3.1142 \(\int \frac{1}{(a+a x)^{5/2} (c-c x)^{5/2}} \, dx\)

Optimal. Leaf size=61 \[ \frac{2 x}{3 a^2 c^2 \sqrt{a x+a} \sqrt{c-c x}}+\frac{x}{3 a c (a x+a)^{3/2} (c-c x)^{3/2}} \]

[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])

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Rubi [A]  time = 0.0491331, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{2 x}{3 a^2 c^2 \sqrt{a x+a} \sqrt{c-c x}}+\frac{x}{3 a c (a x+a)^{3/2} (c-c x)^{3/2}} \]

Antiderivative was successfully verified.

[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])

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Rubi in Sympy [A]  time = 8.36274, size = 51, normalized size = 0.84 \[ \frac{x}{3 a c \left (a x + a\right )^{\frac{3}{2}} \left (- c x + c\right )^{\frac{3}{2}}} + \frac{2 x}{3 a^{2} c^{2} \sqrt{a x + a} \sqrt{- c x + c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a*x+a)**(5/2)/(-c*x+c)**(5/2),x)

[Out]

x/(3*a*c*(a*x + a)**(3/2)*(-c*x + c)**(3/2)) + 2*x/(3*a**2*c**2*sqrt(a*x + a)*sq
rt(-c*x + c))

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Mathematica [A]  time = 0.0609904, size = 42, normalized size = 0.69 \[ -\frac{x (x+1) \left (2 x^2-3\right ) \sqrt{c-c x}}{3 c^3 (x-1)^2 (a (x+1))^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 32, normalized size = 0.5 \[{\frac{ \left ( 1+x \right ) \left ( -1+x \right ) x \left ( 2\,{x}^{2}-3 \right ) }{3} \left ( ax+a \right ) ^{-{\frac{5}{2}}} \left ( -cx+c \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.34875, size = 61, normalized size = 1. \[ \frac{x}{3 \,{\left (-a c x^{2} + a c\right )}^{\frac{3}{2}} a c} + \frac{2 \, x}{3 \, \sqrt{-a c x^{2} + a c} a^{2} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 0.210542, size = 77, normalized size = 1.26 \[ -\frac{{\left (2 \, x^{3} - 3 \, x\right )} \sqrt{a x + a} \sqrt{-c x + c}}{3 \,{\left (a^{3} c^{3} x^{4} - 2 \, a^{3} c^{3} x^{2} + a^{3} c^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.263013, size = 320, normalized size = 5.25 \[ -\frac{\sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c} \sqrt{a x + a}{\left (\frac{4 \,{\left (a x + a\right )}{\left | a \right |}}{a^{2} c} - \frac{9 \,{\left | a \right |}}{a c}\right )}}{12 \,{\left ({\left (a x + a\right )} a c - 2 \, a^{2} c\right )}^{2}} - \frac{16 \, \sqrt{-a c} a^{4} c^{2} - 18 \, \sqrt{-a c}{\left (\sqrt{-a c} \sqrt{a x + a} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{2} a^{2} c + 3 \, \sqrt{-a c}{\left (\sqrt{-a c} \sqrt{a x + a} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{4}}{3 \,{\left (2 \, a^{2} c -{\left (\sqrt{-a c} \sqrt{a x + a} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{2}\right )}^{3} c^{2}{\left | a \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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*s
qrt(-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 - (sqrt(-a*c)*sqrt(a*x + a) - sqrt(-(a*x + a)*a*c + 2*a^2*c))^2)^3*c^2*abs
(a))