)
Leaf count of result is larger than twice the leaf count of optimal. 277 vs. \(2 (91) = 182\).
Time = 0.31 (sec) , antiderivative size = 277, normalized size of antiderivative = 2.54 \[ \int x^5 \sqrt {a+b x} \sqrt {a c-b c x} \, dx=-\frac {\frac {1050 \, a^{7} c \log \left ({\left | -\sqrt {b x + a} \sqrt {-c} + \sqrt {-{\left (b x + a\right )} c + 2 \, a c} \right |}\right )}{\sqrt {-c}} - {\left (2835 \, a^{6} - {\left (6335 \, a^{5} - 2 \, {\left (4781 \, a^{4} - {\left (4551 \, a^{3} - 4 \, {\left (5 \, {\left (6 \, b x - 37 \, a\right )} {\left (b x + a\right )} + 661 \, a^{2}\right )} {\left (b x + a\right )}\right )} {\left (b x + a\right )}\right )} {\left (b x + a\right )}\right )} {\left (b x + a\right )}\right )} \sqrt {-{\left (b x + a\right )} c + 2 \, a c} \sqrt {b x + a} - 7 \, {\left (\frac {150 \, a^{6} c \log \left ({\left | -\sqrt {b x + a} \sqrt {-c} + \sqrt {-{\left (b x + a\right )} c + 2 \, a c} \right |}\right )}{\sqrt {-c}} - {\left (405 \, a^{5} - {\left (745 \, a^{4} - 2 \, {\left (451 \, a^{3} - {\left (4 \, {\left (5 \, b x - 26 \, a\right )} {\left (b x + a\right )} + 321 \, a^{2}\right )} {\left (b x + a\right )}\right )} {\left (b x + a\right )}\right )} {\left (b x + a\right )}\right )} \sqrt {-{\left (b x + a\right )} c + 2 \, a c} \sqrt {b x + a}\right )} a}{1680 \, b^{6}} \] Input:
integrate(x^5*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2),x, algorithm="giac")
Output:
-1/1680*(1050*a^7*c*log(abs(-sqrt(b*x + a)*sqrt(-c) + sqrt(-(b*x + a)*c + 2*a*c)))/sqrt(-c) - (2835*a^6 - (6335*a^5 - 2*(4781*a^4 - (4551*a^3 - 4*(5 *(6*b*x - 37*a)*(b*x + a) + 661*a^2)*(b*x + a))*(b*x + a))*(b*x + a))*(b*x + a))*sqrt(-(b*x + a)*c + 2*a*c)*sqrt(b*x + a) - 7*(150*a^6*c*log(abs(-sq rt(b*x + a)*sqrt(-c) + sqrt(-(b*x + a)*c + 2*a*c)))/sqrt(-c) - (405*a^5 - (745*a^4 - 2*(451*a^3 - (4*(5*b*x - 26*a)*(b*x + a) + 321*a^2)*(b*x + a))* (b*x + a))*(b*x + a))*sqrt(-(b*x + a)*c + 2*a*c)*sqrt(b*x + a))*a)/b^6
Time = 0.33 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.71 \[ \int x^5 \sqrt {a+b x} \sqrt {a c-b c x} \, dx=-\sqrt {a\,c-b\,c\,x}\,\left (\frac {8\,a^6\,\sqrt {a+b\,x}}{105\,b^6}-\frac {x^6\,\sqrt {a+b\,x}}{7}+\frac {a^2\,x^4\,\sqrt {a+b\,x}}{35\,b^2}+\frac {4\,a^4\,x^2\,\sqrt {a+b\,x}}{105\,b^4}\right ) \] Input:
int(x^5*(a*c - b*c*x)^(1/2)*(a + b*x)^(1/2),x)
Output:
-(a*c - b*c*x)^(1/2)*((8*a^6*(a + b*x)^(1/2))/(105*b^6) - (x^6*(a + b*x)^( 1/2))/7 + (a^2*x^4*(a + b*x)^(1/2))/(35*b^2) + (4*a^4*x^2*(a + b*x)^(1/2)) /(105*b^4))
Time = 0.16 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.51 \[ \int x^5 \sqrt {a+b x} \sqrt {a c-b c x} \, dx=\frac {\sqrt {c}\, \sqrt {b x +a}\, \sqrt {-b x +a}\, \left (15 b^{6} x^{6}-3 a^{2} b^{4} x^{4}-4 a^{4} b^{2} x^{2}-8 a^{6}\right )}{105 b^{6}} \] Input:
int(x^5*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2),x)
Output:
(sqrt(c)*sqrt(a + b*x)*sqrt(a - b*x)*( - 8*a**6 - 4*a**4*b**2*x**2 - 3*a** 2*b**4*x**4 + 15*b**6*x**6))/(105*b**6)