\(\int \frac {x^{15/2}}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx\) [184]

Optimal result

Integrand size = 28, antiderivative size = 227 \[ \int \frac {x^{15/2}}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=\frac {x^{13/2}}{3 b^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}-\frac {13 x^{9/2}}{6 b^4 c^2 \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {65 a^2 \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}{14 b^8 c^3}-\frac {39 x^{5/2} \sqrt {a+b x} \sqrt {a c-b c x}}{14 b^6 c^3}+\frac {65 a^{9/2} \sqrt {1-\frac {b^2 x^2}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),-1\right )}{14 b^{17/2} c^2 \sqrt {a+b x} \sqrt {a c-b c x}} \] Output:

1/3*x^(13/2)/b^2/c/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2)-13/6*x^(9/2)/b^4/c^2/( 
b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)-65/14*a^2*x^(1/2)*(b*x+a)^(1/2)*(-b*c*x+a* 
c)^(1/2)/b^8/c^3-39/14*x^(5/2)*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/b^6/c^3+65 
/14*a^(9/2)*(1-b^2*x^2/a^2)^(1/2)*EllipticF(b^(1/2)*x^(1/2)/a^(1/2),I)/b^( 
17/2)/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 = 10.07 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.62 \[ \int \frac {x^{15/2}}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=\frac {\sqrt {x} \left (195 a^6-273 a^4 b^2 x^2+52 a^2 b^4 x^4+12 b^6 x^6-195 a^4 \left (a^2-b^2 x^2\right ) \sqrt {1-\frac {b^2 x^2}{a^2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {b^2 x^2}{a^2}\right )\right )}{42 b^8 c^2 \sqrt {c (a-b x)} \sqrt {a+b x} \left (-a^2+b^2 x^2\right )} \] Input:

Integrate[x^(15/2)/((a + b*x)^(5/2)*(a*c - b*c*x)^(5/2)),x]
 

Output:

(Sqrt[x]*(195*a^6 - 273*a^4*b^2*x^2 + 52*a^2*b^4*x^4 + 12*b^6*x^6 - 195*a^ 
4*(a^2 - b^2*x^2)*Sqrt[1 - (b^2*x^2)/a^2]*Hypergeometric2F1[1/4, 1/2, 5/4, 
 (b^2*x^2)/a^2]))/(42*b^8*c^2*Sqrt[c*(a - b*x)]*Sqrt[a + b*x]*(-a^2 + b^2* 
x^2))
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.14, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {109, 27, 35, 109, 27, 35, 113, 27, 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.

\(\displaystyle \int \frac {x^{15/2}}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx\)

\(\Big \downarrow \) 109

\(\displaystyle \frac {x^{13/2}}{3 b^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}-\frac {\int \frac {13 a c x^{11/2} (a-b x)}{2 (a+b x)^{3/2} (a c-b c x)^{5/2}}dx}{3 a b^2 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x^{13/2}}{3 b^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}-\frac {13 \int \frac {x^{11/2} (a-b x)}{(a+b x)^{3/2} (a c-b c x)^{5/2}}dx}{6 b^2}\)

\(\Big \downarrow \) 35

\(\displaystyle \frac {x^{13/2}}{3 b^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}-\frac {13 \int \frac {x^{11/2}}{(a+b x)^{3/2} (a c-b c x)^{3/2}}dx}{6 b^2 c}\)

\(\Big \downarrow \) 109

\(\displaystyle \frac {x^{13/2}}{3 b^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}-\frac {13 \left (\frac {x^{9/2}}{b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {\int \frac {9 a c x^{7/2} (a-b x)}{2 \sqrt {a+b x} (a c-b c x)^{3/2}}dx}{a b^2 c}\right )}{6 b^2 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x^{13/2}}{3 b^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}-\frac {13 \left (\frac {x^{9/2}}{b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {9 \int \frac {x^{7/2} (a-b x)}{\sqrt {a+b x} (a c-b c x)^{3/2}}dx}{2 b^2}\right )}{6 b^2 c}\)

\(\Big \downarrow \) 35

\(\displaystyle \frac {x^{13/2}}{3 b^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}-\frac {13 \left (\frac {x^{9/2}}{b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {9 \int \frac {x^{7/2}}{\sqrt {a+b x} \sqrt {a c-b c x}}dx}{2 b^2 c}\right )}{6 b^2 c}\)

