Integrand size = 28, antiderivative size = 142 \[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^{9/2}} \, dx=\frac {4 b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}{7 x^{3/2}}-\frac {2 (a+b x)^{3/2} (a c-b c x)^{3/2}}{7 x^{7/2}}+\frac {8 \sqrt {a} b^{7/2} c^2 \sqrt {1-\frac {b^2 x^2}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),-1\right )}{7 \sqrt {a+b x} \sqrt {a c-b c x}} \] Output:
4/7*b^2*c*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/x^(3/2)-2/7*(b*x+a)^(3/2)*(-b*c *x+a*c)^(3/2)/x^(7/2)+8/7*a^(1/2)*b^(7/2)*c^2*(1-b^2*x^2/a^2)^(1/2)*Ellipt icF(b^(1/2)*x^(1/2)/a^(1/2),I)/(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 = 7.26 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.50 \[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^{9/2}} \, dx=-\frac {2 a^2 c \sqrt {c (a-b x)} \sqrt {a+b x} \operatorname {Hypergeometric2F1}\left (-\frac {7}{4},-\frac {3}{2},-\frac {3}{4},\frac {b^2 x^2}{a^2}\right )}{7 x^{7/2} \sqrt {1-\frac {b^2 x^2}{a^2}}} \] Input:
Integrate[((a + b*x)^(3/2)*(a*c - b*c*x)^(3/2))/x^(9/2),x]
Output:
(-2*a^2*c*Sqrt[c*(a - b*x)]*Sqrt[a + b*x]*Hypergeometric2F1[-7/4, -3/2, -3 /4, (b^2*x^2)/a^2])/(7*x^(7/2)*Sqrt[1 - (b^2*x^2)/a^2])
Time = 0.23 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {108, 27, 108, 25, 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.
| \(\displaystyle \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^{9/2}} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {2}{7} \int -\frac {3 b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}{x^{5/2}}dx-\frac {2 (a+b x)^{3/2} (a c-b c x)^{3/2}}{7 x^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {6}{7} b^2 c \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^{5/2}}dx-\frac {2 (a+b x)^{3/2} (a c-b c x)^{3/2}}{7 x^{7/2}}\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle -\frac {6}{7} b^2 c \left (\frac {2}{3} \int -\frac {b^2 c}{\sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}dx-\frac {2 \sqrt {a+b x} \sqrt {a c-b c x}}{3 x^{3/2}}\right )-\frac {2 (a+b x)^{3/2} (a c-b c x)^{3/2}}{7 x^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {6}{7} b^2 c \left (-\frac {2}{3} \int \frac {b^2 c}{\sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}dx-\frac {2 \sqrt {a+b x} \sqrt {a c-b c x}}{3 x^{3/2}}\right )-\frac {2 (a+b x)^{3/2} (a c-b c x)^{3/2}}{7 x^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {6}{7} b^2 c \left (-\frac {2}{3} b^2 c \int \frac {1}{\sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}dx-\frac {2 \sqrt {a+b x} \sqrt {a c-b c x}}{3 x^{3/2}}\right )-\frac {2 (a+b x)^{3/2} (a c-b c x)^{3/2}}{7 x^{7/2}}\) |
| \(\Big \downarrow \) 127 |
| \(\displaystyle -\frac {6}{7} b^2 c \left (-\frac {2 b^2 c \sqrt {1-\frac {b x}{a}} \sqrt {\frac {b x}{a}+1} \int \frac {1}{\sqrt {x} \sqrt {1-\frac {b x}{a}} \sqrt {\frac {b x}{a}+1}}dx}{3 \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {2 \sqrt {a+b x} \sqrt {a c-b c x}}{3 x^{3/2}}\right )-\frac {2 (a+b x)^{3/2} (a c-b c x)^{3/2}}{7 x^{7/2}}\) |
| \(\Big \downarrow \) 126 |
| \(\displaystyle -\frac {6}{7} b^2 c \left (-\frac {4 \sqrt {a} b^{3/2} c \sqrt {1-\frac {b x}{a}} \sqrt {\frac {b x}{a}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),-1\right )}{3 \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {2 \sqrt {a+b x} \sqrt {a c-b c x}}{3 x^{3/2}}\right )-\frac {2 (a+b x)^{3/2} (a c-b c x)^{3/2}}{7 x^{7/2}}\) |
Input:
Int[((a + b*x)^(3/2)*(a*c - b*c*x)^(3/2))/x^(9/2),x]
Output:
(-2*(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2))/(7*x^(7/2)) - (6*b^2*c*((-2*Sqrt[ a + b*x]*Sqrt[a*c - b*c*x])/(3*x^(3/2)) - (4*Sqrt[a]*b^(3/2)*c*Sqrt[1 - (b *x)/a]*Sqrt[1 + (b*x)/a]*EllipticF[ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[a]], -1]) /(3*Sqrt[a + b*x]*Sqrt[a*c - b*c*x])))/7
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/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2 *n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
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 = 1.16 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.87
| method | result | size |
