Integrand size = 28, antiderivative size = 180 \[ \int \frac {\sqrt {x}}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\frac {x^{3/2}}{a^2 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {\sqrt {1-\frac {b^2 x^2}{a^2}} E\left (\left .\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )\right |-1\right )}{\sqrt {a} b^{3/2} c \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {\sqrt {1-\frac {b^2 x^2}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),-1\right )}{\sqrt {a} b^{3/2} c \sqrt {a+b x} \sqrt {a c-b c x}} \] Output:
x^(3/2)/a^2/c/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)-(1-b^2*x^2/a^2)^(1/2)*Ellip ticE(b^(1/2)*x^(1/2)/a^(1/2),I)/a^(1/2)/b^(3/2)/c/(b*x+a)^(1/2)/(-b*c*x+a* c)^(1/2)+(1-b^2*x^2/a^2)^(1/2)*EllipticF(b^(1/2)*x^(1/2)/a^(1/2),I)/a^(1/2 )/b^(3/2)/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 = 5.63 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.41 \[ \int \frac {\sqrt {x}}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\frac {2 x^{3/2} \sqrt {1-\frac {b^2 x^2}{a^2}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},\frac {b^2 x^2}{a^2}\right )}{3 a^2 c \sqrt {c (a-b x)} \sqrt {a+b x}} \] Input:
Integrate[Sqrt[x]/((a + b*x)^(3/2)*(a*c - b*c*x)^(3/2)),x]
Output:
(2*x^(3/2)*Sqrt[1 - (b^2*x^2)/a^2]*Hypergeometric2F1[3/4, 3/2, 7/4, (b^2*x ^2)/a^2])/(3*a^2*c*Sqrt[c*(a - b*x)]*Sqrt[a + b*x])
Time = 0.24 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.89, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {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.
\(\displaystyle \int \frac {\sqrt {x}}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 110 |
\(\displaystyle \frac {\int \frac {c \sqrt {a+b x}}{2 \sqrt {x} (a c-b c x)^{3/2}}dx}{a b c}-\frac {\sqrt {x}}{a b c \sqrt {a+b x} \sqrt {a c-b c x}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {\sqrt {a+b x}}{\sqrt {x} (a c-b c x)^{3/2}}dx}{2 a b}-\frac {\sqrt {x}}{a b c \sqrt {a+b x} \sqrt {a c-b c x}}\) |
\(\Big \downarrow \) 110 |
\(\displaystyle \frac {\frac {2 \sqrt {x} \sqrt {a+b x}}{a c \sqrt {a c-b c x}}-\frac {2 \int \frac {b \sqrt {x}}{2 \sqrt {a+b x} \sqrt {a c-b c x}}dx}{a c}}{2 a b}-\frac {\sqrt {x}}{a b c \sqrt {a+b x} \sqrt {a c-b c x}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {2 \sqrt {x} \sqrt {a+b x}}{a c \sqrt {a c-b c x}}-\frac {b \int \frac {\sqrt {x}}{\sqrt {a+b x} \sqrt {a c-b c x}}dx}{a c}}{2 a b}-\frac {\sqrt {x}}{a b c \sqrt {a+b x} \sqrt {a c-b c x}}\) |
\(\Big \downarrow \) 124 |
\(\displaystyle \frac {\frac {2 \sqrt {x} \sqrt {a+b x}}{a c \sqrt {a c-b c x}}-\frac {b \sqrt {x} \sqrt {\frac {a-b x}{a}} \int \frac {\sqrt {2} \sqrt {-\frac {b x}{a}}}{\sqrt {a+b x} \sqrt {1-\frac {b x}{a}}}dx}{\sqrt {2} a c \sqrt {-\frac {b x}{a}} \sqrt {a c-b c x}}}{2 a b}-\frac {\sqrt {x}}{a b c \sqrt {a+b x} \sqrt {a c-b c x}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {2 \sqrt {x} \sqrt {a+b x}}{a c \sqrt {a c-b c x}}-\frac {b \sqrt {x} \sqrt {\frac {a-b x}{a}} \int \frac {\sqrt {-\frac {b x}{a}}}{\sqrt {a+b x} \sqrt {1-\frac {b x}{a}}}dx}{a c \sqrt {-\frac {b x}{a}} \sqrt {a c-b c x}}}{2 a b}-\frac {\sqrt {x}}{a b c \sqrt {a+b x} \sqrt {a c-b c x}}\) |
\(\Big \downarrow \) 123 |
\(\displaystyle \frac {\frac {2 \sqrt {x} \sqrt {a+b x}}{a c \sqrt {a c-b c x}}-\frac {2 \sqrt {x} \sqrt {\frac {a-b x}{a}} E\left (\left .\arcsin \left (\frac {\sqrt {a+b x}}{\sqrt {2} \sqrt {a}}\right )\right |2\right )}{\sqrt {a} c \sqrt {-\frac {b x}{a}} \sqrt {a c-b c x}}}{2 a b}-\frac {\sqrt {x}}{a b c \sqrt {a+b x} \sqrt {a c-b c x}}\) |
Input:
Int[Sqrt[x]/((a + b*x)^(3/2)*(a*c - b*c*x)^(3/2)),x]
Output:
-(Sqrt[x]/(a*b*c*Sqrt[a + b*x]*Sqrt[a*c - b*c*x])) + ((2*Sqrt[x]*Sqrt[a + b*x])/(a*c*Sqrt[a*c - b*c*x]) - (2*Sqrt[x]*Sqrt[(a - b*x)/a]*EllipticE[Arc Sin[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)
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[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 = 1.36 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.93
method | result | size |
