Integrand size = 26, antiderivative size = 105 \[ \int \frac {x^5}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=-\frac {8 a^4}{3 b^6 c (a+b x)^{3/2} (a c-b c x)^{3/2}}+\frac {4 a^2 x^2}{b^4 c (a+b x)^{3/2} (a c-b c x)^{3/2}}-\frac {x^4}{b^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}} \] Output:
-8/3*a^4/b^6/c/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2)+4*a^2*x^2/b^4/c/(b*x+a)^(3 /2)/(-b*c*x+a*c)^(3/2)-x^4/b^2/c/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2)
Time = 0.11 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.53 \[ \int \frac {x^5}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=\frac {-8 a^4+12 a^2 b^2 x^2-3 b^4 x^4}{3 b^6 c (c (a-b x))^{3/2} (a+b x)^{3/2}} \] Input:
Integrate[x^5/((a + b*x)^(5/2)*(a*c - b*c*x)^(5/2)),x]
Output:
(-8*a^4 + 12*a^2*b^2*x^2 - 3*b^4*x^4)/(3*b^6*c*(c*(a - b*x))^(3/2)*(a + b* x)^(3/2))
Time = 0.20 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.04, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {109, 27, 35, 109, 27, 35, 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.
\(\displaystyle \int \frac {x^5}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx\) |
\(\Big \downarrow \) 109 |
\(\displaystyle \frac {x^4}{3 b^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}-\frac {\int \frac {4 a c x^3 (a-b x)}{(a+b x)^{3/2} (a c-b c x)^{5/2}}dx}{3 a b^2 c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {x^4}{3 b^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}-\frac {4 \int \frac {x^3 (a-b x)}{(a+b x)^{3/2} (a c-b c x)^{5/2}}dx}{3 b^2}\) |
\(\Big \downarrow \) 35 |
\(\displaystyle \frac {x^4}{3 b^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}-\frac {4 \int \frac {x^3}{(a+b x)^{3/2} (a c-b c x)^{3/2}}dx}{3 b^2 c}\) |
\(\Big \downarrow \) 109 |
\(\displaystyle \frac {x^4}{3 b^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}-\frac {4 \left (\frac {x^2}{b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {\int \frac {2 a c x (a-b x)}{\sqrt {a+b x} (a c-b c x)^{3/2}}dx}{a b^2 c}\right )}{3 b^2 c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {x^4}{3 b^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}-\frac {4 \left (\frac {x^2}{b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {2 \int \frac {x (a-b x)}{\sqrt {a+b x} (a c-b c x)^{3/2}}dx}{b^2}\right )}{3 b^2 c}\) |
\(\Big \downarrow \) 35 |
\(\displaystyle \frac {x^4}{3 b^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}-\frac {4 \left (\frac {x^2}{b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {2 \int \frac {x}{\sqrt {a+b x} \sqrt {a c-b c x}}dx}{b^2 c}\right )}{3 b^2 c}\) |
\(\Big \downarrow \) 83 |
\(\displaystyle \frac {x^4}{3 b^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}-\frac {4 \left (\frac {2 \sqrt {a+b x} \sqrt {a c-b c x}}{b^4 c^2}+\frac {x^2}{b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}\right )}{3 b^2 c}\) |
Input:
Int[x^5/((a + b*x)^(5/2)*(a*c - b*c*x)^(5/2)),x]
Output:
x^4/(3*b^2*c*(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2)) - (4*(x^2/(b^2*c*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]) + (2*Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/(b^4*c^2)) )/(3*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])
Time = 0.30 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.52
method | result | size |
gosper | \(-\frac {\left (-b x +a \right ) \left (3 b^{4} x^{4}-12 a^{2} b^{2} x^{2}+8 a^{4}\right )}{3 \left (b x +a \right )^{\frac {3}{2}} b^{6} \left (-b c x +a c \right )^{\frac {5}{2}}}\) | \(55\) |
