Integrand size = 28, antiderivative size = 107 \[ \int \frac {1}{\sqrt {x} (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\frac {\sqrt {x}}{a^2 c \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {\sqrt {1-\frac {b^2 x^2}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),-1\right )}{a^{3/2} \sqrt {b} c \sqrt {a+b x} \sqrt {a c-b c x}} \] Output:
x^(1/2)/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 ticF(b^(1/2)*x^(1/2)/a^(1/2),I)/a^(3/2)/b^(1/2)/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 = 5.73 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.68 \[ \int \frac {1}{\sqrt {x} (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\frac {\sqrt {x} \left (1+\sqrt {1-\frac {b^2 x^2}{a^2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {b^2 x^2}{a^2}\right )\right )}{a^2 c \sqrt {c (a-b x)} \sqrt {a+b x}} \] Input:
Integrate[1/(Sqrt[x]*(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2)),x]
Output:
(Sqrt[x]*(1 + Sqrt[1 - (b^2*x^2)/a^2]*Hypergeometric2F1[1/4, 1/2, 5/4, (b^ 2*x^2)/a^2]))/(a^2*c*Sqrt[c*(a - b*x)]*Sqrt[a + b*x])
Time = 0.20 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {115, 27, 35, 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.
\(\displaystyle \int \frac {1}{\sqrt {x} (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 115 |
\(\displaystyle \frac {\int \frac {b c (a-b x)}{2 \sqrt {x} \sqrt {a+b x} (a c-b c x)^{3/2}}dx}{a^2 b c}+\frac {\sqrt {x}}{a^2 c \sqrt {a+b x} \sqrt {a c-b c x}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {a-b x}{\sqrt {x} \sqrt {a+b x} (a c-b c x)^{3/2}}dx}{2 a^2}+\frac {\sqrt {x}}{a^2 c \sqrt {a+b x} \sqrt {a c-b c x}}\) |
\(\Big \downarrow \) 35 |
\(\displaystyle \frac {\int \frac {1}{\sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}dx}{2 a^2 c}+\frac {\sqrt {x}}{a^2 c \sqrt {a+b x} \sqrt {a c-b c x}}\) |
\(\Big \downarrow \) 127 |
\(\displaystyle \frac {\sqrt {1-\frac {b x}{a}} \sqrt {\frac {b x}{a}+1} \int \frac {1}{\sqrt {x} \sqrt {1-\frac {b x}{a}} \sqrt {\frac {b x}{a}+1}}dx}{2 a^2 c \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {\sqrt {x}}{a^2 c \sqrt {a+b x} \sqrt {a c-b c x}}\) |
\(\Big \downarrow \) 126 |
\(\displaystyle \frac {\sqrt {1-\frac {b x}{a}} \sqrt {\frac {b x}{a}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),-1\right )}{a^{3/2} \sqrt {b} c \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {\sqrt {x}}{a^2 c \sqrt {a+b x} \sqrt {a c-b c x}}\) |
Input:
Int[1/(Sqrt[x]*(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2)),x]
Output:
Sqrt[x]/(a^2*c*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]) + (Sqrt[1 - (b*x)/a]*Sqrt[ 1 + (b*x)/a]*EllipticF[ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[a]], -1])/(a^(3/2)*Sq rt[b]*c*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[(u_.)*((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.), x_Symbol] :> Simp[(b/d)^m Int[u*(c + d*x)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n} , x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && !(IntegerQ[n] && SimplerQ[a + b*x, c + d*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 = 1.60 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.98
method | result | size |
default | \(\frac {\sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}\, \left (\sqrt {2}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right ) a \sqrt {\frac {b x +a}{a}}\, \sqrt {\frac {-b x +a}{a}}\, \sqrt {-\frac {b x}{a}}+2 b x \right )}{2 \sqrt {x}\, b \,a^{2} c^{2} \left (-b^{2} x^{2}+a^{2}\right )}\) | \(105\) |
elliptic | \(\frac {\sqrt {c x \left (-b^{2} x^{2}+a^{2}\right )}\, \left (\frac {x}{c \,a^{2} \sqrt {-\left (x^{2}-\frac {a^{2}}{b^{2}}\right ) b^{2} c x}}+\frac {\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {2 \left (x -\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{2 a c b \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}\right )}{\sqrt {x}\, \sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}}\) | \(157\) |
Input:
int(1/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)*(2^(1/2)*EllipticF(((b*x+a)/a)^(1/2), 1/2*2^(1/2))*a*((b*x+a)/a)^(1/2)*((-b*x+a)/a)^(1/2)*(-b*x/a)^(1/2)+2*b*x)/ x^(1/2)/b/a^2/c^2/(-b^2*x^2+a^2)
Time = 0.08 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.82 \[ \int \frac {1}{\sqrt {x} (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=-\frac {\sqrt {-b c x + a c} \sqrt {b x + a} b^{2} \sqrt {x} + {\left (b^{2} x^{2} - a^{2}\right )} \sqrt {-b^{2} c} {\rm weierstrassPInverse}\left (\frac {4 \, a^{2}}{b^{2}}, 0, x\right )}{a^{2} b^{4} c^{2} x^{2} - a^{4} b^{2} c^{2}} \] Input:
integrate(1/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*sqrt(x) + (b^2*x^2 - a^2)*sqrt(-b^2 *c)*weierstrassPInverse(4*a^2/b^2, 0, x))/(a^2*b^4*c^2*x^2 - a^4*b^2*c^2)
Timed out. \[ \int \frac {1}{\sqrt {x} (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(1/x**(1/2)/(b*x+a)**(3/2)/(-b*c*x+a*c)**(3/2),x)
Output:
Timed out
\[ \int \frac {1}{\sqrt {x} (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\int { \frac {1}{{\left (-b c x + a c\right )}^{\frac {3}{2}} {\left (b x + a\right )}^{\frac {3}{2}} \sqrt {x}} \,d x } \] Input:
integrate(1/x^(1/2)/(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)*sqrt(x)), x)
Timed out. \[ \int \frac {1}{\sqrt {x} (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(1/x^(1/2)/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {1}{\sqrt {x} (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\int \frac {1}{\sqrt {x}\,{\left (a\,c-b\,c\,x\right )}^{3/2}\,{\left (a+b\,x\right )}^{3/2}} \,d x \] Input:
int(1/(x^(1/2)*(a*c - b*c*x)^(3/2)*(a + b*x)^(3/2)),x)
Output:
int(1/(x^(1/2)*(a*c - b*c*x)^(3/2)*(a + b*x)^(3/2)), x)
\[ \int \frac {1}{\sqrt {x} (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\frac {\sqrt {c}\, \left (\int \frac {\sqrt {x}\, \sqrt {b x +a}\, \sqrt {-b x +a}}{b^{4} x^{5}-2 a^{2} b^{2} x^{3}+a^{4} x}d x \right )}{c^{2}} \] Input:
int(1/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*x - 2*a**2*b**2*x **3 + b**4*x**5),x))/c**2