Integrand size = 26, antiderivative size = 147 \[ \int x^7 (a+b x)^{3/2} (a c-b c x)^{3/2} \, dx=-\frac {16 a^6 (a+b x)^{5/2} (a c-b c x)^{5/2}}{1155 b^8 c}-\frac {8 a^4 x^2 (a+b x)^{5/2} (a c-b c x)^{5/2}}{231 b^6 c}-\frac {2 a^2 x^4 (a+b x)^{5/2} (a c-b c x)^{5/2}}{33 b^4 c}-\frac {x^6 (a+b x)^{5/2} (a c-b c x)^{5/2}}{11 b^2 c} \] Output:
-16/1155*a^6*(b*x+a)^(5/2)*(-b*c*x+a*c)^(5/2)/b^8/c-8/231*a^4*x^2*(b*x+a)^ (5/2)*(-b*c*x+a*c)^(5/2)/b^6/c-2/33*a^2*x^4*(b*x+a)^(5/2)*(-b*c*x+a*c)^(5/ 2)/b^4/c-1/11*x^6*(b*x+a)^(5/2)*(-b*c*x+a*c)^(5/2)/b^2/c
Time = 0.22 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.46 \[ \int x^7 (a+b x)^{3/2} (a c-b c x)^{3/2} \, dx=-\frac {(c (a-b x))^{5/2} (a+b x)^{5/2} \left (16 a^6+40 a^4 b^2 x^2+70 a^2 b^4 x^4+105 b^6 x^6\right )}{1155 b^8 c} \] Input:
Integrate[x^7*(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2),x]
Output:
-1/1155*((c*(a - b*x))^(5/2)*(a + b*x)^(5/2)*(16*a^6 + 40*a^4*b^2*x^2 + 70 *a^2*b^4*x^4 + 105*b^6*x^6))/(b^8*c)
Time = 0.22 (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.
\(\displaystyle \int x^7 (a+b x)^{3/2} (a c-b c x)^{3/2} \, dx\) |
\(\Big \downarrow \) 111 |
\(\displaystyle -\frac {\int -6 a^2 c x^5 (a+b x)^{3/2} (a c-b c x)^{3/2}dx}{11 b^2 c}-\frac {x^6 (a+b x)^{5/2} (a c-b c x)^{5/2}}{11 b^2 c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {6 a^2 \int x^5 (a+b x)^{3/2} (a c-b c x)^{3/2}dx}{11 b^2}-\frac {x^6 (a+b x)^{5/2} (a c-b c x)^{5/2}}{11 b^2 c}\) |
\(\Big \downarrow \) 111 |
\(\displaystyle \frac {6 a^2 \left (-\frac {\int -4 a^2 c x^3 (a+b x)^{3/2} (a c-b c x)^{3/2}dx}{9 b^2 c}-\frac {x^4 (a+b x)^{5/2} (a c-b c x)^{5/2}}{9 b^2 c}\right )}{11 b^2}-\frac {x^6 (a+b x)^{5/2} (a c-b c x)^{5/2}}{11 b^2 c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {6 a^2 \left (\frac {4 a^2 \int x^3 (a+b x)^{3/2} (a c-b c x)^{3/2}dx}{9 b^2}-\frac {x^4 (a+b x)^{5/2} (a c-b c x)^{5/2}}{9 b^2 c}\right )}{11 b^2}-\frac {x^6 (a+b x)^{5/2} (a c-b c x)^{5/2}}{11 b^2 c}\) |
\(\Big \downarrow \) 111 |
\(\displaystyle \frac {6 a^2 \left (\frac {4 a^2 \left (-\frac {\int -2 a^2 c x (a+b x)^{3/2} (a c-b c x)^{3/2}dx}{7 b^2 c}-\frac {x^2 (a+b x)^{5/2} (a c-b c x)^{5/2}}{7 b^2 c}\right )}{9 b^2}-\frac {x^4 (a+b x)^{5/2} (a c-b c x)^{5/2}}{9 b^2 c}\right )}{11 b^2}-\frac {x^6 (a+b x)^{5/2} (a c-b c x)^{5/2}}{11 b^2 c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {6 a^2 \left (\frac {4 a^2 \left (\frac {2 a^2 \int x (a+b x)^{3/2} (a c-b c x)^{3/2}dx}{7 b^2}-\frac {x^2 (a+b x)^{5/2} (a c-b c x)^{5/2}}{7 b^2 c}\right )}{9 b^2}-\frac {x^4 (a+b x)^{5/2} (a c-b c x)^{5/2}}{9 b^2 c}\right )}{11 b^2}-\frac {x^6 (a+b x)^{5/2} (a c-b c x)^{5/2}}{11 b^2 c}\) |
\(\Big \downarrow \) 83 |
\(\displaystyle \frac {6 a^2 \left (\frac {4 a^2 \left (-\frac {2 a^2 (a+b x)^{5/2} (a c-b c x)^{5/2}}{35 b^4 c}-\frac {x^2 (a+b x)^{5/2} (a c-b c x)^{5/2}}{7 b^2 c}\right )}{9 b^2}-\frac {x^4 (a+b x)^{5/2} (a c-b c x)^{5/2}}{9 b^2 c}\right )}{11 b^2}-\frac {x^6 (a+b x)^{5/2} (a c-b c x)^{5/2}}{11 b^2 c}\) |
Input:
Int[x^7*(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2),x]
Output:
