\(\int \frac {1}{(a+a x)^{5/2} (c-c x)^{5/2}} \, dx\) [1142]

Optimal result

Integrand size = 20, antiderivative size = 61 \[ \int \frac {1}{(a+a x)^{5/2} (c-c x)^{5/2}} \, dx=\frac {x}{3 a c (a+a x)^{3/2} (c-c x)^{3/2}}+\frac {2 x}{3 a^2 c^2 \sqrt {a+a x} \sqrt {c-c x}} \]

[Out]

1/3*x/a/c/(a*x+a)^(3/2)/(-c*x+c)^(3/2)+2/3*x/a^2/c^2/(a*x+a)^(1/2)/(-c*x+c)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {40, 39} \[ \int \frac {1}{(a+a x)^{5/2} (c-c x)^{5/2}} \, dx=\frac {2 x}{3 a^2 c^2 \sqrt {a x+a} \sqrt {c-c x}}+\frac {x}{3 a c (a x+a)^{3/2} (c-c x)^{3/2}} \]

[In]

Int[1/((a + a*x)^(5/2)*(c - c*x)^(5/2)),x]

[Out]

x/(3*a*c*(a + a*x)^(3/2)*(c - c*x)^(3/2)) + (2*x)/(3*a^2*c^2*Sqrt[a + a*x]*Sqrt[c - 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}{3 a c (a+a x)^{3/2} (c-c x)^{3/2}}+\frac {2 \int \frac {1}{(a+a x)^{3/2} (c-c x)^{3/2}} \, dx}{3 a c} \\ & = \frac {x}{3 a c (a+a x)^{3/2} (c-c x)^{3/2}}+\frac {2 x}{3 a^2 c^2 \sqrt {a+a x} \sqrt {c-c x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.69 \[ \int \frac {1}{(a+a x)^{5/2} (c-c x)^{5/2}} \, dx=\frac {x (1+x) \left (-3+2 x^2\right )}{3 c^2 (-1+x) (a (1+x))^{5/2} \sqrt {c-c x}} \]

[In]

Integrate[1/((a + a*x)^(5/2)*(c - c*x)^(5/2)),x]

[Out]

(x*(1 + x)*(-3 + 2*x^2))/(3*c^2*(-1 + x)*(a*(1 + x))^(5/2)*Sqrt[c - c*x])

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.52

method	result	size
gosper	\(\frac {\left (-1+x \right ) \left (1+x \right ) x \left (2 x^{2}-3\right )}{3 \left (a x +a \right )^{\frac {5}{2}} \left (-c x +c \right )^{\frac {5}{2}}}\)	\(32\)
default	\(-\frac {1}{3 a c \left (a x +a \right )^{\frac {3}{2}} \left (-c x +c \right )^{\frac {3}{2}}}+\frac {-\frac {1}{a c \sqrt {a x +a}\, \left (-c x +c \right )^{\frac {3}{2}}}+\frac {\frac {2 \sqrt {a x +a}}{3 a c \left (-c x +c \right )^{\frac {3}{2}}}+\frac {2 \sqrt {a x +a}}{3 a \,c^{2} \sqrt {-c x +c}}}{a}}{a}\)	\(105\)

[In]

int(1/(a*x+a)^(5/2)/(-c*x+c)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/3*(-1+x)*(1+x)*x*(2*x^2-3)/(a*x+a)^(5/2)/(-c*x+c)^(5/2)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.93 \[ \int \frac {1}{(a+a x)^{5/2} (c-c x)^{5/2}} \, dx=-\frac {{\left (2 \, x^{3} - 3 \, x\right )} \sqrt {a x + a} \sqrt {-c x + c}}{3 \, {\left (a^{3} c^{3} x^{4} - 2 \, a^{3} c^{3} x^{2} + a^{3} c^{3}\right )}} \]

[In]

integrate(1/(a*x+a)^(5/2)/(-c*x+c)^(5/2),x, algorithm="fricas")

[Out]

-1/3*(2*x^3 - 3*x)*sqrt(a*x + a)*sqrt(-c*x + c)/(a^3*c^3*x^4 - 2*a^3*c^3*x^2 + a^3*c^3)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 10.45 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.34 \[ \int \frac {1}{(a+a x)^{5/2} (c-c x)^{5/2}} \, dx=\frac {i {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {5}{4}, \frac {7}{4}, 1 & \frac {1}{2}, \frac {5}{2}, 3 \\\frac {5}{4}, \frac {7}{4}, 2, \frac {5}{2}, 3 & 0 \end {matrix} \middle | {\frac {1}{x^{2}}} \right )}}{3 \pi ^{\frac {3}{2}} a^{\frac {5}{2}} c^{\frac {5}{2}}} + \frac {{G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{2}, 0, \frac {1}{2}, \frac {3}{4}, \frac {5}{4}, 1 & \\\frac {3}{4}, \frac {5}{4} & - \frac {1}{2}, 0, 2, 0 \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{x^{2}}} \right )}}{3 \pi ^{\frac {3}{2}} a^{\frac {5}{2}} c^{\frac {5}{2}}} \]

