Integrand size = 28, antiderivative size = 216 \[ \int \frac {1}{x^{3/2} (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\frac {1}{a^2 c \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {3 \sqrt {a+b x} \sqrt {a c-b c x}}{a^4 c^2 \sqrt {x}}-\frac {3 \sqrt {b} \sqrt {1-\frac {b^2 x^2}{a^2}} E\left (\left .\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )\right |-1\right )}{a^{5/2} c \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {3 \sqrt {b} \sqrt {1-\frac {b^2 x^2}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),-1\right )}{a^{5/2} c \sqrt {a+b x} \sqrt {a c-b c x}} \] Output:
1/a^2/c/x^(1/2)/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)-3*(b*x+a)^(1/2)*(-b*c*x+a *c)^(1/2)/a^4/c^2/x^(1/2)-3*b^(1/2)*(1-b^2*x^2/a^2)^(1/2)*EllipticE(b^(1/2 )*x^(1/2)/a^(1/2),I)/a^(5/2)/c/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)+3*b^(1/2)* (1-b^2*x^2/a^2)^(1/2)*EllipticF(b^(1/2)*x^(1/2)/a^(1/2),I)/a^(5/2)/c/(b*x+ a)^(1/2)/(-b*c*x+a*c)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 7.22 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.33 \[ \int \frac {1}{x^{3/2} (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=-\frac {2 \sqrt {1-\frac {b^2 x^2}{a^2}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {3}{2},\frac {3}{4},\frac {b^2 x^2}{a^2}\right )}{a^2 c \sqrt {x} \sqrt {c (a-b x)} \sqrt {a+b x}} \] Input:
Integrate[1/(x^(3/2)*(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2)),x]
Output:
(-2*Sqrt[1 - (b^2*x^2)/a^2]*Hypergeometric2F1[-1/4, 3/2, 3/4, (b^2*x^2)/a^ 2])/(a^2*c*Sqrt[x]*Sqrt[c*(a - b*x)]*Sqrt[a + b*x])
Time = 0.26 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.95, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {115, 27, 110, 27, 110, 27, 124, 27, 123}
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.
| \(\displaystyle \int \frac {1}{x^{3/2} (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 115 |
| \(\displaystyle -\frac {2 \int -\frac {3 b^2 c \sqrt {x}}{2 (a+b x)^{3/2} (a c-b c x)^{3/2}}dx}{a^2 c}-\frac {2}{a^2 c \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 b^2 \int \frac {\sqrt {x}}{(a+b x)^{3/2} (a c-b c x)^{3/2}}dx}{a^2}-\frac {2}{a^2 c \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}\) |
| \(\Big \downarrow \) 110 |
| \(\displaystyle \frac {3 b^2 \left (\frac {\int \frac {c \sqrt {a+b x}}{2 \sqrt {x} (a c-b c x)^{3/2}}dx}{a b c}-\frac {\sqrt {x}}{a b c \sqrt {a+b x} \sqrt {a c-b c x}}\right )}{a^2}-\frac {2}{a^2 c \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 b^2 \left (\frac {\int \frac {\sqrt {a+b x}}{\sqrt {x} (a c-b c x)^{3/2}}dx}{2 a b}-\frac {\sqrt {x}}{a b c \sqrt {a+b x} \sqrt {a c-b c x}}\right )}{a^2}-\frac {2}{a^2 c \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}\) |
| \(\Big \downarrow \) 110 |
| \(\displaystyle \frac {3 b^2 \left (\frac {\frac {2 \sqrt {x} \sqrt {a+b x}}{a c \sqrt {a c-b c x}}-\frac {2 \int \frac {b \sqrt {x}}{2 \sqrt {a+b x} \sqrt {a c-b c x}}dx}{a c}}{2 a b}-\frac {\sqrt {x}}{a b c \sqrt {a+b x} \sqrt {a c-b c x}}\right )}{a^2}-\frac {2}{a^2 c \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 b^2 \left (\frac {\frac {2 \sqrt {x} \sqrt {a+b x}}{a c \sqrt {a c-b c x}}-\frac {b \int \frac {\sqrt {x}}{\sqrt {a+b x} \sqrt {a c-b c x}}dx}{a c}}{2 a b}-\frac {\sqrt {x}}{a b c \sqrt {a+b x} \sqrt {a c-b c x}}\right )}{a^2}-\frac {2}{a^2 c \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}\) |
| \(\Big \downarrow \) 124 |
| \(\displaystyle \frac {3 b^2 \left (\frac {\frac {2 \sqrt {x} \sqrt {a+b x}}{a c \sqrt {a c-b c x}}-\frac {b \sqrt {x} \sqrt {\frac {a-b x}{a}} \int \frac {\sqrt {2} \sqrt {-\frac {b x}{a}}}{\sqrt {a+b x} \sqrt {1-\frac {b x}{a}}}dx}{\sqrt {2} a c \sqrt {-\frac {b x}{a}} \sqrt {a c-b c x}}}{2 a b}-\frac {\sqrt {x}}{a b c \sqrt {a+b x} \sqrt {a c-b c x}}\right )}{a^2}-\frac {2}{a^2 c \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 b^2 \left (\frac {\frac {2 \sqrt {x} \sqrt {a+b x}}{a c \sqrt {a c-b c x}}-\frac {b \sqrt {x} \sqrt {\frac {a-b x}{a}} \int \frac {\sqrt {-\frac {b x}{a}}}{\sqrt {a+b x} \sqrt {1-\frac {b x}{a}}}dx}{a c \sqrt {-\frac {b x}{a}} \sqrt {a c-b c x}}}{2 a b}-\frac {\sqrt {x}}{a b c \sqrt {a+b x} \sqrt {a c-b c x}}\right )}{a^2}-\frac {2}{a^2 c \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {3 b^2 \left (\frac {\frac {2 \sqrt {x} \sqrt {a+b x}}{a c \sqrt {a c-b c x}}-\frac {2 \sqrt {x} \sqrt {\frac {a-b x}{a}} E\left (\left .