3.1150 \(\int \frac{1}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx\)

Optimal. Leaf size=67 \[ \frac{2 x}{3 a^4 c^2 \sqrt{a+b x} \sqrt{a c-b c x}}+\frac{x}{3 a^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0622127, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{2 x}{3 a^4 c^2 \sqrt{a+b x} \sqrt{a c-b c x}}+\frac{x}{3 a^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.1355, size = 60, normalized size = 0.9 \[ \frac{x}{3 a^{2} c \left (a + b x\right )^{\frac{3}{2}} \left (a c - b c x\right )^{\frac{3}{2}}} + \frac{2 x}{3 a^{4} c^{2} \sqrt{a + b x} \sqrt{a c - b c x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0545734, size = 46, normalized size = 0.69 \[ \frac{3 a^2 x-2 b^2 x^3}{3 a^4 c (a+b x)^{3/2} (c (a-b x))^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 45, normalized size = 0.7 \[{\frac{ \left ( -bx+a \right ) x \left ( -2\,{b}^{2}{x}^{2}+3\,{a}^{2} \right ) }{3\,{a}^{4}} \left ( bx+a \right ) ^{-{\frac{3}{2}}} \left ( -bcx+ac \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(-b*x+a)*x*(-2*b^2*x^2+3*a^2)/(b*x+a)^(3/2)/a^4/(-b*c*x+a*c)^(5/2)

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Maxima [A]  time = 1.32085, size = 72, normalized size = 1.07 \[ \frac{x}{3 \,{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac{3}{2}} a^{2} c} + \frac{2 \, x}{3 \, \sqrt{-b^{2} c x^{2} + a^{2} c} a^{4} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.216363, size = 97, normalized size = 1.45 \[ -\frac{{\left (2 \, b^{2} x^{3} - 3 \, a^{2} x\right )} \sqrt{-b c x + a c} \sqrt{b x + a}}{3 \,{\left (a^{4} b^{4} c^{3} x^{4} - 2 \, a^{6} b^{2} c^{3} x^{2} + a^{8} c^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.257858, size = 339, normalized size = 5.06 \[ -\frac{\sqrt{-b c x + a c}{\left (\frac{9 \,{\left | c \right |}}{a^{3} b c} + \frac{4 \,{\left (b c x - a c\right )}{\left | c \right |}}{a^{4} b c^{2}}\right )}}{12 \,{\left (2 \, a c^{2} +{\left (b c x - a c\right )} c\right )}^{\frac{3}{2}}} + \frac{16 \, a^{2} \sqrt{-c} c^{4} - 18 \, a{\left (\sqrt{-b c x + a c} \sqrt{-c} - \sqrt{2 \, a c^{2} +{\left (b c x - a c\right )} c}\right )}^{2} \sqrt{-c} c^{2} + 3 \,{\left (\sqrt{-b c x + a c} \sqrt{-c} - \sqrt{2 \, a c^{2} +{\left (b c x - a c\right )} c}\right )}^{4} \sqrt{-c}}{3 \,{\left (2 \, a c^{2} -{\left (\sqrt{-b c x + a c} \sqrt{-c} - \sqrt{2 \, a c^{2} +{\left (b c x - a c\right )} c}\right )}^{2}\right )}^{3} a^{3} b{\left | c \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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