Optimal. Leaf size=33 \[ \frac{b x \log (x)}{c \sqrt{c x^2}}-\frac{a}{c \sqrt{c x^2}} \]
[Out]
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Rubi [A] time = 0.0203989, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{b x \log (x)}{c \sqrt{c x^2}}-\frac{a}{c \sqrt{c x^2}} \]
Antiderivative was successfully verified.
[In] Int[(x*(a + b*x))/(c*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x \left (a + b x\right )}{\left (c x^{2}\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(b*x+a)/(c*x**2)**(3/2),x)
[Out]
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Mathematica [A] time = 0.00433833, size = 22, normalized size = 0.67 \[ \frac{x^2 (b x \log (x)-a)}{\left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(x*(a + b*x))/(c*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.007, size = 21, normalized size = 0.6 \[{{x}^{2} \left ( b\ln \left ( x \right ) x-a \right ) \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(b*x+a)/(c*x^2)^(3/2),x)
[Out]
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Maxima [A] time = 1.34042, size = 28, normalized size = 0.85 \[ \frac{b \log \left (x\right )}{c^{\frac{3}{2}}} - \frac{a}{\sqrt{c x^{2}} c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*x/(c*x^2)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.20606, size = 31, normalized size = 0.94 \[ \frac{\sqrt{c x^{2}}{\left (b x \log \left (x\right ) - a\right )}}{c^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*x/(c*x^2)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x \left (a + b x\right )}{\left (c x^{2}\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(b*x+a)/(c*x**2)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.214517, size = 63, normalized size = 1.91 \[ -\frac{b{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2}} \right |}\right ) - \frac{2 \, a \sqrt{c}}{\sqrt{c} x - \sqrt{c x^{2}}}}{c^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*x/(c*x^2)^(3/2),x, algorithm="giac")
[Out]