Integrand size = 26, antiderivative size = 110 \[ \int \frac {1}{x^4 (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=-\frac {1}{3 a^2 c x^3 \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {4 b^2}{3 a^4 c x \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {8 b^4 x}{3 a^6 c \sqrt {a+b x} \sqrt {a c-b c x}} \] Output:
-1/3/a^2/c/x^3/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)-4/3*b^2/a^4/c/x/(b*x+a)^(1 /2)/(-b*c*x+a*c)^(1/2)+8/3*b^4*x/a^6/c/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)
Time = 0.09 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.54 \[ \int \frac {1}{x^4 (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\frac {-a^4-4 a^2 b^2 x^2+8 b^4 x^4}{3 a^6 c x^3 \sqrt {c (a-b x)} \sqrt {a+b x}} \] Input:
Integrate[1/(x^4*(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2)),x]
Output:
(-a^4 - 4*a^2*b^2*x^2 + 8*b^4*x^4)/(3*a^6*c*x^3*Sqrt[c*(a - b*x)]*Sqrt[a + b*x])
Time = 0.19 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {114, 27, 114, 27, 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.
\(\displaystyle \int \frac {1}{x^4 (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 114 |
\(\displaystyle -\frac {\int -\frac {4 b^2 c}{x^2 (a+b x)^{3/2} (a c-b c x)^{3/2}}dx}{3 a^2 c}-\frac {1}{3 a^2 c x^3 \sqrt {a+b x} \sqrt {a c-b c x}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {4 b^2 \int \frac {1}{x^2 (a+b x)^{3/2} (a c-b c x)^{3/2}}dx}{3 a^2}-\frac {1}{3 a^2 c x^3 \sqrt {a+b x} \sqrt {a c-b c x}}\) |
\(\Big \downarrow \) 114 |
\(\displaystyle \frac {4 b^2 \left (-\frac {\int -\frac {2 b^2 c}{(a+b x)^{3/2} (a c-b c x)^{3/2}}dx}{a^2 c}-\frac {1}{a^2 c x \sqrt {a+b x} \sqrt {a c-b c x}}\right )}{3 a^2}-\frac {1}{3 a^2 c x^3 \sqrt {a+b x} \sqrt {a c-b c x}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {4 b^2 \left (\frac {2 b^2 \int \frac {1}{(a+b x)^{3/2} (a c-b c x)^{3/2}}dx}{a^2}-\frac {1}{a^2 c x \sqrt {a+b x} \sqrt {a c-b c x}}\right )}{3 a^2}-\frac {1}{3 a^2 c x^3 \sqrt {a+b x} \sqrt {a c-b c x}}\) |
\(\Big \downarrow \) 41 |
\(\displaystyle \frac {4 b^2 \left (\frac {2 b^2 x}{a^4 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {1}{a^2 c x \sqrt {a+b x} \sqrt {a c-b c x}}\right )}{3 a^2}-\frac {1}{3 a^2 c x^3 \sqrt {a+b x} \sqrt {a c-b c x}}\) |
Input:
Int[1/(x^4*(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2)),x]
Output:
-1/3*1/(a^2*c*x^3*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]) + (4*b^2*(-(1/(a^2*c*x* Sqrt[a + b*x]*Sqrt[a*c - b*c*x])) + (2*b^2*x)/(a^4*c*Sqrt[a + b*x]*Sqrt[a* c - b*c*x])))/(3*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_))^(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 = 0.30 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.51
method | result | size |
gosper | \(-\frac {\left (-b x +a \right ) \left (-8 b^{4} x^{4}+4 a^{2} b^{2} x^{2}+a^{4}\right )}{3 x^{3} \sqrt {b x +a}\, a^{6} \left (-b c x +a c \right )^{\frac {3}{2}}}\) | \(56\) |
orering | \(-\frac {\left (-b x +a \right ) \left (-8 b^{4} x^{4}+4 a^{2} b^{2} x^{2}+a^{4}\right )}{3 x^{3} \sqrt {b x +a}\, a^{6} \left (-b c x +a c \right )^{\frac {3}{2}}}\) | \(56\) |
default | \(-\frac {\sqrt {c \left (-b x +a \right )}\, \left (-8 b^{4} x^{4}+4 a^{2} b^{2} x^{2}+a^{4}\right )}{3 c^{2} a^{6} \left (-b x +a \right ) x^{3} \sqrt {b x +a}}\) | \(60\) |
risch | \(-\frac {\left (-b x +a \right ) \sqrt {b x +a}\, \left (5 b^{2} x^{2}+a^{2}\right )}{3 a^{6} x^{3} \sqrt {-c \left (b x -a \right )}\, c}+\frac {b^{4} x \sqrt {-\left (b x +a \right ) c \left (b x -a \right )}}{\sqrt {\left (b x +a \right ) c \left (-b x +a \right )}\, a^{6} \sqrt {b x +a}\, \sqrt {-c \left (b x -a \right )}\, c}\) | \(112\) |
Input:
int(1/x^4/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/3*(-b*x+a)*(-8*b^4*x^4+4*a^2*b^2*x^2+a^4)/x^3/(b*x+a)^(1/2)/a^6/(-b*c*x +a*c)^(3/2)
Time = 0.09 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.65 \[ \int \frac {1}{x^4 (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=-\frac {{\left (8 \, b^{4} x^{4} - 4 \, a^{2} b^{2} x^{2} - a^{4}\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{3 \, {\left (a^{6} b^{2} c^{2} x^{5} - a^{8} c^{2} x^{3}\right )}} \] Input:
integrate(1/x^4/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2),x, algorithm="fricas")
Output:
-1/3*(8*b^4*x^4 - 4*a^2*b^2*x^2 - a^4)*sqrt(-b*c*x + a*c)*sqrt(b*x + a)/(a ^6*b^2*c^2*x^5 - a^8*c^2*x^3)
Result contains complex when optimal does not.
