\(\int \frac {x^{7/2}}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx\) [186]

Optimal result

Integrand size = 28, antiderivative size = 150 \[ \int \frac {x^{7/2}}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=\frac {x^{5/2}}{3 b^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}-\frac {5 \sqrt {x}}{6 b^4 c^2 \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {5 \sqrt {a} \sqrt {1-\frac {b^2 x^2}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),-1\right )}{6 b^{9/2} c^2 \sqrt {a+b x} \sqrt {a c-b c x}} \] Output:

1/3*x^(5/2)/b^2/c/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2)-5/6*x^(1/2)/b^4/c^2/(b* 
x+a)^(1/2)/(-b*c*x+a*c)^(1/2)+5/6*a^(1/2)*(1-b^2*x^2/a^2)^(1/2)*EllipticF( 
b^(1/2)*x^(1/2)/a^(1/2),I)/b^(9/2)/c^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 = 9.58 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.77 \[ \int \frac {x^{7/2}}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=\frac {\sqrt {x} \left (5 a^2-7 b^2 x^2-5 \left (a^2-b^2 x^2\right ) \sqrt {1-\frac {b^2 x^2}{a^2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {b^2 x^2}{a^2}\right )\right )}{6 b^4 c^2 \sqrt {c (a-b x)} \sqrt {a+b x} \left (-a^2+b^2 x^2\right )} \] Input:

Integrate[x^(7/2)/((a + b*x)^(5/2)*(a*c - b*c*x)^(5/2)),x]
 

Output:

(Sqrt[x]*(5*a^2 - 7*b^2*x^2 - 5*(a^2 - b^2*x^2)*Sqrt[1 - (b^2*x^2)/a^2]*Hy 
pergeometric2F1[1/4, 1/2, 5/4, (b^2*x^2)/a^2]))/(6*b^4*c^2*Sqrt[c*(a - b*x 
)]*Sqrt[a + b*x]*(-a^2 + b^2*x^2))
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.09, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {109, 27, 35, 109, 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.

\(\displaystyle \int \frac {x^{7/2}}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx\)

\(\Big \downarrow \) 109

\(\displaystyle \frac {x^{5/2}}{3 b^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}-\frac {\int \frac {5 a c x^{3/2} (a-b x)}{2 (a+b x)^{3/2} (a c-b c x)^{5/2}}dx}{3 a b^2 c}\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 35

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

\(\Big \downarrow \) 109

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 35

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

\(\Big \downarrow \) 127

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

\(\Big \downarrow \) 126

\(\displaystyle \frac {x^{5/2}}{3 b^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}-\frac {5 \left (\frac {\sqrt {x}}{b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {\sqrt {a} \sqrt {1-\frac {b x}{a}} \sqrt {\frac {b x}{a}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),-1\right )}{b^{5/2} c \sqrt {a+b x} \sqrt {a c-b c x}}\right )}{6 b^2 c}\)

Input:

Int[x^(7/2)/((a + b*x)^(5/2)*(a*c - b*c*x)^(5/2)),x]
 

Output:

x^(5/2)/(3*b^2*c*(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2)) - (5*(Sqrt[x]/(b^2*c 
*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]) - (Sqrt[a]*Sqrt[1 - (b*x)/a]*Sqrt[1 + (b 
*x)/a]*EllipticF[ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[a]], -1])/(b^(5/2)*c*Sqrt[a 
 + b*x]*Sqrt[a*c - b*c*x])))/(6*b^2*c)
 

Defintions of rubi rules used

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 35

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])
 

rule 109

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])
 

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 = 2.06 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.14