orering | \(-\frac {\left (-b x +a \right ) \left (3 b^{4} x^{4}-12 a^{2} b^{2} x^{2}+8 a^{4}\right )}{3 \left (b x +a \right )^{\frac {3}{2}} b^{6} \left (-b c x +a c \right )^{\frac {5}{2}}}\) | \(55\) |
default | \(-\frac {\sqrt {c \left (-b x +a \right )}\, \left (3 b^{4} x^{4}-12 a^{2} b^{2} x^{2}+8 a^{4}\right )}{3 \left (b x +a \right )^{\frac {3}{2}} b^{6} \left (-b x +a \right )^{2} c^{3}}\) | \(59\) |
risch | \(-\frac {\left (-b x +a \right ) \sqrt {b x +a}}{b^{6} \sqrt {-c \left (b x -a \right )}\, c^{2}}-\frac {a^{2} \left (-6 b^{2} x^{2}+5 a^{2}\right ) \sqrt {-\left (b x +a \right ) c \left (b x -a \right )}}{3 b^{6} \sqrt {\left (b x +a \right ) c \left (-b x +a \right )}\, \left (-b^{2} x^{2}+a^{2}\right ) \sqrt {b x +a}\, \sqrt {-c \left (b x -a \right )}\, c^{2}}\) | \(125\) |
Input:
int(x^5/(b*x+a)^(5/2)/(-b*c*x+a*c)^(5/2),x,method=_RETURNVERBOSE)
Output:
-1/3*(-b*x+a)*(3*b^4*x^4-12*a^2*b^2*x^2+8*a^4)/(b*x+a)^(3/2)/b^6/(-b*c*x+a *c)^(5/2)
Time = 0.08 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.78 \[ \int \frac {x^5}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=-\frac {{\left (3 \, b^{4} x^{4} - 12 \, a^{2} b^{2} x^{2} + 8 \, a^{4}\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{3 \, {\left (b^{10} c^{3} x^{4} - 2 \, a^{2} b^{8} c^{3} x^{2} + a^{4} b^{6} c^{3}\right )}} \] Input:
integrate(x^5/(b*x+a)^(5/2)/(-b*c*x+a*c)^(5/2),x, algorithm="fricas")
Output:
-1/3*(3*b^4*x^4 - 12*a^2*b^2*x^2 + 8*a^4)*sqrt(-b*c*x + a*c)*sqrt(b*x + a) /(b^10*c^3*x^4 - 2*a^2*b^8*c^3*x^2 + a^4*b^6*c^3)
Result contains complex when optimal does not.
Time = 9.59 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.13 \[ \int \frac {x^5}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=- \frac {i a {G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {5}{4}, - \frac {3}{4} & -2, 0, \frac {1}{2}, 1 \\- \frac {5}{4}, - \frac {3}{4}, - \frac {1}{2}, 0, \frac {1}{2}, 0 & \end {matrix} \middle | {\frac {a^{2}}{b^{2} x^{2}}} \right )}}{3 \pi ^{\frac {3}{2}} b^{6} c^{\frac {5}{2}}} - \frac {a {G_{6, 6}^{2, 6}\left (\begin {matrix} -3, - \frac {5}{2}, -2, - \frac {7}{4}, - \frac {5}{4}, 1 & \\- \frac {7}{4}, - \frac {5}{4} & -3, - \frac {5}{2}, - \frac {1}{2}, 0 \end {matrix} \middle | {\frac {a^{2} e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{3 \pi ^{\frac {3}{2}} b^{6} c^{\frac {5}{2}}} \] Input:
integrate(x**5/(b*x+a)**(5/2)/(-b*c*x+a*c)**(5/2),x)
Output:
-I*a*meijerg(((-5/4, -3/4), (-2, 0, 1/2, 1)), ((-5/4, -3/4, -1/2, 0, 1/2, 0), ()), a**2/(b**2*x**2))/(3*pi**(3/2)*b**6*c**(5/2)) - a*meijerg(((-3, - 5/2, -2, -7/4, -5/4, 1), ()), ((-7/4, -5/4), (-3, -5/2, -1/2, 0)), a**2*ex p_polar(-2*I*pi)/(b**2*x**2))/(3*pi**(3/2)*b**6*c**(5/2))
Time = 0.11 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.84 \[ \int \frac {x^5}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=-\frac {x^{4}}{{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} b^{2} c} + \frac {4 \, a^{2} x^{2}}{{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} b^{4} c} - \frac {8 \, a^{4}}{3 \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} b^{6} c} \] Input:
integrate(x^5/(b*x+a)^(5/2)/(-b*c*x+a*c)^(5/2),x, algorithm="maxima")
Output:
-x^4/((-b^2*c*x^2 + a^2*c)^(3/2)*b^2*c) + 4*a^2*x^2/((-b^2*c*x^2 + a^2*c)^ (3/2)*b^4*c) - 8/3*a^4/((-b^2*c*x^2 + a^2*c)^(3/2)*b^6*c)
Leaf count of result is larger than twice the leaf count of optimal. 223 vs. \(2 (91) = 182\).