-1/11*(x^6*(a + b*x)^(5/2)*(a*c - b*c*x)^(5/2))/(b^2*c) + (6*a^2*(-1/9*(x^ 4*(a + b*x)^(5/2)*(a*c - b*c*x)^(5/2))/(b^2*c) + (4*a^2*((-2*a^2*(a + b*x) ^(5/2)*(a*c - b*c*x)^(5/2))/(35*b^4*c) - (x^2*(a + b*x)^(5/2)*(a*c - b*c*x )^(5/2))/(7*b^2*c)))/(9*b^2)))/(11*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 | \(-\frac {\left (-b x +a \right ) \left (b x +a \right )^{\frac {5}{2}} \left (105 b^{6} x^{6}+70 a^{2} x^{4} b^{4}+40 a^{4} x^{2} b^{2}+16 a^{6}\right ) \left (-b c x +a c \right )^{\frac {3}{2}}}{1155 b^{8}}\) | \(66\) |
orering | \(-\frac {\left (-b x +a \right ) \left (b x +a \right )^{\frac {5}{2}} \left (105 b^{6} x^{6}+70 a^{2} x^{4} b^{4}+40 a^{4} x^{2} b^{2}+16 a^{6}\right ) \left (-b c x +a c \right )^{\frac {3}{2}}}{1155 b^{8}}\) | \(66\) |
default | \(-\frac {\sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}\, c \left (-b^{2} x^{2}+a^{2}\right ) \left (-105 b^{8} x^{8}+35 a^{2} x^{6} b^{6}+30 a^{4} x^{4} b^{4}+24 a^{6} x^{2} b^{2}+16 a^{8}\right )}{1155 b^{8}}\) | \(83\) |
risch | \(-\frac {\sqrt {b x +a}\, c^{2} \left (105 b^{10} x^{10}-140 a^{2} b^{8} x^{8}+5 a^{4} b^{6} x^{6}+6 a^{6} b^{4} x^{4}+8 a^{8} b^{2} x^{2}+16 a^{10}\right ) \left (-b x +a \right )}{1155 \sqrt {-c \left (b x -a \right )}\, b^{8}}\) | \(92\) |
Input:
int(x^7*(b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/1155*(-b*x+a)*(b*x+a)^(5/2)*(105*b^6*x^6+70*a^2*b^4*x^4+40*a^4*b^2*x^2+ 16*a^6)*(-b*c*x+a*c)^(3/2)/b^8
Time = 0.08 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.59 \[ \int x^7 (a+b x)^{3/2} (a c-b c x)^{3/2} \, dx=-\frac {{\left (105 \, b^{10} c x^{10} - 140 \, a^{2} b^{8} c x^{8} + 5 \, a^{4} b^{6} c x^{6} + 6 \, a^{6} b^{4} c x^{4} + 8 \, a^{8} b^{2} c x^{2} + 16 \, a^{10} c\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{1155 \, b^{8}} \] Input:
integrate(x^7*(b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2),x, algorithm="fricas")
Output:
-1/1155*(105*b^10*c*x^10 - 140*a^2*b^8*c*x^8 + 5*a^4*b^6*c*x^6 + 6*a^6*b^4 *c*x^4 + 8*a^8*b^2*c*x^2 + 16*a^10*c)*sqrt(-b*c*x + a*c)*sqrt(b*x + a)/b^8
\[ \int x^7 (a+b x)^{3/2} (a c-b c x)^{3/2} \, dx=\int x^{7} \left (- c \left (- a + b x\right )\right )^{\frac {3}{2}} \left (a + b x\right )^{\frac {3}{2}}\, dx \] Input:
integrate(x**7*(b*x+a)**(3/2)*(-b*c*x+a*c)**(3/2),x)
Output:
Integral(x**7*(-c*(-a + b*x))**(3/2)*(a + b*x)**(3/2), x)
Time = 0.12 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.81 \[ \int x^7 (a+b x)^{3/2} (a c-b c x)^{3/2} \, dx=-\frac {{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {5}{2}} x^{6}}{11 \, b^{2} c} - \frac {2 \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {5}{2}} a^{2} x^{4}}{33 \, b^{4} c} - \frac {8 \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {5}{2}} a^{4} x^{2}}{231 \, b^{6} c} - \frac {16 \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {5}{2}} a^{6}}{1155 \, b^{8} c} \] Input:
integrate(x^7*(b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2),x, algorithm="maxima")
Output:
-1/11*(-b^2*c*x^2 + a^2*c)^(5/2)*x^6/(b^2*c) - 2/33*(-b^2*c*x^2 + a^2*c)^( 5/2)*a^2*x^4/(b^4*c) - 8/231*(-b^2*c*x^2 + a^2*c)^(5/2)*a^4*x^2/(b^6*c) - 16/1155*(-b^2*c*x^2 + a^2*c)^(5/2)*a^6/(b^8*c)
Leaf count of result is larger than twice the leaf count of optimal. 718 vs. \(2 (123) = 246\).
