\(\int \frac {1}{(a+b x)^{9/2} (a c-b c x)^{9/2}} \, dx\) [1152]

Optimal result

Integrand size = 23, antiderivative size = 133 \[ \int \frac {1}{(a+b x)^{9/2} (a c-b c x)^{9/2}} \, dx=\frac {x}{7 a^2 c (a+b x)^{7/2} (a c-b c x)^{7/2}}+\frac {6 x}{35 a^4 c^2 (a+b x)^{5/2} (a c-b c x)^{5/2}}+\frac {8 x}{35 a^6 c^3 (a+b x)^{3/2} (a c-b c x)^{3/2}}+\frac {16 x}{35 a^8 c^4 \sqrt {a+b x} \sqrt {a c-b c x}} \]

[Out]

1/7*x/a^2/c/(b*x+a)^(7/2)/(-b*c*x+a*c)^(7/2)+6/35*x/a^4/c^2/(b*x+a)^(5/2)/(-b*c*x+a*c)^(5/2)+8/35*x/a^6/c^3/(b
*x+a)^(3/2)/(-b*c*x+a*c)^(3/2)+16/35*x/a^8/c^4/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {40, 39} \[ \int \frac {1}{(a+b x)^{9/2} (a c-b c x)^{9/2}} \, dx=\frac {16 x}{35 a^8 c^4 \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {8 x}{35 a^6 c^3 (a+b x)^{3/2} (a c-b c x)^{3/2}}+\frac {6 x}{35 a^4 c^2 (a+b x)^{5/2} (a c-b c x)^{5/2}}+\frac {x}{7 a^2 c (a+b x)^{7/2} (a c-b c x)^{7/2}} \]

[In]

Int[1/((a + b*x)^(9/2)*(a*c - b*c*x)^(9/2)),x]

[Out]

x/(7*a^2*c*(a + b*x)^(7/2)*(a*c - b*c*x)^(7/2)) + (6*x)/(35*a^4*c^2*(a + b*x)^(5/2)*(a*c - b*c*x)^(5/2)) + (8*
x)/(35*a^6*c^3*(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2)) + (16*x)/(35*a^8*c^4*Sqrt[a + b*x]*Sqrt[a*c - b*c*x])

Rule 39

Int[1/(((a_) + (b_.)*(x_))^(3/2)*((c_) + (d_.)*(x_))^(3/2)), x_Symbol] :> Simp[x/(a*c*Sqrt[a + b*x]*Sqrt[c + d
*x]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0]

Rule 40

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(-x)*(a + b*x)^(m + 1)*((c + d*x)^(m
+ 1)/(2*a*c*(m + 1))), x] + Dist[(2*m + 3)/(2*a*c*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(m + 1), x], x] /;
FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && ILtQ[m + 3/2, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {x}{7 a^2 c (a+b x)^{7/2} (a c-b c x)^{7/2}}+\frac {6 \int \frac {1}{(a+b x)^{7/2} (a c-b c x)^{7/2}} \, dx}{7 a^2 c} \\ & = \frac {x}{7 a^2 c (a+b x)^{7/2} (a c-b c x)^{7/2}}+\frac {6 x}{35 a^4 c^2 (a+b x)^{5/2} (a c-b c x)^{5/2}}+\frac {24 \int \frac {1}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx}{35 a^4 c^2} \\ & = \frac {x}{7 a^2 c (a+b x)^{7/2} (a c-b c x)^{7/2}}+\frac {6 x}{35 a^4 c^2 (a+b x)^{5/2} (a c-b c x)^{5/2}}+\frac {8 x}{35 a^6 c^3 (a+b x)^{3/2} (a c-b c x)^{3/2}}+\frac {16 \int \frac {1}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx}{35 a^6 c^3} \\ & = \frac {x}{7 a^2 c (a+b x)^{7/2} (a c-b c x)^{7/2}}+\frac {6 x}{35 a^4 c^2 (a+b x)^{5/2} (a c-b c x)^{5/2}}+\frac {8 x}{35 a^6 c^3 (a+b x)^{3/2} (a c-b c x)^{3/2}}+\frac {16 x}{35 a^8 c^4 \sqrt {a+b x} \sqrt {a c-b c x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.51 \[ \int \frac {1}{(a+b x)^{9/2} (a c-b c x)^{9/2}} \, dx=\frac {35 a^6 x-70 a^4 b^2 x^3+56 a^2 b^4 x^5-16 b^6 x^7}{35 a^8 c (c (a-b x))^{7/2} (a+b x)^{7/2}} \]

[In]