Time = 0.19 (sec) , antiderivative size = 223, normalized size of antiderivative = 2.12 \[ \int \frac {x^5}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=-\frac {\frac {\sqrt {-{\left (b x + a\right )} c + 2 \, a c} \sqrt {b x + a} {\left ({\left (b x + a\right )} {\left (\frac {12 \, {\left (b x + a\right )}}{b^{5} c} - \frac {59 \, a}{b^{5} c}\right )} + \frac {69 \, a^{2}}{b^{5} c}\right )}}{{\left ({\left (b x + a\right )} c - 2 \, a c\right )}^{2}} + \frac {8 \, {\left (6 \, a^{2} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{4} - 21 \, a^{3} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2} c + 22 \, a^{4} c^{2}\right )}}{{\left ({\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2} - 2 \, a c\right )}^{3} b^{5} \sqrt {-c} c}}{12 \, b} \] Input:
integrate(x^5/(b*x+a)^(5/2)/(-b*c*x+a*c)^(5/2),x, algorithm="giac")
Output:
-1/12*(sqrt(-(b*x + a)*c + 2*a*c)*sqrt(b*x + a)*((b*x + a)*(12*(b*x + a)/( b^5*c) - 59*a/(b^5*c)) + 69*a^2/(b^5*c))/((b*x + a)*c - 2*a*c)^2 + 8*(6*a^ 2*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^4 - 21*a^3*(sqrt(b *x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^2*c + 22*a^4*c^2)/(((sqrt(b *x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^2 - 2*a*c)^3*b^5*sqrt(-c)*c ))/b
Time = 0.70 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.96 \[ \int \frac {x^5}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=\frac {8\,a^4\,\sqrt {a\,c-b\,c\,x}+3\,b^4\,x^4\,\sqrt {a\,c-b\,c\,x}-12\,a^2\,b^2\,x^2\,\sqrt {a\,c-b\,c\,x}}{{\left (a\,c-b\,c\,x\right )}^2\,\left (3\,b^6\,\left (a\,c-b\,c\,x\right )-6\,a\,b^6\,c\right )\,\sqrt {a+b\,x}} \] Input:
int(x^5/((a*c - b*c*x)^(5/2)*(a + b*x)^(5/2)),x)
Output:
(8*a^4*(a*c - b*c*x)^(1/2) + 3*b^4*x^4*(a*c - b*c*x)^(1/2) - 12*a^2*b^2*x^ 2*(a*c - b*c*x)^(1/2))/((a*c - b*c*x)^2*(3*b^6*(a*c - b*c*x) - 6*a*b^6*c)* (a + b*x)^(1/2))
Time = 0.16 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.63 \[ \int \frac {x^5}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=\frac {\sqrt {c}\, \left (-3 b^{4} x^{4}+12 a^{2} b^{2} x^{2}-8 a^{4}\right )}{3 \sqrt {b x +a}\, \sqrt {-b x +a}\, b^{6} c^{3} \left (-b^{2} x^{2}+a^{2}\right )} \] Input:
int(x^5/(b*x+a)^(5/2)/(-b*c*x+a*c)^(5/2),x)
Output:
(sqrt(c)*( - 8*a**4 + 12*a**2*b**2*x**2 - 3*b**4*x**4))/(3*sqrt(a + b*x)*s qrt(a - b*x)*b**6*c**3*(a**2 - b**2*x**2))