3.1143 \(\int \frac{1}{(a+a x)^{7/2} (c-c x)^{7/2}} \, dx\)

Optimal. Leaf size=91 \[ \frac{8 x}{15 a^3 c^3 \sqrt{a x+a} \sqrt{c-c x}}+\frac{4 x}{15 a^2 c^2 (a x+a)^{3/2} (c-c x)^{3/2}}+\frac{x}{5 a c (a x+a)^{5/2} (c-c x)^{5/2}} \]

[Out]

x/(5*a*c*(a + a*x)^(5/2)*(c - c*x)^(5/2)) + (4*x)/(15*a^2*c^2*(a + a*x)^(3/2)*(c
- c*x)^(3/2)) + (8*x)/(15*a^3*c^3*Sqrt[a + a*x]*Sqrt[c - c*x])

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Rubi [A]  time = 0.0783818, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{8 x}{15 a^3 c^3 \sqrt{a x+a} \sqrt{c-c x}}+\frac{4 x}{15 a^2 c^2 (a x+a)^{3/2} (c-c x)^{3/2}}+\frac{x}{5 a c (a x+a)^{5/2} (c-c x)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + a*x)^(7/2)*(c - c*x)^(7/2)),x]

[Out]

x/(5*a*c*(a + a*x)^(5/2)*(c - c*x)^(5/2)) + (4*x)/(15*a^2*c^2*(a + a*x)^(3/2)*(c
- c*x)^(3/2)) + (8*x)/(15*a^3*c^3*Sqrt[a + a*x]*Sqrt[c - c*x])

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Rubi in Sympy [A]  time = 13.1547, size = 80, normalized size = 0.88 \[ \frac{x}{5 a c \left (a x + a\right )^{\frac{5}{2}} \left (- c x + c\right )^{\frac{5}{2}}} + \frac{4 x}{15 a^{2} c^{2} \left (a x + a\right )^{\frac{3}{2}} \left (- c x + c\right )^{\frac{3}{2}}} + \frac{8 x}{15 a^{3} c^{3} \sqrt{a x + a} \sqrt{- c x + c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a*x+a)**(7/2)/(-c*x+c)**(7/2),x)

[Out]

x/(5*a*c*(a*x + a)**(5/2)*(-c*x + c)**(5/2)) + 4*x/(15*a**2*c**2*(a*x + a)**(3/2
)*(-c*x + c)**(3/2)) + 8*x/(15*a**3*c**3*sqrt(a*x + a)*sqrt(-c*x + c))

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Mathematica [A]  time = 0.077784, size = 47, normalized size = 0.52 \[ -\frac{x (x+1) \left (8 x^4-20 x^2+15\right ) \sqrt{c-c x}}{15 c^4 (x-1)^3 (a (x+1))^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + a*x)^(7/2)*(c - c*x)^(7/2)),x]

[Out]

-(x*(1 + x)*Sqrt[c - c*x]*(15 - 20*x^2 + 8*x^4))/(15*c^4*(-1 + x)^3*(a*(1 + x))^
(7/2))

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Maple [A]  time = 0.004, size = 37, normalized size = 0.4 \[ -{\frac{ \left ( 1+x \right ) \left ( -1+x \right ) x \left ( 8\,{x}^{4}-20\,{x}^{2}+15 \right ) }{15} \left ( ax+a \right ) ^{-{\frac{7}{2}}} \left ( -cx+c \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a*x+a)^(7/2)/(-c*x+c)^(7/2),x)

[Out]

-1/15*(1+x)*(-1+x)*x*(8*x^4-20*x^2+15)/(a*x+a)^(7/2)/(-c*x+c)^(7/2)

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Maxima [A]  time = 1.34905, size = 90, normalized size = 0.99 \[ \frac{x}{5 \,{\left (-a c x^{2} + a c\right )}^{\frac{5}{2}} a c} + \frac{4 \, x}{15 \,{\left (-a c x^{2} + a c\right )}^{\frac{3}{2}} a^{2} c^{2}} + \frac{8 \, x}{15 \, \sqrt{-a c x^{2} + a c} a^{3} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a*x + a)^(7/2)*(-c*x + c)^(7/2)),x, algorithm="maxima")

