Optimal. Leaf size=30 \[ \frac{2 (a+b x)^2}{5 b c \sqrt{\frac{c}{a+b x}}} \]
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Rubi [A] time = 0.02392, antiderivative size = 30, 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 \[ \frac{2 (a+b x)^2}{5 b c \sqrt{\frac{c}{a+b x}}} \]
Antiderivative was successfully verified.
[In] Int[(c/(a + b*x))^(-3/2),x]
[Out]
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Rubi in Sympy [A] time = 2.56321, size = 24, normalized size = 0.8 \[ \frac{2 \sqrt{\frac{c}{a + b x}} \left (a + b x\right )^{3}}{5 b c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(c/(b*x+a))**(3/2),x)
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Mathematica [A] time = 0.0221646, size = 21, normalized size = 0.7 \[ \frac{2 c}{5 b \left (\frac{c}{a+b x}\right )^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(c/(a + b*x))^(-3/2),x]
[Out]
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Maple [A] time = 0.004, size = 22, normalized size = 0.7 \[{\frac{2\,bx+2\,a}{5\,b} \left ({\frac{c}{bx+a}} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(c/(b*x+a))^(3/2),x)
[Out]
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Maxima [A] time = 1.34105, size = 23, normalized size = 0.77 \[ \frac{2 \, c}{5 \, b \left (\frac{c}{b x + a}\right )^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c/(b*x + a))^(-3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.215612, size = 47, normalized size = 1.57 \[ \frac{2 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}}{5 \, b c \sqrt{\frac{c}{b x + a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c/(b*x + a))^(-3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.11551, size = 49, normalized size = 1.63 \[ \begin{cases} \frac{2 a}{5 b c^{\frac{3}{2}} \left (\frac{1}{a + b x}\right )^{\frac{3}{2}}} + \frac{2 x}{5 c^{\frac{3}{2}} \left (\frac{1}{a + b x}\right )^{\frac{3}{2}}} & \text{for}\: b \neq 0 \\\frac{x}{\left (\frac{c}{a}\right )^{\frac{3}{2}}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(c/(b*x+a))**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.217871, size = 35, normalized size = 1.17 \[ \frac{2 \,{\left (b x + a\right )}^{2}}{5 \, b c \sqrt{\frac{c}{b x + a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c/(b*x + a))^(-3/2),x, algorithm="giac")
[Out]