Integrand size = 28, antiderivative size = 105 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^{5/2}} \, dx=-\frac {2 \sqrt {a+b x} \sqrt {a c-b c x}}{3 x^{3/2}}-\frac {4 \sqrt {a} b^{3/2} c \sqrt {1-\frac {b^2 x^2}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),-1\right )}{3 \sqrt {a+b x} \sqrt {a c-b c x}} \] Output:
-2/3*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/x^(3/2)-4/3*a^(1/2)*b^(3/2)*c*(1-b^2 *x^2/a^2)^(1/2)*EllipticF(b^(1/2)*x^(1/2)/a^(1/2),I)/(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 = 1.32 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.64 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^{5/2}} \, dx=-\frac {2 \sqrt {c (a-b x)} \sqrt {a+b x} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},-\frac {1}{2},\frac {1}{4},\frac {b^2 x^2}{a^2}\right )}{3 x^{3/2} \sqrt {1-\frac {b^2 x^2}{a^2}}} \] Input:
Integrate[(Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/x^(5/2),x]
Output:
(-2*Sqrt[c*(a - b*x)]*Sqrt[a + b*x]*Hypergeometric2F1[-3/4, -1/2, 1/4, (b^ 2*x^2)/a^2])/(3*x^(3/2)*Sqrt[1 - (b^2*x^2)/a^2])
Time = 0.20 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {108, 25, 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.
| \(\displaystyle \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^{5/2}} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {2}{3} \int -\frac {b^2 c}{\sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}dx-\frac {2 \sqrt {a+b x} \sqrt {a c-b c x}}{3 x^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2}{3} \int \frac {b^2 c}{\sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}dx-\frac {2 \sqrt {a+b x} \sqrt {a c-b c x}}{3 x^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2}{3} b^2 c \int \frac {1}{\sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}dx-\frac {2 \sqrt {a+b x} \sqrt {a c-b c x}}{3 x^{3/2}}\) |
| \(\Big \downarrow \) 127 |
| \(\displaystyle -\frac {2 b^2 c \sqrt {1-\frac {b x}{a}} \sqrt {\frac {b x}{a}+1} \int \frac {1}{\sqrt {x} \sqrt {1-\frac {b x}{a}} \sqrt {\frac {b x}{a}+1}}dx}{3 \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {2 \sqrt {a+b x} \sqrt {a c-b c x}}{3 x^{3/2}}\) |
| \(\Big \downarrow \) 126 |
| \(\displaystyle -\frac {4 \sqrt {a} b^{3/2} c \sqrt {1-\frac {b x}{a}} \sqrt {\frac {b x}{a}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),-1\right )}{3 \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {2 \sqrt {a+b x} \sqrt {a c-b c x}}{3 x^{3/2}}\) |
Input:
Int[(Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/x^(5/2),x]
Output:
(-2*Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/(3*x^(3/2)) - (4*Sqrt[a]*b^(3/2)*c*Sq rt[1 - (b*x)/a]*Sqrt[1 + (b*x)/a]*EllipticF[ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[ a]], -1])/(3*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[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2 *n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
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 = 0.39 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00
| method | result | size |
| default | \(-\frac {2 \sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}\, \left (\sqrt {\frac {b x +a}{a}}\, \sqrt {2}\, \sqrt {\frac {-b x +a}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right ) a b x -b^{2} x^{2}+a^{2}\right )}{3 x^{\frac {3}{2}} \left (-b^{2} x^{2}+a^{2}\right )}\) | \(105\) |
| risch | \(-\frac {2 \left (-b x +a \right ) \sqrt {b x +a}\, c}{3 x^{\frac {3}{2}} \sqrt {-c \left (b x -a \right )}}-\frac {2 b a \sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {2 \left (x -\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-x \left (b x +a \right ) c \left (b x -a \right )}\, c}{3 \sqrt {-b^{2} c \,x^{3}+a^{2} c x}\, \sqrt {x}\, \sqrt {b x +a}\, \sqrt {-c \left (b x -a \right )}}\) | \(155\) |
| elliptic | \(\frac {\sqrt {c \left (-b x +a \right )}\, \sqrt {c x \left (-b^{2} x^{2}+a^{2}\right )}\, \left (-\frac {2 \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}{3 x^{2}}-\frac {2 b c a \sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {2 \left (x -\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{3 \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}\right )}{\sqrt {x}\, \sqrt {b x +a}\, c \left (-b x +a \right )}\) | \(156\) |
Input:
int((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/x^(5/2),x,method=_RETURNVERBOSE)
Output:
-2/3*(b*x+a)^(1/2)*(c*(-b*x+a))^(1/2)*(((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-b^ 2*x^2+a^2)/x^(3/2)/(-b^2*x^2+a^2)
Time = 0.08 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.50 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^{5/2}} \, dx=\frac {2 \, {\left (2 \, \sqrt {-b^{2} c} x^{2} {\rm weierstrassPInverse}\left (\frac {4 \, a^{2}}{b^{2}}, 0, x\right ) - \sqrt {-b c x + a c} \sqrt {b x + a} \sqrt {x}\right )}}{3 \, x^{2}} \] Input:
integrate((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/x^(5/2),x, algorithm="fricas")
Output:
2/3*(2*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))/x^2
\[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^{5/2}} \, dx=\int \frac {\sqrt {- c \left (- a + b x\right )} \sqrt {a + b x}}{x^{\frac {5}{2}}}\, dx \] Input:
integrate((b*x+a)**(1/2)*(-b*c*x+a*c)**(1/2)/x**(5/2),x)
Output:
Integral(sqrt(-c*(-a + b*x))*sqrt(a + b*x)/x**(5/2), x)
\[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^{5/2}} \, dx=\int { \frac {\sqrt {-b c x + a c} \sqrt {b x + a}}{x^{\frac {5}{2}}} \,d x } \] Input:
integrate((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/x^(5/2),x, algorithm="maxima")
Output:
integrate(sqrt(-b*c*x + a*c)*sqrt(b*x + a)/x^(5/2), x)
Timed out. \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/x^(5/2),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^{5/2}} \, dx=\int \frac {\sqrt {a\,c-b\,c\,x}\,\sqrt {a+b\,x}}{x^{5/2}} \,d x \] Input:
int(((a*c - b*c*x)^(1/2)*(a + b*x)^(1/2))/x^(5/2),x)
Output:
int(((a*c - b*c*x)^(1/2)*(a + b*x)^(1/2))/x^(5/2), x)
\[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^{5/2}} \, dx=-\frac {2 \sqrt {c}\, \left (\sqrt {b x +a}\, \sqrt {-b x +a}+\sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {b x +a}\, \sqrt {-b x +a}}{-b^{2} x^{5}+a^{2} x^{3}}d x \right ) a^{2} x \right )}{\sqrt {x}\, x} \] Input:
int((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/x^(5/2),x)
Output:
( - 2*sqrt(c)*(sqrt(a + b*x)*sqrt(a - b*x) + sqrt(x)*int((sqrt(x)*sqrt(a + b*x)*sqrt(a - b*x))/(a**2*x**3 - b**2*x**5),x)*a**2*x))/(sqrt(x)*x)