Integrand size = 28, antiderivative size = 227 \[ \int \frac {1}{x^{9/2} (a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=\frac {1}{3 a^2 c x^{7/2} (a+b x)^{3/2} (a c-b c x)^{3/2}}+\frac {13}{6 a^4 c^2 x^{7/2} \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {39 \sqrt {a+b x} \sqrt {a c-b c x}}{14 a^6 c^3 x^{7/2}}-\frac {65 b^2 \sqrt {a+b x} \sqrt {a c-b c x}}{14 a^8 c^3 x^{3/2}}+\frac {65 b^{7/2} \sqrt {1-\frac {b^2 x^2}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),-1\right )}{14 a^{15/2} c^2 \sqrt {a+b x} \sqrt {a c-b c x}} \] Output:
1/3/a^2/c/x^(7/2)/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2)+13/6/a^4/c^2/x^(7/2)/(b *x+a)^(1/2)/(-b*c*x+a*c)^(1/2)-39/14*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/a^6/ c^3/x^(7/2)-65/14*b^2*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/a^8/c^3/x^(3/2)+65/ 14*b^(7/2)*(1-b^2*x^2/a^2)^(1/2)*EllipticF(b^(1/2)*x^(1/2)/a^(1/2),I)/a^(1 5/2)/c^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 = 10.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.32 \[ \int \frac {1}{x^{9/2} (a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=-\frac {2 \sqrt {1-\frac {b^2 x^2}{a^2}} \operatorname {Hypergeometric2F1}\left (-\frac {7}{4},\frac {5}{2},-\frac {3}{4},\frac {b^2 x^2}{a^2}\right )}{7 a^4 c^2 x^{7/2} \sqrt {c (a-b x)} \sqrt {a+b x}} \] Input:
Integrate[1/(x^(9/2)*(a + b*x)^(5/2)*(a*c - b*c*x)^(5/2)),x]
Output:
(-2*Sqrt[1 - (b^2*x^2)/a^2]*Hypergeometric2F1[-7/4, 5/2, -3/4, (b^2*x^2)/a ^2])/(7*a^4*c^2*x^(7/2)*Sqrt[c*(a - b*x)]*Sqrt[a + b*x])
Time = 0.30 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.13, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {115, 27, 115, 27, 115, 27, 35, 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.
| \(\displaystyle \int \frac {1}{x^{9/2} (a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 115 |
| \(\displaystyle -\frac {2 \int -\frac {13 b^2 c}{2 x^{5/2} (a+b x)^{5/2} (a c-b c x)^{5/2}}dx}{7 a^2 c}-\frac {2}{7 a^2 c x^{7/2} (a+b x)^{3/2} (a c-b c x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {13 b^2 \int \frac {1}{x^{5/2} (a+b x)^{5/2} (a c-b c x)^{5/2}}dx}{7 a^2}-\frac {2}{7 a^2 c x^{7/2} (a+b x)^{3/2} (a c-b c x)^{3/2}}\) |
| \(\Big \downarrow \) 115 |
| \(\displaystyle \frac {13 b^2 \left (-\frac {2 \int -\frac {9 b^2 c}{2 \sqrt {x} (a+b x)^{5/2} (a c-b c x)^{5/2}}dx}{3 a^2 c}-\frac {2}{3 a^2 c x^{3/2} (a+b x)^{3/2} (a c-b c x)^{3/2}}\right )}{7 a^2}-\frac {2}{7 a^2 c x^{7/2} (a+b x)^{3/2} (a c-b c x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {13 b^2 \left (\frac {3 b^2 \int \frac {1}{\sqrt {x} (a+b x)^{5/2} (a c-b c x)^{5/2}}dx}{a^2}-\frac {2}{3 a^2 c x^{3/2} (a+b x)^{3/2} (a c-b c x)^{3/2}}\right )}{7 a^2}-\frac {2}{7 a^2 c x^{7/2} (a+b x)^{3/2} (a c-b c x)^{3/2}}\) |
| \(\Big \downarrow \) 115 |