method	result	size
default	\(\frac {\left (-5 \operatorname {EllipticF}\left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a \,b^{2} x^{2} \sqrt {\frac {b x +a}{a}}\, \sqrt {\frac {-b x +a}{a}}\, \sqrt {-\frac {b x}{a}}+5 \sqrt {\frac {b x +a}{a}}\, \sqrt {2}\, \sqrt {\frac {-b x +a}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right ) a^{3}+14 b^{3} x^{3}-10 a^{2} b x \right ) \sqrt {c \left (-b x +a \right )}}{12 c^{3} \sqrt {x}\, \left (-b x +a \right )^{2} b^{5} \left (b x +a \right )^{\frac {3}{2}}}\)	\(171\)
elliptic	\(\frac {\sqrt {c x \left (-b^{2} x^{2}+a^{2}\right )}\, \left (\frac {a^{2} \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}{3 c^{3} b^{8} \left (x^{2}-\frac {a^{2}}{b^{2}}\right )^{2}}-\frac {7 x}{6 b^{4} c^{2} \sqrt {-\left (x^{2}-\frac {a^{2}}{b^{2}}\right ) b^{2} c x}}+\frac {5 a \sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {2 \left (x -\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{12 b^{5} c^{2} \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}\right )}{\sqrt {x}\, \sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}}\)	\(199\)

Input:

int(x^(7/2)/(b*x+a)^(5/2)/(-b*c*x+a*c)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/12*(-5*EllipticF(((b*x+a)/a)^(1/2),1/2*2^(1/2))*2^(1/2)*a*b^2*x^2*((b*x+ 
a)/a)^(1/2)*((-b*x+a)/a)^(1/2)*(-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+14*b^3*x^3-10*a^2*b*x)*(c*(-b*x+a))^(1/2)/c^3/x^(1/2)/(-b*x+a)^2/b^5/ 
(b*x+a)^(3/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.81 \[ \int \frac {x^{7/2}}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=\frac {{\left (7 \, b^{4} x^{2} - 5 \, a^{2} b^{2}\right )} \sqrt {-b c x + a c} \sqrt {b x + a} \sqrt {x} - 5 \, {\left (b^{4} x^{4} - 2 \, a^{2} b^{2} x^{2} + a^{4}\right )} \sqrt {-b^{2} c} {\rm weierstrassPInverse}\left (\frac {4 \, a^{2}}{b^{2}}, 0, x\right )}{6 \, {\left (b^{10} c^{3} x^{4} - 2 \, a^{2} b^{8} c^{3} x^{2} + a^{4} b^{6} c^{3}\right )}} \] Input:

integrate(x^(7/2)/(b*x+a)^(5/2)/(-b*c*x+a*c)^(5/2),x, algorithm="fricas")
 

Output:

1/6*((7*b^4*x^2 - 5*a^2*b^2)*sqrt(-b*c*x + a*c)*sqrt(b*x + a)*sqrt(x) - 5* 
(b^4*x^4 - 2*a^2*b^2*x^2 + a^4)*sqrt(-b^2*c)*weierstrassPInverse(4*a^2/b^2 
, 0, x))/(b^10*c^3*x^4 - 2*a^2*b^8*c^3*x^2 + a^4*b^6*c^3)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^{7/2}}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(x**(7/2)/(b*x+a)**(5/2)/(-b*c*x+a*c)**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {x^{7/2}}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=\int { \frac {x^{\frac {7}{2}}}{{\left (-b c x + a c\right )}^{\frac {5}{2}} {\left (b x + a\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(x^(7/2)/(b*x+a)^(5/2)/(-b*c*x+a*c)^(5/2),x, algorithm="maxima")
 

Output:

integrate(x^(7/2)/((-b*c*x + a*c)^(5/2)*(b*x + a)^(5/2)), x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {x^{7/2}}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(x^(7/2)/(b*x+a)^(5/2)/(-b*c*x+a*c)^(5/2),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{7/2}}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=\int \frac {x^{7/2}}{{\left (a\,c-b\,c\,x\right )}^{5/2}\,{\left (a+b\,x\right )}^{5/2}} \,d x \] Input:

int(x^(7/2)/((a*c - b*c*x)^(5/2)*(a + b*x)^(5/2)),x)
 

Output:

int(x^(7/2)/((a*c - b*c*x)^(5/2)*(a + b*x)^(5/2)), x)
 

Reduce [F]

\[ \int \frac {x^{7/2}}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=\int \frac {x^{\frac {7}{2}}}{\left (b x +a \right )^{\frac {5}{2}} \left (-b c x +a c \right )^{\frac {5}{2}}}d x \] Input:

int(x^(7/2)/(b*x+a)^(5/2)/(-b*c*x+a*c)^(5/2),x)
 

Output:

int(x^(7/2)/(b*x+a)^(5/2)/(-b*c*x+a*c)^(5/2),x)