Integrand size = 28, antiderivative size = 110 \[ \int \frac {1}{x^{5/2} \sqrt {a+b x} \sqrt {a c-b c x}} \, dx=-\frac {2 \sqrt {a+b x} \sqrt {a c-b c x}}{3 a^2 c x^{3/2}}+\frac {2 b^{3/2} \sqrt {1-\frac {b^2 x^2}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),-1\right )}{3 a^{3/2} \sqrt {a+b x} \sqrt {a c-b c x}} \] Output:
-2/3*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/a^2/c/x^(3/2)+2/3*b^(3/2)*(1-b^2*x^2 /a^2)^(1/2)*EllipticF(b^(1/2)*x^(1/2)/a^(1/2),I)/a^(3/2)/(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 = 1.72 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.61 \[ \int \frac {1}{x^{5/2} \sqrt {a+b x} \sqrt {a c-b c x}} \, dx=-\frac {2 \sqrt {1-\frac {b^2 x^2}{a^2}} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {1}{2},\frac {1}{4},\frac {b^2 x^2}{a^2}\right )}{3 x^{3/2} \sqrt {c (a-b x)} \sqrt {a+b x}} \] Input:
Integrate[1/(x^(5/2)*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]),x]
Output:
(-2*Sqrt[1 - (b^2*x^2)/a^2]*Hypergeometric2F1[-3/4, 1/2, 1/4, (b^2*x^2)/a^ 2])/(3*x^(3/2)*Sqrt[c*(a - b*x)]*Sqrt[a + b*x])
Time = 0.19 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.07, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {115, 27, 127, 126}
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.
\(\displaystyle \int \frac {1}{x^{5/2} \sqrt {a+b x} \sqrt {a c-b c x}} \, dx\) |
\(\Big \downarrow \) 115 |
\(\displaystyle -\frac {2 \int -\frac {b^2 c}{2 \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}dx}{3 a^2 c}-\frac {2 \sqrt {a+b x} \sqrt {a c-b c x}}{3 a^2 c x^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {b^2 \int \frac {1}{\sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}dx}{3 a^2}-\frac {2 \sqrt {a+b x} \sqrt {a c-b c x}}{3 a^2 c x^{3/2}}\) |
\(\Big \downarrow \) 127 |
\(\displaystyle \frac {b^2 \sqrt {1-\frac {b x}{a}} \sqrt {\frac {b x}{a}+1} \int \frac {1}{\sqrt {x} \sqrt {1-\frac {b x}{a}} \sqrt {\frac {b x}{a}+1}}dx}{3 a^2 \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {2 \sqrt {a+b x} \sqrt {a c-b c x}}{3 a^2 c x^{3/2}}\) |
\(\Big \downarrow \) 126 |
\(\displaystyle \frac {2 b^{3/2} \sqrt {1-\frac {b x}{a}} \sqrt {\frac {b x}{a}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),-1\right )}{3 a^{3/2} \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {2 \sqrt {a+b x} \sqrt {a c-b c x}}{3 a^2 c x^{3/2}}\) |
Input:
Int[1/(x^(5/2)*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]),x]
Output:
(-2*Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/(3*a^2*c*x^(3/2)) + (2*b^(3/2)*Sqrt[1 - (b*x)/a]*Sqrt[1 + (b*x)/a]*EllipticF[ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[a]], -1])/(3*a^(3/2)*Sqrt[a + b*x]*Sqrt[a*c - b*c*x])
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[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[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x _] :> Simp[(2/(b*Sqrt[e]))*Rt[-b/d, 2]*EllipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]* Rt[-b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[c, 0] & & GtQ[e, 0] && (PosQ[-b/d] || NegQ[-b/f])
Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x _] :> Simp[Sqrt[1 + d*(x/c)]*(Sqrt[1 + f*(x/e)]/(Sqrt[c + d*x]*Sqrt[e + f*x ])) Int[1/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]*Sqrt[1 + f*(x/e)]), x], x] /; Free Q[{b, c, d, e, f}, x] && !(GtQ[c, 0] && GtQ[e, 0])
