Integrand size = 26, antiderivative size = 147 \[ \int x^7 \sqrt {a+b x} \sqrt {a c-b c x} \, dx=-\frac {16 a^6 (a+b x)^{3/2} (a c-b c x)^{3/2}}{315 b^8 c}-\frac {8 a^4 x^2 (a+b x)^{3/2} (a c-b c x)^{3/2}}{105 b^6 c}-\frac {2 a^2 x^4 (a+b x)^{3/2} (a c-b c x)^{3/2}}{21 b^4 c}-\frac {x^6 (a+b x)^{3/2} (a c-b c x)^{3/2}}{9 b^2 c} \] Output:
-16/315*a^6*(b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2)/b^8/c-8/105*a^4*x^2*(b*x+a)^( 3/2)*(-b*c*x+a*c)^(3/2)/b^6/c-2/21*a^2*x^4*(b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2 )/b^4/c-1/9*x^6*(b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2)/b^2/c
Time = 0.17 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.46 \[ \int x^7 \sqrt {a+b x} \sqrt {a c-b c x} \, dx=-\frac {(c (a-b x))^{3/2} (a+b x)^{3/2} \left (16 a^6+24 a^4 b^2 x^2+30 a^2 b^4 x^4+35 b^6 x^6\right )}{315 b^8 c} \] Input:
Integrate[x^7*Sqrt[a + b*x]*Sqrt[a*c - b*c*x],x]
Output:
-1/315*((c*(a - b*x))^(3/2)*(a + b*x)^(3/2)*(16*a^6 + 24*a^4*b^2*x^2 + 30* a^2*b^4*x^4 + 35*b^6*x^6))/(b^8*c)
Time = 0.23 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {111, 27, 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.
\(\displaystyle \int x^7 \sqrt {a+b x} \sqrt {a c-b c x} \, dx\) |
\(\Big \downarrow \) 111 |
\(\displaystyle -\frac {\int -6 a^2 c x^5 \sqrt {a+b x} \sqrt {a c-b c x}dx}{9 b^2 c}-\frac {x^6 (a+b x)^{3/2} (a c-b c x)^{3/2}}{9 b^2 c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2 a^2 \int x^5 \sqrt {a+b x} \sqrt {a c-b c x}dx}{3 b^2}-\frac {x^6 (a+b x)^{3/2} (a c-b c x)^{3/2}}{9 b^2 c}\) |
\(\Big \downarrow \) 111 |
\(\displaystyle \frac {2 a^2 \left (-\frac {\int -4 a^2 c x^3 \sqrt {a+b x} \sqrt {a c-b c x}dx}{7 b^2 c}-\frac {x^4 (a+b x)^{3/2} (a c-b c x)^{3/2}}{7 b^2 c}\right )}{3 b^2}-\frac {x^6 (a+b x)^{3/2} (a c-b c x)^{3/2}}{9 b^2 c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2 a^2 \left (\frac {4 a^2 \int x^3 \sqrt {a+b x} \sqrt {a c-b c x}dx}{7 b^2}-\frac {x^4 (a+b x)^{3/2} (a c-b c x)^{3/2}}{7 b^2 c}\right )}{3 b^2}-\frac {x^6 (a+b x)^{3/2} (a c-b c x)^{3/2}}{9 b^2 c}\) |
\(\Big \downarrow \) 111 |
\(\displaystyle \frac {2 a^2 \left (\frac {4 a^2 \left (-\frac {\int -2 a^2 c x \sqrt {a+b x} \sqrt {a c-b c x}dx}{5 b^2 c}-\frac {x^2 (a+b x)^{3/2} (a c-b c x)^{3/2}}{5 b^2 c}\right )}{7 b^2}-\frac {x^4 (a+b x)^{3/2} (a c-b c x)^{3/2}}{7 b^2 c}\right )}{3 b^2}-\frac {x^6 (a+b x)^{3/2} (a c-b c x)^{3/2}}{9 b^2 c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2 a^2 \left (\frac {4 a^2 \left (\frac {2 a^2 \int x \sqrt {a+b x} \sqrt {a c-b c x}dx}{5 b^2}-\frac {x^2 (a+b x)^{3/2} (a c-b c x)^{3/2}}{5 b^2 c}\right )}{7 b^2}-\frac {x^4 (a+b x)^{3/2} (a c-b c x)^{3/2}}{7 b^2 c}\right )}{3 b^2}-\frac {x^6 (a+b x)^{3/2} (a c-b c x)^{3/2}}{9 b^2 c}\) |
\(\Big \downarrow \) 83 |
\(\displaystyle \frac {2 a^2 \left (\frac {4 a^2 \left (-\frac {2 a^2 (a+b x)^{3/2} (a c-b c x)^{3/2}}{15 b^4 c}-\frac {x^2 (a+b x)^{3/2} (a c-b c x)^{3/2}}{5 b^2 c}\right )}{7 b^2}-\frac {x^4 (a+b x)^{3/2} (a c-b c x)^{3/2}}{7 b^2 c}\right )}{3 b^2}-\frac {x^6 (a+b x)^{3/2} (a c-b c x)^{3/2}}{9 b^2 c}\) |
