Integrand size = 26, antiderivative size = 109 \[ \int \frac {x^5}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\frac {8 a^4}{3 b^6 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {4 a^2 x^2}{3 b^4 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {x^4}{3 b^2 c \sqrt {a+b x} \sqrt {a c-b c x}} \] Output:
8/3*a^4/b^6/c/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)-4/3*a^2*x^2/b^4/c/(b*x+a)^( 1/2)/(-b*c*x+a*c)^(1/2)-1/3*x^4/b^2/c/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)
Time = 0.11 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.51 \[ \int \frac {x^5}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\frac {8 a^4-4 a^2 b^2 x^2-b^4 x^4}{3 b^6 c \sqrt {c (a-b x)} \sqrt {a+b x}} \] Input:
Integrate[x^5/((a + b*x)^(3/2)*(a*c - b*c*x)^(3/2)),x]
Output:
(8*a^4 - 4*a^2*b^2*x^2 - b^4*x^4)/(3*b^6*c*Sqrt[c*(a - b*x)]*Sqrt[a + b*x] )
Time = 0.20 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {109, 27, 35, 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.
\(\displaystyle \int \frac {x^5}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 109 |
\(\displaystyle \frac {x^4}{b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {\int \frac {4 a c x^3 (a-b x)}{\sqrt {a+b x} (a c-b c x)^{3/2}}dx}{a b^2 c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {x^4}{b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {4 \int \frac {x^3 (a-b x)}{\sqrt {a+b x} (a c-b c x)^{3/2}}dx}{b^2}\) |
\(\Big \downarrow \) 35 |
\(\displaystyle \frac {x^4}{b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {4 \int \frac {x^3}{\sqrt {a+b x} \sqrt {a c-b c x}}dx}{b^2 c}\) |
\(\Big \downarrow \) 111 |
\(\displaystyle \frac {x^4}{b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {4 \left (-\frac {\int -\frac {2 a^2 c x}{\sqrt {a+b x} \sqrt {a c-b c x}}dx}{3 b^2 c}-\frac {x^2 \sqrt {a+b x} \sqrt {a c-b c x}}{3 b^2 c}\right )}{b^2 c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {x^4}{b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {4 \left (\frac {2 a^2 \int \frac {x}{\sqrt {a+b x} \sqrt {a c-b c x}}dx}{3 b^2}-\frac {x^2 \sqrt {a+b x} \sqrt {a c-b c x}}{3 b^2 c}\right )}{b^2 c}\) |
\(\Big \downarrow \) 83 |
\(\displaystyle \frac {x^4}{b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {4 \left (-\frac {2 a^2 \sqrt {a+b x} \sqrt {a c-b c x}}{3 b^4 c}-\frac {x^2 \sqrt {a+b x} \sqrt {a c-b c x}}{3 b^2 c}\right )}{b^2 c}\) |
Input:
Int[x^5/((a + b*x)^(3/2)*(a*c - b*c*x)^(3/2)),x]
Output:
x^4/(b^2*c*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]) - (4*((-2*a^2*Sqrt[a + b*x]*Sq rt[a*c - b*c*x])/(3*b^4*c) - (x^2*Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/(3*b^2* c)))/(b^2*c)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_.)*((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.), x_Symbol] :> Simp[(b/d)^m Int[u*(c + d*x)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n} , x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && !(IntegerQ[n] && SimplerQ[a + b*x, c + d*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*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f *x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
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 = 0.30 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.50
method | result | size |
gosper | \(\frac {\left (-b x +a \right ) \left (-b^{4} x^{4}-4 a^{2} b^{2} x^{2}+8 a^{4}\right )}{3 \sqrt {b x +a}\, b^{6} \left (-b c x +a c \right )^{\frac {3}{2}}}\) | \(55\) |
orering | \(\frac {\left (-b x +a \right ) \left (-b^{4} x^{4}-4 a^{2} b^{2} x^{2}+8 a^{4}\right )}{3 \sqrt {b x +a}\, b^{6} \left (-b c x +a c \right )^{\frac {3}{2}}}\) | \(55\) |
default | \(\frac {\sqrt {c \left (-b x +a \right )}\, \left (-b^{4} x^{4}-4 a^{2} b^{2} x^{2}+8 a^{4}\right )}{3 c^{2} \left (-b x +a \right ) b^{6} \sqrt {b x +a}}\) | \(59\) |
risch | \(\frac {\left (b^{2} x^{2}+5 a^{2}\right ) \left (-b x +a \right ) \sqrt {b x +a}}{3 b^{6} \sqrt {-c \left (b x -a \right )}\, c}+\frac {a^{4} \sqrt {-\left (b x +a \right ) c \left (b x -a \right )}}{b^{6} \sqrt {\left (b x +a \right ) c \left (-b x +a \right )}\, \sqrt {b x +a}\, \sqrt {-c \left (b x -a \right )}\, c}\) | \(109\) |
Input:
int(x^5/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/3*(-b*x+a)*(-b^4*x^4-4*a^2*b^2*x^2+8*a^4)/(b*x+a)^(1/2)/b^6/(-b*c*x+a*c) ^(3/2)
Time = 0.13 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.62 \[ \int \frac {x^5}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\frac {{\left (b^{4} x^{4} + 4 \, a^{2} b^{2} x^{2} - 8 \, a^{4}\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{3 \, {\left (b^{8} c^{2} x^{2} - a^{2} b^{6} c^{2}\right )}} \] Input:
integrate(x^5/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2),x, algorithm="fricas")
Output:
1/3*(b^4*x^4 + 4*a^2*b^2*x^2 - 8*a^4)*sqrt(-b*c*x + a*c)*sqrt(b*x + a)/(b^ 8*c^2*x^2 - a^2*b^6*c^2)
Result contains complex when optimal does not.