\(\Big \downarrow \) 113

\(\displaystyle \frac {x^{13/2}}{3 b^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}-\frac {13 \left (\frac {x^{9/2}}{b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {9 \left (-\frac {2 \int -\frac {5 a^2 c x^{3/2}}{2 \sqrt {a+b x} \sqrt {a c-b c x}}dx}{7 b^2 c}-\frac {2 x^{5/2} \sqrt {a+b x} \sqrt {a c-b c x}}{7 b^2 c}\right )}{2 b^2 c}\right )}{6 b^2 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x^{13/2}}{3 b^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}-\frac {13 \left (\frac {x^{9/2}}{b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {9 \left (\frac {5 a^2 \int \frac {x^{3/2}}{\sqrt {a+b x} \sqrt {a c-b c x}}dx}{7 b^2}-\frac {2 x^{5/2} \sqrt {a+b x} \sqrt {a c-b c x}}{7 b^2 c}\right )}{2 b^2 c}\right )}{6 b^2 c}\)

\(\Big \downarrow \) 113

\(\displaystyle \frac {x^{13/2}}{3 b^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}-\frac {13 \left (\frac {x^{9/2}}{b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {9 \left (\frac {5 a^2 \left (-\frac {2 \int -\frac {a^2 c}{2 \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}dx}{3 b^2 c}-\frac {2 \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}{3 b^2 c}\right )}{7 b^2}-\frac {2 x^{5/2} \sqrt {a+b x} \sqrt {a c-b c x}}{7 b^2 c}\right )}{2 b^2 c}\right )}{6 b^2 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x^{13/2}}{3 b^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}-\frac {13 \left (\frac {x^{9/2}}{b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {9 \left (\frac {5 a^2 \left (\frac {a^2 \int \frac {1}{\sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}dx}{3 b^2}-\frac {2 \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}{3 b^2 c}\right )}{7 b^2}-\frac {2 x^{5/2} \sqrt {a+b x} \sqrt {a c-b c x}}{7 b^2 c}\right )}{2 b^2 c}\right )}{6 b^2 c}\)

\(\Big \downarrow \) 127

\(\displaystyle \frac {x^{13/2}}{3 b^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}-\frac {13 \left (\frac {x^{9/2}}{b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {9 \left (\frac {5 a^2 \left (\frac {a^2 \sqrt {1-\frac {b x}{a}} \sqrt {\frac {b x}{a}+1} \int \frac {1}{\sqrt {x} \sqrt {1-\frac {b x}{a}} \sqrt {\frac {b x}{a}+1}}dx}{3 b^2 \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {2 \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}{3 b^2 c}\right )}{7 b^2}-\frac {2 x^{5/2} \sqrt {a+b x} \sqrt {a c-b c x}}{7 b^2 c}\right )}{2 b^2 c}\right )}{6 b^2 c}\)

\(\Big \downarrow \) 126

\(\displaystyle \frac {x^{13/2}}{3 b^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}-\frac {13 \left (\frac {x^{9/2}}{b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {9 \left (\frac {5 a^2 \left (\frac {2 a^{5/2} \sqrt {1-\frac {b x}{a}} \sqrt {\frac {b x}{a}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),-1\right )}{3 b^{5/2} \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {2 \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}{3 b^2 c}\right )}{7 b^2}-\frac {2 x^{5/2} \sqrt {a+b x} \sqrt {a c-b c x}}{7 b^2 c}\right )}{2 b^2 c}\right )}{6 b^2 c}\)

Input:

Int[x^(15/2)/((a + b*x)^(5/2)*(a*c - b*c*x)^(5/2)),x]
 

Output:

x^(13/2)/(3*b^2*c*(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2)) - (13*(x^(9/2)/(b^2 
*c*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]) - (9*((-2*x^(5/2)*Sqrt[a + b*x]*Sqrt[a 
*c - b*c*x])/(7*b^2*c) + (5*a^2*((-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[A 
rcSin[(Sqrt[b]*Sqrt[x])/Sqrt[a]], -1])/(3*b^(5/2)*Sqrt[a + b*x]*Sqrt[a*c - 
 b*c*x])))/(7*b^2)))/(2*b^2*c)))/(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 113

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]
 

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 = 4.48 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.86

method	result	size
default	\(\frac {\left (-195 \operatorname {EllipticF}\left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a^{5} b^{2} x^{2} \sqrt {\frac {b x +a}{a}}\, \sqrt {\frac {-b x +a}{a}}\, \sqrt {-\frac {b x}{a}}-24 b^{7} x^{7}+195 \sqrt {\frac {b x +a}{a}}\, \sqrt {2}\, \sqrt {\frac {-b x +a}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right ) a^{7}-104 a^{2} b^{5} x^{5}+546 a^{4} b^{3} x^{3}-390 a^{6} b x \right ) \sqrt {c \left (-b x +a \right )}}{84 c^{3} \sqrt {x}\, \left (-b x +a \right )^{2} b^{9} \left (b x +a \right )^{\frac {3}{2}}}\)	\(195\)
elliptic	\(\frac {\sqrt {c x \left (-b^{2} x^{2}+a^{2}\right )}\, \left (\frac {a^{6} \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}{3 c^{3} b^{12} \left (x^{2}-\frac {a^{2}}{b^{2}}\right )^{2}}-\frac {19 x \,a^{4}}{6 b^{8} c^{2} \sqrt {-\left (x^{2}-\frac {a^{2}}{b^{2}}\right ) b^{2} c x}}-\frac {2 x^{2} \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}{7 b^{6} c^{3}}-\frac {38 a^{2} \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}{21 b^{8} c^{3}}+\frac {65 a^{5} \sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {2 \left (x -\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{28 c^{2} b^{9} \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}\right )}{\sqrt {x}\, \sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}}\)	\(262\)
risch	\(\text {Expression too large to display}\)	\(1281\)