Input:
Int[x^7*Sqrt[a + b*x]*Sqrt[a*c - b*c*x],x]
Output:
-1/9*(x^6*(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2))/(b^2*c) + (2*a^2*(-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)))/(3*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 = 0.22 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.45
method | result | size |
gosper | \(-\frac {\left (-b x +a \right ) \left (b x +a \right )^{\frac {3}{2}} \left (35 b^{6} x^{6}+30 a^{2} x^{4} b^{4}+24 a^{4} x^{2} b^{2}+16 a^{6}\right ) \sqrt {-b c x +a c}}{315 b^{8}}\) | \(66\) |
orering | \(-\frac {\left (-b x +a \right ) \left (b x +a \right )^{\frac {3}{2}} \left (35 b^{6} x^{6}+30 a^{2} x^{4} b^{4}+24 a^{4} x^{2} b^{2}+16 a^{6}\right ) \sqrt {-b c x +a c}}{315 b^{8}}\) | \(66\) |
default | \(-\frac {\sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}\, \left (-b^{2} x^{2}+a^{2}\right ) \left (35 b^{6} x^{6}+30 a^{2} x^{4} b^{4}+24 a^{4} x^{2} b^{2}+16 a^{6}\right )}{315 b^{8}}\) | \(71\) |
risch | \(-\frac {\sqrt {b x +a}\, c \left (-35 b^{8} x^{8}+5 a^{2} x^{6} b^{6}+6 a^{4} x^{4} b^{4}+8 a^{6} x^{2} b^{2}+16 a^{8}\right ) \left (-b x +a \right )}{315 \sqrt {-c \left (b x -a \right )}\, b^{8}}\) | \(79\) |
Input:
int(x^7*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/315*(-b*x+a)*(b*x+a)^(3/2)*(35*b^6*x^6+30*a^2*b^4*x^4+24*a^4*b^2*x^2+16 *a^6)*(-b*c*x+a*c)^(1/2)/b^8
Time = 0.09 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.48 \[ \int x^7 \sqrt {a+b x} \sqrt {a c-b c x} \, dx=\frac {{\left (35 \, b^{8} x^{8} - 5 \, a^{2} b^{6} x^{6} - 6 \, a^{4} b^{4} x^{4} - 8 \, a^{6} b^{2} x^{2} - 16 \, a^{8}\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{315 \, b^{8}} \] Input:
integrate(x^7*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2),x, algorithm="fricas")
Output:
1/315*(35*b^8*x^8 - 5*a^2*b^6*x^6 - 6*a^4*b^4*x^4 - 8*a^6*b^2*x^2 - 16*a^8 )*sqrt(-b*c*x + a*c)*sqrt(b*x + a)/b^8
\[ \int x^7 \sqrt {a+b x} \sqrt {a c-b c x} \, dx=\int x^{7} \sqrt {- c \left (- a + b x\right )} \sqrt {a + b x}\, dx \] Input:
integrate(x**7*(b*x+a)**(1/2)*(-b*c*x+a*c)**(1/2),x)
Output:
Integral(x**7*sqrt(-c*(-a + b*x))*sqrt(a + b*x), x)
Time = 0.13 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.81 \[ \int x^7 \sqrt {a+b x} \sqrt {a c-b c x} \, dx=-\frac {{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} x^{6}}{9 \, b^{2} c} - \frac {2 \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} a^{2} x^{4}}{21 \, b^{4} c} - \frac {8 \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} a^{4} x^{2}}{105 \, b^{6} c} - \frac {16 \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} a^{6}}{315 \, b^{8} c} \] Input:
integrate(x^7*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2),x, algorithm="maxima")
Output:
-1/9*(-b^2*c*x^2 + a^2*c)^(3/2)*x^6/(b^2*c) - 2/21*(-b^2*c*x^2 + a^2*c)^(3 /2)*a^2*x^4/(b^4*c) - 8/105*(-b^2*c*x^2 + a^2*c)^(3/2)*a^4*x^2/(b^6*c) - 1 6/315*(-b^2*c*x^2 + a^2*c)^(3/2)*a^6/(b^8*c)
Leaf count of result is larger than twice the leaf count of optimal. 329 vs. \(2 (123) = 246\).