| \(\displaystyle \frac {13 b^2 \left (\frac {3 b^2 \left (\frac {\int \frac {5 b c (a-b x)}{2 \sqrt {x} (a+b x)^{3/2} (a c-b c x)^{5/2}}dx}{3 a^2 b c}+\frac {\sqrt {x}}{3 a^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}\right )}{a^2}-\frac {2}{3 a^2 c x^{3/2} (a+b x)^{3/2} (a c-b c x)^{3/2}}\right )}{7 a^2}-\frac {2}{7 a^2 c x^{7/2} (a+b x)^{3/2} (a c-b c x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {13 b^2 \left (\frac {3 b^2 \left (\frac {5 \int \frac {a-b x}{\sqrt {x} (a+b x)^{3/2} (a c-b c x)^{5/2}}dx}{6 a^2}+\frac {\sqrt {x}}{3 a^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}\right )}{a^2}-\frac {2}{3 a^2 c x^{3/2} (a+b x)^{3/2} (a c-b c x)^{3/2}}\right )}{7 a^2}-\frac {2}{7 a^2 c x^{7/2} (a+b x)^{3/2} (a c-b c x)^{3/2}}\) |
| \(\Big \downarrow \) 35 |
| \(\displaystyle \frac {13 b^2 \left (\frac {3 b^2 \left (\frac {5 \int \frac {1}{\sqrt {x} (a+b x)^{3/2} (a c-b c x)^{3/2}}dx}{6 a^2 c}+\frac {\sqrt {x}}{3 a^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}\right )}{a^2}-\frac {2}{3 a^2 c x^{3/2} (a+b x)^{3/2} (a c-b c x)^{3/2}}\right )}{7 a^2}-\frac {2}{7 a^2 c x^{7/2} (a+b x)^{3/2} (a c-b c x)^{3/2}}\) |
| \(\Big \downarrow \) 115 |
| \(\displaystyle \frac {13 b^2 \left (\frac {3 b^2 \left (\frac {5 \left (\frac {\int \frac {b c (a-b x)}{2 \sqrt {x} \sqrt {a+b x} (a c-b c x)^{3/2}}dx}{a^2 b c}+\frac {\sqrt {x}}{a^2 c \sqrt {a+b x} \sqrt {a c-b c x}}\right )}{6 a^2 c}+\frac {\sqrt {x}}{3 a^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}\right )}{a^2}-\frac {2}{3 a^2 c x^{3/2} (a+b x)^{3/2} (a c-b c x)^{3/2}}\right )}{7 a^2}-\frac {2}{7 a^2 c x^{7/2} (a+b x)^{3/2} (a c-b c x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {13 b^2 \left (\frac {3 b^2 \left (\frac {5 \left (\frac {\int \frac {a-b x}{\sqrt {x} \sqrt {a+b x} (a c-b c x)^{3/2}}dx}{2 a^2}+\frac {\sqrt {x}}{a^2 c \sqrt {a+b x} \sqrt {a c-b c x}}\right )}{6 a^2 c}+\frac {\sqrt {x}}{3 a^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}\right )}{a^2}-\frac {2}{3 a^2 c x^{3/2} (a+b x)^{3/2} (a c-b c x)^{3/2}}\right )}{7 a^2}-\frac {2}{7 a^2 c x^{7/2} (a+b x)^{3/2} (a c-b c x)^{3/2}}\) |
| \(\Big \downarrow \) 35 |
| \(\displaystyle \frac {13 b^2 \left (\frac {3 b^2 \left (\frac {5 \left (\frac {\int \frac {1}{\sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}dx}{2 a^2 c}+\frac {\sqrt {x}}{a^2 c \sqrt {a+b x} \sqrt {a c-b c x}}\right )}{6 a^2 c}+\frac {\sqrt {x}}{3 a^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}\right )}{a^2}-\frac {2}{3 a^2 c x^{3/2} (a+b x)^{3/2} (a c-b c x)^{3/2}}\right )}{7 a^2}-\frac {2}{7 a^2 c x^{7/2} (a+b x)^{3/2} (a c-b c x)^{3/2}}\) |
| \(\Big \downarrow \) 127 |
