Integrand size = 26, antiderivative size = 144 \[ \int \frac {1}{x^4 (a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=-\frac {1}{3 a^2 c x^3 (a+b x)^{3/2} (a c-b c x)^{3/2}}-\frac {2 b^2}{a^4 c x (a+b x)^{3/2} (a c-b c x)^{3/2}}+\frac {8 b^4 x}{3 a^6 c (a+b x)^{3/2} (a c-b c x)^{3/2}}+\frac {16 b^4 x}{3 a^8 c^2 \sqrt {a+b x} \sqrt {a c-b c x}} \] Output:
-1/3/a^2/c/x^3/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2)-2*b^2/a^4/c/x/(b*x+a)^(3/2 )/(-b*c*x+a*c)^(3/2)+8/3*b^4*x/a^6/c/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2)+16/3 *b^4*x/a^8/c^2/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)
Time = 0.11 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.49 \[ \int \frac {1}{x^4 (a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=\frac {-a^6-6 a^4 b^2 x^2+24 a^2 b^4 x^4-16 b^6 x^6}{3 a^8 c x^3 (c (a-b x))^{3/2} (a+b x)^{3/2}} \] Input:
Integrate[1/(x^4*(a + b*x)^(5/2)*(a*c - b*c*x)^(5/2)),x]
Output:
(-a^6 - 6*a^4*b^2*x^2 + 24*a^2*b^4*x^4 - 16*b^6*x^6)/(3*a^8*c*x^3*(c*(a - b*x))^(3/2)*(a + b*x)^(3/2))
Time = 0.21 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {114, 27, 114, 27, 42, 41}
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.
\(\displaystyle \int \frac {1}{x^4 (a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx\) |
\(\Big \downarrow \) 114 |
\(\displaystyle -\frac {\int -\frac {6 b^2 c}{x^2 (a+b x)^{5/2} (a c-b c x)^{5/2}}dx}{3 a^2 c}-\frac {1}{3 a^2 c x^3 (a+b x)^{3/2} (a c-b c x)^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2 b^2 \int \frac {1}{x^2 (a+b x)^{5/2} (a c-b c x)^{5/2}}dx}{a^2}-\frac {1}{3 a^2 c x^3 (a+b x)^{3/2} (a c-b c x)^{3/2}}\) |
\(\Big \downarrow \) 114 |
\(\displaystyle \frac {2 b^2 \left (-\frac {\int -\frac {4 b^2 c}{(a+b x)^{5/2} (a c-b c x)^{5/2}}dx}{a^2 c}-\frac {1}{a^2 c x (a+b x)^{3/2} (a c-b c x)^{3/2}}\right )}{a^2}-\frac {1}{3 a^2 c x^3 (a+b x)^{3/2} (a c-b c x)^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2 b^2 \left (\frac {4 b^2 \int \frac {1}{(a+b x)^{5/2} (a c-b c x)^{5/2}}dx}{a^2}-\frac {1}{a^2 c x (a+b x)^{3/2} (a c-b c x)^{3/2}}\right )}{a^2}-\frac {1}{3 a^2 c x^3 (a+b x)^{3/2} (a c-b c x)^{3/2}}\) |
\(\Big \downarrow \) 42 |
\(\displaystyle \frac {2 b^2 \left (\frac {4 b^2 \left (\frac {2 \int \frac {1}{(a+b x)^{3/2} (a c-b c x)^{3/2}}dx}{3 a^2 c}+\frac {x}{3 a^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}\right )}{a^2}-\frac {1}{a^2 c x (a+b x)^{3/2} (a c-b c x)^{3/2}}\right )}{a^2}-\frac {1}{3 a^2 c x^3 (a+b x)^{3/2} (a c-b c x)^{3/2}}\) |
\(\Big \downarrow \) 41 |
\(\displaystyle \frac {2 b^2 \left (\frac {4 b^2 \left (\frac {2 x}{3 a^4 c^2 \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {x}{3 a^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}\right )}{a^2}-\frac {1}{a^2 c x (a+b x)^{3/2} (a c-b c x)^{3/2}}\right )}{a^2}-\frac {1}{3 a^2 c x^3 (a+b x)^{3/2} (a c-b c x)^{3/2}}\) |
Input:
Int[1/(x^4*(a + b*x)^(5/2)*(a*c - b*c*x)^(5/2)),x]
Output:
