Optimal. Leaf size=27 \[ \frac{a+b x}{2 b \left (\frac{c}{(a+b x)^{3/2}}\right )^{2/3}} \]
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Rubi [A] time = 0.0266699, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{a+b x}{2 b \left (\frac{c}{(a+b x)^{3/2}}\right )^{2/3}} \]
Antiderivative was successfully verified.
[In] Int[(c/(a + b*x)^(3/2))^(-2/3),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{\sqrt [3]{\frac{c}{\left (a + b x\right )^{\frac{3}{2}}}} \sqrt{a + b x} \int ^{a + b x} x\, dx}{b c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(c/(b*x+a)**(3/2))**(2/3),x)
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Mathematica [A] time = 0.0345671, size = 34, normalized size = 1.26 \[ \frac{x (2 a+b x)}{2 (a+b x) \left (\frac{c}{(a+b x)^{3/2}}\right )^{2/3}} \]
Antiderivative was successfully verified.
[In] Integrate[(c/(a + b*x)^(3/2))^(-2/3),x]
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Maple [A] time = 0.002, size = 29, normalized size = 1.1 \[{\frac{x \left ( bx+2\,a \right ) }{2\,bx+2\,a} \left ({c \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \right ) ^{-{\frac{2}{3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(c/(b*x+a)^(3/2))^(2/3),x)
[Out]
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Maxima [A] time = 1.33623, size = 28, normalized size = 1.04 \[ \frac{b x + a}{2 \, b \left (\frac{c}{{\left (b x + a\right )}^{\frac{3}{2}}}\right )^{\frac{2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c/(b*x + a)^(3/2))^(-2/3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.336873, size = 59, normalized size = 2.19 \[ \frac{{\left (b^{2} x^{3} + 3 \, a b x^{2} + 2 \, a^{2} x\right )} \left (\frac{c}{{\left (b x + a\right )}^{\frac{3}{2}}}\right )^{\frac{1}{3}}}{2 \, \sqrt{b x + a} c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c/(b*x + a)^(3/2))^(-2/3),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (\frac{c}{\left (a + b x\right )^{\frac{3}{2}}}\right )^{\frac{2}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(c/(b*x+a)**(3/2))**(2/3),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (\frac{c}{{\left (b x + a\right )}^{\frac{3}{2}}}\right )^{\frac{2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c/(b*x + a)^(3/2))^(-2/3),x, algorithm="giac")
[Out]