Integrate[1/((a + b*x)^(9/2)*(a*c - b*c*x)^(9/2)),x]

[Out]

(35*a^6*x - 70*a^4*b^2*x^3 + 56*a^2*b^4*x^5 - 16*b^6*x^7)/(35*a^8*c*(c*(a - b*x))^(7/2)*(a + b*x)^(7/2))

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.50

method	result	size
gosper	\(\frac {\left (-b x +a \right ) x \left (-16 b^{6} x^{6}+56 a^{2} x^{4} b^{4}-70 a^{4} x^{2} b^{2}+35 a^{6}\right )}{35 \left (b x +a \right )^{\frac {7}{2}} a^{8} \left (-b c x +a c \right )^{\frac {9}{2}}}\)	\(67\)
default	\(-\frac {1}{7 b a c \left (b x +a \right )^{\frac {7}{2}} \left (-b c x +a c \right )^{\frac {7}{2}}}+\frac {-\frac {1}{5 b a c \left (b x +a \right )^{\frac {5}{2}} \left (-b c x +a c \right )^{\frac {7}{2}}}+\frac {-\frac {2}{5 b a c \left (b x +a \right )^{\frac {3}{2}} \left (-b c x +a c \right )^{\frac {7}{2}}}+\frac {6 \left (-\frac {5}{3 b a c \sqrt {b x +a}\, \left (-b c x +a c \right )^{\frac {7}{2}}}+\frac {5 \left (\frac {4 \sqrt {b x +a}}{7 b a c \left (-b c x +a c \right )^{\frac {7}{2}}}+\frac {4 \left (\frac {3 \sqrt {b x +a}}{35 b a c \left (-b c x +a c \right )^{\frac {5}{2}}}+\frac {3 \left (\frac {2 \sqrt {b x +a}}{15 b a c \left (-b c x +a c \right )^{\frac {3}{2}}}+\frac {2 \sqrt {b x +a}}{15 b \,a^{2} c^{2} \sqrt {-b c x +a c}}\right )}{7 a c}\right )}{a c}\right )}{3 a}\right )}{5 a}}{a}}{a}\)	\(275\)

[In]

int(1/(b*x+a)^(9/2)/(-b*c*x+a*c)^(9/2),x,method=_RETURNVERBOSE)

[Out]

1/35*(-b*x+a)*x*(-16*b^6*x^6+56*a^2*b^4*x^4-70*a^4*b^2*x^2+35*a^6)/(b*x+a)^(7/2)/a^8/(-b*c*x+a*c)^(9/2)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.92 \[ \int \frac {1}{(a+b x)^{9/2} (a c-b c x)^{9/2}} \, dx=-\frac {{\left (16 \, b^{6} x^{7} - 56 \, a^{2} b^{4} x^{5} + 70 \, a^{4} b^{2} x^{3} - 35 \, a^{6} x\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{35 \, {\left (a^{8} b^{8} c^{5} x^{8} - 4 \, a^{10} b^{6} c^{5} x^{6} + 6 \, a^{12} b^{4} c^{5} x^{4} - 4 \, a^{14} b^{2} c^{5} x^{2} + a^{16} c^{5}\right )}} \]

[In]

integrate(1/(b*x+a)^(9/2)/(-b*c*x+a*c)^(9/2),x, algorithm="fricas")

[Out]

-1/35*(16*b^6*x^7 - 56*a^2*b^4*x^5 + 70*a^4*b^2*x^3 - 35*a^6*x)*sqrt(-b*c*x + a*c)*sqrt(b*x + a)/(a^8*b^8*c^5*
x^8 - 4*a^10*b^6*c^5*x^6 + 6*a^12*b^4*c^5*x^4 - 4*a^14*b^2*c^5*x^2 + a^16*c^5)

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(a+b x)^{9/2} (a c-b c x)^{9/2}} \, dx=\text {Timed out} \]

[In]

integrate(1/(b*x+a)**(9/2)/(-b*c*x+a*c)**(9/2),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.79 \[ \int \frac {1}{(a+b x)^{9/2} (a c-b c x)^{9/2}} \, dx=\frac {x}{7 \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {7}{2}} a^{2} c} + \frac {6 \, x}{35 \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {5}{2}} a^{4} c^{2}} + \frac {8 \, x}{35 \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} a^{6} c^{3}} + \frac {16 \, x}{35 \, \sqrt {-b^{2} c x^{2} + a^{2} c} a^{8} c^{4}} \]

[In]

integrate(1/(b*x+a)^(9/2)/(-b*c*x+a*c)^(9/2),x, algorithm="maxima")

[Out]

1/7*x/((-b^2*c*x^2 + a^2*c)^(7/2)*a^2*c) + 6/35*x/((-b^2*c*x^2 + a^2*c)^(5/2)*a^4*c^2) + 8/35*x/((-b^2*c*x^2 +
a^2*c)^(3/2)*a^6*c^3) + 16/35*x/(sqrt(-b^2*c*x^2 + a^2*c)*a^8*c^4)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 393 vs. \(2 (109) = 218\).

