Optimal. Leaf size=25 \[ \frac{2 (a+b x)}{5 b \sqrt{\frac{c}{(a+b x)^3}}} \]
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Rubi [A] time = 0.0220779, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{2 (a+b x)}{5 b \sqrt{\frac{c}{(a+b x)^3}}} \]
Antiderivative was successfully verified.
[In] Int[1/Sqrt[c/(a + b*x)^3],x]
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Rubi in Sympy [A] time = 2.52962, size = 24, normalized size = 0.96 \[ \frac{2 \sqrt{\frac{c}{\left (a + b x\right )^{3}}} \left (a + b x\right )^{4}}{5 b c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(c/(b*x+a)**3)**(1/2),x)
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Mathematica [A] time = 0.025573, size = 25, normalized size = 1. \[ \frac{2 (a+b x)}{5 b \sqrt{\frac{c}{(a+b x)^3}}} \]
Antiderivative was successfully verified.
[In] Integrate[1/Sqrt[c/(a + b*x)^3],x]
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Maple [A] time = 0.003, size = 22, normalized size = 0.9 \[{\frac{2\,bx+2\,a}{5\,b}{\frac{1}{\sqrt{{\frac{c}{ \left ( bx+a \right ) ^{3}}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(c/(b*x+a)^3)^(1/2),x)
[Out]
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Maxima [A] time = 1.54344, size = 36, normalized size = 1.44 \[ \frac{2 \,{\left (b \sqrt{c} x + a \sqrt{c}\right )}{\left (b x + a\right )}^{\frac{3}{2}}}{5 \, b c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(c/(b*x + a)^3),x, algorithm="maxima")
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Fricas [A] time = 0.217922, size = 107, normalized size = 4.28 \[ \frac{2 \,{\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}\right )} \sqrt{\frac{c}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}}}{5 \, b c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(c/(b*x + a)^3),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{\frac{c}{\left (a + b x\right )^{3}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(c/(b*x+a)**3)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.219584, size = 217, normalized size = 8.68 \[ \frac{2 \,{\left (15 \, \sqrt{b c x + a c} a^{2} - \frac{10 \,{\left (3 \, \sqrt{b c x + a c} a c -{\left (b c x + a c\right )}^{\frac{3}{2}}\right )} a}{c} + \frac{15 \, \sqrt{b c x + a c} a^{2} b^{8} c^{10} - 10 \,{\left (b c x + a c\right )}^{\frac{3}{2}} a b^{8} c^{9} + 3 \,{\left (b c x + a c\right )}^{\frac{5}{2}} b^{8} c^{8}}{b^{8} c^{10}}\right )}}{15 \, b c{\rm sign}\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right ){\rm sign}\left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(c/(b*x + a)^3),x, algorithm="giac")
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