[In]

integrate(1/(a*x+a)**(5/2)/(-c*x+c)**(5/2),x)

[Out]

I*meijerg(((5/4, 7/4, 1), (1/2, 5/2, 3)), ((5/4, 7/4, 2, 5/2, 3), (0,)), x**(-2))/(3*pi**(3/2)*a**(5/2)*c**(5/
2)) + meijerg(((-1/2, 0, 1/2, 3/4, 5/4, 1), ()), ((3/4, 5/4), (-1/2, 0, 2, 0)), exp_polar(-2*I*pi)/x**2)/(3*pi
**(3/2)*a**(5/2)*c**(5/2))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.74 \[ \int \frac {1}{(a+a x)^{5/2} (c-c x)^{5/2}} \, dx=\frac {x}{3 \, {\left (-a c x^{2} + a c\right )}^{\frac {3}{2}} a c} + \frac {2 \, x}{3 \, \sqrt {-a c x^{2} + a c} a^{2} c^{2}} \]

[In]

integrate(1/(a*x+a)^(5/2)/(-c*x+c)^(5/2),x, algorithm="maxima")

[Out]

1/3*x/((-a*c*x^2 + a*c)^(3/2)*a*c) + 2/3*x/(sqrt(-a*c*x^2 + a*c)*a^2*c^2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 237 vs. \(2 (49) = 98\).

Time = 0.32 (sec) , antiderivative size = 237, normalized size of antiderivative = 3.89 \[ \int \frac {1}{(a+a x)^{5/2} (c-c x)^{5/2}} \, dx=-\frac {\sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c} \sqrt {a x + a} {\left (\frac {4 \, {\left (a x + a\right )} {\left | a \right |}}{a^{2} c} - \frac {9 \, {\left | a \right |}}{a c}\right )}}{12 \, {\left ({\left (a x + a\right )} a c - 2 \, a^{2} c\right )}^{2}} - \frac {16 \, \sqrt {-a c} a^{4} c^{2} - 18 \, \sqrt {-a c} {\left (\sqrt {-a c} \sqrt {a x + a} - \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{2} a^{2} c + 3 \, \sqrt {-a c} {\left (\sqrt {-a c} \sqrt {a x + a} - \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{4}}{3 \, {\left (2 \, a^{2} c - {\left (\sqrt {-a c} \sqrt {a x + a} - \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{2}\right )}^{3} c^{2} {\left | a \right |}} \]

[In]

integrate(1/(a*x+a)^(5/2)/(-c*x+c)^(5/2),x, algorithm="giac")

[Out]

-1/12*sqrt(-(a*x + a)*a*c + 2*a^2*c)*sqrt(a*x + a)*(4*(a*x + a)*abs(a)/(a^2*c) - 9*abs(a)/(a*c))/((a*x + a)*a*
c - 2*a^2*c)^2 - 1/3*(16*sqrt(-a*c)*a^4*c^2 - 18*sqrt(-a*c)*(sqrt(-a*c)*sqrt(a*x + a) - sqrt(-(a*x + a)*a*c +
2*a^2*c))^2*a^2*c + 3*sqrt(-a*c)*(sqrt(-a*c)*sqrt(a*x + a) - sqrt(-(a*x + a)*a*c + 2*a^2*c))^4)/((2*a^2*c - (s
qrt(-a*c)*sqrt(a*x + a) - sqrt(-(a*x + a)*a*c + 2*a^2*c))^2)^3*c^2*abs(a))

Mupad [B] (verification not implemented)

Time = 0.47 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.02 \[ \int \frac {1}{(a+a x)^{5/2} (c-c x)^{5/2}} \, dx=-\frac {3\,x\,\sqrt {c-c\,x}-2\,x^3\,\sqrt {c-c\,x}}{\sqrt {a+a\,x}\,{\left (c-c\,x\right )}^2\,\left (3\,a^2\,\left (c-c\,x\right )-6\,a^2\,c\right )} \]

[In]

int(1/((a + a*x)^(5/2)*(c - c*x)^(5/2)),x)

[Out]

-(3*x*(c - c*x)^(1/2) - 2*x^3*(c - c*x)^(1/2))/((a + a*x)^(1/2)*(c - c*x)^2*(3*a^2*(c - c*x) - 6*a^2*c))