Integrand size = 28, antiderivative size = 147 \[ \int \frac {x^{7/2}}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\frac {x^{5/2}}{b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {5 \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}{3 b^4 c^2}-\frac {5 a^{5/2} \sqrt {1-\frac {b^2 x^2}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),-1\right )}{3 b^{9/2} c \sqrt {a+b x} \sqrt {a c-b c x}} \] Output:
x^(5/2)/b^2/c/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)+5/3*x^(1/2)*(b*x+a)^(1/2)*( -b*c*x+a*c)^(1/2)/b^4/c^2-5/3*a^(5/2)*(1-b^2*x^2/a^2)^(1/2)*EllipticF(b^(1 /2)*x^(1/2)/a^(1/2),I)/b^(9/2)/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 = 7.64 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.63 \[ \int \frac {x^{7/2}}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\frac {\sqrt {x} \left (5 a^2-2 b^2 x^2-5 a^2 \sqrt {1-\frac {b^2 x^2}{a^2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {b^2 x^2}{a^2}\right )\right )}{3 b^4 c \sqrt {c (a-b x)} \sqrt {a+b x}} \] Input:
Integrate[x^(7/2)/((a + b*x)^(3/2)*(a*c - b*c*x)^(3/2)),x]
Output:
(Sqrt[x]*(5*a^2 - 2*b^2*x^2 - 5*a^2*Sqrt[1 - (b^2*x^2)/a^2]*Hypergeometric 2F1[1/4, 1/2, 5/4, (b^2*x^2)/a^2]))/(3*b^4*c*Sqrt[c*(a - b*x)]*Sqrt[a + b* x])
Time = 0.23 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {109, 27, 35, 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.
| \(\displaystyle \int \frac {x^{7/2}}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {x^{5/2}}{b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {\int \frac {5 a c x^{3/2} (a-b x)}{2 \sqrt {a+b x} (a c-b c x)^{3/2}}dx}{a b^2 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^{5/2}}{b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {5 \int \frac {x^{3/2} (a-b x)}{\sqrt {a+b x} (a c-b c x)^{3/2}}dx}{2 b^2}\) |
| \(\Big \downarrow \) 35 |
| \(\displaystyle \frac {x^{5/2}}{b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {5 \int \frac {x^{3/2}}{\sqrt {a+b x} \sqrt {a c-b c x}}dx}{2 b^2 c}\) |
| \(\Big \downarrow \) 113 |
| \(\displaystyle \frac {x^{5/2}}{b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {5 \left (-\frac {2 \int -\frac {a^2 c}{2 \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}dx}{3 b^2 c}-\frac {2 \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}{3 b^2 c}\right )}{2 b^2 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^{5/2}}{b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {5 \left (\frac {a^2 \int \frac {1}{\sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}dx}{3 b^2}-\frac {2 \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}{3 b^2 c}\right )}{2 b^2 c}\) |
| \(\Big \downarrow \) 127 |
| \(\displaystyle \frac {x^{5/2}}{b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {5 \left (\frac {a^2 \sqrt {1-\frac {b x}{a}} \sqrt {\frac {b x}{a}+1} \int \frac {1}{\sqrt {x} \sqrt {1-\frac {b x}{a}} \sqrt {\frac {b x}{a}+1}}dx}{3 b^2 \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {2 \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}{3 b^2 c}\right )}{2 b^2 c}\) |
| \(\Big \downarrow \) 126 |
| \(\displaystyle \frac {x^{5/2}}{b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {5 \left (\frac {2 a^{5/2} \sqrt {1-\frac {b x}{a}} \sqrt {\frac {b x}{a}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),-1\right )}{3 b^{5/2} \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {2 \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}{3 b^2 c}\right )}{2 b^2 c}\) |
Input:
Int[x^(7/2)/((a + b*x)^(3/2)*(a*c - b*c*x)^(3/2)),x]
Output:
x^(5/2)/(b^2*c*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]) - (5*((-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)*Sqr t[a + b*x]*Sqrt[a*c - b*c*x])))/(2*b^2*c)
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*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f *x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
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 = 2.29 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.79
| method | result | size |
| default | \(-\frac {\sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}\, \left (5 \sqrt {\frac {b x +a}{a}}\, \sqrt {2}\, \sqrt {\frac {-b x +a}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right ) a^{3}+4 b^{3} x^{3}-10 a^{2} b x \right )}{6 \sqrt {x}\, c^{2} \left (-b^{2} x^{2}+a^{2}\right ) b^{5}}\) | \(116\) |