Time = 0.46 (sec) , antiderivative size = 393, normalized size of antiderivative = 2.95 \[ \int \frac {1}{(a+b x)^{9/2} (a c-b c x)^{9/2}} \, dx=-\frac {\frac {{\left ({\left (b x + a\right )} {\left ({\left (b x + a\right )} {\left (\frac {256 \, {\left (b x + a\right )}}{a^{8} c} - \frac {1617}{a^{7} c}\right )} + \frac {3430}{a^{6} c}\right )} - \frac {2450}{a^{5} c}\right )} \sqrt {-{\left (b x + a\right )} c + 2 \, a c} \sqrt {b x + a}}{{\left ({\left (b x + a\right )} c - 2 \, a c\right )}^{4}} + \frac {4 \, {\left (175 \, {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{12} - 2450 \, a {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{10} c + 14280 \, a^{2} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{8} c^{2} - 43120 \, a^{3} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{6} c^{3} + 66416 \, a^{4} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{4} c^{4} - 51744 \, a^{5} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2} c^{5} + 16384 \, a^{6} c^{6}\right )}}{{\left ({\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2} - 2 \, a c\right )}^{7} a^{7} \sqrt {-c} c^{3}}}{1120 \, b} \]

[In]

integrate(1/(b*x+a)^(9/2)/(-b*c*x+a*c)^(9/2),x, algorithm="giac")

[Out]

-1/1120*(((b*x + a)*((b*x + a)*(256*(b*x + a)/(a^8*c) - 1617/(a^7*c)) + 3430/(a^6*c)) - 2450/(a^5*c))*sqrt(-(b
*x + a)*c + 2*a*c)*sqrt(b*x + a)/((b*x + a)*c - 2*a*c)^4 + 4*(175*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c
+ 2*a*c))^12 - 2450*a*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^10*c + 14280*a^2*(sqrt(b*x + a)*sq
rt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^8*c^2 - 43120*a^3*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^6
*c^3 + 66416*a^4*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^4*c^4 - 51744*a^5*(sqrt(b*x + a)*sqrt(-
c) - sqrt(-(b*x + a)*c + 2*a*c))^2*c^5 + 16384*a^6*c^6)/(((sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c)
)^2 - 2*a*c)^7*a^7*sqrt(-c)*c^3))/b

Mupad [B] (verification not implemented)

Time = 0.78 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.28 \[ \int \frac {1}{(a+b x)^{9/2} (a c-b c x)^{9/2}} \, dx=-\frac {35\,a^6\,x\,\sqrt {a\,c-b\,c\,x}-16\,b^6\,x^7\,\sqrt {a\,c-b\,c\,x}-70\,a^4\,b^2\,x^3\,\sqrt {a\,c-b\,c\,x}+56\,a^2\,b^4\,x^5\,\sqrt {a\,c-b\,c\,x}}{\left (\left (70\,a^9\,{\left (a\,c-b\,c\,x\right )}^5+35\,a^8\,{\left (a\,c-b\,c\,x\right )}^5\,\left (a+b\,x\right )\right )\,\left (a+b\,x\right )+{\left (a\,c-b\,c\,x\right )}^4\,\left (140\,a^{10}\,\left (a\,c-b\,c\,x\right )-280\,a^{11}\,c\right )\right )\,\sqrt {a+b\,x}} \]

[In]

int(1/((a*c - b*c*x)^(9/2)*(a + b*x)^(9/2)),x)

[Out]

-(35*a^6*x*(a*c - b*c*x)^(1/2) - 16*b^6*x^7*(a*c - b*c*x)^(1/2) - 70*a^4*b^2*x^3*(a*c - b*c*x)^(1/2) + 56*a^2*
b^4*x^5*(a*c - b*c*x)^(1/2))/(((70*a^9*(a*c - b*c*x)^5 + 35*a^8*(a*c - b*c*x)^5*(a + b*x))*(a + b*x) + (a*c -
b*c*x)^4*(140*a^10*(a*c - b*c*x) - 280*a^11*c))*(a + b*x)^(1/2))