-1/3*1/(a^2*c*x^3*(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2)) + (2*b^2*(-(1/(a^2* c*x*(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2))) + (4*b^2*(x/(3*a^2*c*(a + b*x)^( 3/2)*(a*c - b*c*x)^(3/2)) + (2*x)/(3*a^4*c^2*Sqrt[a + b*x]*Sqrt[a*c - b*c* x])))/a^2))/a^2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(((a_) + (b_.)*(x_))^(3/2)*((c_) + (d_.)*(x_))^(3/2)), x_Symbol] :> S imp[x/(a*c*Sqrt[a + b*x]*Sqrt[c + d*x]), x] /; FreeQ[{a, b, c, d}, x] && Eq Q[b*c + a*d, 0]
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(- x)*(a + b*x)^(m + 1)*((c + d*x)^(m + 1)/(2*a*c*(m + 1))), x] + Simp[(2*m + 3)/(2*a*c*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(m + 1), x], x] /; Fre eQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && ILtQ[m + 3/2, 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 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
Time = 0.31 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.47
method | result | size |
gosper | \(-\frac {\left (-b x +a \right ) \left (16 b^{6} x^{6}-24 a^{2} x^{4} b^{4}+6 a^{4} x^{2} b^{2}+a^{6}\right )}{3 x^{3} \left (b x +a \right )^{\frac {3}{2}} a^{8} \left (-b c x +a c \right )^{\frac {5}{2}}}\) | \(67\) |
orering | \(-\frac {\left (-b x +a \right ) \left (16 b^{6} x^{6}-24 a^{2} x^{4} b^{4}+6 a^{4} x^{2} b^{2}+a^{6}\right )}{3 x^{3} \left (b x +a \right )^{\frac {3}{2}} a^{8} \left (-b c x +a c \right )^{\frac {5}{2}}}\) | \(67\) |
default | \(-\frac {\sqrt {c \left (-b x +a \right )}\, \left (16 b^{6} x^{6}-24 a^{2} x^{4} b^{4}+6 a^{4} x^{2} b^{2}+a^{6}\right )}{3 c^{3} a^{8} \left (-b x +a \right )^{2} x^{3} \left (b x +a \right )^{\frac {3}{2}}}\) | \(71\) |
risch | \(-\frac {\left (-b x +a \right ) \sqrt {b x +a}\, \left (8 b^{2} x^{2}+a^{2}\right )}{3 a^{8} x^{3} \sqrt {-c \left (b x -a \right )}\, c^{2}}+\frac {b^{4} x \left (-8 b^{2} x^{2}+9 a^{2}\right ) \sqrt {-\left (b x +a \right ) c \left (b x -a \right )}}{3 \sqrt {\left (b x +a \right ) c \left (-b x +a \right )}\, \left (-b^{2} x^{2}+a^{2}\right ) a^{8} \sqrt {b x +a}\, \sqrt {-c \left (b x -a \right )}\, c^{2}}\) | \(141\) |
Input:
int(1/x^4/(b*x+a)^(5/2)/(-b*c*x+a*c)^(5/2),x,method=_RETURNVERBOSE)
Output:
-1/3*(-b*x+a)*(16*b^6*x^6-24*a^2*b^4*x^4+6*a^4*b^2*x^2+a^6)/x^3/(b*x+a)^(3 /2)/a^8/(-b*c*x+a*c)^(5/2)
Time = 0.11 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.65 \[ \int \frac {1}{x^4 (a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=-\frac {{\left (16 \, b^{6} x^{6} - 24 \, a^{2} b^{4} x^{4} + 6 \, a^{4} b^{2} x^{2} + a^{6}\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{3 \, {\left (a^{8} b^{4} c^{3} x^{7} - 2 \, a^{10} b^{2} c^{3} x^{5} + a^{12} c^{3} x^{3}\right )}} \] Input:
integrate(1/x^4/(b*x+a)^(5/2)/(-b*c*x+a*c)^(5/2),x, algorithm="fricas")
Output:
-1/3*(16*b^6*x^6 - 24*a^2*b^4*x^4 + 6*a^4*b^2*x^2 + a^6)*sqrt(-b*c*x + a*c )*sqrt(b*x + a)/(a^8*b^4*c^3*x^7 - 2*a^10*b^2*c^3*x^5 + a^12*c^3*x^3)
Result contains complex when optimal does not.