Time = 10.45 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x^4 (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=- \frac {i b^{3} {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {11}{4}, \frac {13}{4}, 1 & \frac {5}{2}, \frac {7}{2}, 4 \\\frac {11}{4}, 3, \frac {13}{4}, \frac {7}{2}, 4 & 0 \end {matrix} \middle | {\frac {a^{2}}{b^{2} x^{2}}} \right )}}{2 \pi ^{\frac {3}{2}} a^{6} c^{\frac {3}{2}}} + \frac {b^{3} {G_{6, 6}^{2, 6}\left (\begin {matrix} \frac {3}{2}, 2, \frac {9}{4}, \frac {5}{2}, \frac {11}{4}, 1 & \\\frac {9}{4}, \frac {11}{4} & \frac {3}{2}, 2, 3, 0 \end {matrix} \middle | {\frac {a^{2} e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{2 \pi ^{\frac {3}{2}} a^{6} c^{\frac {3}{2}}} \] Input:
integrate(1/x**4/(b*x+a)**(3/2)/(-b*c*x+a*c)**(3/2),x)
Output:
-I*b**3*meijerg(((11/4, 13/4, 1), (5/2, 7/2, 4)), ((11/4, 3, 13/4, 7/2, 4) , (0,)), a**2/(b**2*x**2))/(2*pi**(3/2)*a**6*c**(3/2)) + b**3*meijerg(((3/ 2, 2, 9/4, 5/2, 11/4, 1), ()), ((9/4, 11/4), (3/2, 2, 3, 0)), a**2*exp_pol ar(-2*I*pi)/(b**2*x**2))/(2*pi**(3/2)*a**6*c**(3/2))
Time = 0.04 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.81 \[ \int \frac {1}{x^4 (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\frac {8 \, b^{4} x}{3 \, \sqrt {-b^{2} c x^{2} + a^{2} c} a^{6} c} - \frac {4 \, b^{2}}{3 \, \sqrt {-b^{2} c x^{2} + a^{2} c} a^{4} c x} - \frac {1}{3 \, \sqrt {-b^{2} c x^{2} + a^{2} c} a^{2} c x^{3}} \] Input:
integrate(1/x^4/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2),x, algorithm="maxima")
Output:
8/3*b^4*x/(sqrt(-b^2*c*x^2 + a^2*c)*a^6*c) - 4/3*b^2/(sqrt(-b^2*c*x^2 + a^ 2*c)*a^4*c*x) - 1/3/(sqrt(-b^2*c*x^2 + a^2*c)*a^2*c*x^3)
Leaf count of result is larger than twice the leaf count of optimal. 249 vs. \(2 (92) = 184\).
Time = 0.20 (sec) , antiderivative size = 249, normalized size of antiderivative = 2.26 \[ \int \frac {1}{x^4 (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=-\frac {1}{6} \, b^{3} {\left (\frac {16 \, {\left (3 \, {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{8} \sqrt {-c} + 48 \, a^{2} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{4} \sqrt {-c} c^{2} + 80 \, a^{4} \sqrt {-c} c^{4}\right )}}{{\left ({\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{4} + 4 \, a^{2} c^{2}\right )}^{3} a^{4}} + \frac {12}{{\left ({\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2} - 2 \, a c\right )} a^{5} \sqrt {-c}} + \frac {3 \, \sqrt {-{\left (b x + a\right )} c + 2 \, a c} \sqrt {b x + a}}{{\left ({\left (b x + a\right )} c - 2 \, a c\right )} a^{6} c}\right )} \] Input:
integrate(1/x^4/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2),x, algorithm="giac")
Output:
-1/6*b^3*(16*(3*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^8*sq rt(-c) + 48*a^2*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^4*sq rt(-c)*c^2 + 80*a^4*sqrt(-c)*c^4)/(((sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^4 + 4*a^2*c^2)^3*a^4) + 12/(((sqrt(b*x + a)*sqrt(-c) - sqr t(-(b*x + a)*c + 2*a*c))^2 - 2*a*c)*a^5*sqrt(-c)) + 3*sqrt(-(b*x + a)*c + 2*a*c)*sqrt(b*x + a)/(((b*x + a)*c - 2*a*c)*a^6*c))
Time = 0.73 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.47 \[ \int \frac {1}{x^4 (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=-\frac {a^4+4\,a^2\,b^2\,x^2-8\,b^4\,x^4}{3\,a^6\,c\,x^3\,\sqrt {a\,c-b\,c\,x}\,\sqrt {a+b\,x}} \] Input:
int(1/(x^4*(a*c - b*c*x)^(3/2)*(a + b*x)^(3/2)),x)
Output:
-(a^4 - 8*b^4*x^4 + 4*a^2*b^2*x^2)/(3*a^6*c*x^3*(a*c - b*c*x)^(1/2)*(a + b *x)^(1/2))
Time = 0.18 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.50 \[ \int \frac {1}{x^4 (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\frac {\sqrt {c}\, \left (8 b^{4} x^{4}-4 a^{2} b^{2} x^{2}-a^{4}\right )}{3 \sqrt {b x +a}\, \sqrt {-b x +a}\, a^{6} c^{2} x^{3}} \] Input:
int(1/x^4/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2),x)
Output:
(sqrt(c)*( - a**4 - 4*a**2*b**2*x**2 + 8*b**4*x**4))/(3*sqrt(a + b*x)*sqrt (a - b*x)*a**6*c**2*x**3)