Optimal. Leaf size=38 \[ \frac{a x^2}{c \sqrt{c x^2}}+\frac{b x^3}{2 c \sqrt{c x^2}} \]
[Out]
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Rubi [A] time = 0.0163118, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ \frac{a x^2}{c \sqrt{c x^2}}+\frac{b x^3}{2 c \sqrt{c x^2}} \]
Antiderivative was successfully verified.
[In] Int[(x^3*(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. \[ \frac{b \sqrt{c x^{2}} \int x\, dx}{c^{2} x} + \frac{\sqrt{c x^{2}} \int a\, dx}{c^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(b*x+a)/(c*x**2)**(3/2),x)
[Out]
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Mathematica [A] time = 0.00729913, size = 23, normalized size = 0.61 \[ \frac{x^4 (2 a+b x)}{2 \left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^3*(a + b*x))/(c*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.005, size = 20, normalized size = 0.5 \[{\frac{{x}^{4} \left ( bx+2\,a \right ) }{2} \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(b*x+a)/(c*x^2)^(3/2),x)
[Out]
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Maxima [A] time = 1.33841, size = 43, normalized size = 1.13 \[ \frac{b x^{3}}{2 \, \sqrt{c x^{2}} c} + \frac{a x^{2}}{\sqrt{c x^{2}} c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*x^3/(c*x^2)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.206474, size = 26, normalized size = 0.68 \[ \frac{\sqrt{c x^{2}}{\left (b x + 2 \, a\right )}}{2 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*x^3/(c*x^2)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.27244, size = 34, normalized size = 0.89 \[ \frac{a x^{4}}{c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} + \frac{b x^{5}}{2 c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3*(b*x+a)/(c*x**2)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.210118, size = 34, normalized size = 0.89 \[ \frac{\sqrt{c x^{2}}{\left (\frac{b x}{c} + \frac{2 \, a}{c}\right )}}{2 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*x^3/(c*x^2)^(3/2),x, algorithm="giac")
[Out]