| elliptic | \(\frac {\sqrt {c x \left (-b^{2} x^{2}+a^{2}\right )}\, \left (\frac {x \,a^{2}}{b^{4} c \sqrt {-\left (x^{2}-\frac {a^{2}}{b^{2}}\right ) b^{2} c x}}+\frac {2 \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}{3 b^{4} c^{2}}-\frac {5 a^{3} \sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {2 \left (x -\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{6 b^{5} c \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}\right )}{\sqrt {x}\, \sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}}\) | \(186\) |
| risch | \(\frac {2 \sqrt {x}\, \sqrt {b x +a}\, \left (-b x +a \right )}{3 b^{4} \sqrt {-c \left (b x -a \right )}\, c}-\frac {a^{2} \left (\frac {4 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 )}{b \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}+\frac {3 a \left (\frac {-b^{2} c \,x^{2}-a b c x}{a^{2} c b \sqrt {\left (x -\frac {a}{b}\right ) \left (-b^{2} c \,x^{2}-a b c x \right )}}-\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 b \sqrt {-b^{2} c \,x^{3}+a^{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 \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}\right )}{2}-\frac {3 a \left (\frac {-b^{2} c \,x^{2}+a b c x}{a^{2} c b \sqrt {\left (x +\frac {a}{b}\right ) \left (-b^{2} c \,x^{2}+a b c x \right )}}+\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 b \sqrt {-b^{2} c \,x^{3}+a^{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 \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}\right )}{2}\right ) \sqrt {-x \left (b x +a \right ) c \left (b x -a \right )}}{3 b^{4} \sqrt {x}\, \sqrt {b x +a}\, \sqrt {-c \left (b x -a \right )}\, c}\) | \(676\) |
Input:
int(x^(7/2)/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/6*(b*x+a)^(1/2)*(c*(-b*x+a))^(1/2)*(5*((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+4* b^3*x^3-10*a^2*b*x)/x^(1/2)/c^2/(-b^2*x^2+a^2)/b^5
Time = 0.09 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.70 \[ \int \frac {x^{7/2}}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\frac {{\left (2 \, b^{4} x^{2} - 5 \, a^{2} b^{2}\right )} \sqrt {-b c x + a c} \sqrt {b x + a} \sqrt {x} + 5 \, {\left (a^{2} b^{2} x^{2} - a^{4}\right )} \sqrt {-b^{2} c} {\rm weierstrassPInverse}\left (\frac {4 \, a^{2}}{b^{2}}, 0, x\right )}{3 \, {\left (b^{8} c^{2} x^{2} - a^{2} b^{6} c^{2}\right )}} \] Input:
integrate(x^(7/2)/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2),x, algorithm="fricas")
Output:
1/3*((2*b^4*x^2 - 5*a^2*b^2)*sqrt(-b*c*x + a*c)*sqrt(b*x + a)*sqrt(x) + 5* (a^2*b^2*x^2 - a^4)*sqrt(-b^2*c)*weierstrassPInverse(4*a^2/b^2, 0, x))/(b^ 8*c^2*x^2 - a^2*b^6*c^2)
Timed out. \[ \int \frac {x^{7/2}}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(x**(7/2)/(b*x+a)**(3/2)/(-b*c*x+a*c)**(3/2),x)
Output:
Timed out
\[ \int \frac {x^{7/2}}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\int { \frac {x^{\frac {7}{2}}}{{\left (-b c x + a c\right )}^{\frac {3}{2}} {\left (b x + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^(7/2)/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2),x, algorithm="maxima")
Output:
integrate(x^(7/2)/((-b*c*x + a*c)^(3/2)*(b*x + a)^(3/2)), x)
Timed out. \[ \int \frac {x^{7/2}}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(x^(7/2)/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {x^{7/2}}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\int \frac {x^{7/2}}{{\left (a\,c-b\,c\,x\right )}^{3/2}\,{\left (a+b\,x\right )}^{3/2}} \,d x \] Input:
int(x^(7/2)/((a*c - b*c*x)^(3/2)*(a + b*x)^(3/2)),x)
Output:
int(x^(7/2)/((a*c - b*c*x)^(3/2)*(a + b*x)^(3/2)), x)
\[ \int \frac {x^{7/2}}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\frac {\sqrt {c}\, \left (10 \sqrt {x}\, \sqrt {b x +a}\, \sqrt {-b x +a}\, a^{2}-2 \sqrt {x}\, \sqrt {b x +a}\, \sqrt {-b x +a}\, b^{2} x^{2}-5 \left (\int \frac {\sqrt {x}\, \sqrt {b x +a}\, \sqrt {-b x +a}}{b^{4} x^{5}-2 a^{2} b^{2} x^{3}+a^{4} x}d x \right ) a^{6}+5 \left (\int \frac {\sqrt {x}\, \sqrt {b x +a}\, \sqrt {-b x +a}}{b^{4} x^{5}-2 a^{2} b^{2} x^{3}+a^{4} x}d x \right ) a^{4} b^{2} x^{2}\right )}{3 b^{4} c^{2} \left (-b^{2} x^{2}+a^{2}\right )} \] Input:
int(x^(7/2)/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2),x)
Output:
(sqrt(c)*(10*sqrt(x)*sqrt(a + b*x)*sqrt(a - b*x)*a**2 - 2*sqrt(x)*sqrt(a + b*x)*sqrt(a - b*x)*b**2*x**2 - 5*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)*a**6 + 5*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)*a**4*b**2 *x**2))/(3*b**4*c**2*(a**2 - b**2*x**2))