\(\int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^{13/2}} \, dx\) [156]

Optimal result

Integrand size = 28, antiderivative size = 180 \[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^{13/2}} \, dx=\frac {12 b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}{77 x^{7/2}}-\frac {8 b^4 c \sqrt {a+b x} \sqrt {a c-b c x}}{77 a^2 x^{3/2}}-\frac {2 (a+b x)^{3/2} (a c-b c x)^{3/2}}{11 x^{11/2}}+\frac {8 b^{11/2} c^2 \sqrt {1-\frac {b^2 x^2}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),-1\right )}{77 a^{3/2} \sqrt {a+b x} \sqrt {a c-b c x}} \] Output:

12/77*b^2*c*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/x^(7/2)-8/77*b^4*c*(b*x+a)^(1 
/2)*(-b*c*x+a*c)^(1/2)/a^2/x^(3/2)-2/11*(b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2)/x 
^(11/2)+8/77*b^(11/2)*c^2*(1-b^2*x^2/a^2)^(1/2)*EllipticF(b^(1/2)*x^(1/2)/ 
a^(1/2),I)/a^(3/2)/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 8.29 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.39 \[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^{13/2}} \, dx=-\frac {2 a^2 c \sqrt {c (a-b x)} \sqrt {a+b x} \operatorname {Hypergeometric2F1}\left (-\frac {11}{4},-\frac {3}{2},-\frac {7}{4},\frac {b^2 x^2}{a^2}\right )}{11 x^{11/2} \sqrt {1-\frac {b^2 x^2}{a^2}}} \] Input:

Integrate[((a + b*x)^(3/2)*(a*c - b*c*x)^(3/2))/x^(13/2),x]
 

Output:

(-2*a^2*c*Sqrt[c*(a - b*x)]*Sqrt[a + b*x]*Hypergeometric2F1[-11/4, -3/2, - 
7/4, (b^2*x^2)/a^2])/(11*x^(11/2)*Sqrt[1 - (b^2*x^2)/a^2])
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.10, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {108, 27, 108, 25, 27, 115, 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.

\(\displaystyle \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^{13/2}} \, dx\)

\(\Big \downarrow \) 108

\(\displaystyle \frac {2}{11} \int -\frac {3 b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}{x^{9/2}}dx-\frac {2 (a+b x)^{3/2} (a c-b c x)^{3/2}}{11 x^{11/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {6}{11} b^2 c \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^{9/2}}dx-\frac {2 (a+b x)^{3/2} (a c-b c x)^{3/2}}{11 x^{11/2}}\)

\(\Big \downarrow \) 108

\(\displaystyle -\frac {6}{11} b^2 c \left (\frac {2}{7} \int -\frac {b^2 c}{x^{5/2} \sqrt {a+b x} \sqrt {a c-b c x}}dx-\frac {2 \sqrt {a+b x} \sqrt {a c-b c x}}{7 x^{7/2}}\right )-\frac {2 (a+b x)^{3/2} (a c-b c x)^{3/2}}{11 x^{11/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {6}{11} b^2 c \left (-\frac {2}{7} \int \frac {b^2 c}{x^{5/2} \sqrt {a+b x} \sqrt {a c-b c x}}dx-\frac {2 \sqrt {a+b x} \sqrt {a c-b c x}}{7 x^{7/2}}\right )-\frac {2 (a+b x)^{3/2} (a c-b c x)^{3/2}}{11 x^{11/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {6}{11} b^2 c \left (-\frac {2}{7} b^2 c \int \frac {1}{x^{5/2} \sqrt {a+b x} \sqrt {a c-b c x}}dx-\frac {2 \sqrt {a+b x} \sqrt {a c-b c x}}{7 x^{7/2}}\right )-\frac {2 (a+b x)^{3/2} (a c-b c x)^{3/2}}{11 x^{11/2}}\)