\arcsin \left (\frac {\sqrt {a+b x}}{\sqrt {2} \sqrt {a}}\right )\right |2\right )}{\sqrt {a} c \sqrt {-\frac {b x}{a}} \sqrt {a c-b c x}}}{2 a b}-\frac {\sqrt {x}}{a b c \sqrt {a+b x} \sqrt {a c-b c x}}\right )}{a^2}-\frac {2}{a^2 c \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}\) |
Input:
Int[1/(x^(3/2)*(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2)),x]
Output:
-2/(a^2*c*Sqrt[x]*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]) + (3*b^2*(-(Sqrt[x]/(a* b*c*Sqrt[a + b*x]*Sqrt[a*c - b*c*x])) + ((2*Sqrt[x]*Sqrt[a + b*x])/(a*c*Sq rt[a*c - b*c*x]) - (2*Sqrt[x]*Sqrt[(a - b*x)/a]*EllipticE[ArcSin[Sqrt[a + b*x]/(Sqrt[2]*Sqrt[a])], 2])/(Sqrt[a]*c*Sqrt[-((b*x)/a)]*Sqrt[a*c - b*c*x] ))/(2*a*b)))/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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[1/((m + 1)*(b*e - a*f)) Int[(a + b*x)^(m + 1) *(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && Gt Q[n, 0] && (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 )/((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] && LtQ[m, -1] && IntegersQ[2*m, 2 *n, 2*p]
Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ )]), x_] :> Simp[(2/b)*Rt[-(b*e - a*f)/d, 2]*EllipticE[ArcSin[Sqrt[a + b*x] /Rt[-(b*c - a*d)/d, 2]], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] && !L tQ[-(b*c - a*d)/d, 0] && !(SimplerQ[c + d*x, a + b*x] && GtQ[-d/(b*c - a*d ), 0] && GtQ[d/(d*e - c*f), 0] && !LtQ[(b*c - a*d)/b, 0])
Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ )]), x_] :> Simp[Sqrt[e + f*x]*(Sqrt[b*((c + d*x)/(b*c - a*d))]/(Sqrt[c + d *x]*Sqrt[b*((e + f*x)/(b*e - a*f))])) Int[Sqrt[b*(e/(b*e - a*f)) + b*f*(x /(b*e - a*f))]/(Sqrt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))] ), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !(GtQ[b/(b*c - a*d), 0] && Gt Q[b/(b*e - a*f), 0]) && !LtQ[-(b*c - a*d)/d, 0]
Time = 2.62 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.79
| method | result | size |
| default | \(\frac {\left (6 \sqrt {\frac {b x +a}{a}}\, \sqrt {2}\, \sqrt {\frac {-b x +a}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right ) a^{2}-3 \sqrt {\frac {b x +a}{a}}\, \sqrt {2}\, \sqrt {\frac {-b x +a}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right ) a^{2}+6 b^{2} x^{2}-4 a^{2}\right ) \sqrt {c \left (-b x +a \right )}\, \sqrt {b x +a}}{2 c^{2} \left (-b^{2} x^{2}+a^{2}\right ) a^{4} \sqrt {x}}\) | \(171\) |
| elliptic | \(\frac {\sqrt {c x \left (-b^{2} x^{2}+a^{2}\right )}\, \left (\frac {b^{2} x^{2}}{c \,a^{4} \sqrt {-\left (x^{2}-\frac {a^{2}}{b^{2}}\right ) b^{2} c x}}-\frac {2 \left (-b^{2} c \,x^{2}+a^{2} c \right )}{a^{4} c^{2} \sqrt {x \left (-b^{2} c \,x^{2}+a^{2} c \right )}}-\frac {3 b \sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {2 \left (x -\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {b x}{a}}\, \left (-\frac {2 a \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {a \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{2 a^{3} c \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}\right )}{\sqrt {x}\, \sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}}\) | \(234\) |