Time = 4.70 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.17 \[ \int \frac {x^5}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\frac {i a^{3} {G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {7}{4}, - \frac {5}{4} & -2, -1, - \frac {1}{2}, 1 \\- \frac {7}{4}, - \frac {3}{2}, - \frac {5}{4}, -1, - \frac {1}{2}, 0 & \end {matrix} \middle | {\frac {a^{2}}{b^{2} x^{2}}} \right )}}{2 \pi ^{\frac {3}{2}} b^{6} c^{\frac {3}{2}}} - \frac {a^{3} {G_{6, 6}^{2, 6}\left (\begin {matrix} -3, - \frac {5}{2}, - \frac {9}{4}, -2, - \frac {7}{4}, 1 & \\- \frac {9}{4}, - \frac {7}{4} & -3, - \frac {5}{2}, - \frac {3}{2}, 0 \end {matrix} \middle | {\frac {a^{2} e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{2 \pi ^{\frac {3}{2}} b^{6} c^{\frac {3}{2}}} \] Input:
integrate(x**5/(b*x+a)**(3/2)/(-b*c*x+a*c)**(3/2),x)
Output:
I*a**3*meijerg(((-7/4, -5/4), (-2, -1, -1/2, 1)), ((-7/4, -3/2, -5/4, -1, -1/2, 0), ()), a**2/(b**2*x**2))/(2*pi**(3/2)*b**6*c**(3/2)) - a**3*meijer g(((-3, -5/2, -9/4, -2, -7/4, 1), ()), ((-9/4, -7/4), (-3, -5/2, -3/2, 0)) , a**2*exp_polar(-2*I*pi)/(b**2*x**2))/(2*pi**(3/2)*b**6*c**(3/2))
Time = 0.12 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.81 \[ \int \frac {x^5}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=-\frac {x^{4}}{3 \, \sqrt {-b^{2} c x^{2} + a^{2} c} b^{2} c} - \frac {4 \, a^{2} x^{2}}{3 \, \sqrt {-b^{2} c x^{2} + a^{2} c} b^{4} c} + \frac {8 \, a^{4}}{3 \, \sqrt {-b^{2} c x^{2} + a^{2} c} b^{6} c} \] Input:
integrate(x^5/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2),x, algorithm="maxima")
Output:
-1/3*x^4/(sqrt(-b^2*c*x^2 + a^2*c)*b^2*c) - 4/3*a^2*x^2/(sqrt(-b^2*c*x^2 + a^2*c)*b^4*c) + 8/3*a^4/(sqrt(-b^2*c*x^2 + a^2*c)*b^6*c)
Time = 0.18 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.45 \[ \int \frac {x^5}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\frac {\frac {{\left (2 \, {\left (b x + a\right )} {\left ({\left (b x + a\right )} {\left (\frac {b x + a}{b^{5} c} - \frac {4 \, a}{b^{5} c}\right )} + \frac {10 \, a^{2}}{b^{5} c}\right )} - \frac {27 \, a^{3}}{b^{5} c}\right )} \sqrt {-{\left (b x + a\right )} c + 2 \, a c} \sqrt {b x + a}}{{\left (b x + a\right )} c - 2 \, a c} - \frac {12 \, a^{4} \sqrt {-c}}{{\left ({\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2} - 2 \, a c\right )} b^{5} c}}{6 \, b} \] Input:
integrate(x^5/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2),x, algorithm="giac")
Output:
1/6*((2*(b*x + a)*((b*x + a)*((b*x + a)/(b^5*c) - 4*a/(b^5*c)) + 10*a^2/(b ^5*c)) - 27*a^3/(b^5*c))*sqrt(-(b*x + a)*c + 2*a*c)*sqrt(b*x + a)/((b*x + a)*c - 2*a*c) - 12*a^4*sqrt(-c)/(((sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a )*c + 2*a*c))^2 - 2*a*c)*b^5*c))/b
Time = 0.65 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.46 \[ \int \frac {x^5}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=-\frac {-8\,a^4+4\,a^2\,b^2\,x^2+b^4\,x^4}{3\,b^6\,c\,\sqrt {a\,c-b\,c\,x}\,\sqrt {a+b\,x}} \] Input:
int(x^5/((a*c - b*c*x)^(3/2)*(a + b*x)^(3/2)),x)
Output:
-(b^4*x^4 - 8*a^4 + 4*a^2*b^2*x^2)/(3*b^6*c*(a*c - b*c*x)^(1/2)*(a + b*x)^ (1/2))
Time = 0.16 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.48 \[ \int \frac {x^5}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\frac {\sqrt {c}\, \left (-b^{4} x^{4}-4 a^{2} b^{2} x^{2}+8 a^{4}\right )}{3 \sqrt {b x +a}\, \sqrt {-b x +a}\, b^{6} c^{2}} \] Input:
int(x^5/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2),x)
Output:
(sqrt(c)*(8*a**4 - 4*a**2*b**2*x**2 - b**4*x**4))/(3*sqrt(a + b*x)*sqrt(a - b*x)*b**6*c**2)