Time = 0.52 (sec) , antiderivative size = 718, normalized size of antiderivative = 4.88 \[ \int x^7 (a+b x)^{3/2} (a c-b c x)^{3/2} \, dx =\text {Too large to display} \] Input:
integrate(x^7*(b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2),x, algorithm="giac")
Output:
1/887040*(66*(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^3*c - 22*(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))*a^2*c - 11*(39690*a^10*c* log(abs(-sqrt(b*x + a)*sqrt(-c) + sqrt(-(b*x + a)*c + 2*a*c)))/sqrt(-c) - (141435*a^9 - (463575*a^8 - 2*(560469*a^7 - (934839*a^6 - 4*(272449*a^5 - (222545*a^4 - 2*(62625*a^3 - 7*(8*(9*b*x - 82*a)*(b*x + a) + 3313*a^2)*(b* x + a))*(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))*a*c + (436590*a^11*c*log(abs(-sq rt(b*x + a)*sqrt(-c) + sqrt(-(b*x + a)*c + 2*a*c)))/sqrt(-c) - (1555785*a^ 10 - (5690685*a^9 - 2*(7732263*a^8 - (14743773*a^7 - 4*(5014603*a^6 - (491 7755*a^5 - 2*(1730955*a^4 - 7*(122203*a^3 - 8*(9*(10*b*x - 101*a)*(b*x + a ) + 5041*a^2)*(b*x + a))*(b*x + a))*(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))*c)/b ^8
Time = 0.51 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.81 \[ \int x^7 (a+b x)^{3/2} (a c-b c x)^{3/2} \, dx=-\sqrt {a\,c-b\,c\,x}\,\left (\frac {16\,a^{10}\,c\,\sqrt {a+b\,x}}{1155\,b^8}-\frac {4\,a^2\,c\,x^8\,\sqrt {a+b\,x}}{33}+\frac {b^2\,c\,x^{10}\,\sqrt {a+b\,x}}{11}+\frac {a^4\,c\,x^6\,\sqrt {a+b\,x}}{231\,b^2}+\frac {2\,a^6\,c\,x^4\,\sqrt {a+b\,x}}{385\,b^4}+\frac {8\,a^8\,c\,x^2\,\sqrt {a+b\,x}}{1155\,b^6}\right ) \] Input:
int(x^7*(a*c - b*c*x)^(3/2)*(a + b*x)^(3/2),x)
Output:
-(a*c - b*c*x)^(1/2)*((16*a^10*c*(a + b*x)^(1/2))/(1155*b^8) - (4*a^2*c*x^ 8*(a + b*x)^(1/2))/33 + (b^2*c*x^10*(a + b*x)^(1/2))/11 + (a^4*c*x^6*(a + b*x)^(1/2))/(231*b^2) + (2*a^6*c*x^4*(a + b*x)^(1/2))/(385*b^4) + (8*a^8*c *x^2*(a + b*x)^(1/2))/(1155*b^6))
Time = 0.16 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.54 \[ \int x^7 (a+b x)^{3/2} (a c-b c x)^{3/2} \, dx=\frac {\sqrt {c}\, \sqrt {b x +a}\, \sqrt {-b x +a}\, c \left (-105 b^{10} x^{10}+140 a^{2} b^{8} x^{8}-5 a^{4} b^{6} x^{6}-6 a^{6} b^{4} x^{4}-8 a^{8} b^{2} x^{2}-16 a^{10}\right )}{1155 b^{8}} \] Input:
int(x^7*(b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2),x)
Output:
(sqrt(c)*sqrt(a + b*x)*sqrt(a - b*x)*c*( - 16*a**10 - 8*a**8*b**2*x**2 - 6 *a**6*b**4*x**4 - 5*a**4*b**6*x**6 + 140*a**2*b**8*x**8 - 105*b**10*x**10) )/(1155*b**8)