\(\Big \downarrow \) 115

\(\displaystyle -\frac {6}{11} b^2 c \left (-\frac {2}{7} b^2 c \left (-\frac {2 \int -\frac {b^2 c}{2 \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}dx}{3 a^2 c}-\frac {2 \sqrt {a+b x} \sqrt {a c-b c x}}{3 a^2 c x^{3/2}}\right )-\frac {2 \sqrt {a+b x} \sqrt {a c-b c x}}{7 x^{7/2}}\right )-\frac {2 (a+b x)^{3/2} (a c-b c x)^{3/2}}{11 x^{11/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {6}{11} b^2 c \left (-\frac {2}{7} b^2 c \left (\frac {b^2 \int \frac {1}{\sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}dx}{3 a^2}-\frac {2 \sqrt {a+b x} \sqrt {a c-b c x}}{3 a^2 c x^{3/2}}\right )-\frac {2 \sqrt {a+b x} \sqrt {a c-b c x}}{7 x^{7/2}}\right )-\frac {2 (a+b x)^{3/2} (a c-b c x)^{3/2}}{11 x^{11/2}}\)

\(\Big \downarrow \) 127

\(\displaystyle -\frac {6}{11} b^2 c \left (-\frac {2}{7} b^2 c \left (\frac {b^2 \sqrt {1-\frac {b x}{a}} \sqrt {\frac {b x}{a}+1} \int \frac {1}{\sqrt {x} \sqrt {1-\frac {b x}{a}} \sqrt {\frac {b x}{a}+1}}dx}{3 a^2 \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {2 \sqrt {a+b x} \sqrt {a c-b c x}}{3 a^2 c x^{3/2}}\right )-\frac {2 \sqrt {a+b x} \sqrt {a c-b c x}}{7 x^{7/2}}\right )-\frac {2 (a+b x)^{3/2} (a c-b c x)^{3/2}}{11 x^{11/2}}\)

\(\Big \downarrow \) 126

\(\displaystyle -\frac {6}{11} b^2 c \left (-\frac {2}{7} b^2 c \left (\frac {2 b^{3/2} \sqrt {1-\frac {b x}{a}} \sqrt {\frac {b x}{a}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),-1\right )}{3 a^{3/2} \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {2 \sqrt {a+b x} \sqrt {a c-b c x}}{3 a^2 c x^{3/2}}\right )-\frac {2 \sqrt {a+b x} \sqrt {a c-b c x}}{7 x^{7/2}}\right )-\frac {2 (a+b x)^{3/2} (a c-b c x)^{3/2}}{11 x^{11/2}}\)

Input:

Int[((a + b*x)^(3/2)*(a*c - b*c*x)^(3/2))/x^(13/2),x]
 

Output:

(-2*(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2))/(11*x^(11/2)) - (6*b^2*c*((-2*Sqr 
t[a + b*x]*Sqrt[a*c - b*c*x])/(7*x^(7/2)) - (2*b^2*c*((-2*Sqrt[a + b*x]*Sq 
rt[a*c - b*c*x])/(3*a^2*c*x^(3/2)) + (2*b^(3/2)*Sqrt[1 - (b*x)/a]*Sqrt[1 + 
 (b*x)/a]*EllipticF[ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[a]], -1])/(3*a^(3/2)*Sqr 
t[a + b*x]*Sqrt[a*c - b*c*x])))/7))/11
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 108

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])
 

rule 115

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]
 

rule 126

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])
 

rule 127

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])
 

Maple [A] (verified)

Time = 1.60 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.77

method	result	size
default	\(\frac {2 \sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}\, c \left (2 \sqrt {2}\, \sqrt {\frac {b x +a}{a}}\, \sqrt {\frac {-b x +a}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right ) a \,b^{5} x^{5}+4 b^{6} x^{6}-17 a^{2} x^{4} b^{4}+20 a^{4} x^{2} b^{2}-7 a^{6}\right )}{77 x^{\frac {11}{2}} a^{2} \left (-b^{2} x^{2}+a^{2}\right )}\)	\(138\)
risch	\(-\frac {2 \left (-b x +a \right ) \sqrt {b x +a}\, \left (4 b^{4} x^{4}-13 a^{2} b^{2} x^{2}+7 a^{4}\right ) c^{2}}{77 x^{\frac {11}{2}} a^{2} \sqrt {-c \left (b x -a \right )}}+\frac {4 b^{5} \sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {2 \left (x -\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-x \left (b x +a \right ) c \left (b x -a \right )}\, c^{2}}{77 a \sqrt {-b^{2} c \,x^{3}+a^{2} c x}\, \sqrt {x}\, \sqrt {b x +a}\, \sqrt {-c \left (b x -a \right )}}\)	\(191\)
elliptic	\(\frac {\sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}\, \sqrt {c x \left (-b^{2} x^{2}+a^{2}\right )}\, \left (-\frac {2 a^{2} c \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}{11 x^{6}}+\frac {26 b^{2} c \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}{77 x^{4}}-\frac {8 b^{4} c \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}{77 a^{2} x^{2}}+\frac {4 b^{5} c^{2} \sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {2 \left (x -\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{77 a \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}\right )}{\sqrt {x}\, c \left (-b^{2} x^{2}+a^{2}\right )}\)	\(229\)