[Out]

1/5*x/((-a*c*x^2 + a*c)^(5/2)*a*c) + 4/15*x/((-a*c*x^2 + a*c)^(3/2)*a^2*c^2) + 8
/15*x/(sqrt(-a*c*x^2 + a*c)*a^3*c^3)

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Fricas [A]  time = 0.206526, size = 100, normalized size = 1.1 \[ -\frac{{\left (8 \, x^{5} - 20 \, x^{3} + 15 \, x\right )} \sqrt{a x + a} \sqrt{-c x + c}}{15 \,{\left (a^{4} c^{4} x^{6} - 3 \, a^{4} c^{4} x^{4} + 3 \, a^{4} c^{4} x^{2} - a^{4} c^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a*x + a)^(7/2)*(-c*x + c)^(7/2)),x, algorithm="fricas")

[Out]

-1/15*(8*x^5 - 20*x^3 + 15*x)*sqrt(a*x + a)*sqrt(-c*x + c)/(a^4*c^4*x^6 - 3*a^4*
c^4*x^4 + 3*a^4*c^4*x^2 - a^4*c^4)

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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)**(7/2)/(-c*x+c)**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.307291, size = 450, normalized size = 4.95 \[ -\frac{\sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c} \sqrt{a x + a}{\left ({\left (a x + a\right )}{\left (\frac{64 \,{\left (a x + a\right )}}{c{\left | a \right |}} - \frac{275 \, a}{c{\left | a \right |}}\right )} + \frac{300 \, a^{2}}{c{\left | a \right |}}\right )}}{240 \,{\left ({\left (a x + a\right )} a c - 2 \, a^{2} c\right )}^{3}} + \frac{1024 \, a^{8} c^{4} - 2200 \,{\left (\sqrt{-a c} \sqrt{a x + a} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{2} a^{6} c^{3} + 1660 \,{\left (\sqrt{-a c} \sqrt{a x + a} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{4} a^{4} c^{2} - 450 \,{\left (\sqrt{-a c} \sqrt{a x + a} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{6} a^{2} c + 45 \,{\left (\sqrt{-a c} \sqrt{a x + a} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{8}}{60 \,{\left (2 \, a^{2} c -{\left (\sqrt{-a c} \sqrt{a x + a} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{2}\right )}^{5} \sqrt{-a c} c^{2}{\left | a \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a*x + a)^(7/2)*(-c*x + c)^(7/2)),x, algorithm="giac")

[Out]

-1/240*sqrt(-(a*x + a)*a*c + 2*a^2*c)*sqrt(a*x + a)*((a*x + a)*(64*(a*x + a)/(c*
abs(a)) - 275*a/(c*abs(a))) + 300*a^2/(c*abs(a)))/((a*x + a)*a*c - 2*a^2*c)^3 +
1/60*(1024*a^8*c^4 - 2200*(sqrt(-a*c)*sqrt(a*x + a) - sqrt(-(a*x + a)*a*c + 2*a^
2*c))^2*a^6*c^3 + 1660*(sqrt(-a*c)*sqrt(a*x + a) - sqrt(-(a*x + a)*a*c + 2*a^2*c
))^4*a^4*c^2 - 450*(sqrt(-a*c)*sqrt(a*x + a) - sqrt(-(a*x + a)*a*c + 2*a^2*c))^6
*a^2*c + 45*(sqrt(-a*c)*sqrt(a*x + a) - sqrt(-(a*x + a)*a*c + 2*a^2*c))^8)/((2*a
^2*c - (sqrt(-a*c)*sqrt(a*x + a) - sqrt(-(a*x + a)*a*c + 2*a^2*c))^2)^5*sqrt(-a*
c)*c^2*abs(a))