| \(\displaystyle \frac {13 b^2 \left (\frac {3 b^2 \left (\frac {5 \left (\frac {\sqrt {1-\frac {b x}{a}} \sqrt {\frac {b x}{a}+1} \int \frac {1}{\sqrt {x} \sqrt {1-\frac {b x}{a}} \sqrt {\frac {b x}{a}+1}}dx}{2 a^2 c \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {\sqrt {x}}{a^2 c \sqrt {a+b x} \sqrt {a c-b c x}}\right )}{6 a^2 c}+\frac {\sqrt {x}}{3 a^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}\right )}{a^2}-\frac {2}{3 a^2 c x^{3/2} (a+b x)^{3/2} (a c-b c x)^{3/2}}\right )}{7 a^2}-\frac {2}{7 a^2 c x^{7/2} (a+b x)^{3/2} (a c-b c x)^{3/2}}\) |
| \(\Big \downarrow \) 126 |
| \(\displaystyle \frac {13 b^2 \left (\frac {3 b^2 \left (\frac {\sqrt {x}}{3 a^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}+\frac {5 \left (\frac {\sqrt {1-\frac {b x}{a}} \sqrt {\frac {b x}{a}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),-1\right )}{a^{3/2} \sqrt {b} c \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {\sqrt {x}}{a^2 c \sqrt {a+b x} \sqrt {a c-b c x}}\right )}{6 a^2 c}\right )}{a^2}-\frac {2}{3 a^2 c x^{3/2} (a+b x)^{3/2} (a c-b c x)^{3/2}}\right )}{7 a^2}-\frac {2}{7 a^2 c x^{7/2} (a+b x)^{3/2} (a c-b c x)^{3/2}}\) |
Input:
Int[1/(x^(9/2)*(a + b*x)^(5/2)*(a*c - b*c*x)^(5/2)),x]
Output:
-2/(7*a^2*c*x^(7/2)*(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2)) + (13*b^2*(-2/(3* a^2*c*x^(3/2)*(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2)) + (3*b^2*(Sqrt[x]/(3*a^ 2*c*(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2)) + (5*(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[ArcS in[(Sqrt[b]*Sqrt[x])/Sqrt[a]], -1])/(a^(3/2)*Sqrt[b]*c*Sqrt[a + b*x]*Sqrt[ a*c - b*c*x])))/(6*a^2*c)))/a^2))/(7*a^2)
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 = 6.21 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.87
| method | result | size |
| default | \(\frac {\left (-195 \sqrt {2}\, \sqrt {\frac {b x +a}{a}}\, \sqrt {\frac {-b x +a}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right ) a \,b^{5} x^{5}+195 \operatorname {EllipticF}\left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a^{3} b^{3} x^{3} \sqrt {\frac {b x +a}{a}}\, \sqrt {\frac {-b x +a}{a}}\, \sqrt {-\frac {b x}{a}}-390 b^{6} x^{6}+546 a^{2} x^{4} b^{4}-104 a^{4} x^{2} b^{2}-24 a^{6}\right ) \sqrt {c \left (-b x +a \right )}}{84 c^{3} \left (-b x +a \right )^{2} a^{8} \left (b x +a \right )^{\frac {3}{2}} x^{\frac {7}{2}}}\) | \(197\) |
| elliptic | \(\frac {\sqrt {c x \left (-b^{2} x^{2}+a^{2}\right )}\, \left (-\frac {2 \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}{7 a^{6} c^{3} x^{4}}-\frac {38 b^{2} \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}{21 a^{8} c^{3} x^{2}}+\frac {\sqrt {-b^{2} c \,x^{3}+a^{2} c x}}{3 a^{6} c^{3} \left (x^{2}-\frac {a^{2}}{b^{2}}\right )^{2}}+\frac {17 b^{4} x}{6 c^{2} a^{8} \sqrt {-\left (x^{2}-\frac {a^{2}}{b^{2}}\right ) b^{2} c x}}+\frac {65 b^{3} \sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {2 \left (x -\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{28 a^{7} c^{2} \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}\right )}{\sqrt {x}\, \sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}}\) | \(262\) |
