Optimal. Leaf size=25 \[ -\frac{x}{b c \sqrt{c x^2} (a+b x)} \]
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Rubi [A] time = 0.0140044, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{x}{b c \sqrt{c x^2} (a+b x)} \]
Antiderivative was successfully verified.
[In] Int[x^3/((c*x^2)^(3/2)*(a + b*x)^2),x]
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Rubi in Sympy [A] time = 13.9533, size = 20, normalized size = 0.8 \[ - \frac{\sqrt{c x^{2}}}{b c^{2} x \left (a + b x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3/(c*x**2)**(3/2)/(b*x+a)**2,x)
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Mathematica [A] time = 0.0083106, size = 24, normalized size = 0.96 \[ -\frac{x^3}{b \left (c x^2\right )^{3/2} (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[x^3/((c*x^2)^(3/2)*(a + b*x)^2),x]
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Maple [A] time = 0.004, size = 23, normalized size = 0.9 \[ -{\frac{{x}^{3}}{ \left ( bx+a \right ) b} \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3/(c*x^2)^(3/2)/(b*x+a)^2,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/((c*x^2)^(3/2)*(b*x + a)^2),x, algorithm="maxima")
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Fricas [A] time = 0.20543, size = 39, normalized size = 1.56 \[ -\frac{\sqrt{c x^{2}}}{b^{2} c^{2} x^{2} + a b c^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/((c*x^2)^(3/2)*(b*x + a)^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 6.14921, size = 90, normalized size = 3.6 \[ \begin{cases} \frac{\tilde{\infty } x^{2}}{c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{x^{2}}{b^{2} c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} & \text{for}\: a = 0 \\\frac{\tilde{\infty } x^{4}}{c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} & \text{for}\: b = - \frac{a}{x} \\\frac{x^{4}}{a^{2} c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}} + a b c^{\frac{3}{2}} x \left (x^{2}\right )^{\frac{3}{2}}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3/(c*x**2)**(3/2)/(b*x+a)**2,x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{\left (c x^{2}\right )^{\frac{3}{2}}{\left (b x + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/((c*x^2)^(3/2)*(b*x + a)^2),x, algorithm="giac")
[Out]