| default | \(\frac {2 \sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}\, c \left (2 \sqrt {\frac {b x +a}{a}}\, \sqrt {2}\, \sqrt {\frac {-b x +a}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right ) a \,b^{3} x^{3}-3 b^{4} x^{4}+4 a^{2} b^{2} x^{2}-a^{4}\right )}{7 x^{\frac {7}{2}} \left (-b^{2} x^{2}+a^{2}\right )}\) | \(124\) |
| risch | \(-\frac {2 \left (-b x +a \right ) \sqrt {b x +a}\, \left (-3 b^{2} x^{2}+a^{2}\right ) c^{2}}{7 x^{\frac {7}{2}} \sqrt {-c \left (b x -a \right )}}+\frac {4 b^{3} a \sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {2 \left (x -\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-x \left (b x +a \right ) c \left (b x -a \right )}\, c^{2}}{7 \sqrt {-b^{2} c \,x^{3}+a^{2} c x}\, \sqrt {x}\, \sqrt {b x +a}\, \sqrt {-c \left (b x -a \right )}}\) | \(173\) |
| elliptic | \(\frac {\sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}\, \sqrt {c x \left (-b^{2} x^{2}+a^{2}\right )}\, \left (-\frac {2 a^{2} c \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}{7 x^{4}}+\frac {6 b^{2} c \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}{7 x^{2}}+\frac {4 b^{3} c^{2} a \sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {2 \left (x -\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{7 \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}\right )}{\sqrt {x}\, c \left (-b^{2} x^{2}+a^{2}\right )}\) | \(197\) |
Input:
int((b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2)/x^(9/2),x,method=_RETURNVERBOSE)
Output:
2/7*(b*x+a)^(1/2)*(c*(-b*x+a))^(1/2)*c*(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*b^3 *x^3-3*b^4*x^4+4*a^2*b^2*x^2-a^4)/x^(7/2)/(-b^2*x^2+a^2)
Time = 0.09 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.51 \[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^{9/2}} \, dx=-\frac {2 \, {\left (4 \, \sqrt {-b^{2} c} b^{2} c x^{4} {\rm weierstrassPInverse}\left (\frac {4 \, a^{2}}{b^{2}}, 0, x\right ) - {\left (3 \, b^{2} c x^{2} - a^{2} c\right )} \sqrt {-b c x + a c} \sqrt {b x + a} \sqrt {x}\right )}}{7 \, x^{4}} \] Input:
integrate((b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2)/x^(9/2),x, algorithm="fricas")
Output:
-2/7*(4*sqrt(-b^2*c)*b^2*c*x^4*weierstrassPInverse(4*a^2/b^2, 0, x) - (3*b ^2*c*x^2 - a^2*c)*sqrt(-b*c*x + a*c)*sqrt(b*x + a)*sqrt(x))/x^4
Timed out. \[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^{9/2}} \, dx=\text {Timed out} \] Input:
integrate((b*x+a)**(3/2)*(-b*c*x+a*c)**(3/2)/x**(9/2),x)
Output:
Timed out
\[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^{9/2}} \, dx=\int { \frac {{\left (-b c x + a c\right )}^{\frac {3}{2}} {\left (b x + a\right )}^{\frac {3}{2}}}{x^{\frac {9}{2}}} \,d x } \] Input:
integrate((b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2)/x^(9/2),x, algorithm="maxima")
Output:
integrate((-b*c*x + a*c)^(3/2)*(b*x + a)^(3/2)/x^(9/2), x)
\[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^{9/2}} \, dx=\int { \frac {{\left (-b c x + a c\right )}^{\frac {3}{2}} {\left (b x + a\right )}^{\frac {3}{2}}}{x^{\frac {9}{2}}} \,d x } \] Input:
integrate((b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2)/x^(9/2),x, algorithm="giac")
Output:
integrate((-b*c*x + a*c)^(3/2)*(b*x + a)^(3/2)/x^(9/2), x)
Timed out. \[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^{9/2}} \, dx=\int \frac {{\left (a\,c-b\,c\,x\right )}^{3/2}\,{\left (a+b\,x\right )}^{3/2}}{x^{9/2}} \,d x \] Input:
int(((a*c - b*c*x)^(3/2)*(a + b*x)^(3/2))/x^(9/2),x)
Output:
int(((a*c - b*c*x)^(3/2)*(a + b*x)^(3/2))/x^(9/2), x)
\[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^{9/2}} \, dx=\frac {2 \sqrt {c}\, c \left (\sqrt {b x +a}\, \sqrt {-b x +a}\, a^{2}+5 \sqrt {b x +a}\, \sqrt {-b x +a}\, b^{2} x^{2}+6 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {b x +a}\, \sqrt {-b x +a}}{-b^{2} x^{7}+a^{2} x^{5}}d x \right ) a^{4} x^{3}\right )}{5 \sqrt {x}\, x^{3}} \] Input:
int((b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2)/x^(9/2),x)
Output:
(2*sqrt(c)*c*(sqrt(a + b*x)*sqrt(a - b*x)*a**2 + 5*sqrt(a + b*x)*sqrt(a - b*x)*b**2*x**2 + 6*sqrt(x)*int((sqrt(x)*sqrt(a + b*x)*sqrt(a - b*x))/(a**2 *x**5 - b**2*x**7),x)*a**4*x**3))/(5*sqrt(x)*x**3)