Time = 28.56 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.65 \[ \int \frac {1}{x^4 (a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=\frac {i b^{3} {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {13}{4}, \frac {15}{4}, 1 & \frac {5}{2}, \frac {9}{2}, 5 \\\frac {13}{4}, \frac {15}{4}, 4, \frac {9}{2}, 5 & 0 \end {matrix} \middle | {\frac {a^{2}}{b^{2} x^{2}}} \right )}}{3 \pi ^{\frac {3}{2}} a^{8} c^{\frac {5}{2}}} + \frac {b^{3} {G_{6, 6}^{2, 6}\left (\begin {matrix} \frac {3}{2}, 2, \frac {5}{2}, \frac {11}{4}, \frac {13}{4}, 1 & \\\frac {11}{4}, \frac {13}{4} & \frac {3}{2}, 2, 4, 0 \end {matrix} \middle | {\frac {a^{2} e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{3 \pi ^{\frac {3}{2}} a^{8} c^{\frac {5}{2}}} \] Input:
integrate(1/x**4/(b*x+a)**(5/2)/(-b*c*x+a*c)**(5/2),x)
Output:
I*b**3*meijerg(((13/4, 15/4, 1), (5/2, 9/2, 5)), ((13/4, 15/4, 4, 9/2, 5), (0,)), a**2/(b**2*x**2))/(3*pi**(3/2)*a**8*c**(5/2)) + b**3*meijerg(((3/2 , 2, 5/2, 11/4, 13/4, 1), ()), ((11/4, 13/4), (3/2, 2, 4, 0)), a**2*exp_po lar(-2*I*pi)/(b**2*x**2))/(3*pi**(3/2)*a**8*c**(5/2))
Time = 0.04 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.82 \[ \int \frac {1}{x^4 (a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=\frac {8 \, b^{4} x}{3 \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} a^{6} c} - \frac {2 \, b^{2}}{{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} a^{4} c x} + \frac {16 \, b^{4} x}{3 \, \sqrt {-b^{2} c x^{2} + a^{2} c} a^{8} c^{2}} - \frac {1}{3 \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} a^{2} c x^{3}} \] Input:
integrate(1/x^4/(b*x+a)^(5/2)/(-b*c*x+a*c)^(5/2),x, algorithm="maxima")
Output:
8/3*b^4*x/((-b^2*c*x^2 + a^2*c)^(3/2)*a^6*c) - 2*b^2/((-b^2*c*x^2 + a^2*c) ^(3/2)*a^4*c*x) + 16/3*b^4*x/(sqrt(-b^2*c*x^2 + a^2*c)*a^8*c^2) - 1/3/((-b ^2*c*x^2 + a^2*c)^(3/2)*a^2*c*x^3)
Leaf count of result is larger than twice the leaf count of optimal. 527 vs. \(2 (122) = 244\).
