Optimal. Leaf size=100 \[ \frac{8 x}{15 a^6 c^3 \sqrt{a+b x} \sqrt{a c-b c x}}+\frac{4 x}{15 a^4 c^2 (a+b x)^{3/2} (a c-b c x)^{3/2}}+\frac{x}{5 a^2 c (a+b x)^{5/2} (a c-b c x)^{5/2}} \]
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Rubi [A] time = 0.0986098, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{8 x}{15 a^6 c^3 \sqrt{a+b x} \sqrt{a c-b c x}}+\frac{4 x}{15 a^4 c^2 (a+b x)^{3/2} (a c-b c x)^{3/2}}+\frac{x}{5 a^2 c (a+b x)^{5/2} (a c-b c x)^{5/2}} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*x)^(7/2)*(a*c - b*c*x)^(7/2)),x]
[Out]
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Rubi in Sympy [A] time = 17.6681, size = 92, normalized size = 0.92 \[ \frac{x}{5 a^{2} c \left (a + b x\right )^{\frac{5}{2}} \left (a c - b c x\right )^{\frac{5}{2}}} + \frac{4 x}{15 a^{4} c^{2} \left (a + b x\right )^{\frac{3}{2}} \left (a c - b c x\right )^{\frac{3}{2}}} + \frac{8 x}{15 a^{6} c^{3} \sqrt{a + b x} \sqrt{a c - b c x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x+a)**(7/2)/(-b*c*x+a*c)**(7/2),x)
[Out]
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Mathematica [A] time = 0.0713898, size = 57, normalized size = 0.57 \[ \frac{15 a^4 x-20 a^2 b^2 x^3+8 b^4 x^5}{15 a^6 c (a+b x)^{5/2} (c (a-b x))^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b*x)^(7/2)*(a*c - b*c*x)^(7/2)),x]
[Out]
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Maple [A] time = 0.006, size = 56, normalized size = 0.6 \[{\frac{ \left ( -bx+a \right ) x \left ( 8\,{x}^{4}{b}^{4}-20\,{x}^{2}{a}^{2}{b}^{2}+15\,{a}^{4} \right ) }{15\,{a}^{6}} \left ( bx+a \right ) ^{-{\frac{5}{2}}} \left ( -bcx+ac \right ) ^{-{\frac{7}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x+a)^(7/2)/(-b*c*x+a*c)^(7/2),x)
[Out]
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Maxima [A] time = 1.33626, size = 107, normalized size = 1.07 \[ \frac{x}{5 \,{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac{5}{2}} a^{2} c} + \frac{4 \, x}{15 \,{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac{3}{2}} a^{4} c^{2}} + \frac{8 \, x}{15 \, \sqrt{-b^{2} c x^{2} + a^{2} c} a^{6} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((-b*c*x + a*c)^(7/2)*(b*x + a)^(7/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.242713, size = 132, normalized size = 1.32 \[ -\frac{{\left (8 \, b^{4} x^{5} - 20 \, a^{2} b^{2} x^{3} + 15 \, a^{4} x\right )} \sqrt{-b c x + a c} \sqrt{b x + a}}{15 \,{\left (a^{6} b^{6} c^{4} x^{6} - 3 \, a^{8} b^{4} c^{4} x^{4} + 3 \, a^{10} b^{2} c^{4} x^{2} - a^{12} c^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((-b*c*x + a*c)^(7/2)*(b*x + a)^(7/2)),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)**(7/2)/(-b*c*x+a*c)**(7/2),x)
[Out]
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GIAC/XCAS [A] time = 0.321724, size = 494, normalized size = 4.94 \[ -\frac{\sqrt{-b c x + a c}{\left ({\left (b c x - a c\right )}{\left (\frac{275 \, c}{a^{5} b{\left | c \right |}} + \frac{64 \,{\left (b c x - a c\right )}}{a^{6} b{\left | c \right |}}\right )} + \frac{300 \, c^{2}}{a^{4} b{\left | c \right |}}\right )}}{240 \,{\left (2 \, a c^{2} +{\left (b c x - a c\right )} c\right )}^{\frac{5}{2}}} - \frac{1024 \, a^{4} c^{8} - 2200 \, a^{3}{\left (\sqrt{-b c x + a c} \sqrt{-c} - \sqrt{2 \, a c^{2} +{\left (b c x - a c\right )} c}\right )}^{2} c^{6} + 1660 \, a^{2}{\left (\sqrt{-b c x + a c} \sqrt{-c} - \sqrt{2 \, a c^{2} +{\left (b c x - a c\right )} c}\right )}^{4} c^{4} - 450 \, a{\left (\sqrt{-b c x + a c} \sqrt{-c} - \sqrt{2 \, a c^{2} +{\left (b c x - a c\right )} c}\right )}^{6} c^{2} + 45 \,{\left (\sqrt{-b c x + a c} \sqrt{-c} - \sqrt{2 \, a c^{2} +{\left (b c x - a c\right )} c}\right )}^{8}}{60 \,{\left (2 \, a c^{2} -{\left (\sqrt{-b c x + a c} \sqrt{-c} - \sqrt{2 \, a c^{2} +{\left (b c x - a c\right )} c}\right )}^{2}\right )}^{5} a^{5} b \sqrt{-c}{\left | c \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((-b*c*x + a*c)^(7/2)*(b*x + a)^(7/2)),x, algorithm="giac")
[Out]