Input:

int(x^(15/2)/(b*x+a)^(5/2)/(-b*c*x+a*c)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/84*(-195*EllipticF(((b*x+a)/a)^(1/2),1/2*2^(1/2))*2^(1/2)*a^5*b^2*x^2*(( 
b*x+a)/a)^(1/2)*((-b*x+a)/a)^(1/2)*(-b*x/a)^(1/2)-24*b^7*x^7+195*((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^7-104*a^2*b^5*x^5+546*a^4*b^3*x^3-390*a^6*b*x)*(c*(-b* 
x+a))^(1/2)/c^3/x^(1/2)/(-b*x+a)^2/b^9/(b*x+a)^(3/2)
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.65 \[ \int \frac {x^{15/2}}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=-\frac {{\left (12 \, b^{8} x^{6} + 52 \, a^{2} b^{6} x^{4} - 273 \, a^{4} b^{4} x^{2} + 195 \, a^{6} b^{2}\right )} \sqrt {-b c x + a c} \sqrt {b x + a} \sqrt {x} + 195 \, {\left (a^{4} b^{4} x^{4} - 2 \, a^{6} b^{2} x^{2} + a^{8}\right )} \sqrt {-b^{2} c} {\rm weierstrassPInverse}\left (\frac {4 \, a^{2}}{b^{2}}, 0, x\right )}{42 \, {\left (b^{14} c^{3} x^{4} - 2 \, a^{2} b^{12} c^{3} x^{2} + a^{4} b^{10} c^{3}\right )}} \] Input:

integrate(x^(15/2)/(b*x+a)^(5/2)/(-b*c*x+a*c)^(5/2),x, algorithm="fricas")
 

Output:

-1/42*((12*b^8*x^6 + 52*a^2*b^6*x^4 - 273*a^4*b^4*x^2 + 195*a^6*b^2)*sqrt( 
-b*c*x + a*c)*sqrt(b*x + a)*sqrt(x) + 195*(a^4*b^4*x^4 - 2*a^6*b^2*x^2 + a 
^8)*sqrt(-b^2*c)*weierstrassPInverse(4*a^2/b^2, 0, x))/(b^14*c^3*x^4 - 2*a 
^2*b^12*c^3*x^2 + a^4*b^10*c^3)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^{15/2}}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(x**(15/2)/(b*x+a)**(5/2)/(-b*c*x+a*c)**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {x^{15/2}}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=\int { \frac {x^{\frac {15}{2}}}{{\left (-b c x + a c\right )}^{\frac {5}{2}} {\left (b x + a\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(x^(15/2)/(b*x+a)^(5/2)/(-b*c*x+a*c)^(5/2),x, algorithm="maxima")
 

Output:

integrate(x^(15/2)/((-b*c*x + a*c)^(5/2)*(b*x + a)^(5/2)), x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {x^{15/2}}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(x^(15/2)/(b*x+a)^(5/2)/(-b*c*x+a*c)^(5/2),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{15/2}}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=\int \frac {x^{15/2}}{{\left (a\,c-b\,c\,x\right )}^{5/2}\,{\left (a+b\,x\right )}^{5/2}} \,d x \] Input:

int(x^(15/2)/((a*c - b*c*x)^(5/2)*(a + b*x)^(5/2)),x)
 

Output:

int(x^(15/2)/((a*c - b*c*x)^(5/2)*(a + b*x)^(5/2)), x)
 

Reduce [F]

\[ \int \frac {x^{15/2}}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=\int \frac {x^{\frac {15}{2}}}{\left (b x +a \right )^{\frac {5}{2}} \left (-b c x +a c \right )^{\frac {5}{2}}}d x \] Input:

int(x^(15/2)/(b*x+a)^(5/2)/(-b*c*x+a*c)^(5/2),x)
 

Output:

int(x^(15/2)/(b*x+a)^(5/2)/(-b*c*x+a*c)^(5/2),x)