Time = 0.26 (sec) , antiderivative size = 527, normalized size of antiderivative = 3.66 \[ \int \frac {1}{x^4 (a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=-\frac {1}{12} \, b^{3} {\left (\frac {\sqrt {-{\left (b x + a\right )} c + 2 \, a c} \sqrt {b x + a} {\left (\frac {16 \, {\left (b x + a\right )}}{a^{8} c} - \frac {33}{a^{7} c}\right )}}{{\left ({\left (b x + a\right )} c - 2 \, a c\right )}^{2}} + \frac {4 \, {\left (15 \, {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{16} - 114 \, a {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{14} c + 532 \, a^{2} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{12} c^{2} - 1944 \, a^{3} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{10} c^{3} + 5328 \, a^{4} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{8} c^{4} - 11104 \, a^{5} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{6} c^{5} + 14784 \, a^{6} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{4} c^{6} - 16512 \, a^{7} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2} c^{7} + 12288 \, a^{8} c^{8}\right )}}{{\left ({\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{6} - 2 \, a {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{4} c + 4 \, a^{2} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2} c^{2} - 8 \, a^{3} c^{3}\right )}^{3} a^{7} \sqrt {-c} c}\right )} \] Input:
integrate(1/x^4/(b*x+a)^(5/2)/(-b*c*x+a*c)^(5/2),x, algorithm="giac")
Output:
-1/12*b^3*(sqrt(-(b*x + a)*c + 2*a*c)*sqrt(b*x + a)*(16*(b*x + a)/(a^8*c) - 33/(a^7*c))/((b*x + a)*c - 2*a*c)^2 + 4*(15*(sqrt(b*x + a)*sqrt(-c) - sq rt(-(b*x + a)*c + 2*a*c))^16 - 114*a*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^14*c + 532*a^2*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)* c + 2*a*c))^12*c^2 - 1944*a^3*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^10*c^3 + 5328*a^4*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^8*c^4 - 11104*a^5*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2* a*c))^6*c^5 + 14784*a^6*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a* c))^4*c^6 - 16512*a^7*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c) )^2*c^7 + 12288*a^8*c^8)/(((sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2 *a*c))^6 - 2*a*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^4*c + 4*a^2*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^2*c^2 - 8*a^3 *c^3)^3*a^7*sqrt(-c)*c))
Time = 0.75 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x^4 (a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=\frac {a^6\,\sqrt {a\,c-b\,c\,x}+16\,b^6\,x^6\,\sqrt {a\,c-b\,c\,x}+6\,a^4\,b^2\,x^2\,\sqrt {a\,c-b\,c\,x}-24\,a^2\,b^4\,x^4\,\sqrt {a\,c-b\,c\,x}}{{\left (a\,c-b\,c\,x\right )}^2\,\left (3\,a^8\,x^3\,\left (a\,c-b\,c\,x\right )-6\,a^9\,c\,x^3\right )\,\sqrt {a+b\,x}} \] Input:
int(1/(x^4*(a*c - b*c*x)^(5/2)*(a + b*x)^(5/2)),x)
Output:
(a^6*(a*c - b*c*x)^(1/2) + 16*b^6*x^6*(a*c - b*c*x)^(1/2) + 6*a^4*b^2*x^2* (a*c - b*c*x)^(1/2) - 24*a^2*b^4*x^4*(a*c - b*c*x)^(1/2))/((a*c - b*c*x)^2 *(3*a^8*x^3*(a*c - b*c*x) - 6*a^9*c*x^3)*(a + b*x)^(1/2))
Time = 0.19 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.56 \[ \int \frac {1}{x^4 (a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=\frac {\sqrt {c}\, \left (-16 b^{6} x^{6}+24 a^{2} b^{4} x^{4}-6 a^{4} b^{2} x^{2}-a^{6}\right )}{3 \sqrt {b x +a}\, \sqrt {-b x +a}\, a^{8} c^{3} x^{3} \left (-b^{2} x^{2}+a^{2}\right )} \] Input:
int(1/x^4/(b*x+a)^(5/2)/(-b*c*x+a*c)^(5/2),x)
Output:
(sqrt(c)*( - a**6 - 6*a**4*b**2*x**2 + 24*a**2*b**4*x**4 - 16*b**6*x**6))/ (3*sqrt(a + b*x)*sqrt(a - b*x)*a**8*c**3*x**3*(a**2 - b**2*x**2))