| risch | \(-\frac {2 \left (-b x +a \right ) \sqrt {b x +a}}{a^{4} \sqrt {x}\, \sqrt {-c \left (b x -a \right )}\, c}-\frac {b^{4} \left (\frac {a \sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {2 \left (x -\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {b x}{a}}\, \left (-\frac {2 a \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {a \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{b^{3} \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}+\frac {a^{2} \left (\frac {-b^{2} c \,x^{2}-a b c x}{a^{2} c b \sqrt {\left (x -\frac {a}{b}\right ) \left (-b^{2} c \,x^{2}-a b c x \right )}}-\frac {\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {2 \left (x -\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{2 b \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}+\frac {\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {2 \left (x -\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {b x}{a}}\, \left (-\frac {2 a \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {a \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{2 a \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}\right )}{2 b^{3}}+\frac {a^{2} \left (\frac {-b^{2} c \,x^{2}+a b c x}{a^{2} c b \sqrt {\left (x +\frac {a}{b}\right ) \left (-b^{2} c \,x^{2}+a b c x \right )}}+\frac {\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {2 \left (x -\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{2 b \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}+\frac {\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {2 \left (x -\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {b x}{a}}\, \left (-\frac {2 a \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {a \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{2 a \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}\right )}{2 b^{3}}\right ) \sqrt {-x \left (b x +a \right ) c \left (b x -a \right )}}{a^{4} \sqrt {x}\, \sqrt {b x +a}\, \sqrt {-c \left (b x -a \right )}\, c}\) | \(717\) |
Input:
int(1/x^(3/2)/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/2*(6*((b*x+a)/a)^(1/2)*2^(1/2)*((-b*x+a)/a)^(1/2)*(-b*x/a)^(1/2)*Ellipti cE(((b*x+a)/a)^(1/2),1/2*2^(1/2))*a^2-3*((b*x+a)/a)^(1/2)*2^(1/2)*((-b*x+a )/a)^(1/2)*(-b*x/a)^(1/2)*EllipticF(((b*x+a)/a)^(1/2),1/2*2^(1/2))*a^2+6*b ^2*x^2-4*a^2)/c^2*(c*(-b*x+a))^(1/2)*(b*x+a)^(1/2)/(-b^2*x^2+a^2)/a^4/x^(1 /2)
Time = 0.10 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.50 \[ \int \frac {1}{x^{3/2} (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=-\frac {{\left (3 \, b^{2} x^{2} - 2 \, a^{2}\right )} \sqrt {-b c x + a c} \sqrt {b x + a} \sqrt {x} + 3 \, {\left (b^{2} x^{3} - a^{2} x\right )} \sqrt {-b^{2} c} {\rm weierstrassZeta}\left (\frac {4 \, a^{2}}{b^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {4 \, a^{2}}{b^{2}}, 0, x\right )\right )}{a^{4} b^{2} c^{2} x^{3} - a^{6} c^{2} x} \] Input:
integrate(1/x^(3/2)/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2),x, algorithm="fricas" )
Output:
-((3*b^2*x^2 - 2*a^2)*sqrt(-b*c*x + a*c)*sqrt(b*x + a)*sqrt(x) + 3*(b^2*x^ 3 - a^2*x)*sqrt(-b^2*c)*weierstrassZeta(4*a^2/b^2, 0, weierstrassPInverse( 4*a^2/b^2, 0, x)))/(a^4*b^2*c^2*x^3 - a^6*c^2*x)
Timed out. \[ \int \frac {1}{x^{3/2} (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(1/x**(3/2)/(b*x+a)**(3/2)/(-b*c*x+a*c)**(3/2),x)
Output:
Timed out
\[ \int \frac {1}{x^{3/2} (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\int { \frac {1}{{\left (-b c x + a c\right )}^{\frac {3}{2}} {\left (b x + a\right )}^{\frac {3}{2}} x^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/x^(3/2)/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2),x, algorithm="maxima" )
Output:
integrate(1/((-b*c*x + a*c)^(3/2)*(b*x + a)^(3/2)*x^(3/2)), x)
Timed out. \[ \int \frac {1}{x^{3/2} (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(1/x^(3/2)/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {1}{x^{3/2} (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\int \frac {1}{x^{3/2}\,{\left (a\,c-b\,c\,x\right )}^{3/2}\,{\left (a+b\,x\right )}^{3/2}} \,d x \] Input:
int(1/(x^(3/2)*(a*c - b*c*x)^(3/2)*(a + b*x)^(3/2)),x)
Output:
int(1/(x^(3/2)*(a*c - b*c*x)^(3/2)*(a + b*x)^(3/2)), x)
\[ \int \frac {1}{x^{3/2} (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\frac {\sqrt {c}\, \left (\int \frac {\sqrt {x}\, \sqrt {b x +a}\, \sqrt {-b x +a}}{b^{4} x^{6}-2 a^{2} b^{2} x^{4}+a^{4} x^{2}}d x \right )}{c^{2}} \] Input:
int(1/x^(3/2)/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2),x)
Output:
(sqrt(c)*int((sqrt(x)*sqrt(a + b*x)*sqrt(a - b*x))/(a**4*x**2 - 2*a**2*b** 2*x**4 + b**4*x**6),x))/c**2