3.1144 \(\int \frac{1}{(a+a x)^{9/2} (c-c x)^{9/2}} \, dx\)

Optimal. Leaf size=121 \[ \frac{16 x}{35 a^4 c^4 \sqrt{a x+a} \sqrt{c-c x}}+\frac{8 x}{35 a^3 c^3 (a x+a)^{3/2} (c-c x)^{3/2}}+\frac{6 x}{35 a^2 c^2 (a x+a)^{5/2} (c-c x)^{5/2}}+\frac{x}{7 a c (a x+a)^{7/2} (c-c x)^{7/2}} \]

[Out]

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

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Rubi [A]  time = 0.108639, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{16 x}{35 a^4 c^4 \sqrt{a x+a} \sqrt{c-c x}}+\frac{8 x}{35 a^3 c^3 (a x+a)^{3/2} (c-c x)^{3/2}}+\frac{6 x}{35 a^2 c^2 (a x+a)^{5/2} (c-c x)^{5/2}}+\frac{x}{7 a c (a x+a)^{7/2} (c-c x)^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 19.853, size = 109, normalized size = 0.9 \[ \frac{x}{7 a c \left (a x + a\right )^{\frac{7}{2}} \left (- c x + c\right )^{\frac{7}{2}}} + \frac{6 x}{35 a^{2} c^{2} \left (a x + a\right )^{\frac{5}{2}} \left (- c x + c\right )^{\frac{5}{2}}} + \frac{8 x}{35 a^{3} c^{3} \left (a x + a\right )^{\frac{3}{2}} \left (- c x + c\right )^{\frac{3}{2}}} + \frac{16 x}{35 a^{4} c^{4} \sqrt{a x + a} \sqrt{- c x + c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a*x+a)**(9/2)/(-c*x+c)**(9/2),x)

[Out]

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

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Mathematica [A]  time = 0.0938597, size = 54, normalized size = 0.45 \[ -\frac{x \left (16 x^6-56 x^4+70 x^2-35\right ) \sqrt{a (x+1)} \sqrt{c-c x}}{35 a^5 c^5 \left (x^2-1\right )^4} \]

Antiderivative was successfully verified.

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

[Out]

-(x*Sqrt[a*(1 + x)]*Sqrt[c - c*x]*(-35 + 70*x^2 - 56*x^4 + 16*x^6))/(35*a^5*c^5*
(-1 + x^2)^4)

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Maple [A]  time = 0.006, size = 42, normalized size = 0.4 \[{\frac{ \left ( 1+x \right ) \left ( -1+x \right ) x \left ( 16\,{x}^{6}-56\,{x}^{4}+70\,{x}^{2}-35 \right ) }{35} \left ( ax+a \right ) ^{-{\frac{9}{2}}} \left ( -cx+c \right ) ^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.33319, size = 120, normalized size = 0.99 \[ \frac{x}{7 \,{\left (-a c x^{2} + a c\right )}^{\frac{7}{2}} a c} + \frac{6 \, x}{35 \,{\left (-a c x^{2} + a c\right )}^{\frac{5}{2}} a^{2} c^{2}} + \frac{8 \, x}{35 \,{\left (-a c x^{2} + a c\right )}^{\frac{3}{2}} a^{3} c^{3}} + \frac{16 \, x}{35 \, \sqrt{-a c x^{2} + a c} a^{4} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.209836, size = 120, normalized size = 0.99 \[ -\frac{{\left (16 \, x^{7} - 56 \, x^{5} + 70 \, x^{3} - 35 \, x\right )} \sqrt{a x + a} \sqrt{-c x + c}}{35 \,{\left (a^{5} c^{5} x^{8} - 4 \, a^{5} c^{5} x^{6} + 6 \, a^{5} c^{5} x^{4} - 4 \, a^{5} c^{5} x^{2} + a^{5} c^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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)**(9/2)/(-c*x+c)**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.414531, size = 590, normalized size = 4.88 \[ -\frac{\sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}{\left ({\left (a x + a\right )}{\left ({\left (a x + a\right )}{\left (\frac{256 \,{\left (a x + a\right )}{\left | a \right |}}{a^{2} c} - \frac{1617 \,{\left | a \right |}}{a c}\right )} + \frac{3430 \,{\left | a \right |}}{c}\right )} - \frac{2450 \, a{\left | a \right |}}{c}\right )} \sqrt{a x + a}}{1120 \,{\left ({\left (a x + a\right )} a c - 2 \, a^{2} c\right )}^{4}} + \frac{16384 \, a^{12} c^{6} - 51744 \,{\left (\sqrt{-a c} \sqrt{a x + a} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{2} a^{10} c^{5} + 66416 \,{\left (\sqrt{-a c} \sqrt{a x + a} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{4} a^{8} c^{4} - 43120 \,{\left (\sqrt{-a c} \sqrt{a x + a} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{6} a^{6} c^{3} + 14280 \,{\left (\sqrt{-a c} \sqrt{a x + a} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{8} a^{4} c^{2} - 2450 \,{\left (\sqrt{-a c} \sqrt{a x + a} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{10} a^{2} c + 175 \,{\left (\sqrt{-a c} \sqrt{a x + a} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{12}}{280 \,{\left (2 \, a^{2} c -{\left (\sqrt{-a c} \sqrt{a x + a} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{2}\right )}^{7} \sqrt{-a c} a c^{3}{\left | a \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1120*sqrt(-(a*x + a)*a*c + 2*a^2*c)*((a*x + a)*((a*x + a)*(256*(a*x + a)*abs(
a)/(a^2*c) - 1617*abs(a)/(a*c)) + 3430*abs(a)/c) - 2450*a*abs(a)/c)*sqrt(a*x + a
)/((a*x + a)*a*c - 2*a^2*c)^4 + 1/280*(16384*a^12*c^6 - 51744*(sqrt(-a*c)*sqrt(a
*x + a) - sqrt(-(a*x + a)*a*c + 2*a^2*c))^2*a^10*c^5 + 66416*(sqrt(-a*c)*sqrt(a*
x + a) - sqrt(-(a*x + a)*a*c + 2*a^2*c))^4*a^8*c^4 - 43120*(sqrt(-a*c)*sqrt(a*x
+ a) - sqrt(-(a*x + a)*a*c + 2*a^2*c))^6*a^6*c^3 + 14280*(sqrt(-a*c)*sqrt(a*x +
a) - sqrt(-(a*x + a)*a*c + 2*a^2*c))^8*a^4*c^2 - 2450*(sqrt(-a*c)*sqrt(a*x + a)
- sqrt(-(a*x + a)*a*c + 2*a^2*c))^10*a^2*c + 175*(sqrt(-a*c)*sqrt(a*x + a) - sqr
t(-(a*x + a)*a*c + 2*a^2*c))^12)/((2*a^2*c - (sqrt(-a*c)*sqrt(a*x + a) - sqrt(-(
a*x + a)*a*c + 2*a^2*c))^2)^7*sqrt(-a*c)*a*c^3*abs(a))