Input:

int((b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2)/x^(13/2),x,method=_RETURNVERBOSE)
 

Output:

2/77*(b*x+a)^(1/2)*(c*(-b*x+a))^(1/2)*c*(2*2^(1/2)*((b*x+a)/a)^(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^ 
5*x^5+4*b^6*x^6-17*a^2*x^4*b^4+20*a^4*x^2*b^2-7*a^6)/x^(11/2)/a^2/(-b^2*x^ 
2+a^2)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.48 \[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^{13/2}} \, dx=-\frac {2 \, {\left (4 \, \sqrt {-b^{2} c} b^{4} c x^{6} {\rm weierstrassPInverse}\left (\frac {4 \, a^{2}}{b^{2}}, 0, x\right ) + {\left (4 \, b^{4} c x^{4} - 13 \, a^{2} b^{2} c x^{2} + 7 \, a^{4} c\right )} \sqrt {-b c x + a c} \sqrt {b x + a} \sqrt {x}\right )}}{77 \, a^{2} x^{6}} \] Input:

integrate((b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2)/x^(13/2),x, algorithm="fricas")
 

Output:

-2/77*(4*sqrt(-b^2*c)*b^4*c*x^6*weierstrassPInverse(4*a^2/b^2, 0, x) + (4* 
b^4*c*x^4 - 13*a^2*b^2*c*x^2 + 7*a^4*c)*sqrt(-b*c*x + a*c)*sqrt(b*x + a)*s 
qrt(x))/(a^2*x^6)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^{13/2}} \, dx=\text {Timed out} \] Input:

integrate((b*x+a)**(3/2)*(-b*c*x+a*c)**(3/2)/x**(13/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^{13/2}} \, dx=\int { \frac {{\left (-b c x + a c\right )}^{\frac {3}{2}} {\left (b x + a\right )}^{\frac {3}{2}}}{x^{\frac {13}{2}}} \,d x } \] Input:

integrate((b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2)/x^(13/2),x, algorithm="maxima")
 

Output:

integrate((-b*c*x + a*c)^(3/2)*(b*x + a)^(3/2)/x^(13/2), x)
 

Giac [F]

\[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^{13/2}} \, dx=\int { \frac {{\left (-b c x + a c\right )}^{\frac {3}{2}} {\left (b x + a\right )}^{\frac {3}{2}}}{x^{\frac {13}{2}}} \,d x } \] Input:

integrate((b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2)/x^(13/2),x, algorithm="giac")
 

Output:

integrate((-b*c*x + a*c)^(3/2)*(b*x + a)^(3/2)/x^(13/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^{13/2}} \, dx=\int \frac {{\left (a\,c-b\,c\,x\right )}^{3/2}\,{\left (a+b\,x\right )}^{3/2}}{x^{13/2}} \,d x \] Input:

int(((a*c - b*c*x)^(3/2)*(a + b*x)^(3/2))/x^(13/2),x)
 

Output:

int(((a*c - b*c*x)^(3/2)*(a + b*x)^(3/2))/x^(13/2), x)
 

Reduce [F]

\[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^{13/2}} \, dx=\int \frac {\left (b x +a \right )^{\frac {3}{2}} \left (-b c x +a c \right )^{\frac {3}{2}}}{x^{\frac {13}{2}}}d x \] Input:

int((b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2)/x^(13/2),x)
 

Output:

int((b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2)/x^(13/2),x)