Integrand size = 28, antiderivative size = 110 \[ \int \frac {x^{3/2}}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=-\frac {2 \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}{3 b^2 c}+\frac {2 a^{5/2} \sqrt {1-\frac {b^2 x^2}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),-1\right )}{3 b^{5/2} \sqrt {a+b x} \sqrt {a c-b c x}} \] Output:
-2/3*x^(1/2)*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/b^2/c+2/3*a^(5/2)*(1-b^2*x^2 /a^2)^(1/2)*EllipticF(b^(1/2)*x^(1/2)/a^(1/2),I)/b^(5/2)/(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 = 1.23 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.79 \[ \int \frac {x^{3/2}}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=\frac {2 \sqrt {x} \left (-a^2+b^2 x^2+a^2 \sqrt {1-\frac {b^2 x^2}{a^2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {b^2 x^2}{a^2}\right )\right )}{3 b^2 \sqrt {c (a-b x)} \sqrt {a+b x}} \] Input:
Integrate[x^(3/2)/(Sqrt[a + b*x]*Sqrt[a*c - b*c*x]),x]
Output:
(2*Sqrt[x]*(-a^2 + b^2*x^2 + a^2*Sqrt[1 - (b^2*x^2)/a^2]*Hypergeometric2F1 [1/4, 1/2, 5/4, (b^2*x^2)/a^2]))/(3*b^2*Sqrt[c*(a - b*x)]*Sqrt[a + b*x])
Time = 0.21 (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 = {113, 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.
\(\displaystyle \int \frac {x^{3/2}}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx\) |
\(\Big \downarrow \) 113 |
\(\displaystyle -\frac {2 \int -\frac {a^2 c}{2 \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}dx}{3 b^2 c}-\frac {2 \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}{3 b^2 c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {a^2 \int \frac {1}{\sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}dx}{3 b^2}-\frac {2 \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}{3 b^2 c}\) |
\(\Big \downarrow \) 127 |
\(\displaystyle \frac {a^2 \sqrt {1-\frac {b x}{a}} \sqrt {\frac {b x}{a}+1} \int \frac {1}{\sqrt {x} \sqrt {1-\frac {b x}{a}} \sqrt {\frac {b x}{a}+1}}dx}{3 b^2 \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {2 \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}{3 b^2 c}\) |
\(\Big \downarrow \) 126 |
\(\displaystyle \frac {2 a^{5/2} \sqrt {1-\frac {b x}{a}} \sqrt {\frac {b x}{a}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),-1\right )}{3 b^{5/2} \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {2 \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}{3 b^2 c}\) |
Input:
Int[x^(3/2)/(Sqrt[a + b*x]*Sqrt[a*c - b*c*x]),x]
Output:
(-2*Sqrt[x]*Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/(3*b^2*c) + (2*a^(5/2)*Sqrt[1 - (b*x)/a]*Sqrt[1 + (b*x)/a]*EllipticF[ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[a]], -1])/(3*b^(5/2)*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 )/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1)) Int[(a + b*x) ^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] & & GtQ[m, 1] && NeQ[m + n + p + 1, 0] && 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 = 0.77 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.05
method | result | size |
default | \(\frac {\sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}\, \left (\sqrt {\frac {b x +a}{a}}\, \sqrt {2}\, \sqrt {\frac {-b x +a}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right ) a^{3}+2 b^{3} x^{3}-2 a^{2} b x \right )}{3 \sqrt {x}\, c \left (-b^{2} x^{2}+a^{2}\right ) b^{3}}\) | \(115\) |
elliptic | \(\frac {\sqrt {c x \left (-b^{2} x^{2}+a^{2}\right )}\, \left (-\frac {2 \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}{3 b^{2} c}+\frac {a^{3} \sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {2 \left (x -\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{3 b^{3} \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}\right )}{\sqrt {x}\, \sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}}\) | \(151\) |
risch | \(-\frac {2 \sqrt {x}\, \sqrt {b x +a}\, \left (-b x +a \right )}{3 b^{2} \sqrt {-c \left (b x -a \right )}}+\frac {a^{3} \sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {2 \left (x -\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-x \left (b x +a \right ) c \left (b x -a \right )}}{3 b^{3} \sqrt {-b^{2} c \,x^{3}+a^{2} c x}\, \sqrt {x}\, \sqrt {b x +a}\, \sqrt {-c \left (b x -a \right )}}\) | \(160\) |
Input:
int(x^(3/2)/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/3/x^(1/2)*(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^3+2*b^3*x^3-2*a^2*b*x)/(-b^2*x^2+a^2)/b^3
Time = 0.08 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.52 \[ \int \frac {x^{3/2}}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=-\frac {2 \, {\left (\sqrt {-b c x + a c} \sqrt {b x + a} b^{2} \sqrt {x} + \sqrt {-b^{2} c} a^{2} {\rm weierstrassPInverse}\left (\frac {4 \, a^{2}}{b^{2}}, 0, x\right )\right )}}{3 \, b^{4} c} \] Input:
integrate(x^(3/2)/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2),x, algorithm="fricas")
Output:
-2/3*(sqrt(-b*c*x + a*c)*sqrt(b*x + a)*b^2*sqrt(x) + sqrt(-b^2*c)*a^2*weie rstrassPInverse(4*a^2/b^2, 0, x))/(b^4*c)
Timed out. \[ \int \frac {x^{3/2}}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=\text {Timed out} \] Input:
integrate(x**(3/2)/(b*x+a)**(1/2)/(-b*c*x+a*c)**(1/2),x)
Output:
Timed out
\[ \int \frac {x^{3/2}}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=\int { \frac {x^{\frac {3}{2}}}{\sqrt {-b c x + a c} \sqrt {b x + a}} \,d x } \] Input:
integrate(x^(3/2)/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2),x, algorithm="maxima")
Output:
integrate(x^(3/2)/(sqrt(-b*c*x + a*c)*sqrt(b*x + a)), x)
Timed out. \[ \int \frac {x^{3/2}}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=\text {Timed out} \] Input:
integrate(x^(3/2)/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {x^{3/2}}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=\int \frac {x^{3/2}}{\sqrt {a\,c-b\,c\,x}\,\sqrt {a+b\,x}} \,d x \] Input:
int(x^(3/2)/((a*c - b*c*x)^(1/2)*(a + b*x)^(1/2)),x)
Output:
int(x^(3/2)/((a*c - b*c*x)^(1/2)*(a + b*x)^(1/2)), x)
\[ \int \frac {x^{3/2}}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=\frac {\sqrt {c}\, \left (-2 \sqrt {x}\, \sqrt {b x +a}\, \sqrt {-b x +a}+\left (\int \frac {\sqrt {x}\, \sqrt {b x +a}\, \sqrt {-b x +a}}{-b^{2} x^{3}+a^{2} x}d x \right ) a^{2}\right )}{3 b^{2} c} \] Input:
int(x^(3/2)/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2),x)
Output:
(sqrt(c)*( - 2*sqrt(x)*sqrt(a + b*x)*sqrt(a - b*x) + int((sqrt(x)*sqrt(a + b*x)*sqrt(a - b*x))/(a**2*x - b**2*x**3),x)*a**2))/(3*b**2*c)