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