Time = 0.31 (sec) , antiderivative size = 329, normalized size of antiderivative = 2.24 \[ \int x^7 \sqrt {a+b x} \sqrt {a c-b c x} \, dx=-\frac {\frac {22050 \, a^{9} c \log \left ({\left | -\sqrt {b x + a} \sqrt {-c} + \sqrt {-{\left (b x + a\right )} c + 2 \, a c} \right |}\right )}{\sqrt {-c}} - {\left (69615 \, a^{8} - {\left (205275 \, a^{7} - 2 \, {\left (216993 \, a^{6} - {\left (310203 \, a^{5} - 4 \, {\left (75293 \, a^{4} - 5 \, {\left (9833 \, a^{3} - 2 \, {\left (7 \, {\left (8 \, b x - 65 \, a\right )} {\left (b x + a\right )} + 2073 \, a^{2}\right )} {\left (b x + a\right )}\right )} {\left (b x + a\right )}\right )} {\left (b x + a\right )}\right )} {\left (b x + a\right )}\right )} {\left (b x + a\right )}\right )} {\left (b x + a\right )}\right )} \sqrt {-{\left (b x + a\right )} c + 2 \, a c} \sqrt {b x + a} - 3 \, {\left (\frac {7350 \, a^{8} c \log \left ({\left | -\sqrt {b x + a} \sqrt {-c} + \sqrt {-{\left (b x + a\right )} c + 2 \, a c} \right |}\right )}{\sqrt {-c}} - {\left (23205 \, a^{7} - {\left (59465 \, a^{6} - 2 \, {\left (53963 \, a^{5} - {\left (64233 \, a^{4} - 4 \, {\left (12463 \, a^{3} - 5 \, {\left (6 \, {\left (7 \, b x - 50 \, a\right )} {\left (b x + a\right )} + 1219 \, a^{2}\right )} {\left (b x + a\right )}\right )} {\left (b x + a\right )}\right )} {\left (b x + a\right )}\right )} {\left (b x + a\right )}\right )} {\left (b x + a\right )}\right )} \sqrt {-{\left (b x + a\right )} c + 2 \, a c} \sqrt {b x + a}\right )} a}{40320 \, b^{8}} \] Input:
integrate(x^7*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2),x, algorithm="giac")
Output:
-1/40320*(22050*a^9*c*log(abs(-sqrt(b*x + a)*sqrt(-c) + sqrt(-(b*x + a)*c + 2*a*c)))/sqrt(-c) - (69615*a^8 - (205275*a^7 - 2*(216993*a^6 - (310203*a ^5 - 4*(75293*a^4 - 5*(9833*a^3 - 2*(7*(8*b*x - 65*a)*(b*x + a) + 2073*a^2 )*(b*x + a))*(b*x + a))*(b*x + a))*(b*x + a))*(b*x + a))*(b*x + a))*sqrt(- (b*x + a)*c + 2*a*c)*sqrt(b*x + a) - 3*(7350*a^8*c*log(abs(-sqrt(b*x + a)* sqrt(-c) + sqrt(-(b*x + a)*c + 2*a*c)))/sqrt(-c) - (23205*a^7 - (59465*a^6 - 2*(53963*a^5 - (64233*a^4 - 4*(12463*a^3 - 5*(6*(7*b*x - 50*a)*(b*x + a ) + 1219*a^2)*(b*x + a))*(b*x + a))*(b*x + a))*(b*x + a))*(b*x + a))*sqrt( -(b*x + a)*c + 2*a*c)*sqrt(b*x + a))*a)/b^8
Time = 0.35 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.65 \[ \int x^7 \sqrt {a+b x} \sqrt {a c-b c x} \, dx=-\sqrt {a\,c-b\,c\,x}\,\left (\frac {16\,a^8\,\sqrt {a+b\,x}}{315\,b^8}-\frac {x^8\,\sqrt {a+b\,x}}{9}+\frac {a^2\,x^6\,\sqrt {a+b\,x}}{63\,b^2}+\frac {2\,a^4\,x^4\,\sqrt {a+b\,x}}{105\,b^4}+\frac {8\,a^6\,x^2\,\sqrt {a+b\,x}}{315\,b^6}\right ) \] Input:
int(x^7*(a*c - b*c*x)^(1/2)*(a + b*x)^(1/2),x)
Output:
-(a*c - b*c*x)^(1/2)*((16*a^8*(a + b*x)^(1/2))/(315*b^8) - (x^8*(a + b*x)^ (1/2))/9 + (a^2*x^6*(a + b*x)^(1/2))/(63*b^2) + (2*a^4*x^4*(a + b*x)^(1/2) )/(105*b^4) + (8*a^6*x^2*(a + b*x)^(1/2))/(315*b^6))
Time = 0.16 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.46 \[ \int x^7 \sqrt {a+b x} \sqrt {a c-b c x} \, dx=\frac {\sqrt {c}\, \sqrt {b x +a}\, \sqrt {-b x +a}\, \left (35 b^{8} x^{8}-5 a^{2} b^{6} x^{6}-6 a^{4} b^{4} x^{4}-8 a^{6} b^{2} x^{2}-16 a^{8}\right )}{315 b^{8}} \] Input:
int(x^7*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2),x)
Output:
(sqrt(c)*sqrt(a + b*x)*sqrt(a - b*x)*( - 16*a**8 - 8*a**6*b**2*x**2 - 6*a* *4*b**4*x**4 - 5*a**2*b**6*x**6 + 35*b**8*x**8))/(315*b**8)