| risch | \(\text {Expression too large to display}\) | \(1281\) |
Input:
int(1/x^(9/2)/(b*x+a)^(5/2)/(-b*c*x+a*c)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/84*(-195*2^(1/2)*((b*x+a)/a)^(1/2)*((-b*x+a)/a)^(1/2)*(-b*x/a)^(1/2)*Ell ipticF(((b*x+a)/a)^(1/2),1/2*2^(1/2))*a*b^5*x^5+195*EllipticF(((b*x+a)/a)^ (1/2),1/2*2^(1/2))*2^(1/2)*a^3*b^3*x^3*((b*x+a)/a)^(1/2)*((-b*x+a)/a)^(1/2 )*(-b*x/a)^(1/2)-390*b^6*x^6+546*a^2*x^4*b^4-104*a^4*x^2*b^2-24*a^6)*(c*(- b*x+a))^(1/2)/c^3/(-b*x+a)^2/a^8/(b*x+a)^(3/2)/x^(7/2)
Time = 0.11 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.67 \[ \int \frac {1}{x^{9/2} (a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=-\frac {{\left (195 \, b^{6} x^{6} - 273 \, a^{2} b^{4} x^{4} + 52 \, a^{4} b^{2} x^{2} + 12 \, a^{6}\right )} \sqrt {-b c x + a c} \sqrt {b x + a} \sqrt {x} + 195 \, {\left (b^{6} x^{8} - 2 \, a^{2} b^{4} x^{6} + a^{4} b^{2} x^{4}\right )} \sqrt {-b^{2} c} {\rm weierstrassPInverse}\left (\frac {4 \, a^{2}}{b^{2}}, 0, x\right )}{42 \, {\left (a^{8} b^{4} c^{3} x^{8} - 2 \, a^{10} b^{2} c^{3} x^{6} + a^{12} c^{3} x^{4}\right )}} \] Input:
integrate(1/x^(9/2)/(b*x+a)^(5/2)/(-b*c*x+a*c)^(5/2),x, algorithm="fricas" )
Output:
-1/42*((195*b^6*x^6 - 273*a^2*b^4*x^4 + 52*a^4*b^2*x^2 + 12*a^6)*sqrt(-b*c *x + a*c)*sqrt(b*x + a)*sqrt(x) + 195*(b^6*x^8 - 2*a^2*b^4*x^6 + a^4*b^2*x ^4)*sqrt(-b^2*c)*weierstrassPInverse(4*a^2/b^2, 0, x))/(a^8*b^4*c^3*x^8 - 2*a^10*b^2*c^3*x^6 + a^12*c^3*x^4)
Timed out. \[ \int \frac {1}{x^{9/2} (a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=\text {Timed out} \] Input:
integrate(1/x**(9/2)/(b*x+a)**(5/2)/(-b*c*x+a*c)**(5/2),x)
Output:
Timed out
\[ \int \frac {1}{x^{9/2} (a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=\int { \frac {1}{{\left (-b c x + a c\right )}^{\frac {5}{2}} {\left (b x + a\right )}^{\frac {5}{2}} x^{\frac {9}{2}}} \,d x } \] Input:
integrate(1/x^(9/2)/(b*x+a)^(5/2)/(-b*c*x+a*c)^(5/2),x, algorithm="maxima" )
Output:
integrate(1/((-b*c*x + a*c)^(5/2)*(b*x + a)^(5/2)*x^(9/2)), x)
Timed out. \[ \int \frac {1}{x^{9/2} (a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=\text {Timed out} \] Input:
integrate(1/x^(9/2)/(b*x+a)^(5/2)/(-b*c*x+a*c)^(5/2),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {1}{x^{9/2} (a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=\int \frac {1}{x^{9/2}\,{\left (a\,c-b\,c\,x\right )}^{5/2}\,{\left (a+b\,x\right )}^{5/2}} \,d x \] Input:
int(1/(x^(9/2)*(a*c - b*c*x)^(5/2)*(a + b*x)^(5/2)),x)
Output:
int(1/(x^(9/2)*(a*c - b*c*x)^(5/2)*(a + b*x)^(5/2)), x)
\[ \int \frac {1}{x^{9/2} (a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=\int \frac {1}{x^{\frac {9}{2}} \left (b x +a \right )^{\frac {5}{2}} \left (-b c x +a c \right )^{\frac {5}{2}}}d x \] Input:
int(1/x^(9/2)/(b*x+a)^(5/2)/(-b*c*x+a*c)^(5/2),x)
Output:
int(1/x^(9/2)/(b*x+a)^(5/2)/(-b*c*x+a*c)^(5/2),x)