Time = 1.34 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.03
method | result | size |
default | \(\frac {\sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}\, \left (\sqrt {\frac {b x +a}{a}}\, \sqrt {2}\, \sqrt {\frac {-b x +a}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right ) a b x +2 b^{2} x^{2}-2 a^{2}\right )}{3 c \,x^{\frac {3}{2}} a^{2} \left (-b^{2} x^{2}+a^{2}\right )}\) | \(113\) |
elliptic | \(\frac {\sqrt {c x \left (-b^{2} x^{2}+a^{2}\right )}\, \left (-\frac {2 \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}{3 a^{2} c \,x^{2}}+\frac {b \sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {2 \left (x -\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{3 a \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}\right )}{\sqrt {x}\, \sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}}\) | \(152\) |
risch | \(-\frac {2 \left (-b x +a \right ) \sqrt {b x +a}}{3 a^{2} x^{\frac {3}{2}} \sqrt {-c \left (b x -a \right )}}+\frac {b \sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {2 \left (x -\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-x \left (b x +a \right ) c \left (b x -a \right )}}{3 a \sqrt {-b^{2} c \,x^{3}+a^{2} c x}\, \sqrt {x}\, \sqrt {b x +a}\, \sqrt {-c \left (b x -a \right )}}\) | \(158\) |
Input:
int(1/x^(5/2)/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/3*(b*x+a)^(1/2)*(c*(-b*x+a))^(1/2)/c*(((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*b*x+2 *b^2*x^2-2*a^2)/x^(3/2)/a^2/(-b^2*x^2+a^2)
Time = 0.09 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.52 \[ \int \frac {1}{x^{5/2} \sqrt {a+b x} \sqrt {a c-b c x}} \, dx=-\frac {2 \, {\left (\sqrt {-b^{2} c} x^{2} {\rm weierstrassPInverse}\left (\frac {4 \, a^{2}}{b^{2}}, 0, x\right ) + \sqrt {-b c x + a c} \sqrt {b x + a} \sqrt {x}\right )}}{3 \, a^{2} c x^{2}} \] Input:
integrate(1/x^(5/2)/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2),x, algorithm="fricas" )
Output:
-2/3*(sqrt(-b^2*c)*x^2*weierstrassPInverse(4*a^2/b^2, 0, x) + sqrt(-b*c*x + a*c)*sqrt(b*x + a)*sqrt(x))/(a^2*c*x^2)
Timed out. \[ \int \frac {1}{x^{5/2} \sqrt {a+b x} \sqrt {a c-b c x}} \, dx=\text {Timed out} \] Input:
integrate(1/x**(5/2)/(b*x+a)**(1/2)/(-b*c*x+a*c)**(1/2),x)
Output:
Timed out
\[ \int \frac {1}{x^{5/2} \sqrt {a+b x} \sqrt {a c-b c x}} \, dx=\int { \frac {1}{\sqrt {-b c x + a c} \sqrt {b x + a} x^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/x^(5/2)/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2),x, algorithm="maxima" )
Output:
integrate(1/(sqrt(-b*c*x + a*c)*sqrt(b*x + a)*x^(5/2)), x)
Timed out. \[ \int \frac {1}{x^{5/2} \sqrt {a+b x} \sqrt {a c-b c x}} \, dx=\text {Timed out} \] Input:
integrate(1/x^(5/2)/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {1}{x^{5/2} \sqrt {a+b x} \sqrt {a c-b c x}} \, dx=\int \frac {1}{x^{5/2}\,\sqrt {a\,c-b\,c\,x}\,\sqrt {a+b\,x}} \,d x \] Input:
int(1/(x^(5/2)*(a*c - b*c*x)^(1/2)*(a + b*x)^(1/2)),x)
Output:
int(1/(x^(5/2)*(a*c - b*c*x)^(1/2)*(a + b*x)^(1/2)), x)
\[ \int \frac {1}{x^{5/2} \sqrt {a+b x} \sqrt {a c-b c x}} \, dx=\frac {\sqrt {c}\, \left (\int \frac {\sqrt {x}\, \sqrt {b x +a}\, \sqrt {-b x +a}}{-b^{2} x^{5}+a^{2} x^{3}}d x \right )}{c} \] Input:
int(1/x^(5/2)/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2),x)
Output:
(sqrt(c)*int((sqrt(x)*sqrt(a + b*x)*sqrt(a - b*x))/(a**2*x**3 - b**2*x**5) ,x))/c