default | \(-\frac {\sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}\, \left (\sqrt {\frac {b x +a}{a}}\, \sqrt {2}\, \sqrt {\frac {-b x +a}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right ) a^{2}-2 \sqrt {\frac {b x +a}{a}}\, \sqrt {2}\, \sqrt {\frac {-b x +a}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right ) a^{2}-2 b^{2} x^{2}\right )}{2 \sqrt {x}\, a^{2} c^{2} \left (-b^{2} x^{2}+a^{2}\right ) b^{2}}\) | \(168\) |
elliptic | \(\frac {\sqrt {c x \left (-b^{2} x^{2}+a^{2}\right )}\, \left (\frac {x^{2}}{c \,a^{2} \sqrt {-\left (x^{2}-\frac {a^{2}}{b^{2}}\right ) b^{2} c x}}-\frac {\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {2 \left (x -\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {b x}{a}}\, \left (-\frac {2 a \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {a \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{2 a c b \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}\right )}{\sqrt {x}\, \sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}}\) | \(191\) |
Input:
int(x^(1/2)/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/2*(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^2-2*(( b*x+a)/a)^(1/2)*2^(1/2)*((-b*x+a)/a)^(1/2)*(-b*x/a)^(1/2)*EllipticE(((b*x+ a)/a)^(1/2),1/2*2^(1/2))*a^2-2*b^2*x^2)/x^(1/2)/a^2/c^2/(-b^2*x^2+a^2)/b^2
Time = 0.09 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.54 \[ \int \frac {\sqrt {x}}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=-\frac {\sqrt {-b c x + a c} \sqrt {b x + a} b^{2} x^{\frac {3}{2}} + {\left (b^{2} x^{2} - a^{2}\right )} \sqrt {-b^{2} c} {\rm weierstrassZeta}\left (\frac {4 \, a^{2}}{b^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {4 \, a^{2}}{b^{2}}, 0, x\right )\right )}{a^{2} b^{4} c^{2} x^{2} - a^{4} b^{2} c^{2}} \] Input:
integrate(x^(1/2)/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2),x, algorithm="fricas")
Output:
-(sqrt(-b*c*x + a*c)*sqrt(b*x + a)*b^2*x^(3/2) + (b^2*x^2 - a^2)*sqrt(-b^2 *c)*weierstrassZeta(4*a^2/b^2, 0, weierstrassPInverse(4*a^2/b^2, 0, x)))/( a^2*b^4*c^2*x^2 - a^4*b^2*c^2)
Time = 30.31 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.62 \[ \int \frac {\sqrt {x}}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=- \frac {i {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {1}{2}, 1, 1 & \frac {1}{4}, \frac {5}{4}, \frac {7}{4} \\\frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4}, \frac {7}{4} & 0 \end {matrix} \middle | {\frac {a^{2}}{b^{2} x^{2}}} \right )}}{2 \pi ^{\frac {3}{2}} a^{\frac {3}{2}} b^{\frac {3}{2}} c^{\frac {3}{2}}} + \frac {i {G_{6, 6}^{3, 5}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2} & 1 \\0, \frac {1}{2}, 0 & - \frac {3}{4}, - \frac {1}{4}, \frac {3}{4} \end {matrix} \middle | {\frac {a^{2} e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{2 \pi ^{\frac {3}{2}} a^{\frac {3}{2}} b^{\frac {3}{2}} c^{\frac {3}{2}}} \] Input:
integrate(x**(1/2)/(b*x+a)**(3/2)/(-b*c*x+a*c)**(3/2),x)
Output:
-I*meijerg(((1/2, 1, 1), (1/4, 5/4, 7/4)), ((1/2, 3/4, 1, 5/4, 7/4), (0,)) , a**2/(b**2*x**2))/(2*pi**(3/2)*a**(3/2)*b**(3/2)*c**(3/2)) + I*meijerg(( (-3/4, -1/4, 0, 1/4, 1/2), (1,)), ((0, 1/2, 0), (-3/4, -1/4, 3/4)), a**2*e xp_polar(-2*I*pi)/(b**2*x**2))/(2*pi**(3/2)*a**(3/2)*b**(3/2)*c**(3/2))
\[ \int \frac {\sqrt {x}}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\int { \frac {\sqrt {x}}{{\left (-b c x + a c\right )}^{\frac {3}{2}} {\left (b x + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^(1/2)/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2),x, algorithm="maxima")
Output:
integrate(sqrt(x)/((-b*c*x + a*c)^(3/2)*(b*x + a)^(3/2)), x)
Timed out. \[ \int \frac {\sqrt {x}}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(x^(1/2)/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {\sqrt {x}}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\int \frac {\sqrt {x}}{{\left (a\,c-b\,c\,x\right )}^{3/2}\,{\left (a+b\,x\right )}^{3/2}} \,d x \] Input:
int(x^(1/2)/((a*c - b*c*x)^(3/2)*(a + b*x)^(3/2)),x)
Output:
int(x^(1/2)/((a*c - b*c*x)^(3/2)*(a + b*x)^(3/2)), x)
\[ \int \frac {\sqrt {x}}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\frac {\sqrt {c}\, \left (\int \frac {\sqrt {x}\, \sqrt {b x +a}\, \sqrt {-b x +a}}{b^{4} x^{4}-2 a^{2} b^{2} x^{2}+a^{4}}d x \right )}{c^{2}} \] Input:
int(x^(1/2)/(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 - 2*a**2*b**2*x** 2 + b**4*x**4),x))/c**2